Piston seals play a crucial role in preventing fluid leakage or flow across the piston. Most piston seals are single-acting, focusing pressure on one side of the piston. This pressure build-up allows the piston to move smoothly within the cylinder and enables the cylinder to operate with maximum mechanical efficiency. As a result, dynamic piston seals are vital for the effective functioning of hydraulic systems. Double-acting piston seals, on the other hand, apply pressure to both sides of the piston, driving the attached ram. In contrast, static piston seals address the gap between the piston and the piston rod, rather than between the piston and the cylinder bore.

Figure 6-5 Cylindrical cam and end cam 6.2.4 Constraints on the Follower Gravity constraint: The weight of the follower system is sufficient to maintain contact. Spring constraint: The spring must be properly designed to maintain contact. Positive mechanical constraint: A groove maintains positive action. (Figure 6-4 and Figure 6-5a) For the cam in Figure 6-6, the follower has two rollers, separated by a fixed distance, which act as the constraint; the mating cam in such an arrangement is often called a constant-diameter cam. Figure 6-6 Constant diameter cam A mechanical constraint cam also be introduced by employing a dual or conjugate cam in arrangement similar to what shown in Figure 6-7. Each cam has its own roller, but the rollers are mounted on the same reciprocating or oscillating follower. Figure 6-7 Dual cam 6.2.5 Examples in SimDesign Rotating Cam, Translating Follower Figure 6-8 SimDesign translating cam Load the SimDesign file simdesign/cam.translating.sim. If you turn the cam, the follower will move. The weight of the follower keeps them in contact. This is called a gravity constraint cam. Rotating Cam/Rotating Follower Figure 6-9 SimDesign oscillating cam The SimDesign file is simdesign/cam.oscillating.sim. Notice that a roller is used at the end of the follower. In addition, a spring is used to maintain the contact of the cam and the roller. If you try to calculate the degrees of freedom (DOF) of the mechanism, you must imagine that the roller is welded onto the follower because turning the roller does not influence the motion of the follower. 6.3 Cam Nomenclature Figure 6-10 illustrates some cam nomenclature: Figure 6-10 Cam nomenclature Trace point: A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller. Pitch curve: The path generated by the trace point at the follower is rotated about a stationary cam. Working curve: The working surface of a cam in contact with the follower. For the knife-edge follower of the plate cam, the pitch curve and the working curves coincide. In a close or grooved cam there is an inner profile and an outer working curve. Pitch circle: A circle from the cam center through the pitch point. The pitch circle radius is used to calculate a cam of minimum size for a given pressure angle. Prime circle (reference circle): The smallest circle from the cam center through the pitch curve. Base circle: The smallest circle from the cam center through the cam profile curve. Stroke or throw:The greatest distance or angle through which the follower moves or rotates. Follower displacement: The position of the follower from a specific zero or rest position (usually its the position when the f ollower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam. Pressure angle: The angle at any point between the normal to the pitch curve and the instantaneous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. 6.4 Motion events When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

They are integral to a wide array of industries, each with its own unique set of applications. These seals play a crucial role in construction equipment, agricultural machinery, brake systems, clean rooms, conveyors, mixers, presses, valves, and testing equipment.

CamandfollowerDiagram

(6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

A cam may be defined as a machine element having a curved outline or a curved groove, which, by its oscillation or rotation motion, gives a predetermined specified motion to another element called the follower . The cam has a very important function in the operation of many classes of machines, especially those of the automatic type, such as printing presses, shoe machinery, textile machinery, gear-cutting machines, and screw machines. In any class of machinery in which automatic control and accurate timing are paramount, the cam is an indispensable part of mechanism. The possible applications of cams are unlimited, and their shapes occur in great variety. Some of the most common forms will be considered in this chapter. 6.2 Classification of Cam Mechanisms We can classify cam mechanisms by the modes of input/output motion, the configuration and arrangement of the follower, and the shape of the cam. We can also classify cams by the different types of motion events of the follower and by means of a great variety of the motion characteristics of the cam profile. (Chen 82) Figure 6-2 Classification of cam mechanisms 4.2.1 Modes of Input/Output Motion Rotating cam-translating follower. (Figure 6-2a,b,c,d,e) Rotating follower (Figure 6-2f): The follower arm swings or oscillates in a circular arc with respect to the follower pivot. Translating cam-translating follower (Figure 6-3). Stationary cam-rotating follower: The follower system revolves with respect to the center line of the vertical shaft. Figure 6-3 Translating cam - translating follower 6.2.1 Follower Configuration Knife-edge follower (Figure 6-2a) Roller follower (Figure 6-2b,e,f) Flat-faced follower (Figure 6-2c) Oblique flat-faced follower Spherical-faced follower (Figure 6-2d) 6.2.2 Follower Arrangement In-line follower: The center line of the follower passes through the center line of the camshaft. Offset follower: The center line of the follower does not pass through the center line of the cam shaft. The amount of offset is the distance between these two center lines. The offset causes a reduction of the side thrust present in the roller follower. 6.2.3 Cam Shape Plate cam or disk cam: The follower moves in a plane perpendicular to the axis of rotation of the camshaft. A translating or a swing arm follower must be constrained to maintain contact with the cam profile. Grooved cam or closed cam (Figure 6-4): This is a plate cam with the follower riding in a groove in the face of the cam. Figure 6-4 Grooved cam Cylindrical cam or barrel cam (Figure 6-5a): The roller follower operates in a groove cut on the periphery of a cylinder. The follower may translate or oscillate. If the cylindrical surface is replaced by a conical one, a conical cam results. End cam (Figure 6-5b): This cam has a rotating portion of a cylinder. The follower translates or oscillates, whereas the cam usually rotates. The end cam is rarely used because of the cost and the difficulty in cutting its contour. Figure 6-5 Cylindrical cam and end cam 6.2.4 Constraints on the Follower Gravity constraint: The weight of the follower system is sufficient to maintain contact. Spring constraint: The spring must be properly designed to maintain contact. Positive mechanical constraint: A groove maintains positive action. (Figure 6-4 and Figure 6-5a) For the cam in Figure 6-6, the follower has two rollers, separated by a fixed distance, which act as the constraint; the mating cam in such an arrangement is often called a constant-diameter cam. Figure 6-6 Constant diameter cam A mechanical constraint cam also be introduced by employing a dual or conjugate cam in arrangement similar to what shown in Figure 6-7. Each cam has its own roller, but the rollers are mounted on the same reciprocating or oscillating follower. Figure 6-7 Dual cam 6.2.5 Examples in SimDesign Rotating Cam, Translating Follower Figure 6-8 SimDesign translating cam Load the SimDesign file simdesign/cam.translating.sim. If you turn the cam, the follower will move. The weight of the follower keeps them in contact. This is called a gravity constraint cam. Rotating Cam/Rotating Follower Figure 6-9 SimDesign oscillating cam The SimDesign file is simdesign/cam.oscillating.sim. Notice that a roller is used at the end of the follower. In addition, a spring is used to maintain the contact of the cam and the roller. If you try to calculate the degrees of freedom (DOF) of the mechanism, you must imagine that the roller is welded onto the follower because turning the roller does not influence the motion of the follower. 6.3 Cam Nomenclature Figure 6-10 illustrates some cam nomenclature: Figure 6-10 Cam nomenclature Trace point: A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller. Pitch curve: The path generated by the trace point at the follower is rotated about a stationary cam. Working curve: The working surface of a cam in contact with the follower. For the knife-edge follower of the plate cam, the pitch curve and the working curves coincide. In a close or grooved cam there is an inner profile and an outer working curve. Pitch circle: A circle from the cam center through the pitch point. The pitch circle radius is used to calculate a cam of minimum size for a given pressure angle. Prime circle (reference circle): The smallest circle from the cam center through the pitch curve. Base circle: The smallest circle from the cam center through the cam profile curve. Stroke or throw:The greatest distance or angle through which the follower moves or rotates. Follower displacement: The position of the follower from a specific zero or rest position (usually its the position when the f ollower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam. Pressure angle: The angle at any point between the normal to the pitch curve and the instantaneous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. 6.4 Motion events When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? Figure 6-1 Simple Cam experiment Take a pencil and a book to do an experiment as shown above. Make the book an inclined plane and use the pencil as a slider (use your hand as a guide). When you move the book smoothly upward, what happens to the pencil? It will be pushed up along the guide. By this method, you have transformed one motion into another motion by a very simple device. This is the basic idea of a cam. By rotating the cams in the figure below, the bars will have either translational or oscillatory motion. 6.1.2 Cam Mechanisms The transformation of one of the simple motions, such as rotation, into any other motions is often conveniently accomplished by means of a cam mechanism A cam mechanism usually consists of two moving elements, the cam and the follower, mounted on a fixed frame. Cam devices are versatile, and almost any arbitrarily-specified motion can be obtained. In some instances, they offer the simplest and most compact way to transform motions. A cam may be defined as a machine element having a curved outline or a curved groove, which, by its oscillation or rotation motion, gives a predetermined specified motion to another element called the follower . The cam has a very important function in the operation of many classes of machines, especially those of the automatic type, such as printing presses, shoe machinery, textile machinery, gear-cutting machines, and screw machines. In any class of machinery in which automatic control and accurate timing are paramount, the cam is an indispensable part of mechanism. The possible applications of cams are unlimited, and their shapes occur in great variety. Some of the most common forms will be considered in this chapter. 6.2 Classification of Cam Mechanisms We can classify cam mechanisms by the modes of input/output motion, the configuration and arrangement of the follower, and the shape of the cam. We can also classify cams by the different types of motion events of the follower and by means of a great variety of the motion characteristics of the cam profile. (Chen 82) Figure 6-2 Classification of cam mechanisms 4.2.1 Modes of Input/Output Motion Rotating cam-translating follower. (Figure 6-2a,b,c,d,e) Rotating follower (Figure 6-2f): The follower arm swings or oscillates in a circular arc with respect to the follower pivot. Translating cam-translating follower (Figure 6-3). Stationary cam-rotating follower: The follower system revolves with respect to the center line of the vertical shaft. Figure 6-3 Translating cam - translating follower 6.2.1 Follower Configuration Knife-edge follower (Figure 6-2a) Roller follower (Figure 6-2b,e,f) Flat-faced follower (Figure 6-2c) Oblique flat-faced follower Spherical-faced follower (Figure 6-2d) 6.2.2 Follower Arrangement In-line follower: The center line of the follower passes through the center line of the camshaft. Offset follower: The center line of the follower does not pass through the center line of the cam shaft. The amount of offset is the distance between these two center lines. The offset causes a reduction of the side thrust present in the roller follower. 6.2.3 Cam Shape Plate cam or disk cam: The follower moves in a plane perpendicular to the axis of rotation of the camshaft. A translating or a swing arm follower must be constrained to maintain contact with the cam profile. Grooved cam or closed cam (Figure 6-4): This is a plate cam with the follower riding in a groove in the face of the cam. Figure 6-4 Grooved cam Cylindrical cam or barrel cam (Figure 6-5a): The roller follower operates in a groove cut on the periphery of a cylinder. The follower may translate or oscillate. If the cylindrical surface is replaced by a conical one, a conical cam results. End cam (Figure 6-5b): This cam has a rotating portion of a cylinder. The follower translates or oscillates, whereas the cam usually rotates. The end cam is rarely used because of the cost and the difficulty in cutting its contour. Figure 6-5 Cylindrical cam and end cam 6.2.4 Constraints on the Follower Gravity constraint: The weight of the follower system is sufficient to maintain contact. Spring constraint: The spring must be properly designed to maintain contact. Positive mechanical constraint: A groove maintains positive action. (Figure 6-4 and Figure 6-5a) For the cam in Figure 6-6, the follower has two rollers, separated by a fixed distance, which act as the constraint; the mating cam in such an arrangement is often called a constant-diameter cam. Figure 6-6 Constant diameter cam A mechanical constraint cam also be introduced by employing a dual or conjugate cam in arrangement similar to what shown in Figure 6-7. Each cam has its own roller, but the rollers are mounted on the same reciprocating or oscillating follower. Figure 6-7 Dual cam 6.2.5 Examples in SimDesign Rotating Cam, Translating Follower Figure 6-8 SimDesign translating cam Load the SimDesign file simdesign/cam.translating.sim. If you turn the cam, the follower will move. The weight of the follower keeps them in contact. This is called a gravity constraint cam. Rotating Cam/Rotating Follower Figure 6-9 SimDesign oscillating cam The SimDesign file is simdesign/cam.oscillating.sim. Notice that a roller is used at the end of the follower. In addition, a spring is used to maintain the contact of the cam and the roller. If you try to calculate the degrees of freedom (DOF) of the mechanism, you must imagine that the roller is welded onto the follower because turning the roller does not influence the motion of the follower. 6.3 Cam Nomenclature Figure 6-10 illustrates some cam nomenclature: Figure 6-10 Cam nomenclature Trace point: A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller. Pitch curve: The path generated by the trace point at the follower is rotated about a stationary cam. Working curve: The working surface of a cam in contact with the follower. For the knife-edge follower of the plate cam, the pitch curve and the working curves coincide. In a close or grooved cam there is an inner profile and an outer working curve. Pitch circle: A circle from the cam center through the pitch point. The pitch circle radius is used to calculate a cam of minimum size for a given pressure angle. Prime circle (reference circle): The smallest circle from the cam center through the pitch curve. Base circle: The smallest circle from the cam center through the cam profile curve. Stroke or throw:The greatest distance or angle through which the follower moves or rotates. Follower displacement: The position of the follower from a specific zero or rest position (usually its the position when the f ollower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam. Pressure angle: The angle at any point between the normal to the pitch curve and the instantaneous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. 6.4 Motion events When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Cam followerfunction

