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The dynamic response of a payload to a motion law is a function of the motionlaw itself, the stiffness of the input and output transmissions, its inertia, and its backlash.
When a cam moves a payload, the shape of the cam and the motionover the camfollower defines the command motion function. The cam commands a motion and the payload responds with its motion. The response does not equal the command! 

The image shows the CamFollowers Motion  the command  and an exaggerated diagram of payload's motion  its response 
All payloads have, mass and inertia and transmission stiffness. Hence, a dynamic force or torque is required to move it. That force is transmitted through an elastic transmission system. It is 'elastic' because it has a finite stiffness and real mass and inertia. The elastic transmission system distorts when subjected to the force or torque. The motion of the payload, its response, is a distortion of the command and takes the form of a vibration of approximately constant frequency, but of varying amplitude, superimposed on the command motion. The mechanism can be represented by a simple model of a: •Cam with a camfollower roller of negligible mass. The Cam drives a: •Payload of mass, W ( W for mass(kg), as it avoids confusing with M we use for Torque) Between the Cam and Payload, there is a: •Spring, S, which represents the output transmission, and its Stiffness. 
Motion Response and the progression of the Payload in response to the Motion Command. The Payload's motion starts later than the start of the CamFollower Roller's motion. The command and response can be described thus: •The CamFollower Roller begins to move. The Payload will not move until the force that is imparted to it overcome's the Payload's Inertia. •The Force on the spring compresses the spring by an amount proportional to its stiffness, S. •The motion/position of the Payload lags that of the motion/position of the CamFollower Roller until the force on the Payload from the Spring gives sufficient acceleration to it such that its velocity exceeds that of the CamFollower's velocity. •Then the Spring begins to expand again. The Spring expands until it matches its original length and the Payload's displacement is equal to the CamFollower Roller displacement. •However, at that instant, the Payload is moving faster than the CamFollower Roller. •The Payload overshoots and leads the CamFollower Roller displacement. As the spring continues to expand, the Payload's velocity decreases, relative to that of the CamFollower Roller, until their velocities are the same. •The velocity continues to decrease until the displacement of the Payload and CamFollower become the same. •A similar cycle of events occurs again and again, throughout the stroke, so that the Payload motion is an oscillation of varying amplitude superimpose on that of the CamFollower Roller. 
Note: To prevent confusion with the term Oscillation of swinging arm followers, the oscillation associated with dynamic response is referred to as a Vibration and not Oscillation. The displacement amplitude of the vibration is much less than that illustrated in the image above. It is often difficult to see. The acceleration amplitude, however, is a large part of the total acceleration. Since inertiaforce is proportional to acceleration, the vibration makes a significant contribution to the peak force that acts on the system. The nature of the vibration, its amplitude in particular, is a function of the motionlaw, the duration or motion period, the mass/inertia of the payload, and the stiffness/rigidity of the mechanism. The frequency of the vibration is the natural frequency of the massspring system. All massspring systems, and cam mechanisms, have a natural frequency of vibration. The natural frequency can be measured, or estimated with reasonable accuracy, from the mass and stiffness (or in the case of rotary systems the momentofinertia and torsional rigidity). PeriodRatio (n) is the ratio of the motionperiod (stroke time) to the vibrationperiod (time for one vibration cycle) see Natural Frequency and PeriodRatio below. PeriodRatio is one of the most important parameters in the estimation of the peak load on a cam. For a given motionlaw, the maximum amplitude of the dynamic response is greatly dependent on the PeriodRatio of the system. Also, for any given PeriodRatio, the maximum amplitude of the dynamic response is greatly dependent on the MotionLaw. PeriodRatio and the Application Speed. We can use PeriodRatio to designate the 'speed' of an application. Highspeed applications are those with a low PeriodRatio, say less than 8. Slowspeed applications are those with a high PeriodRatio, say more than 20. The choice of motionlaw is less critical with slowspeed applications. Most cam mechanisms in modern industrial machinery run with a PeriodRatio less than 20. There are few successful cammechanisms with with a PeriodRatio less than 5. EXAMPLES: ModifiedTrapezoidal (ModTrap), Cycloidal and ModifiedSinusoid (ModSine) MotionLaws  Acceleration Plots 

