Dynamic Response: Vibration, Period-Ratio, Damping and Backlash

The dynamic response of a payload to a motion law is a function of the motion-law itself, the stiffness of the input and output transmissions, as well its inertia and backlash.

tog_minusDynamic Response

When a cam moves a payload, the shape of the cam gives 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 Cam-Followers Motion - the command - and an exaggerated diagram of payload's motion - its response

The image shows the Cam-Followers Motion - the command - and an exaggerated diagram of payload's motion - its response

The payload has mass and inertia. 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 a real mass or inertia.

The elastic transmission system distorts when subjected to the force or torque. The motion of the payload, its dynamic response, is a distortion of the command motion 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 cam-follower 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, Stiffness.

Motion Response and Progression of the Payload in response to the Motion Command.

The Payload's motion starts later than the start of that of the Cam-Follower's motion. The command and response can be described thus:

The Cam-Follower begins to move.

The Payload will not move until the force that is imparted to it overcome's the Payload's inertia.

Thus, the force 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 Cam-Follower until the force on the Payload from the Spring gives sufficient acceleration to it such that its velocity eventually exceeds that of the Cam-Follower'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 Cam-Follower displacement.
However, at that instant, the Payload is moving faster than the Cam-Follower.
The Payload overshoots and leads the Cam-Follower displacement. As the spring continues to expand, the Payload's velocity decreases, relative to that of the Cam-Follower, until their velocities are the same.
The velocity continues to decrease until the displacement of the Payload and Cam-Follower 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 Cam-Follower.


Note: To prevent confusion with the term 'oscillating' or 'oscillation' of swinging arm follower mechanisms, the oscillation that is associated with the 'cam command and cam-follower response' is referred to as a 'Vibration' rather than 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 inertia-force is proportional to acceleration, the oscillation 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 motion-law, 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 mass-spring system. All mass-spring 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 moment-of-inertia and torsional rigidity).

Period-Ratio [n] is the ratio of the motion-period (stroke time) to the vibration-period (time for one vibration cycle) -see 'Natural Frequency and Period-Ratio' below.

Period-Ratio is one of the most important parameters in the estimation of the peak load on a cam.

For a given motion-law, the maximum amplitude of the dynamic response is greatly dependent on the Period-Ratio of the system.

Also, for any given Period-Ratio, the maximum amplitude of the dynamic response is greatly dependent on the Motion-Law.

Period-Ratio and the Application Speed.

We can use Period-Ratio to designate the 'speed' of an application.

High-speed applications are those with a low Period-Ratio, say less than 8.

Slow-speed applications are those with a high Period-Ratio, say more than 20.

The choice of motion-law is less critical with 'slow-speed applications'.

Most cam mechanisms in modern industrial machinery run with a Period-Ratio less than 20. There are few with a Period-Ratio less than 5.

EXAMPLES: Modified-Trapezoidal (Mod-Trap), Cycloidal and Modified-Sinusoid (Mod-Sine) Motion-Laws

Dynamic-Response of a Mechanism to a Modified Trapezoid Motion-Law with Good and Bad Transmission Design

 Mod-Trap Response with High and Low Period Ratio

Dynamic-Response of a Mechanism to a Cycloidal Motion-Law with Good and Bad Transmission Design

Cycloidal Response with High and Low Period Ratio

Dynamic-Response of a Mechanism to a Modified Sinusoid Motion-Law with Good and Bad Transmission Design

 Mod-Sine Response with High and Low Period Ratio

Dynamic-Response 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 motion-laws and two different Period-Ratios. They show the nominal acceleration (system with no vibration) and the acceleration response (the dynamic-response) plotted against time for one motion-period and one dwell period.