The transformation of one of the simple motions, such as rotation, into any other motions is often conveniently accomplished by means of a cam mechanism A cam mechanism usually consists of two moving elements, the cam and the follower, mounted on a fixed frame. Cam devices are versatile, and almost any arbitrarily-specified motion can be obtained. In some instances, they offer the simplest and most compact way to transform motions. A cam may be defined as a machine element having a curved outline or a curved groove, which, by its oscillation or rotation motion, gives a predetermined specified motion to another element called the follower . The cam has a very important function in the operation of many classes of machines, especially those of the automatic type, such as printing presses, shoe machinery, textile machinery, gear-cutting machines, and screw machines. In any class of machinery in which automatic control and accurate timing are paramount, the cam is an indispensable part of mechanism. The possible applications of cams are unlimited, and their shapes occur in great variety. Some of the most common forms will be considered in this chapter. 6.2 Classification of Cam Mechanisms We can classify cam mechanisms by the modes of input/output motion, the configuration and arrangement of the follower, and the shape of the cam. We can also classify cams by the different types of motion events of the follower and by means of a great variety of the motion characteristics of the cam profile. (Chen 82) Figure 6-2 Classification of cam mechanisms 4.2.1 Modes of Input/Output Motion Rotating cam-translating follower. (Figure 6-2a,b,c,d,e) Rotating follower (Figure 6-2f): The follower arm swings or oscillates in a circular arc with respect to the follower pivot. Translating cam-translating follower (Figure 6-3). Stationary cam-rotating follower: The follower system revolves with respect to the center line of the vertical shaft. Figure 6-3 Translating cam - translating follower 6.2.1 Follower Configuration Knife-edge follower (Figure 6-2a) Roller follower (Figure 6-2b,e,f) Flat-faced follower (Figure 6-2c) Oblique flat-faced follower Spherical-faced follower (Figure 6-2d) 6.2.2 Follower Arrangement In-line follower: The center line of the follower passes through the center line of the camshaft. Offset follower: The center line of the follower does not pass through the center line of the cam shaft. The amount of offset is the distance between these two center lines. The offset causes a reduction of the side thrust present in the roller follower. 6.2.3 Cam Shape Plate cam or disk cam: The follower moves in a plane perpendicular to the axis of rotation of the camshaft. A translating or a swing arm follower must be constrained to maintain contact with the cam profile. Grooved cam or closed cam (Figure 6-4): This is a plate cam with the follower riding in a groove in the face of the cam. Figure 6-4 Grooved cam Cylindrical cam or barrel cam (Figure 6-5a): The roller follower operates in a groove cut on the periphery of a cylinder. The follower may translate or oscillate. If the cylindrical surface is replaced by a conical one, a conical cam results. End cam (Figure 6-5b): This cam has a rotating portion of a cylinder. The follower translates or oscillates, whereas the cam usually rotates. The end cam is rarely used because of the cost and the difficulty in cutting its contour. Figure 6-5 Cylindrical cam and end cam 6.2.4 Constraints on the Follower Gravity constraint: The weight of the follower system is sufficient to maintain contact. Spring constraint: The spring must be properly designed to maintain contact. Positive mechanical constraint: A groove maintains positive action. (Figure 6-4 and Figure 6-5a) For the cam in Figure 6-6, the follower has two rollers, separated by a fixed distance, which act as the constraint; the mating cam in such an arrangement is often called a constant-diameter cam. Figure 6-6 Constant diameter cam A mechanical constraint cam also be introduced by employing a dual or conjugate cam in arrangement similar to what shown in Figure 6-7. Each cam has its own roller, but the rollers are mounted on the same reciprocating or oscillating follower. Figure 6-7 Dual cam 6.2.5 Examples in SimDesign Rotating Cam, Translating Follower Figure 6-8 SimDesign translating cam Load the SimDesign file simdesign/cam.translating.sim. If you turn the cam, the follower will move. The weight of the follower keeps them in contact. This is called a gravity constraint cam. Rotating Cam/Rotating Follower Figure 6-9 SimDesign oscillating cam The SimDesign file is simdesign/cam.oscillating.sim. Notice that a roller is used at the end of the follower. In addition, a spring is used to maintain the contact of the cam and the roller. If you try to calculate the degrees of freedom (DOF) of the mechanism, you must imagine that the roller is welded onto the follower because turning the roller does not influence the motion of the follower. 6.3 Cam Nomenclature Figure 6-10 illustrates some cam nomenclature: Figure 6-10 Cam nomenclature Trace point: A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller. Pitch curve: The path generated by the trace point at the follower is rotated about a stationary cam. Working curve: The working surface of a cam in contact with the follower. For the knife-edge follower of the plate cam, the pitch curve and the working curves coincide. In a close or grooved cam there is an inner profile and an outer working curve. Pitch circle: A circle from the cam center through the pitch point. The pitch circle radius is used to calculate a cam of minimum size for a given pressure angle. Prime circle (reference circle): The smallest circle from the cam center through the pitch curve. Base circle: The smallest circle from the cam center through the cam profile curve. Stroke or throw:The greatest distance or angle through which the follower moves or rotates. Follower displacement: The position of the follower from a specific zero or rest position (usually its the position when the f ollower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam. Pressure angle: The angle at any point between the normal to the pitch curve and the instantaneous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. 6.4 Motion events When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Die cutting is the mass fabrication of cut-out shapes by shearing a stock material such as paper and chipboard using tooling called a die. A die is a specialized tool used in manufacturing to cut or shape a material fitted into a press...