ModTrap Response with High and Low Period Ratio 
Cycloidal Response with High and Low Period Ratio 
ModSine Response with High and Low Period Ratio 
DynamicResponse and Residual Vibration with Good Output Transmission Design (top) and Bad Output Transmission Design (bottom). Good Output Transmission = high ratio of Rigidity : Inertia = high value of Period Ratio Bad Output Transmission = low ratio of Rigidity : Inertia = low value of Period Ratio 

The three diagrams above show the dynamic response and residual vibration of a mechanism with three popular motionlaws and two different PeriodRatios. They show the nominal acceleration (system with no vibration) and the acceleration response (the dynamicresponse) plotted against time for one motionperiod and one dwell period. •The diagrams at the top represent a mechanism with a fairly rigid output transmission. The PeriodRatio is 13.5. •The diagrams at the bottom represent the same mechanism, but the PeriodRatio is 3.25. The output transmission rigidity is the only difference between the top and bottom. (Alternatively, the bottom diagrams might be same transmission rigidity but operating 13.5/3.25 times faster). A number of important points are illustrated by these diagrams: 1.All three motionlaws produce a vibration with low amplitude when the PeriodRatio is high. 2.A low PeriodRatio will give a high vibration amplitude (force or torque). It may be higher than the nominal maximum of the motionlaw, by a factor of more that 2. 3.There is a residual vibration in the dwell period which may be damped down to zero before the next motion period if there is some friction load and the vibration amplitude is low, that is, the transmission rigidity is good. The residual vibration of a bad transmission, however, may not be damped out quickly enough and could make the dynamic response of the subsequent motion worse still. 4.If the transmissions are good, the ModTrap motion has a peak load less than either the ModSine or Cycloidal, by virtue of its lower maximum acceleration. If transmissions are bad, however, ModSine is better than ModTrap (note the negative peak of the ModTrap) and Cycloidal is better than the ModSine, especially as regards the residual vibration. All three motionlaws shown are considered to be 'good' laws for highspeed mechanisms. 

Simple Harmonic Motion Response with High and Low Period Ratio 
Parabolic Response with High and Low Period Ratio 
Parabolic and Simple Harmonic Motion For comparison, the dynamic response of mechanisms similar in all respects, except for the motionlaw, are shown to the left with the Parabolic (Constant Acceleration and Retardation) and Simple Harmonic Motions. These motionlaws were popular because of their simple formulation, their low nominal maximum acceleration and the apparently smooth profiles that they produced. However, they do not give a satisfactory dynamic response, even at low speed with good transmission design (high PeriodRatio). Both motionlaws have a stepchange  that is, a discontinuity  in the acceleration profiles. The stepchange in the acceleration 'shocks' the mechanical system, and incites the vibrations in it much more than 'modern' motionlaws that do not have a stepchange in acceleration. The value of peak force or torque in both cases is much the same, at all PeriodRatios. This is because the vibrations are caused mainly by the stepchange in acceleration. The peak force or torque is higher than the peak force or torque of the modern dynamically favorable motionlaws. 
For a simple SpringMass System, as shown in the representation of the camsystem above, the Natural Frequency is: 

for a Linear System. 

for a Rotary System. 

