Cam And Follower Motion

[Práxedes Jamal]

YiZhang

with

Susan Finger

Stephannie BehrensTable of Contents 6 Cams6.1 Introduction6.1.1 A Simple Experiment: What is a Cam?Figure 6-1 Simple Cam experimentTake a pencil and a book to do an experiment as shown above. Make thebook an inclined plane and use the pencil as a slider (use your handas a guide). When you move the book smoothly upward, what happens tothe pencil? It will be pushed up along the guide. By this method, youhave transformed one motion into another motion by a very simpledevice. This is the basic idea of a cam. By rotating the cams in thefigure below, the bars will have either translational or oscillatorymotion. 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 ofa cam mechanism A cam mechanism usually consists of two movingelements, the cam and the follower, mounted on a fixed frame. Camdevices are versatile, and almost any arbitrarily-specified motion canbe obtained. In some instances, they offer the simplest and mostcompact way to transform motions. A cam may be defined as a machine element having a curvedoutline or a curved groove, which, by its oscillation or rotationmotion, gives a predetermined specified motion to another elementcalled the follower . The cam has a very important functioninthe operation of many classes of machines, especially those of theautomatic type, such as printing presses, shoe machinery, textilemachinery, gear-cutting machines, and screw machines. In any class ofmachinery in which automatic control and accurate timing areparamount, the cam is an indispensable part of mechanism. The possibleapplications of cams are unlimited, and their shapes occur in greatvariety. Some of the most common forms will be considered in thischapter.

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 ofthe cam. We can also classify cams by the different types of motionevents of the follower and by means of a great variety of the motioncharacteristics of the cam profile. (Chen 82) Figure 6-2 Classification of cam mechanisms4.2.1 Modes of Input/Output MotionRotating 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 respectto 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 thevertical shaft.Figure 6-3 Translating cam - translating follower6.2.1 Follower ConfigurationKnife-edge follower (Figure 6-2a)Roller follower (Figure 6-2b,e,f)Flat-faced follower (Figure 6-2c)Oblique flat-faced followerSpherical-faced follower (Figure 6-2d)6.2.2 Follower ArrangementIn-line follower:

The center line of the follower passes through the center line of thecamshaft.Offset follower:

The center line of the follower does not pass through the center lineof the cam shaft. The amount of offset is the distance betweenthese two center lines. The offset causes a reduction of the sidethrust present in the roller follower.6.2.3 Cam ShapePlate cam or disk cam:

The follower moves in a plane perpendicular to the axis of rotation ofthe camshaft. A translating or a swing arm follower must beconstrained 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 faceof the cam.Figure 6-4 Grooved cam

Cylindrical cam or barrel cam (Figure6-5a):