(6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Camandfollowerreal life examples

Static seals are commonly positioned in grooves and confined spaces, functioning as gaskets. Here, a gasket refers to a mechanical seal that fills the gaps between stationary mating surfaces, secured by the pressure from tightened bolts. While the number and specific locations of static seals vary depending on the cylinder structure, their primary role is to close the gaps between immobile surfaces. Static seals can be categorized into axial and radial types. Axial static seals are compressed between their top and bottom surfaces to create a secure seal, whereas radial static seals achieve the same effect by being compressed between their inner and outer surfaces.

Hydraulic seals are an integral component of most hydraulic systems. Usually made of a soft, flexible elastomer that provides exceptional water and air sealing capabilities, hydraulic seals are ring-shaped and primarily designed to eliminate or limit the leakage of fluid moving within a hydraulic or pneumatic system. Hydraulic seals also play important roles in excluding contaminants from and appropriately pressurizing the overall hydraulic system. Read More…

The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Yi Zhang with Susan Finger Stephannie Behrens Table of Contents 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? Figure 6-1 Simple Cam experiment Take a pencil and a book to do an experiment as shown above. Make the book an inclined plane and use the pencil as a slider (use your hand as a guide). When you move the book smoothly upward, what happens to the pencil? It will be pushed up along the guide. By this method, you have transformed one motion into another motion by a very simple device. This is the basic idea of a cam. By rotating the cams in the figure below, the bars will have either translational or oscillatory motion. 6.1.2 Cam Mechanisms The transformation of one of the simple motions, such as rotation, into any other motions is often conveniently accomplished by means of a cam mechanism A cam mechanism usually consists of two moving elements, the cam and the follower, mounted on a fixed frame. Cam devices are versatile, and almost any arbitrarily-specified motion can be obtained. In some instances, they offer the simplest and most compact way to transform motions. A cam may be defined as a machine element having a curved outline or a curved groove, which, by its oscillation or rotation motion, gives a predetermined specified motion to another element called the follower . The cam has a very important function in the operation of many classes of machines, especially those of the automatic type, such as printing presses, shoe machinery, textile machinery, gear-cutting machines, and screw machines. In any class of machinery in which automatic control and accurate timing are paramount, the cam is an indispensable part of mechanism. The possible applications of cams are unlimited, and their shapes occur in great variety. Some of the most common forms will be considered in this chapter. 6.2 Classification of Cam Mechanisms We can classify cam mechanisms by the modes of input/output motion, the configuration and arrangement of the follower, and the shape of the cam. We can also classify cams by the different types of motion events of the follower and by means of a great variety of the motion characteristics of the cam profile. (Chen 82) Figure 6-2 Classification of cam mechanisms 4.2.1 Modes of Input/Output Motion Rotating cam-translating follower. (Figure 6-2a,b,c,d,e) Rotating follower (Figure 6-2f): The follower arm swings or oscillates in a circular arc with respect to the follower pivot. Translating cam-translating follower (Figure 6-3). Stationary cam-rotating follower: The follower system revolves with respect to the center line of the vertical shaft. Figure 6-3 Translating cam - translating follower 6.2.1 Follower Configuration Knife-edge follower (Figure 6-2a) Roller follower (Figure 6-2b,e,f) Flat-faced follower (Figure 6-2c) Oblique flat-faced follower Spherical-faced follower (Figure 6-2d) 6.2.2 Follower Arrangement In-line follower: The center line of the follower passes through the center line of the camshaft. Offset follower: The center line of the follower does not pass through the center line of the cam shaft. The amount of offset is the distance between these two center lines. The offset causes a reduction of the side thrust present in the roller follower. 6.2.3 Cam Shape Plate cam or disk cam: The follower moves in a plane perpendicular to the axis of rotation of the camshaft. A translating or a swing arm follower must be constrained to maintain contact with the cam profile. Grooved cam or closed cam (Figure 6-4): This is a plate cam with the follower riding in a groove in the face of the cam. Figure 6-4 Grooved cam Cylindrical cam or barrel cam (Figure 6-5a): The roller follower operates in a groove cut on the periphery of a cylinder. The follower may translate or oscillate. If the cylindrical surface is replaced by a conical one, a conical cam results. End cam (Figure 6-5b): This cam has a rotating portion of a cylinder. The follower translates or oscillates, whereas the cam usually rotates. The end cam is rarely used because of the cost and the difficulty in cutting its contour. Figure 6-5 Cylindrical cam and end cam 6.2.4 Constraints on the Follower Gravity constraint: The weight of the follower system is sufficient to maintain contact. Spring constraint: The spring must be properly designed to maintain contact. Positive mechanical constraint: A groove maintains positive action. (Figure 6-4 and Figure 6-5a) For the cam in Figure 6-6, the follower has two rollers, separated by a fixed distance, which act as the constraint; the mating cam in such an arrangement is often called a constant-diameter cam. Figure 6-6 Constant diameter cam A mechanical constraint cam also be introduced by employing a dual or conjugate cam in arrangement similar to what shown in Figure 6-7. Each cam has its own roller, but the rollers are mounted on the same reciprocating or oscillating follower. Figure 6-7 Dual cam 6.2.5 Examples in SimDesign Rotating Cam, Translating Follower Figure 6-8 SimDesign translating cam Load the SimDesign file simdesign/cam.translating.sim. If you turn the cam, the follower will move. The weight of the follower keeps them in contact. This is called a gravity constraint cam. Rotating Cam/Rotating Follower Figure 6-9 SimDesign oscillating cam The SimDesign file is simdesign/cam.oscillating.sim. Notice that a roller is used at the end of the follower. In addition, a spring is used to maintain the contact of the cam and the roller. If you try to calculate the degrees of freedom (DOF) of the mechanism, you must imagine that the roller is welded onto the follower because turning the roller does not influence the motion of the follower. 6.3 Cam Nomenclature Figure 6-10 illustrates some cam nomenclature: Figure 6-10 Cam nomenclature Trace point: A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller. Pitch curve: The path generated by the trace point at the follower is rotated about a stationary cam. Working curve: The working surface of a cam in contact with the follower. For the knife-edge follower of the plate cam, the pitch curve and the working curves coincide. In a close or grooved cam there is an inner profile and an outer working curve. Pitch circle: A circle from the cam center through the pitch point. The pitch circle radius is used to calculate a cam of minimum size for a given pressure angle. Prime circle (reference circle): The smallest circle from the cam center through the pitch curve. Base circle: The smallest circle from the cam center through the cam profile curve. Stroke or throw:The greatest distance or angle through which the follower moves or rotates. Follower displacement: The position of the follower from a specific zero or rest position (usually its the position when the f ollower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam. Pressure angle: The angle at any point between the normal to the pitch curve and the instantaneous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. 6.4 Motion events When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Hydraulic seals are typically crafted from elastomers and polymers—both natural and synthetic—known for their weak molecular attraction and exceptional elasticity. The primary materials used for hydraulic seals include rubber and plastics like PTFE (Teflon) and polyurethane. Other elastomers employed in seal production are butadiene, nitrile, butyl, and silicone. Most seals made from these elastomeric materials are produced through extrusion, while Teflon seals are uniquely sintered from powder form in an oven.

Oil seals, known as metric oil seals internationally, are designed to withstand repeated exposure to hydraulic oil without breaking down or malfunctioning. Made from durable materials like polyacrylate, silicone, Teflon, and various fluoroelastomers, these oil seals effectively prevent leaks while retaining lubricants in rotary applications.

Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Rod seals are designed to stop external fluid leaks from cylinders. Typically single-acting, they are often complemented by a secondary rod seal for enhanced performance. Dynamic rod seals operate within the space between the piston rod and the cylinder head, while static rod seals close the gaps between the cylinder head and the cylinder bore. Beyond containing hydraulic fluid within the cylinder, rod seals also help regulate lubrication for the rod, the wiper seal, and the rod seal itself.

Cam followervs lifter

ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Cam followerbearing

Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

O-rings stand out in this list because their identification relies mainly on their shape rather than their function or location. True to their name, O-rings are donut-shaped and are available in a range of materials such as rubber, silicone, and fluorocarbon, as well as sizes from under an inch to several meters in diameter. When subjected to mechanical deformation from pressure, O-rings form highly effective sealing barriers. Their affordability and simplicity make them the most prevalent type of hydraulic seal available today.

Our list of inventory has grown since our founding over 20 years ago. We started small, but now offer an extensive product list of hydraulic seals, o rings, and other rubber products.

Hydraulic pumps are mechanisms in hydraulic systems that move hydraulic fluid from point to point initiating the production of hydraulic power. Hydraulic pumps are sometimes incorrectly referred to as "hydrolic" pumps...

