Dynamic Response and Considerations

We use vibration terminology to describe the dynamic performance and response (or output) of a mechanical system to a regular, repeating motion design (or input).

Nominal Motion: The intended [input] motion that you want the mechanical part or system to follow. It is given by your motion design.

Transient Motion: The actual [output] motion that the mechanical part or system has during your motion.

Residual Vibration: The actual [output – input] motion that the mechanical part or system has after your motion. This term usually applies to the dwell period at the end of an indexing motion.

Dynamic Performance

All mechanical systems have a degree of elasticity. In applications, the elasticity means that the resulting actual accelerations at the driven tooling (the output and the point where the motions is actually applied to the product) will be greater than the nominal acceleration as applied by the cam-follower (the input and the point where the motion is intended) and predicted by the Motion-Law and the motion design [as designed in MotionDesigner] . This issue is a reality of all mechanical systems.

Transient oscillations, or vibrations, occur in all mechanical systems when the input motion-law has a discontinuity in the function at any motion derivative, particularly lower derivatives such as velocity and acceleration.

The Dynamic Performance, or Dynamic Response, is a result of many factors.

The speed of the drive shaft (cycle speed) is one of the factors which determine the Period Ratio [speed category] of the mechanism. Others being:

the natural frequency,
the segment duration angle and
the range of movement (lift) - this being the LEAST significant.

Besides the effect of these factors on jerk and the resultant distortion of the actual motion produced, the maximum allowable operating speed is also influenced by the accuracy of profile manufacture. The adverse effect of local profile inaccuracies is aggravated as the speed of operation increases but, in some cases, may be compensated by the flexibility of the system. Very stiff mechanisms are to be preferred in all cases, provided the cam profile is smooth.

Account must be taken of the compliance of the following system.

Residual Vibration

Vibration levels don't stop when the segment is finished! Residual vibration levels should be considered. If high vibration levels are experienced by the system in a dwell immediately after a segment, then frequently the machine tooling may well be out of position as another mechanism tries to interact with it. Some designers then redesign the motion with an even longer dwell (= shorter motion segment) creating even more vibrations and longer to wait. This is not good motion-design.

Period Ratio.

In order to provide an accurate definition of the operating characteristics of a mechanism, including induced vibration of the following system, use the non-dimensional parameter called PERIOD RATIO.

The period ratio is the ratio of the

Period [duration] of the motion segment : Period [duration] of the fundamental vibration of the following system.

Here, period might mean the segments duration

High values of period ratio [>10] indicate stiff, low mass systems, operating at moderate or low shaft speeds.

Low values of period ratio [<5] occur in systems with compliant followers, a large driven mass or high shaft speed which produces a short motion segment duration.

Period Ratio < 5 represents very compliant or 'high speed mechanisms'

Mechanisms operating at moderate speeds with fairly stiff follower systems would be indicated by Period Ratio values of approximately 10.

For Period Ratio values of more than 20, the acceleration at the driven mass approaches the nominal acceleration defined by the Motion-Law and the dynamic response of the follower system may be neglected when determining actual accelerations.

However, it is very important to note:

In the case of the Simple-Harmonic-Motion and the Constant-Acceleration motion-laws, the actual acceleration will always be significantly higher than the nominal value, no matter what the period ratio or machine speed.

Motion-Laws or motions that exhibit discontinuities in acceleration (infinite jerk) at any point in their cam profile, produce particularly severe vibrations at the driven mass.

The actual acceleration/deceleration at the driven mass will be up to 4 times the nominal acceleration when driven with by a (cam) motion with infinite jerk. The kinematically predicted acceleration ignores the flexibility or compliance in the following mechanical system.

Pressure Angle Considerations.

Clearly, this relates to Cam designs, and the influence of Motion-Law on the Pressure Angle of Cams.

The pressure angle and the way in which it varies throughout a DRD motion segment depends upon the basic dimension of the cam, the type of the follower (roller, or flat faced) and, to a lesser extent on the particular cam law employed.

Drive Torque

The operating torques for a cam system depends on the Motion-Law and may influence the choice of the most suitable one.

There are three Torque factors to consider:

Constant Load Factor

This is the component of torque required to overcome the constant component of the external load on the cam follower! The constant load is usually due to the weight of the following system (or the referred weight), the load at the start of the motion due to any spring constraint, and also friction. This is usually the least significant of the three, for a 'normal' cam driven system - but it depends on the other two!

Inertia Torque Factor

This is the component of torque required to accelerate the mass of the follower assembly. It is usually the most significant in normal systems - but that depends on the others!

Spring Stiffness Torque Factor

This component is due to the linear change of spring constraint with the follower movement. This factor is based on the minimum spring force required at the point of maximum deceleration to maintain contact between the follower and the cam. The spring, might be an 'air-cylinder' which might be either a 'constant force' or as a 'fixed air mass'.

The magnitude of the total drive torque gives some indication of the amount of torsional deflection in the drive shaft and therefore the amount of segment motion distortion that may occur. The distortion will tend to attenuate (reduce) the accelerations and amplify (increase) the decelerations of the follower. An abrupt reversal of torque (e.g. due to backlash in the drive train) will result in torsional vibration in the driving shaft which will then be transmitted through the cam to the driven system. Such distortions can be reduced by increasing the torsional stiffness of the shaft and by increasing the mass moment of inertia, especially near the cam.

Motion distortion can result from shaft bending due to cam contact forces. The shaft size and the position of its support bearings should be chosen to minimise any distortion of the cam-shaft.

Motion distortion can also result from the deflection of the cam-follower support shaft due to the pressure angle and cam contact-force.

Jerk

The jerk function is related to the rate-of-change of the strain-energy of the system throughout the motion design.

Jerk should not be considered in isolation; it must be considered with system rigidity/stiffness and operating speed of the machine system.

The machine designer is usually most interested in jerk:

at the start and end of a motion or segment
the peak value of jerk
the value of jerk as the acceleration is changing sign from positive to negative, or vice versa. This is called the 'crossover jerk'. Low values crossover jerk are beneficial for systems with backlash. Backlash will typically traverse once the velocity has reached its peak and begins to reduce. Standard 'Rise' Motions, between dwells, with low 'crossover jerk' will typically have low peak velocity.  A low peak velocity will mean a reduced impact as the backlash traversal is completed.
for many of the Traditional Motion-Laws, jerk will change instantaneously from zero to some finite value immediately after the motion segment starts. This will tend to induce vibrations in the mechanical system being driven.
other motions will start and end with 'infinite jerk' - this is an even worse condition.
however, motions with zero jerk at the start, will have a large maximum acceleration. The mechanical system will 'nominally' [without vibrational considerations] strain more when the acceleration is increased.

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