The diagrams at the top represent a mechanism with a fairly rigid output transmission. The Period-Ratio is 13.5.
The diagrams at the bottom represent the same mechanism, but the Period-Ratio 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 motion-laws produce a vibration with low amplitude when the Period-Ratio is high.
2.A low Period-Ratio will give a high vibration amplitude (force or torque). It may be higher than the nominal maximum of the motion-law, 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 Mod-Trap motion has a peak load less than either the Mod-Sine or Cycloidal, by virtue of its lower maximum acceleration. If transmissions are bad, however, Mod-Sine is better than Mod-Trap (note the negative peak of the Mod-Trap) and Cycloidal is better than the Mod-Sine, especially as regards the residual vibration.

All three motion-laws shown are considered to be 'good' laws for high-speed mechanisms.

Dynamic-Response of a Mechanism to a Simple Harmonic Motion-Law with Good and Bad Transmission Design

Simple Harmonic Motion Response with High and Low Period Ratio

Dynamic-Response of a Mechanism to a Parabolic Motion-Law with Good and Bad Transmission Design

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 motion-law, are shown to the left with the Parabolic (Constant Acceleration and Retardation) and Simple Harmonic Motions.

These motion-laws 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 Period-Ratio).

Both motion-laws have a step-change - that is, a discontinuity - in the acceleration profiles. The step-change in the acceleration 'shocks' the mechanical system, and incites the vibrations in it much more than 'modern' motion-laws that do not have a step-change in acceleration.

The value of peak force or torque in both cases is much the same, at all Period-Ratios.This is because the vibrations are caused mainly by the step-change in acceleration.

The peak force or torque is higher than the peak force or torque of the modern dynamically favorable motion-laws.

tog_minusNatural Frequency and Period-Ratio

For a simple Spring-Mass System, as shown in the representation of the cam-system above, the Natural Frequency is:



for a Linear System.



for a Rotary System.


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]

Period-Ratio is:


- or -

n = Period of Motion / Natural Period of Vibration


T = duration of motion-period[s], the time take of the motion segment in the Machine-Cycle.

tog_minusTorsion Factor

The Peak Torque (positive or negative) from the superimposed 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 Motion-Law is known and Period-Ratio can be estimated.

Torsion Factor, Ct, is the ratio of the peak torque to the nominal peak torque of an undamped cam system with a wholly inertial load.

When a cam-mechanism has the same motion-law, its Torsion-Factor is dependent only on its Period-Ratio.

Torsion Factor Equation

The Torsion-Factor Equation estimates the ratio of peak (or actual) acceleration to nominal (or motion-law) acceleration. You need the Period-Ratio and the Motion-Law.

Curves have been plotted of Torsion-Factor, Ct, against Period-Ratio, n, for the SCCA family of Motion-Laws. Their peak value of Torsion-Factor is empirically given by the equation:

C t = p n q + r :

Torsion-Factor Equation for Cams






Note: The term Torsion-Factor is historical. It was first used with indexing mechanisms where the transmission was always rotary, hence torsion.





Parabolic( Constant Acc-Dec)




Simple Harmonic Motion (SHM)




Modified Trapezoid (Mod-Trap)




Modified Sinusoid (Mod-Sine)




























MSC-20, MSC-33, MSC-50, MSC-66 = Modified Sinusoid with 20%, 33%, 50%, 66% Constant Velocity, all symmetric

CYCC-50 = Cycloidal with Constant Velocity of 50%, symmetric

Period-Ratio > 3


Parabolic Motion-Law: Torsion-Factor Ct=4 at all Period-Ratios. Thus, it is a poor motion for any application, even slow speed, with good transmissions.

Simple Harmonic Motion-Law: Torsion-Factor Ct=2 at all Period-Ratios.

This makes it not as good as the Mod-Trap, Mod-Sine or Cycloidal for many applications.

The high Torsion-Factor of Simple Harmonic Motion is due to the sudden change of acceleration at the 'start' and 'end' of the Motion-Law, when you use this motion-law 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 Cross-Over (transition between acceleration and deceleration) of Simple-Harmonic-Motion also reduces the effects of backlash in the system.

tog_minusResidual Vibration

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 in-motion amplitude might be very large.