Figure 6-3 Translating cam - translating follower 6.2.1 Follower Configuration Knife-edge follower (Figure 6-2a) Roller follower (Figure 6-2b,e,f) Flat-faced follower (Figure 6-2c) Oblique flat-faced follower Spherical-faced follower (Figure 6-2d) 6.2.2 Follower Arrangement In-line follower: The center line of the follower passes through the center line of the camshaft. Offset follower: The center line of the follower does not pass through the center line of the cam shaft. The amount of offset is the distance between these two center lines. The offset causes a reduction of the side thrust present in the roller follower. 6.2.3 Cam Shape Plate cam or disk cam: The follower moves in a plane perpendicular to the axis of rotation of the camshaft. A translating or a swing arm follower must be constrained to maintain contact with the cam profile. Grooved cam or closed cam (Figure 6-4): This is a plate cam with the follower riding in a groove in the face of the cam. Figure 6-4 Grooved cam Cylindrical cam or barrel cam (Figure 6-5a): The roller follower operates in a groove cut on the periphery of a cylinder. The follower may translate or oscillate. If the cylindrical surface is replaced by a conical one, a conical cam results. End cam (Figure 6-5b): This cam has a rotating portion of a cylinder. The follower translates or oscillates, whereas the cam usually rotates. The end cam is rarely used because of the cost and the difficulty in cutting its contour. Figure 6-5 Cylindrical cam and end cam 6.2.4 Constraints on the Follower Gravity constraint: The weight of the follower system is sufficient to maintain contact. Spring constraint: The spring must be properly designed to maintain contact. Positive mechanical constraint: A groove maintains positive action. (Figure 6-4 and Figure 6-5a) For the cam in Figure 6-6, the follower has two rollers, separated by a fixed distance, which act as the constraint; the mating cam in such an arrangement is often called a constant-diameter cam. Figure 6-6 Constant diameter cam A mechanical constraint cam also be introduced by employing a dual or conjugate cam in arrangement similar to what shown in Figure 6-7. Each cam has its own roller, but the rollers are mounted on the same reciprocating or oscillating follower. Figure 6-7 Dual cam 6.2.5 Examples in SimDesign Rotating Cam, Translating Follower Figure 6-8 SimDesign translating cam Load the SimDesign file simdesign/cam.translating.sim. If you turn the cam, the follower will move. The weight of the follower keeps them in contact. This is called a gravity constraint cam. Rotating Cam/Rotating Follower Figure 6-9 SimDesign oscillating cam The SimDesign file is simdesign/cam.oscillating.sim. Notice that a roller is used at the end of the follower. In addition, a spring is used to maintain the contact of the cam and the roller. If you try to calculate the degrees of freedom (DOF) of the mechanism, you must imagine that the roller is welded onto the follower because turning the roller does not influence the motion of the follower. 6.3 Cam Nomenclature Figure 6-10 illustrates some cam nomenclature: Figure 6-10 Cam nomenclature Trace point: A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller. Pitch curve: The path generated by the trace point at the follower is rotated about a stationary cam. Working curve: The working surface of a cam in contact with the follower. For the knife-edge follower of the plate cam, the pitch curve and the working curves coincide. In a close or grooved cam there is an inner profile and an outer working curve. Pitch circle: A circle from the cam center through the pitch point. The pitch circle radius is used to calculate a cam of minimum size for a given pressure angle. Prime circle (reference circle): The smallest circle from the cam center through the pitch curve. Base circle: The smallest circle from the cam center through the cam profile curve. Stroke or throw:The greatest distance or angle through which the follower moves or rotates. Follower displacement: The position of the follower from a specific zero or rest position (usually its the position when the f ollower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam. Pressure angle: The angle at any point between the normal to the pitch curve and the instantaneous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. 6.4 Motion events When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

We can classify cam mechanisms by the modes of input/output motion, the configuration and arrangement of the follower, and the shape of the cam. We can also classify cams by the different types of motion events of the follower and by means of a great variety of the motion characteristics of the cam profile. (Chen 82) Figure 6-2 Classification of cam mechanisms 4.2.1 Modes of Input/Output Motion Rotating cam-translating follower. (Figure 6-2a,b,c,d,e) Rotating follower (Figure 6-2f): The follower arm swings or oscillates in a circular arc with respect to the follower pivot. Translating cam-translating follower (Figure 6-3). Stationary cam-rotating follower: The follower system revolves with respect to the center line of the vertical shaft. Figure 6-3 Translating cam - translating follower 6.2.1 Follower Configuration Knife-edge follower (Figure 6-2a) Roller follower (Figure 6-2b,e,f) Flat-faced follower (Figure 6-2c) Oblique flat-faced follower Spherical-faced follower (Figure 6-2d) 6.2.2 Follower Arrangement In-line follower: The center line of the follower passes through the center line of the camshaft. Offset follower: The center line of the follower does not pass through the center line of the cam shaft. The amount of offset is the distance between these two center lines. The offset causes a reduction of the side thrust present in the roller follower. 6.2.3 Cam Shape Plate cam or disk cam: The follower moves in a plane perpendicular to the axis of rotation of the camshaft. A translating or a swing arm follower must be constrained to maintain contact with the cam profile. Grooved cam or closed cam (Figure 6-4): This is a plate cam with the follower riding in a groove in the face of the cam. Figure 6-4 Grooved cam Cylindrical cam or barrel cam (Figure 6-5a): The roller follower operates in a groove cut on the periphery of a cylinder. The follower may translate or oscillate. If the cylindrical surface is replaced by a conical one, a conical cam results. End cam (Figure 6-5b): This cam has a rotating portion of a cylinder. The follower translates or oscillates, whereas the cam usually rotates. The end cam is rarely used because of the cost and the difficulty in cutting its contour. Figure 6-5 Cylindrical cam and end cam 6.2.4 Constraints on the Follower Gravity constraint: The weight of the follower system is sufficient to maintain contact. Spring constraint: The spring must be properly designed to maintain contact. Positive mechanical constraint: A groove maintains positive action. (Figure 6-4 and Figure 6-5a) For the cam in Figure 6-6, the follower has two rollers, separated by a fixed distance, which act as the constraint; the mating cam in such an arrangement is often called a constant-diameter cam. Figure 6-6 Constant diameter cam A mechanical constraint cam also be introduced by employing a dual or conjugate cam in arrangement similar to what shown in Figure 6-7. Each cam has its own roller, but the rollers are mounted on the same reciprocating or oscillating follower. Figure 6-7 Dual cam 6.2.5 Examples in SimDesign Rotating Cam, Translating Follower Figure 6-8 SimDesign translating cam Load the SimDesign file simdesign/cam.translating.sim. If you turn the cam, the follower will move. The weight of the follower keeps them in contact. This is called a gravity constraint cam. Rotating Cam/Rotating Follower Figure 6-9 SimDesign oscillating cam The SimDesign file is simdesign/cam.oscillating.sim. Notice that a roller is used at the end of the follower. In addition, a spring is used to maintain the contact of the cam and the roller. If you try to calculate the degrees of freedom (DOF) of the mechanism, you must imagine that the roller is welded onto the follower because turning the roller does not influence the motion of the follower. 6.3 Cam Nomenclature Figure 6-10 illustrates some cam nomenclature: Figure 6-10 Cam nomenclature Trace point: A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller. Pitch curve: The path generated by the trace point at the follower is rotated about a stationary cam. Working curve: The working surface of a cam in contact with the follower. For the knife-edge follower of the plate cam, the pitch curve and the working curves coincide. In a close or grooved cam there is an inner profile and an outer working curve. Pitch circle: A circle from the cam center through the pitch point. The pitch circle radius is used to calculate a cam of minimum size for a given pressure angle. Prime circle (reference circle): The smallest circle from the cam center through the pitch curve. Base circle: The smallest circle from the cam center through the cam profile curve. Stroke or throw:The greatest distance or angle through which the follower moves or rotates. Follower displacement: The position of the follower from a specific zero or rest position (usually its the position when the f ollower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam. Pressure angle: The angle at any point between the normal to the pitch curve and the instantaneous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. 6.4 Motion events When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

The SimDesign file is simdesign/cam.oscillating.sim. Notice that a roller is used at the end of the follower. In addition, a spring is used to maintain the contact of the cam and the roller. If you try to calculate the degrees of freedom (DOF) of the mechanism, you must imagine that the roller is welded onto the follower because turning the roller does not influence the motion of the follower. 6.3 Cam Nomenclature Figure 6-10 illustrates some cam nomenclature: Figure 6-10 Cam nomenclature Trace point: A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller. Pitch curve: The path generated by the trace point at the follower is rotated about a stationary cam. Working curve: The working surface of a cam in contact with the follower. For the knife-edge follower of the plate cam, the pitch curve and the working curves coincide. In a close or grooved cam there is an inner profile and an outer working curve. Pitch circle: A circle from the cam center through the pitch point. The pitch circle radius is used to calculate a cam of minimum size for a given pressure angle. Prime circle (reference circle): The smallest circle from the cam center through the pitch curve. Base circle: The smallest circle from the cam center through the cam profile curve. Stroke or throw:The greatest distance or angle through which the follower moves or rotates. Follower displacement: The position of the follower from a specific zero or rest position (usually its the position when the f ollower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam. Pressure angle: The angle at any point between the normal to the pitch curve and the instantaneous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. 6.4 Motion events When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Hydraulic motors are rotary or mechanical actuators that operate by converting hydraulic pressure or fluid energy into torque and angular displacement. The most relevant nomenclature used in hydraulic motors include...

Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Certain hydraulic seals are designed for highly specialized applications. In the aerospace sector, custom seals like silicone sealant strips are essential for sealing hydraulic components in aircraft systems. In the electronics industry, seals such as EMI shielding gaskets are crucial for blocking unwanted electromagnetic interference in various devices.