Where: f = natural frequency (Hz) S = linear stiffness (N/m) W = mass (kg) (we use W to prevent confusion with M for 'Torque (moment), which is used later). R = torsional rigidity (N.m/rad) I = moment of inertia (kg.m2) PeriodRatio is:  or  n = Period of Motion / Natural Period of Vibration Where: T = duration of motionperiod(s), the time take of the motion segment in the MachineCycle. 
The Peak Torque (positive or negative) from the vibration is, of course, a critical factor in estimating the load/life capacity of a cam mechanism.It is possible to estimate the peak torque at the mechanism design stage if the MotionLaw is known and PeriodRatio can be estimated. Torsion Factor, Ct, is the ratio of the peak torque to the nominal peak torque for an undamped cam system with a wholly inertial load. It assumes Spring Loads. Friction Loads, and others, are all zero. Note: The term TorsionFactor is historical. It was first used with indexing mechanisms where the transmission was always rotary, hence torsion. Torsion Factor Equation The TorsionFactor Equation estimates the ratio of peak (or actual) acceleration to nominal (or motionlaw) acceleration. When a cammechanism has the same motionlaw, its TorsionFactor is dependent only on its PeriodRatio. You need the PeriodRatio and the MotionLaw. Curves have been plotted of TorsionFactor, Ct, against PeriodRatio, n, for the SCCA family of MotionLaws. The TorsionFactor is empirically given by the equation: 

${C}_{t}=\frac{p}{{n}^{q}}+r$ : 
TorsionFactor Equation for Cams 





Notes: Parabolic MotionLaw: TorsionFactor Ct=4 at all PeriodRatios. Thus, it is a poor motion for any application, even slow speed, with good transmissions. Simple Harmonic MotionLaw: TorsionFactor Ct=2 at all PeriodRatios when preceded and followed by a Dwell. This makes it not as good as the ModTrap, ModSine or Cycloidal for many applications. The high TorsionFactor of Simple Harmonic Motion is due to the sudden change of acceleration at the 'start' and 'end' of the MotionLaw, when you use this motionlaw between dwell segments. In applications where there are no dwells, and no sudden changes, then Simple Harmonic Motion is the best for minimum vibration. In many cases, a low value of Jerk at CrossOver (transition between acceleration and deceleration) of SimpleHarmonicMotion also reduces the effects of backlash in the system. 
Above, we have seen that in most cases there is a residual vibration at the end of the motion, whose amplitude may be considerable. It is of course a continuation of the vibration that has built up during the motion period. However, it is possible for the residual vibration amplitude to be very small or zero, even though the inmotion amplitude might be very large. It depends on exactly where in a vibration cycle the motion finishes. Slight variations in the value of PeriodRatio, n, cam make an enormous difference to the residual vibration amplitude. The dynamic responses of two similar mechanisms are shown below. A high speed application, low PeriodRatio, has been chosen for the clarity. 