It depends on exactly where in a vibration cycle the motion finishes. Slight variations in the value of Period-Ratio, 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 Period-Ratio, has been chosen for the clarity.

Possible REDUCTION of Residual Vibration with small INCREASE in speed, with certain Period Ratios

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 Period-Ratio in each case. The vibration response of the design on the left side has a Period-Ratio, n=5.25. The one on the right has a Period-Ratio, n= 4.5. Surprisingly, there is an extremely large residual vibration with the better Period-Ratio, while the worse Period-Ratio 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 in-motion vibration is the same for both mechanisms and is of similar amplitude to the worst residual vibration.

The value of Period-Ratio, 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, period-ratio, 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 in-motion 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 : more energy that is used to stress a material than is released when it is unstressed.
Constant, or Coulomb Damping (Friction Damping) : more or less 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.


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, high-speed, 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 in-motion period, is quite small.

A constant friction is often present in the form of friction in bearings, slide-ways 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 in-motion 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 under-shooting or over-shooting its target.

tog_minusSelecting a Motion-Law based on Peak Load Criterion


Ca*Ct vs Period-Ratio for Mod-Trap, Mod-Sine and Cycloidal Motion-Laws

If peak load is used as a criterion for selecting a motion-law [although it is by no means the only criterion], then the product of Torsion Factor and Coefficient of Acceleration [= Ct x Ca], which is one measure of peak inertia load, identifies the best cam for a particular range of Period-Ratios.

Using the equation of Torsion-Factor, with the associated table of parameters for each motion-law, we can plot:

Ct x Ca against Period-Ratio, n

... for Mod-Trap, Mod-Sine and Cycloidal Motion-Laws.

We can use the plot to see which motion-laws is best based on Period-Ratio.


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

Mod-Trap is the best choice for a Period-Ratio above 6.4 and
Cycloidal is best below 6.4.

However, Mod-Sine is by far the most useful of the three cam laws because, although it produces a slightly higher peak load than Mod-Trap, 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 Mod-Sine cam law in preference to the Mod-Trap. Nevertheless, there is a positive advantage to use the Cycloidal for systems with a low Period-Ratio.

It is recommended that Mod-Sine be the first choice for Period-Ratios 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 cam-tracks and worn follower roller bearings. There are many others.

Any backlash, between the cam and follower or between any two force-transmitting 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 cross-over in high inertia applications, there is a moment when the payload is in free-flight 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 free-flight 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 free-flight velocity and the profile velocity. The free-flight 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 cross-over point of the motion.

To minimize the backlash impact force, it is necessary to have a motion-law 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 cross-over.  The best 'low-impact' Traditional Motion-Law is Simple Harmonic, but special low impact motion-laws have been developed which improve on this. Here again, the Mod-Sine offers a very good compromise: it has a lower jerk at cross-over than the Mod-Trap, 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

Bk = Real cumulative [total] backlash referred to the cam output

Y = Output Stroke of the motion, linear or angular

T = The full motion period [time]

W = Mass of the Payload, referred to the cam

Fd = Decelerating Force on the payload, referred to the cam output

Md = Decelerating Torque on the payload, referred to the cam output

I = Inertia of the payload, referred to the cam output

V = Actual impact velocity, linear or angular

The natural deceleration of the payload in free-flight can vary between zero (at cross-over) and the maximum deceleration of the motion-law: a higher deceleration would maintain contact with the positive force cam surface! The higher the natural deceleration, the lower the impact velocity. For any cam-law, a graph can be constructed to show how the normalized impact velocity varies with backlash and natural deceleration.

All the motion-laws have similar impact velocities when the natural deceleration is zero and the backlash is large.

The are considerable differences in impact velocities between motion-laws when the natural deceleration is medium to high.

The motion-laws with zero jerk at cross-over (Simple-Harmonic and Mod-Sine) 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 Motion-Laws, Modified-Sinusoid [Mod-Sine] 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' motion-law - 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.

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