A common question is: Which motionlaw is best?
When your motion requirements are complex, possibly with many segments, and there are different position, velocity and acceleration constraints, we will recommend the Flexible Polynomial MotionLaw . It is the the law that can satisfy many motion requirements and also have motioncontinuity. However, when the motion requirements are simple, it is possible to compare the Traditional MotionLaws, and decide which is 'probably' the best motion for a particular application.
One way to compare the Traditional MotionLaws is to compare their MotionLaw Coefficients.
It is easier to identify differences in MotionLaws when we look at their velocity and acceleration graphs than it is with displacement graphs. Thus, the 'MotionLaw Coefficients' apply mostly to velocity and acceleration curves.
Velocity and Acceleration MotionLaw Coefficients
MotionLaw Coefficients are the maximum velocity, maximum acceleration [and sometimes maximum jerk] of a motionlaw when the output has a stroke of '1' [linear or angular displacement unit] and the time taken has a period of '1' [second]. The coefficients are dimensionless.
•  Velocity Coefficient, Cv = Maximum Velocity of a motionlaw when its stroke is '1' and its period is '1' second. 
•  Acceleration Coefficient, Ca = Maximum Acceleration of a motionlaw when its stroke is '1' and its period is '1' second. 
Input Torque Coefficient
The Input Torque Coefficient, Cc
•  Input Torque Coefficient, Cc = max(vi × ai)/Ca 
Notes
The Torque Coefficient is sometimes called the Power Coefficient  E.g. HeinzAutomation.
Note: Indexer catalogues often use Q [e.g. Sankyo Indexers] rather than Cc . However, later in this section, we use Q for 'inertia ratio'. See 'Cam Mechanism'.
Power Coefficient
Power Coefficient is found from the maximum product of the acceleration and velocity, calculated together at each instant in the motion. Thus,
•  Power Coefficient, Cp= max(vi × ai) 
Note, Cp ≠ Cv × Ca because maximum values of velocity and acceleration are at different phases of the motion.
It is easy to use the MotionLaw Coefficients to calculate the actual maximum velocity and the actual maximum acceleration when the stroke and period are not equal to '1':
•  Actual Maximum Velocity = Cv × Actual Stroke / Actual Period 
•  Actual Maximum Acceleration = Ca × Actual Stroke / Actual Period 2 
Actual Period = Index Angle*N/60
MotionLaw Name 
Nondimensional Maximum Cv 
Nondimensional Maximum Acceleration Ca 
Nondimensional Input Torque Coefficient Cc 
Nondimensional Power Coefficient Cp 

Constant Acceleration, Parabolic 
2 
4 
2 
8 
Simple Harmonic 
1.570796 (π/2) 
4.934803 (π2/2) 
0.785 
3.8758 
Cycloidal 
2 
6.283185 
1.298 
8.1621 
Modified Trapezoid 
2 
4.888124 
1.655 
8.0894 
Polynomial 345 
1.875 
5.773503 
1.159 
6.6925 
Polynomial 4567 
2.1875 
7.5132 
1.431 
10.750 
Modified Sine 
1.759603 
5.527957 
0.987 
5.4575 
Motion Law Name 
Coefficients 
SCCA Segment Parameters 


Velocity Coefficient Cv 
Acceleration Coefficient Ca 
a 
b 
c 

ModifiedSine 
1.760 
5.528 
0.25 
0 
0.75 
ModifiedSine 
1.528 
5.999 
0.2 
0 
0.6 
ModifiedSine 
1.404 
6.616 
0.1667 
0 
0.5 
ModifiedSine 
1.275 
8.0127 
0.125 
0 
0.375 
ModifiedSine 
1.168 
11.009 
0.0833 
0 
0.25 
Cycloidal 
1.333 
8.378 
0.25 
0 
0.25 
Trapezoidal Velocity 
1.5 
4.5 
0 
0.6667 
0 
You can use the Triple Harmonic Motion Law to give alternatives to some of the popular motionlaws. Enter the First and Second Harmonic in the Segment Editor. MechDesigner calculates the Third Harmonic. 