Gaskets are a type of mechanical seal that inhibits leakage by filling the gap between static mating surfaces. It takes advantage of the compressive force that drives the gasket material to plastically flow between the...

Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Consider a hypothetical scenario where we choose O-rings, taking various factors into account. One key characteristic to evaluate is the compression set, which indicates how well an elastomer returns to its original thickness after being compressed. This property is influenced by temperature changes, hydraulic pressure variations, and the specific fluid the O-ring interacts with. For example, very high operating temperatures might lead to significant swelling of the O-ring, necessitating a larger “housing” within the cylinder to accommodate this expansion.

The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Types ofcamandfollowerpdf

Figure 6-10 Cam nomenclature Trace point: A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller. Pitch curve: The path generated by the trace point at the follower is rotated about a stationary cam. Working curve: The working surface of a cam in contact with the follower. For the knife-edge follower of the plate cam, the pitch curve and the working curves coincide. In a close or grooved cam there is an inner profile and an outer working curve. Pitch circle: A circle from the cam center through the pitch point. The pitch circle radius is used to calculate a cam of minimum size for a given pressure angle. Prime circle (reference circle): The smallest circle from the cam center through the pitch curve. Base circle: The smallest circle from the cam center through the cam profile curve. Stroke or throw:The greatest distance or angle through which the follower moves or rotates. Follower displacement: The position of the follower from a specific zero or rest position (usually its the position when the f ollower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam. Pressure angle: The angle at any point between the normal to the pitch curve and the instantaneous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. 6.4 Motion events When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Load the SimDesign file simdesign/cam.translating.sim. If you turn the cam, the follower will move. The weight of the follower keeps them in contact. This is called a gravity constraint cam. Rotating Cam/Rotating Follower Figure 6-9 SimDesign oscillating cam The SimDesign file is simdesign/cam.oscillating.sim. Notice that a roller is used at the end of the follower. In addition, a spring is used to maintain the contact of the cam and the roller. If you try to calculate the degrees of freedom (DOF) of the mechanism, you must imagine that the roller is welded onto the follower because turning the roller does not influence the motion of the follower. 6.3 Cam Nomenclature Figure 6-10 illustrates some cam nomenclature: Figure 6-10 Cam nomenclature Trace point: A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller. Pitch curve: The path generated by the trace point at the follower is rotated about a stationary cam. Working curve: The working surface of a cam in contact with the follower. For the knife-edge follower of the plate cam, the pitch curve and the working curves coincide. In a close or grooved cam there is an inner profile and an outer working curve. Pitch circle: A circle from the cam center through the pitch point. The pitch circle radius is used to calculate a cam of minimum size for a given pressure angle. Prime circle (reference circle): The smallest circle from the cam center through the pitch curve. Base circle: The smallest circle from the cam center through the cam profile curve. Stroke or throw:The greatest distance or angle through which the follower moves or rotates. Follower displacement: The position of the follower from a specific zero or rest position (usually its the position when the f ollower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam. Pressure angle: The angle at any point between the normal to the pitch curve and the instantaneous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. 6.4 Motion events When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Cam mechanism

6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Hydraulic seals play a crucial role in maintaining the efficiency of hydraulic systems, making them essential across various industries. They are utilized in aerospace manufacturing, agriculture, automotive production, chemical processing, defense contracting, food processing, marine product manufacturing, medical and pharmaceutical development, nuclear power, pulp and paper industries, and waste disposal.

Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

© Hydraulic Seal will keep your information private and secure.  We will not share your email address with any third parties.

They are typically positioned on the cylinder head, rod shaft, or within the piston. In these critical locations, they effectively prevent fluid from leaking between the rod and head, escaping to the outside of the cylinder, or flowing across the piston.

Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Seals can also be crafted from non-elastic materials like felt and leather. Certain hydraulic seals, such as bonded seals, utilize metallic components including brass, bronze, aluminum, carbon steel, and stainless steel. These metals may be plated or galvanized to enhance their resistance to oxidation and increase their durability. In bonded seals, a chemical bond forms between the rubber material and the metal, ensuring a strong adhesion.

Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

(6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

(6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Figure 6-10 illustrates some cam nomenclature: Figure 6-10 Cam nomenclature Trace point: A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller. Pitch curve: The path generated by the trace point at the follower is rotated about a stationary cam. Working curve: The working surface of a cam in contact with the follower. For the knife-edge follower of the plate cam, the pitch curve and the working curves coincide. In a close or grooved cam there is an inner profile and an outer working curve. Pitch circle: A circle from the cam center through the pitch point. The pitch circle radius is used to calculate a cam of minimum size for a given pressure angle. Prime circle (reference circle): The smallest circle from the cam center through the pitch curve. Base circle: The smallest circle from the cam center through the cam profile curve. Stroke or throw:The greatest distance or angle through which the follower moves or rotates. Follower displacement: The position of the follower from a specific zero or rest position (usually its the position when the f ollower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam. Pressure angle: The angle at any point between the normal to the pitch curve and the instantaneous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. 6.4 Motion events When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Figure 6-1 Simple Cam experiment Take a pencil and a book to do an experiment as shown above. Make the book an inclined plane and use the pencil as a slider (use your hand as a guide). When you move the book smoothly upward, what happens to the pencil? It will be pushed up along the guide. By this method, you have transformed one motion into another motion by a very simple device. This is the basic idea of a cam. By rotating the cams in the figure below, the bars will have either translational or oscillatory motion. 6.1.2 Cam Mechanisms The transformation of one of the simple motions, such as rotation, into any other motions is often conveniently accomplished by means of a cam mechanism A cam mechanism usually consists of two moving elements, the cam and the follower, mounted on a fixed frame. Cam devices are versatile, and almost any arbitrarily-specified motion can be obtained. In some instances, they offer the simplest and most compact way to transform motions. A cam may be defined as a machine element having a curved outline or a curved groove, which, by its oscillation or rotation motion, gives a predetermined specified motion to another element called the follower . The cam has a very important function in the operation of many classes of machines, especially those of the automatic type, such as printing presses, shoe machinery, textile machinery, gear-cutting machines, and screw machines. In any class of machinery in which automatic control and accurate timing are paramount, the cam is an indispensable part of mechanism. The possible applications of cams are unlimited, and their shapes occur in great variety. Some of the most common forms will be considered in this chapter. 6.2 Classification of Cam Mechanisms We can classify cam mechanisms by the modes of input/output motion, the configuration and arrangement of the follower, and the shape of the cam. We can also classify cams by the different types of motion events of the follower and by means of a great variety of the motion characteristics of the cam profile. (Chen 82) Figure 6-2 Classification of cam mechanisms 4.2.1 Modes of Input/Output Motion Rotating cam-translating follower. (Figure 6-2a,b,c,d,e) Rotating follower (Figure 6-2f): The follower arm swings or oscillates in a circular arc with respect to the follower pivot. Translating cam-translating follower (Figure 6-3). Stationary cam-rotating follower: The follower system revolves with respect to the center line of the vertical shaft. Figure 6-3 Translating cam - translating follower 6.2.1 Follower Configuration Knife-edge follower (Figure 6-2a) Roller follower (Figure 6-2b,e,f) Flat-faced follower (Figure 6-2c) Oblique flat-faced follower Spherical-faced follower (Figure 6-2d) 6.2.2 Follower Arrangement In-line follower: The center line of the follower passes through the center line of the camshaft. Offset follower: The center line of the follower does not pass through the center line of the cam shaft. The amount of offset is the distance between these two center lines. The offset causes a reduction of the side thrust present in the roller follower. 6.2.3 Cam Shape Plate cam or disk cam: The follower moves in a plane perpendicular to the axis of rotation of the camshaft. A translating or a swing arm follower must be constrained to maintain contact with the cam profile. Grooved cam or closed cam (Figure 6-4): This is a plate cam with the follower riding in a groove in the face of the cam. Figure 6-4 Grooved cam Cylindrical cam or barrel cam (Figure 6-5a): The roller follower operates in a groove cut on the periphery of a cylinder. The follower may translate or oscillate. If the cylindrical surface is replaced by a conical one, a conical cam results. End cam (Figure 6-5b): This cam has a rotating portion of a cylinder. The follower translates or oscillates, whereas the cam usually rotates. The end cam is rarely used because of the cost and the difficulty in cutting its contour. Figure 6-5 Cylindrical cam and end cam 6.2.4 Constraints on the Follower Gravity constraint: The weight of the follower system is sufficient to maintain contact. Spring constraint: The spring must be properly designed to maintain contact. Positive mechanical constraint: A groove maintains positive action. (Figure 6-4 and Figure 6-5a) For the cam in Figure 6-6, the follower has two rollers, separated by a fixed distance, which act as the constraint; the mating cam in such an arrangement is often called a constant-diameter cam. Figure 6-6 Constant diameter cam A mechanical constraint cam also be introduced by employing a dual or conjugate cam in arrangement similar to what shown in Figure 6-7. Each cam has its own roller, but the rollers are mounted on the same reciprocating or oscillating follower. Figure 6-7 Dual cam 6.2.5 Examples in SimDesign Rotating Cam, Translating Follower Figure 6-8 SimDesign translating cam Load the SimDesign file simdesign/cam.translating.sim. If you turn the cam, the follower will move. The weight of the follower keeps them in contact. This is called a gravity constraint cam. Rotating Cam/Rotating Follower Figure 6-9 SimDesign oscillating cam The SimDesign file is simdesign/cam.oscillating.sim. Notice that a roller is used at the end of the follower. In addition, a spring is used to maintain the contact of the cam and the roller. If you try to calculate the degrees of freedom (DOF) of the mechanism, you must imagine that the roller is welded onto the follower because turning the roller does not influence the motion of the follower. 6.3 Cam Nomenclature Figure 6-10 illustrates some cam nomenclature: Figure 6-10 Cam nomenclature Trace point: A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller. Pitch curve: The path generated by the trace point at the follower is rotated about a stationary cam. Working curve: The working surface of a cam in contact with the follower. For the knife-edge follower of the plate cam, the pitch curve and the working curves coincide. In a close or grooved cam there is an inner profile and an outer working curve. Pitch circle: A circle from the cam center through the pitch point. The pitch circle radius is used to calculate a cam of minimum size for a given pressure angle. Prime circle (reference circle): The smallest circle from the cam center through the pitch curve. Base circle: The smallest circle from the cam center through the cam profile curve. Stroke or throw:The greatest distance or angle through which the follower moves or rotates. Follower displacement: The position of the follower from a specific zero or rest position (usually its the position when the f ollower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam. Pressure angle: The angle at any point between the normal to the pitch curve and the instantaneous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. 6.4 Motion events When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