The roller follower operates in a groove cut on the periphery of acylinder. The follower may translate or oscillate. If the cylindricalsurface 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 translatesor oscillates, whereas the cam usually rotates. The end cam is rarelyused because of the cost and the difficulty in cutting its contour.Figure 6-5 Cylindrical cam and end cam6.2.4 Constraints on the FollowerGravity 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 fixeddistance, 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 orconjugate cam in arrangement similar to what shown in Figure 6-7.Each cam has its own roller, but the rollers are mounted on the samereciprocating or oscillating follower.Figure 6-7 Dual cam 6.2.5 Examples in SimDesign Rotating Cam, Translating FollowerFigure 6-8 SimDesign translating cam Load the SimDesign file simdesign/cam.translating.sim. If youturn the cam, the follower will move. The weight of the followerkeeps them in contact. This is called a gravity constraint cam.Rotating Cam/Rotating FollowerFigure 6-9 SimDesign oscillating cam The SimDesign file is simdesign/cam.oscillating.sim. Noticethat a roller is used at the end of the follower. In addition, aspring is used to maintain the contact of the cam and the roller. If you try to calculate the degrees offreedom (DOF) of the mechanism, you must imagine that the rolleris welded onto the follower because turning the roller does notinfluence 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 afictitious knife-edge follower. It is used to generate thepitch curve. In the case of a roller follower, the tracepoint is at the center of the roller.Pitch curve: The path generated by the trace point atthe follower is rotated about a stationary cam.Working curve: The working surface ofa cam in contact with the follower. For the knife-edge followerof the plate cam, the pitch curve and the working curvescoincide. In a close or grooved cam there is an innerprofile and an outer working curve.Pitch circle: A circle from the cam center through the pitchpoint. The pitch circle radius is used to calculate a cam of minimum sizefor a given pressure angle.Prime circle (reference circle): The smallest circlefrom the cam center through the pitch curve.Base circle: The smallest circle from the cam center throughthe cam profile curve.Stroke or throw:The greatest distance or angle throughwhichthe follower moves or rotates.Follower displacement: The position of the follower from aspecific zero or rest position (usually its the position when the follower contacts with the base circle of the cam) in relationto time or the rotary angle of the cam.Pressure angle: The angle at any point between the normal tothe pitch curve and the instantaneous direction of the follower motion. Thisangle is important in cam design because it represents the steepness of thecam profile.6.4 Motion events When the cam turns through one motion cycle, the follower executes aseries of events consisting of rises, dwells and returns. Riseis the motion of the follower away from the cam center, dwellis the motion during which the follower is at rest; and returnis the motion of the follower toward the cam center. There are many follower motions that can be used for the rises and thereturns. In this chapter, we describe a number of basic curves. Figure 6-11 Motion eventsNotation : The rotary angle ofthe cam, measured from the beginning of the motion event;: The range of therotary 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 displacementsin equal units of time, i.e., uniform velocity from thebeginning 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. Thediagrams show abrupt changes of velocity, which result in large forcesat the beginning and the end of the stroke. These forces areundesirable, especially when the cam rotates at high velocity. Theconstant velocity motion is therefore only of theoreticalinterest. (6-1)6.4.2 Constant Acceleration Motion Constant acceleration motion is shown in Figure 6-11d, e, f. As indicated in e, the velocityincreases at a uniform rate during the first half of the motion anddecreases at a uniform rate during the second half of the motion. Theacceleration is constant and positive throughout the first half of themotion, as shown in f, and is constant and negative throughout thesecond half. This type of motion gives the follower the smallestvalue of maximum acceleration along the path of motion. In high-speedmachinery this is particularly important because of the forces thatare required to produce the accelerations. When

,

(6-2)