Motion Law Name 
Coefficients 
Harmonic Segment Parameters 

Velocity Coefficient Cv 
Acceleration Coefficient Ca 
1st 
2nd 
3rd 

3Harmonic 
2.0 
5.16 
5.96 
0 
0.9696 
3Harmonic 
1.72 
6.07 
5.1968 
1.7690 
0.6057 
3Harmonic 
2.0 
9.42 
9*Π/4 
0 
3*Π/4 
We can give relative ratings to the most common Traditional MotionLaws. You can use the ratings to help you select a law at the initial stage of a machine design. The ratings range from 1 (relatively bad) to 5 (excellent). The ratings apply to DwellRise Dwell type motions.
If we look at the table, it can be seen that, of the laws listed, that the Modified Sine(MS) is the best for general purposes. Its particular merit is that it is very tolerant of a bad input drive and transmission (elasticity, backlash, wear, low inertia). It is frequently the first choice of cam designers and is almost always used by commercial manufacturers of camoperated indexing mechanisms.
Additionally, you can look at the MotionLaw Coefficients of the common cam motionlaws. These indicate the relative values of their Velocity Coefficient and Acceleration Coefficient.
Cam Law Designation 
Peak Acceleration 
Output Vibration 
Peak Velocity 
Impact 
Input Torque 
Input Vibration 
Residual Vibration 
Parabolic 
5 
1 
2 
1 
1 
1 
1 
Simple Harmonic 
3 
1 
4 
4 
5 
2 
1 
Modified Trapezoid 
3 
3 
2 
2 
2 
3 
3 
Modified Sinusoid 
2 
4 
3 
4 
4 
4 
4 
Cycloidal 
1 
5 
2 
3 
3 
4 
5 
Peak AccelerationThis merit rating applies to the nominal maximum output acceleration during the motion period, calculated by the motionlaw equation. 
Output VibrationOutput vibration is superimposed on the nominal output acceleration, thereby increasing the nominal peak value. The vibration severity depends on the elasticity and operating speed of the mechanism. The merit rating applies to mechanisms of average rigidity running at fairly high speed. 
Peak VelocityPeak Velocity is the nominal maximum output velocity during the motion period, calculated by the motionlaw equation. Its value is also increased by superimposed vibration. 
Impact / BacklashImpact forces occur at the locations of backlash in the mechanism when the changeover from acceleration to deceleration occurs. The severity of the impact depends on how gradually the changeover takes place. That is, how low the jerk is at point of impact. Strictly speaking, it is the changeover from positive to negative force or torque that matters, but in most high speed systems, that almost coincides with the acceleration changeover. 
Input TorqueThe nominal input torque of a mechanism varies throughout the motion period and is a function of the output load profile, and the velocity pattern. The peak acceleration and the peak velocity do not coincide and neither coincides with the peak input torque. MotionLaws with good, that is low, acceleration do not necessarily have good input torque. 
Input VibrationThe elasticity and backlash of the input transmission can cause serious 'overrun' and 'inputvibration'. This is when the sudden reversal of the input torque at the changeover from acceleration to deceleration  or load  causes the cam to jump forwards before it can transmit a decelerating force to the output. The more gradual that the nominal input torque changes sign, the less severe is the overrun and its consequences. 
Residual VibrationResidual Vibration takes place in the dwell period immediately following the motion period in high speed or elastic systems. Its amplitude depends on the vibration generated during the motion period, and the degree of damping present in the output transmission. It is very difficult to add sufficient damping to high speed mechanisms to eliminate residual vibration, so the choice of a motionlaw is vital in some cases. 
This table includes 'Period Ratio' as a parameter with which you can select a MotionLaw. It is from an ESDU* item.
* Engineering Science Data Unit, published in the UK.