Possible REDUCTION of Residual Vibration with small INCREASE in speed, with certain Period Ratios 
The only difference between the two mechanisms is the rigidity of the input transmission. There is a slightly different PeriodRatio in each case. The vibration response of the design on the left side has a PeriodRatio, n=5.25. The one on the right has a PeriodRatio, n= 4.5. Surprisingly, there is an extremely large residual vibration with the better PeriodRatio, while the worse PeriodRatio produced practically no residual vibration. It is important to note that for these mechanism models no damping of any kind was used: the zero residual vibration is entirely due to the vibratory state of the mechanism at the beginning of the dwell period. The inmotion vibration is the same for both mechanisms and is of similar amplitude to the worst residual vibration. 
The value of PeriodRatio, n, can seldom be predicted exactly because of unforeseen errors in mass and rigidity. In any case, there is a great variation in speed, and therefore, periodratio, during the run up to speed, and during the machine shutting down. Although residual vibration may be undesirable for many reasons, it is not in itself the criterion for estimating the load capacity of a cam system. The peak load on the system is always at the peak (positive or negative) of an inmotion vibration cycle. 
Damping absorbs energy by the application of a force or torque. It has the effect of reducing, and possibly eliminating vibration. There are three basic kinds of damping that apply to cam and follower systems: •Viscous Damping : force that is proportional to velocity. •Hysteresis Damping : there is more energy to stress a material than the amount of energy that is returned to the system when it is released or unstressed. •Constant, or Coulomb Damping (Friction Damping) : more or less a constant force that opposes the motion. We find all of these are present, to some extent, in industrial machines. But, the first two, viscous and hysteresis, are usually so low as to have little or no effect on vibration. Why? It is futile to increase viscous damping deliberately to dissipate vibration energy. To damp vibration near to the beginning or the end of the motion where the nominal velocity is very low, or in the dwell period, the damping factor must be very high. In that case, energy will be wasted in the central, highspeed, period of the motion. Only that part of the viscous damping force that is related to the change of velocity contributes to the reduction of vibration, and the relative change of velocity due to vibration, even in the inmotion period, is quite small. A constant friction is often present in the form of friction in bearings, slideways etc. This force, however, has no damping effect at all unless there is a change direction. That is, a reversal of the direction of motion. Such reversals do not take place until the vibration velocity exceeds the nominal follower velocity, that is during and just before the dwell period. This form of damping is very effective in dissipating residual vibration, except in very high speed applications, but does not affect the majority of inmotion vibrations at all. If it is important in a particular application to eliminate vibration then the deliberate introduction of constant friction damping may be justified. However, it has the drawback that the follower can come to rest slightly out of position in the dwell period, where it is being held by the friction force in a strained condition, either undershooting or overshooting its target. 
Ca*Ct vs PeriodRatio for ModTrap, ModSine and Cycloidal MotionLaws 
If peak load is used as a criterion for selecting a motionlaw (although it is by no means the only criterion), then the product of Torsion Factor and Coefficient of Acceleration (= Ct × Ca), which is one measure of peak inertia load, identifies the best cam for a particular range of PeriodRatios. Using the equation of TorsionFactor, with the associated table of parameters for each motionlaw, we can plot: •Ct × Ca against PeriodRatio, n ... for ModTrap, ModSine and Cycloidal MotionLaws. We can use the plot to see which motionlaws is best based on PeriodRatio.

The comparison is only valid where there is good input transmission and the peak load is mainly an inertia load. In these circumstance it can be seen that the •ModTrap is the best choice for a PeriodRatio better than 6.4 •Cycloidal is best below 6.4. However, ModSine is by far the most useful of the three motionlaws because, although it produces a slightly higher peak load than ModTrap, it is also very much more tolerant to an elastic input transmission (low Period Ratio). The transmission systems of most industrial cams are such as to benefit from the use of the ModSine cam law in preference to the ModTrap. Nevertheless, there is a positive advantage to use the Cycloidal for systems with a low PeriodRatio. It is recommended that ModSine be the first choice for PeriodRatios above 6.4, and Cycloidal for ratios lower than that. 
The total backlash in a cam mechanism is the sum of all clearances, play or slack in the input and output transmissions (adjusted, if necessary, by gearing ratios), and in the cam track and cam follower. Typical examples are: slack chain drives, gear tooth clearances, oversize enclosed camtracks and worn follower roller bearings. There are many others. Any backlash, between the cam and follower or between any two forcetransmitting components in the cam system gives rise to a shock load when there is a reversal of force or torque. Backlash can be often be eliminated by applying sufficient external force with a spring or the payload weight, or even a friction force, to ensure that there is not a force reversal at the operating speed. Spring loaded, open cam track cam systems are quite common, but to be fully effective in eliminating backlash the spring must be applied not just to the follower but at a point in the output transmission that closes ALL of the significant clearances. This is illustrated in the image. 