6.4.3 Harmonic Motion A cam mechanism with the basic curve like g in Figure6-7g will impart simple harmonic motion to thefollower. The velocity diagram at h indicates smooth action. Theacceleration, as shown at i, is maximum at the initial position, zeroat the mid-position, and negative maximum at the final position. (6-4)6.5 Cam DesignThe translational or rotational displacement of the follower is a functionof the rotary angle of the cam. A designer can define the functionaccording to the specific requirements in the design. The motionrequirements, listed below, are commonly used in cam profile design.6.5.1 Disk Cam with Knife-Edge Translating FollowerFigure 6-12 is a skeleton diagram of a disk cam with a knife-edgetranslating follower. We assume that the cam mechanism will be usedto realize the displacement relationship between the rotation of thecam and the translation of the follower. Figure 6-12 A Skeleton Diagram of disk cam with knife-edge translationBelow is a list of the essential parameters for the evaluation of thesetypes of cam mechanisms. However, these parameters are adequate onlyto define a knife-edge follower and a translating follower cam mechanism.Parameters:ro: The radius of the basecircle;e: The offset of the follower from the rotarycenter of the cam. Notice: it could be negative.s: The displacement of the follower which is a function ofthe rotary angle of the cam -- . IW: A parameter whose absolute value is 1. It representsthe turning direction of the cam. When the cam turns clockwise:IW=+1, otherwise: IW=-1.Cam profile design principle:The method termed inversion iscommonly used in cam profile design. For example, in a disk cam withtranslating follower mechanism, the followertranslates when the cam turns. This means that the relative motionbetween them is a combination of a relative turning motion and arelative translating motion. Without changing this feature of theirrelative motion, imagine that the cam remains fixed. Now thefollower performs both the relative turning and translatingmotions. We have inverted the mechanism. Furthermore, imagine that the knife-edge of thefollower moves along the fixed cam profile in the inverted mechanism.In other words, the knife edge of the followerdraws the profile of the cam. Thus, the problem of designing the camprofile becomes a problem of calculating the trace of the knife edgeof the follower whose motion is the combination of the relativeturning and the relative translating. Design equations: Figure 6-13 Profile design of translating cam followerIn Figure 6-13, only part of the cam profile AK isdisplayed. Assume the cam turns clockwise. At the beginning ofmotion, the knife edge of the follower contacts the point ofintersection A of the base circle and thecam profile. The coordinates of A are (So, e), andSo can be calculated from equation Suppose the displacement of the follower is S when the angulardisplacement 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 inthe inverted mechanism, turn the follower around the center of the camin the reverse direction through an angle of . The knife edge will beinverted to point K, which corresponds to the point onthe cam profile in the inverted mechanism. Therefore, the coordinatesof point K can be calculated with the following equation: (6-5)Note:The offset e is negative if the followeris 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 FollowerSuppose the cam mechanism will be used to make the knife edge oscillate.We need to compute the coordinates of the cam profile that results inthe required motion of the follower.Figure 6-14 Disk cam with knife-edge oscillating followerThe essential parameters in this kind of cam mechanismsare given below. ro: The radius of the basecircle;a: The distance between the pivot of the cam and the pivot ofthe follower.l: The length of the follower which is a distance from its pivotto its knife edge.: The angulardisplacement of the follower which is a function of the rotary angleof the cam -- .IP: A parameter whose absolute value is 1. It representsthe location of the follower. When the follower is located above thex axis: IP=+1, otherwise: IP=-1.IW: A parameter whose absolute value is 1. It represents the turningdirection of the cam. When the cam turns clockwise: IW=+1, otherwise:IW=-1.Cam profile design principleThe fundamental principle in designing the cam profiles is still inversion, similar to that that fordesigning other cam mechanisms, (e.g., the translating follower cam mechanism). Normally, the followeroscillates when the cam turns. This means that the relative motionbetween them is a combination of a relative turning motion and arelative oscillating motion. Without changing this feature of theirrelative motion, let the cam remain fixed and the follower performsboth the relative turning motion and oscillating motion. By imaginingin this way, we have actually inverted the mechanism. Figure 6-15 Cam profile design for a rotating followerIn Figure 6-15, only part of the cam profile BK is shown. Weassume that the cam turns clockwise. At the beginning of motion, the knife edge of thefollower contacts the point of intersection (B) of the basecircle and the cam profile. The initial angle between the follower(AB) and the line of two pivots (AO) is 0. It can be calculated fromthe triangle OAB. When the angular displacement of the cam is , the oscillating displacementof the follower is whichmeasures from its own initial position. At this moment, the anglebetween the follower and the line passes through two pivots should be+0. The coordinates of the knife edge at this momentwill be(6-6)To get the corresponding knife-edge of the follower in the invertedmechanism, simply turn the follower around the center of the cam inthe reverse direction of the cam rotation through an angle of . The knife edge will beinverted to point K which corresponds to the point on the camprofile in the inverted mechanism. Therefore, the coordinates ofpoint K can be calculated with the following equation: (6-7)Note: When the initial position of the follower is above thex 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 whichenvelope curve will be adopted.RM: inner or outer envelope curve. When it is an inner envelopecurve: RM=+1, otherwise: RM=-1.Design principle: The basic principle of designing a cam profile with the inversion method is still used. However, thecurve is not directly generated by inversion. This procedure has twosteps: Imagine the center of the roller as a knife edge. This concept isimportant in cam profile design and is called the trace point) of follower. Calculate the pitch curve aa, that is, the trace of thepitch point in the inverted mechanism.The cam profile bb is a product of the enveloping motion of aseries 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 theproblem of calculating the tangent points of a sequence of rollers inthe inverted mechanism. At the moment shown Figure 6-17, the tangentpoint is P on the cam profile. Figure 6-17 The tangent point, P, of a roller to the disk camThe calculation of the coordinates of the point P has two steps:Calculate the slope of the tangent tt of point K onpitch curve, aa.Calculate the slope of the normal nn of the curve aa atpoint 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 Contents1 Physical Principles2 Mechanisms and Simple Machines3 More on Machines and Mechanisms4 Basic Kinematics of Constrained Rigid Bodies5 Planar Linkages6 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 OscillatingFollower 6.5.3 Disk Cam with Roller Follower 7 Gears8 Other MechanismsIndexReferences

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