(6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

When choosing the right hydraulic seal solution, it’s crucial to consider more than just its ability to prevent fluid leakage. Hydraulic seals perform multiple essential functions and require proper maintenance to work effectively. Therefore, selecting a seal should involve evaluating the overall operating conditions of your hydraulic system. Aim to choose a seal that excels under these specific conditions, rather than focusing solely on its sealing capabilities.

The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Oscillating seals work with shafts that turn through a limited range of motion around their axis. Due to the frequent rotation of these shafts, oscillating seals are typically crafted from durable materials and feature self-lubricating properties.

Rotating Cam/Rotating Follower Figure 6-9 SimDesign oscillating cam The SimDesign file is simdesign/cam.oscillating.sim. Notice that a roller is used at the end of the follower. In addition, a spring is used to maintain the contact of the cam and the roller. If you try to calculate the degrees of freedom (DOF) of the mechanism, you must imagine that the roller is welded onto the follower because turning the roller does not influence the motion of the follower. 6.3 Cam Nomenclature Figure 6-10 illustrates some cam nomenclature: Figure 6-10 Cam nomenclature Trace point: A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller. Pitch curve: The path generated by the trace point at the follower is rotated about a stationary cam. Working curve: The working surface of a cam in contact with the follower. For the knife-edge follower of the plate cam, the pitch curve and the working curves coincide. In a close or grooved cam there is an inner profile and an outer working curve. Pitch circle: A circle from the cam center through the pitch point. The pitch circle radius is used to calculate a cam of minimum size for a given pressure angle. Prime circle (reference circle): The smallest circle from the cam center through the pitch curve. Base circle: The smallest circle from the cam center through the cam profile curve. Stroke or throw:The greatest distance or angle through which the follower moves or rotates. Follower displacement: The position of the follower from a specific zero or rest position (usually its the position when the f ollower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam. Pressure angle: The angle at any point between the normal to the pitch curve and the instantaneous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. 6.4 Motion events When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Kent Rubber Supply has been a trusted supplier of hydraulic seals since 1946. Focusing in exceeding customer service, we leverage the best equipment and the most knowledge staff to ensure we meet our customers unique needs. With our trusted experts, we determine the best rubber, foam, sponge or PVC material for your specific need. We continue serving customers both offshore and domestic.

(6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Rotating Cam, Translating Follower Figure 6-8 SimDesign translating cam Load the SimDesign file simdesign/cam.translating.sim. If you turn the cam, the follower will move. The weight of the follower keeps them in contact. This is called a gravity constraint cam. Rotating Cam/Rotating Follower Figure 6-9 SimDesign oscillating cam The SimDesign file is simdesign/cam.oscillating.sim. Notice that a roller is used at the end of the follower. In addition, a spring is used to maintain the contact of the cam and the roller. If you try to calculate the degrees of freedom (DOF) of the mechanism, you must imagine that the roller is welded onto the follower because turning the roller does not influence the motion of the follower. 6.3 Cam Nomenclature Figure 6-10 illustrates some cam nomenclature: Figure 6-10 Cam nomenclature Trace point: A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller. Pitch curve: The path generated by the trace point at the follower is rotated about a stationary cam. Working curve: The working surface of a cam in contact with the follower. For the knife-edge follower of the plate cam, the pitch curve and the working curves coincide. In a close or grooved cam there is an inner profile and an outer working curve. Pitch circle: A circle from the cam center through the pitch point. The pitch circle radius is used to calculate a cam of minimum size for a given pressure angle. Prime circle (reference circle): The smallest circle from the cam center through the pitch curve. Base circle: The smallest circle from the cam center through the cam profile curve. Stroke or throw:The greatest distance or angle through which the follower moves or rotates. Follower displacement: The position of the follower from a specific zero or rest position (usually its the position when the f ollower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam. Pressure angle: The angle at any point between the normal to the pitch curve and the instantaneous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. 6.4 Motion events When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

: The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

The function of a hydraulic seal can vary slightly based on its position relative to the hydraulic cylinder. The most prevalent types are piston seals and rod seals. Both of these seals feature a flexible lip that makes contact with the housing or shaft, enhancing the seal’s effectiveness during linear motion. Collectively referred to as lip seals, piston and rod seals play a crucial role in ensuring the smooth operation of rotating equipment and machinery.

If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

At Gasket & Seal Fabricators, Inc., we take pride in our extensive expertise in the design and manufacturing of high-quality gaskets and seals, particularly specializing in hydraulic seals. Our commitment to excellence drives us to deliver reliable solutions that meet the diverse needs of our clients across various industries. Our product range includes an array of hydraulic seals designed to...

With the right considerations in mind, operators across all industries can anticipate dependable and efficient sealing solutions. It’s crucial to find an experienced and reliable hydraulic specialist or supplier to assist in the seal selection process. Top hydraulic suppliers provide services such as custom sealing materials, comprehensive documentation for various seals, and guidance on optimal operating conditions.

Southern Rubber Company, Inc. was chartered in 1925 as the Carolina’s pioneer rubber company. The primary focus in the early years was oil jobber supplies, automobile tires, hoses and belts. As more industries located in the South, Southern Rubber Company, Inc. began to diversify, and replaced their automobile products with industrial rubber products including hose, gaskets and seals.

An essential consideration is the space where the hydraulic seal will be used. This will influence the diameters of both the inner and outer seals, the housing bore, and the outer shaft dimensions. Additionally, it’s crucial for stakeholders to determine whether their application needs an axial or radial cross section and its corresponding diameter.

Buffer seals improve rod seal performance by acting as a protective barrier against excessive internal fluid pressure. Additionally, they safeguard the rod seal from contaminants, like metal chips, that might infiltrate the cylinder.

Figure 6-2 Classification of cam mechanisms 4.2.1 Modes of Input/Output Motion Rotating cam-translating follower. (Figure 6-2a,b,c,d,e) Rotating follower (Figure 6-2f): The follower arm swings or oscillates in a circular arc with respect to the follower pivot. Translating cam-translating follower (Figure 6-3). Stationary cam-rotating follower: The follower system revolves with respect to the center line of the vertical shaft. Figure 6-3 Translating cam - translating follower 6.2.1 Follower Configuration Knife-edge follower (Figure 6-2a) Roller follower (Figure 6-2b,e,f) Flat-faced follower (Figure 6-2c) Oblique flat-faced follower Spherical-faced follower (Figure 6-2d) 6.2.2 Follower Arrangement In-line follower: The center line of the follower passes through the center line of the camshaft. Offset follower: The center line of the follower does not pass through the center line of the cam shaft. The amount of offset is the distance between these two center lines. The offset causes a reduction of the side thrust present in the roller follower. 6.2.3 Cam Shape Plate cam or disk cam: The follower moves in a plane perpendicular to the axis of rotation of the camshaft. A translating or a swing arm follower must be constrained to maintain contact with the cam profile. Grooved cam or closed cam (Figure 6-4): This is a plate cam with the follower riding in a groove in the face of the cam. Figure 6-4 Grooved cam Cylindrical cam or barrel cam (Figure 6-5a): The roller follower operates in a groove cut on the periphery of a cylinder. The follower may translate or oscillate. If the cylindrical surface is replaced by a conical one, a conical cam results. End cam (Figure 6-5b): This cam has a rotating portion of a cylinder. The follower translates or oscillates, whereas the cam usually rotates. The end cam is rarely used because of the cost and the difficulty in cutting its contour. Figure 6-5 Cylindrical cam and end cam 6.2.4 Constraints on the Follower Gravity constraint: The weight of the follower system is sufficient to maintain contact. Spring constraint: The spring must be properly designed to maintain contact. Positive mechanical constraint: A groove maintains positive action. (Figure 6-4 and Figure 6-5a) For the cam in Figure 6-6, the follower has two rollers, separated by a fixed distance, which act as the constraint; the mating cam in such an arrangement is often called a constant-diameter cam. Figure 6-6 Constant diameter cam A mechanical constraint cam also be introduced by employing a dual or conjugate cam in arrangement similar to what shown in Figure 6-7. Each cam has its own roller, but the rollers are mounted on the same reciprocating or oscillating follower. Figure 6-7 Dual cam 6.2.5 Examples in SimDesign Rotating Cam, Translating Follower Figure 6-8 SimDesign translating cam Load the SimDesign file simdesign/cam.translating.sim. If you turn the cam, the follower will move. The weight of the follower keeps them in contact. This is called a gravity constraint cam. Rotating Cam/Rotating Follower Figure 6-9 SimDesign oscillating cam The SimDesign file is simdesign/cam.oscillating.sim. Notice that a roller is used at the end of the follower. In addition, a spring is used to maintain the contact of the cam and the roller. If you try to calculate the degrees of freedom (DOF) of the mechanism, you must imagine that the roller is welded onto the follower because turning the roller does not influence the motion of the follower. 6.3 Cam Nomenclature Figure 6-10 illustrates some cam nomenclature: Figure 6-10 Cam nomenclature Trace point: A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller. Pitch curve: The path generated by the trace point at the follower is rotated about a stationary cam. Working curve: The working surface of a cam in contact with the follower. For the knife-edge follower of the plate cam, the pitch curve and the working curves coincide. In a close or grooved cam there is an inner profile and an outer working curve. Pitch circle: A circle from the cam center through the pitch point. The pitch circle radius is used to calculate a cam of minimum size for a given pressure angle. Prime circle (reference circle): The smallest circle from the cam center through the pitch curve. Base circle: The smallest circle from the cam center through the cam profile curve. Stroke or throw:The greatest distance or angle through which the follower moves or rotates. Follower displacement: The position of the follower from a specific zero or rest position (usually its the position when the f ollower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam. Pressure angle: The angle at any point between the normal to the pitch curve and the instantaneous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. 6.4 Motion events When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Guide rings, often referred to as wear rings, fulfill a dual role: they center the piston and piston rod, guiding them through the cylinder and preventing metal-on-metal contact. These rings are installed at both the rod and piston positions within a hydraulic cylinder.

Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

X rings, also known as quad or square rings, represent an advanced iteration of O-rings. With their four-lobed configuration, X rings deliver up to twice the sealing effectiveness of traditional O-rings while minimizing mechanical deformation. They are versatile enough to serve both static and dynamic sealing applications.

An O-ring is a round elastic loop that is used as a seal for static and dynamic applications. Their main purpose is to serve as a seal between structures such as pipes, tubes, in pistons, and cylinders. O-rings are made of various materials depending on how they will be used and are highly pliable...

6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Take a pencil and a book to do an experiment as shown above. Make the book an inclined plane and use the pencil as a slider (use your hand as a guide). When you move the book smoothly upward, what happens to the pencil? It will be pushed up along the guide. By this method, you have transformed one motion into another motion by a very simple device. This is the basic idea of a cam. By rotating the cams in the figure below, the bars will have either translational or oscillatory motion. 6.1.2 Cam Mechanisms The transformation of one of the simple motions, such as rotation, into any other motions is often conveniently accomplished by means of a cam mechanism A cam mechanism usually consists of two moving elements, the cam and the follower, mounted on a fixed frame. Cam devices are versatile, and almost any arbitrarily-specified motion can be obtained. In some instances, they offer the simplest and most compact way to transform motions. A cam may be defined as a machine element having a curved outline or a curved groove, which, by its oscillation or rotation motion, gives a predetermined specified motion to another element called the follower . The cam has a very important function in the operation of many classes of machines, especially those of the automatic type, such as printing presses, shoe machinery, textile machinery, gear-cutting machines, and screw machines. In any class of machinery in which automatic control and accurate timing are paramount, the cam is an indispensable part of mechanism. The possible applications of cams are unlimited, and their shapes occur in great variety. Some of the most common forms will be considered in this chapter. 6.2 Classification of Cam Mechanisms We can classify cam mechanisms by the modes of input/output motion, the configuration and arrangement of the follower, and the shape of the cam. We can also classify cams by the different types of motion events of the follower and by means of a great variety of the motion characteristics of the cam profile. (Chen 82) Figure 6-2 Classification of cam mechanisms 4.2.1 Modes of Input/Output Motion Rotating cam-translating follower. (Figure 6-2a,b,c,d,e) Rotating follower (Figure 6-2f): The follower arm swings or oscillates in a circular arc with respect to the follower pivot. Translating cam-translating follower (Figure 6-3). Stationary cam-rotating follower: The follower system revolves with respect to the center line of the vertical shaft. Figure 6-3 Translating cam - translating follower 6.2.1 Follower Configuration Knife-edge follower (Figure 6-2a) Roller follower (Figure 6-2b,e,f) Flat-faced follower (Figure 6-2c) Oblique flat-faced follower Spherical-faced follower (Figure 6-2d) 6.2.2 Follower Arrangement In-line follower: The center line of the follower passes through the center line of the camshaft. Offset follower: The center line of the follower does not pass through the center line of the cam shaft. The amount of offset is the distance between these two center lines. The offset causes a reduction of the side thrust present in the roller follower. 6.2.3 Cam Shape Plate cam or disk cam: The follower moves in a plane perpendicular to the axis of rotation of the camshaft. A translating or a swing arm follower must be constrained to maintain contact with the cam profile. Grooved cam or closed cam (Figure 6-4): This is a plate cam with the follower riding in a groove in the face of the cam. Figure 6-4 Grooved cam Cylindrical cam or barrel cam (Figure 6-5a): The roller follower operates in a groove cut on the periphery of a cylinder. The follower may translate or oscillate. If the cylindrical surface is replaced by a conical one, a conical cam results. End cam (Figure 6-5b): This cam has a rotating portion of a cylinder. The follower translates or oscillates, whereas the cam usually rotates. The end cam is rarely used because of the cost and the difficulty in cutting its contour. Figure 6-5 Cylindrical cam and end cam 6.2.4 Constraints on the Follower Gravity constraint: The weight of the follower system is sufficient to maintain contact. Spring constraint: The spring must be properly designed to maintain contact. Positive mechanical constraint: A groove maintains positive action. (Figure 6-4 and Figure 6-5a) For the cam in Figure 6-6, the follower has two rollers, separated by a fixed distance, which act as the constraint; the mating cam in such an arrangement is often called a constant-diameter cam. Figure 6-6 Constant diameter cam A mechanical constraint cam also be introduced by employing a dual or conjugate cam in arrangement similar to what shown in Figure 6-7. Each cam has its own roller, but the rollers are mounted on the same reciprocating or oscillating follower. Figure 6-7 Dual cam 6.2.5 Examples in SimDesign Rotating Cam, Translating Follower Figure 6-8 SimDesign translating cam Load the SimDesign file simdesign/cam.translating.sim. If you turn the cam, the follower will move. The weight of the follower keeps them in contact. This is called a gravity constraint cam. Rotating Cam/Rotating Follower Figure 6-9 SimDesign oscillating cam The SimDesign file is simdesign/cam.oscillating.sim. Notice that a roller is used at the end of the follower. In addition, a spring is used to maintain the contact of the cam and the roller. If you try to calculate the degrees of freedom (DOF) of the mechanism, you must imagine that the roller is welded onto the follower because turning the roller does not influence the motion of the follower. 6.3 Cam Nomenclature Figure 6-10 illustrates some cam nomenclature: Figure 6-10 Cam nomenclature Trace point: A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller. Pitch curve: The path generated by the trace point at the follower is rotated about a stationary cam. Working curve: The working surface of a cam in contact with the follower. For the knife-edge follower of the plate cam, the pitch curve and the working curves coincide. In a close or grooved cam there is an inner profile and an outer working curve. Pitch circle: A circle from the cam center through the pitch point. The pitch circle radius is used to calculate a cam of minimum size for a given pressure angle. Prime circle (reference circle): The smallest circle from the cam center through the pitch curve. Base circle: The smallest circle from the cam center through the cam profile curve. Stroke or throw:The greatest distance or angle through which the follower moves or rotates. Follower displacement: The position of the follower from a specific zero or rest position (usually its the position when the f ollower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam. Pressure angle: The angle at any point between the normal to the pitch curve and the instantaneous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. 6.4 Motion events When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Reciprocating dynamic seals are positioned within glands that accommodate relative movement, traveling along an axis between inner and outer surfaces. These seals are commonly employed to power linear actuators, hydraulic cylinders, and pistons in internal combustion engines.

Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

If you try to calculate the degrees of freedom (DOF) of the mechanism, you must imagine that the roller is welded onto the follower because turning the roller does not influence the motion of the follower. 6.3 Cam Nomenclature Figure 6-10 illustrates some cam nomenclature: Figure 6-10 Cam nomenclature Trace point: A theoretical point on the follower, corresponding to the point of a fictitious knife-edge follower. It is used to generate the pitch curve. In the case of a roller follower, the trace point is at the center of the roller. Pitch curve: The path generated by the trace point at the follower is rotated about a stationary cam. Working curve: The working surface of a cam in contact with the follower. For the knife-edge follower of the plate cam, the pitch curve and the working curves coincide. In a close or grooved cam there is an inner profile and an outer working curve. Pitch circle: A circle from the cam center through the pitch point. The pitch circle radius is used to calculate a cam of minimum size for a given pressure angle. Prime circle (reference circle): The smallest circle from the cam center through the pitch curve. Base circle: The smallest circle from the cam center through the cam profile curve. Stroke or throw:The greatest distance or angle through which the follower moves or rotates. Follower displacement: The position of the follower from a specific zero or rest position (usually its the position when the f ollower contacts with the base circle of the cam) in relation to time or the rotary angle of the cam. Pressure angle: The angle at any point between the normal to the pitch curve and the instantaneous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. 6.4 Motion events When the cam turns through one motion cycle, the follower executes a series of events consisting of rises, dwells and returns. Rise is the motion of the follower away from the cam center, dwell is the motion during which the follower is at rest; and return is the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and the returns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion events Notation : The rotary angle of the cam, measured from the beginning of the motion event; : The range of the rotary angle corresponding to the motion event; h : The stoke of the motion event of the follower; S : Displacement of the follower; V : Velocity of the follower; A : Acceleration of the follower. 6.4.1 Constant Velocity Motion If the motion of the follower were a straight line, Figure 6-11a,b,c, it would have equal displacements in equal units of time, i.e., uniform velocity from the beginning to the end of the stroke, as shown in b. The acceleration, except at the end of the stroke would be zero, as shown in c. The diagrams show abrupt changes of velocity, which result in large forces at the beginning and the end of the stroke. These forces are undesirable, especially when the cam rotates at high velocity. The constant velocity motion is therefore only of theoretical interest. (6-1) 6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocity increases at a uniform rate during the first half of the motion and decreases at a uniform rate during the second half of the motion. The acceleration is constant and positive throughout the first half of the motion, as shown in f, and is constant and negative throughout the second half. This type of motion gives the follower the smallest value of maximum acceleration along the path of motion. In high-speed machinery this is particularly important because of the forces that are required to produce the accelerations. When , (6-2) When , (6-3) 6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Additionally, those buying hydraulic seals should consider the vacuum rating, operating temperatures, chemical compatibility with the hydraulic fluid, maximum and minimum working speeds, and pressure ranges.

Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

It’s important to recognize that while all hydraulic seals play a crucial role, not all of them carry the same level of importance. Prioritizing which seals to use and how to apply them is essential. Maintaining internal pressure and preventing contamination are the top priorities for ensuring a hydraulic cylinder operates effectively, as contamination is the primary cause of hydraulic system failure. Therefore, the rod and wiper seals are especially vital because they help maintain internal pressure and keep contaminants out.

(6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Camandfollower mechanismexamples

The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Dynamic seals, also known as shaft seals, effectively close gaps between two surfaces that experience relative motion. They accommodate various types of motion, including reciprocation, oscillation, and rotation.

Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Hydraulic cylinders often feature various common types of seals. Rod wipers, sometimes called scrapers or wiper seals, prevent contaminants from entering the cylinder. These contaminants, such as dirt and moisture, can compromise the cylinder’s performance. As the wiper seals retract into the cylinder, they effectively remove these foreign particles.

Hydraulic seals are a form of gasket-like rings that are used to fill gaps between hydraulic cylinder components. Many different components are found in hydraulic cylinders, some of which get in...

The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

A cam mechanism with the basic curve like g in Figure 6-7g will impart simple harmonic motion to the follower. The velocity diagram at h indicates smooth action. The acceleration, as shown at i, is maximum at the initial position, zero at the mid-position, and negative maximum at the final position. (6-4) 6.5 Cam Design The translational or rotational displacement of the follower is a function of the rotary angle of the cam. A designer can define the function according to the specific requirements in the design. The motion requirements, listed below, are commonly used in cam profile design. 6.5.1 Disk Cam with Knife-Edge Translating Follower Figure 6-12 is a skeleton diagram of a disk cam with a knife-edge translating follower. We assume that the cam mechanism will be used to realize the displacement relationship between the rotation of the cam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translation Below is a list of the essential parameters for the evaluation of these types of cam mechanisms. However, these parameters are adequate only to define a knife-edge follower and a translating follower cam mechanism. Parameters: ro: The radius of the base circle; e: The offset of the follower from the rotary center of the cam. Notice: it could be negative. s: The displacement of the follower which is a function of the rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle: The method termed inversion is commonly used in cam profile design. For example, in a disk cam with translating follower mechanism, the follower translates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative translating motion. Without changing this feature of their relative motion, imagine that the cam remains fixed. Now the follower performs both the relative turning and translating motions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu

Provide your project details, and we'll connect you to 5-20 qualified hydraulic seal companies in your region. 100% free.

Hydraulic seals are commonly used in hydraulic cylinders, which are mechanical actuators that transform hydraulic pressure—from oil, water, or other pressurized fluids—into unidirectional force. These cylinders are essential for agriculture and forestry vehicles, construction equipment, and various other machinery.

Furthermore, imagine that the knife-edge of the follower moves along the fixed cam profile in the inverted mechanism. In other words, the knife edge of the follower draws the profile of the cam. Thus, the problem of designing the cam profile becomes a problem of calculating the trace of the knife edge of the follower whose motion is the combination of the relative turning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam follower In Figure 6-13, only part of the cam profile AK is displayed. Assume the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection A of the base circle and the cam profile. The coordinates of A are (So, e), and So can be calculated from equation Suppose the displacement of the follower is S when the angular displacement of the cam is . At this moment, the coordinates of the knife edge of the follower should be (So + S, e). To get the corresponding position of the knife edge of the follower in the inverted mechanism, turn the follower around the center of the cam in the reverse direction through an angle of . The knife edge will be inverted to point K, which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-5) Note: The offset e is negative if the follower is located below the x axis. When the rotational direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.2 Disk Cam with Oscillating Knife-Edge Follower Suppose the cam mechanism will be used to make the knife edge oscillate. We need to compute the coordinates of the cam profile that results in the required motion of the follower. Figure 6-14 Disk cam with knife-edge oscillating follower The essential parameters in this kind of cam mechanisms are given below. ro: The radius of the base circle; a: The distance between the pivot of the cam and the pivot of the follower. l: The length of the follower which is a distance from its pivot to its knife edge. : The angular displacement of the follower which is a function of the rotary angle of the cam -- . IP: A parameter whose absolute value is 1. It represents the location of the follower. When the follower is located above the x axis: IP=+1, otherwise: IP=-1. IW: A parameter whose absolute value is 1. It represents the turning direction of the cam. When the cam turns clockwise: IW=+1, otherwise: IW=-1. Cam profile design principle The fundamental principle in designing the cam profiles is still inversion, similar to that that for designing other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the follower oscillates when the cam turns. This means that the relative motion between them is a combination of a relative turning motion and a relative oscillating motion. Without changing this feature of their relative motion, let the cam remain fixed and the follower performs both the relative turning motion and oscillating motion. By imagining in this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating follower In Figure 6-15, only part of the cam profile BK is shown. We assume that the cam turns clockwise. At the beginning of motion, the knife edge of the follower contacts the point of intersection (B) of the base circle and the cam profile. The initial angle between the follower (AB) and the line of two pivots (AO) is 0. It can be calculated from the triangle OAB. When the angular displacement of the cam is , the oscillating displacement of the follower is which measures from its own initial position. At this moment, the angle between the follower and the line passes through two pivots should be +0. The coordinates of the knife edge at this moment will be (6-6) To get the corresponding knife-edge of the follower in the inverted mechanism, simply turn the follower around the center of the cam in the reverse direction of the cam rotation through an angle of . The knife edge will be inverted to point K which corresponds to the point on the cam profile in the inverted mechanism. Therefore, the coordinates of point K can be calculated with the following equation: (6-7) Note: When the initial position of the follower is above the x axis, IP = +1, otherwise: IP = -1. When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. 6.5.3 Disk Cam with Roller Follower Additional parameters: r: the radius of the roller. IM: a parameter whose absolute value is 1, indicating which envelope curve will be adopted. RM: inner or outer envelope curve. When it is an inner envelope curve: RM=+1, otherwise: RM=-1. Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, the curve is not directly generated by inversion. This procedure has two steps: Imagine the center of the roller as a knife edge. This concept is important in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of the pitch point in the inverted mechanism. The cam profile bb is a product of the enveloping motion of a series of rollers. Figure 6-16 The trace point of the follower on a disk cam Design equations: The problem of calculating the coordinates of the cam profile is the problem of calculating the tangent points of a sequence of rollers in the inverted mechanism. At the moment shown Figure 6-17, the tangent point is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk cam The calculation of the coordinates of the point P has two steps: Calculate the slope of the tangent tt of point K on pitch curve, aa. Calculate the slope of the normal nn of the curve aa at point K. Since we have already have the coordinates of point K: (x, y), we can express the coordinates of point P as (6-8) Note: When the rotary direction of the cam is clockwise: IW = +1, otherwise: IW = -1. when the envelope curve (cam profile) lies inside the pitch curve: RM = +1, otherwise: RM = -1. Table of Contents    Complete Table of Contents 1 Physical Principles 2 Mechanisms and Simple Machines 3 More on Machines and Mechanisms 4 Basic Kinematics of Constrained Rigid Bodies 5 Planar Linkages 6 Cams 6.1 Introduction 6.1.1 A Simple Experiment: What is a Cam? 6.1.2 Cam Mechanisms 6.2 Classification of Cam Mechanisms 6.2.1 Follower Configuration 6.2.2 Follower Arrangement 6.2.3 Cam Shape 6.2.4 Constraints on the Follower 6.2.5 Examples in SimDesign 6.3 Cam Nomenclature 6.4 Motion events 6.4.1 Constant Velocity Motion 6.4.2 Constant Acceleration Motion 6.4.3 Harmonic Motion 6.5 Cam Design 6.5.1 Disk Cam with Knife-Edge Translating Follower 6.5.2 Disk Cam with Knife-Edge Oscillating Follower 6.5.3 Disk Cam with Roller Follower 7 Gears 8 Other Mechanisms Index References               sfinger@ri.cmu.edu