When the direction of force reverses, typically at the crossover in high inertia applications, there is a moment when the payload is in freeflight after losing contact with the 'positive force surface' and before making contact with the 'negative force surface'. The magnitude of the force in making contact  the impact force  is related to the impact velocity of the contact surfaces. This, in turn, is related to the values of cam acceleration and jerk at this point in the motion. The process can be seen in the simplified and exaggerated image. All of the cumulative backlash in system is represented by the separation of the two cam profiles and the payload is represented at a concentrated point running along the profiles. The lower profile can only exert a positive contact force on the payload and the upper profile only a negative force. It is assumed for the purpose of this description that dynamic response vibration and any other detrimental effects are not significant compared with the impact force. The payload departs from the lower profile when its contact force becomes zero: this is when the inertia force due to the profile deceleration is about to exceed any other retarding force on the payload, such as friction, gravity. From this point on, the payload takes a freeflight path with a natural deceleration which is determined by external forces  such as friction  until it makes contact with the upper profile. It strikes the profile with an impact velocity  which is the difference between the freeflight velocity and the profile velocity. The freeflight is now a little slower that at the point of departure (due to its deceleration), and the profile velocity is even slower because the impact point is a little further along the profile (past the point of maximum profile velocity). It is obvious from the image that the cam follower separates and impacts in the deceleration phase of the motion and are separated by a time and distance very much dependent on the amount of backlash. How far into the declaration phase that these events occur is dependent on the magnitude of the external force in relation to the payload inertia force. In practice, the possible range of impact positions is fairly wide and occurs just after the nominal crossover point of the motion. To minimize the backlash impact force, it is necessary to have a motionlaw with a low value of acceleration for a long period near the middle of the motion. In general, this condition is fulfilled by motions that have a low jerk at crossover. The best 'lowimpact' Traditional MotionLaw is Simple Harmonic, but special low impact motionlaws have been developed which improve on this. Here again, the ModSine offers a very good compromise: it has a lower jerk at crossover than the ModTrap, or Cycloidal and a much better dynamic response vibration than Simple Harmonic. 

It is difficult to quantify the impact effect of backlash in all circumstances, but an attempt is made here to indicate its importance in certain cases. To take account of various sizes, speeds and loads of all kinds of cam mechanisms it is convenient to normalize the variables on the same basis as: 

: bk= Normalized backlash (from real backlash as a ratio of the output stroke) 

: dn = Normalized 'natural' deceleration  due to friction, for example, not the motion law. 

: Dn = Real natural deceleration of the payload  from linear model 

: Dn = Real natural deceleration of the payload  from angular model 

: v = Normalized impact velocity from real impact velocity and stroke 

The natural deceleration of the payload in freeflight can vary between zero (at crossover) and the maximum deceleration of the motionlaw: a higher deceleration would maintain contact with the positive force cam surface! The higher the natural deceleration, the lower the impact velocity. For any camlaw, a graph can be constructed to show how the normalized impact velocity varies with backlash and natural deceleration. All the motionlaws have similar impact velocities when the natural deceleration is zero and the backlash is large. The are considerable differences in impact velocities between motionlaws when the natural deceleration is medium to high. The motionlaws with zero jerk at crossover (SimpleHarmonic and ModSine) are better than the others at low values of backlash  the sort of backlash values likely to occur in practice. For very high speed mechanisms, the value of the normalized natural deceleration must be low, because it is proportional to the square of the motion period, T. which can be very small. Also, the backlash is in practice kept to a minimum by precision manufacture. Of the Traditional MotionLaws, ModifiedSinusoid (ModSine) gives a good compromise. A shock vibration is set up on impact, the amplitude of which depends on the impact velocity and the natural frequency of the system. We have already seen that natural frequency, f, can be calculated from the rigidity and stiffness of the system and the payload mass or inertia. The impact force or torque at the cam to follower contact point is given by : 

(Note, see above for V, Fd, Md ) In summary, the best way to minimize the detrimental effects of dynamic response is to : •use a 'Good' motionlaw  for example Modified Sinusoid, Cycloidal. •to ensure that the input and output transmissions are rigid / stiff •to ensure that the input and output transmissions are free of backlash. 