﻿ Motion-Law Coefficients

# Motion-Law Coefficients

## MOTION LAW COEFFICIENTS

A common question is: Which motion-law 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 Motion-Law . It is the the law that can satisfy many motion requirements and also have motion-continuity. However, when the motion requirements are simple, it is possible to compare the Traditional Motion-Laws, and decide which is 'probably' the best motion for a particular application.

One way to compare the Traditional Motion-Laws is to compare their Motion-Law Coefficients.

### MOTION-LAW COEFFICIENTS

It is easier to identify differences in Motion-Laws when we look at their velocity and acceleration graphs than it is with displacement graphs. Thus, the 'Motion-Law Coefficients' apply mostly to velocity and acceleration curves.

Velocity and Acceleration Motion-Law Coefficients

Motion-Law Coefficients are the maximum velocity, maximum acceleration [and sometimes maximum jerk] of a motion-law 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 motion-law when its stroke is '1' and its period is '1' second.
 • Acceleration Coefficient, Ca = Maximum Acceleration of a motion-law 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. Heinz-Automation.

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.

### How to use the Motion-Law Coefficient to get the actual Velocity and Acceleration

It is easy to use the Motion-Law 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 Period2

Actual Period = Index Angle*N/60

Motion-Law Name

Non-dimensional Maximum
Velocity

Cv

Non-dimensional Maximum Acceleration

Ca

Non-dimensional Input Torque Coefficient

Cc

Non-dimensional 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

### Sine-Constant-Cosine Acceleration + Constant Velocity [SCCA+CV].

Motion Law Name

Coefficients

SCCA

Segment Parameters

Velocity Coefficient

Cv

Acceleration Coefficient

Ca

a

b

c

Modified-Sine
CV 0%

1.760

5.528

0.25

0

0.75

Modified-Sine
CV 20%

1.528

5.999

0.2

0

0.6

Modified-Sine
CV 33%

1.404

6.616

0.1667

0

0.5

Modified-Sine
CV 50%

1.275

8.0127

0.125

0

0.375

Modified-Sine
CV 66%

1.168

11.009

0.0833

0

0.25

Cycloidal
CV 50%

1.333

8.378

0.25

0

0.25

Trapezoidal Velocity
CV 33%

1.5

4.5

0

0.6667

0

### Three Harmonic Series

You can use the Triple Harmonic Motion Law to give alternatives to some of the popular motion-laws.

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

3-Harmonic
Modified Trapezoidal

2.0

5.16

5.96

0

0.9696

3-Harmonic
Modified Sine

1.72

6.07

5.1968

1.7690

0.6057

3-Harmonic
Zero-Jerk at Crossover

2.0

9.42

9*Π/4

0

-3*Π/4

### MOTION-LAWS COMPARED - for Initial Selection

We can give relative ratings to the most common Traditional Motion-Laws. 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 Dwell-Rise 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 cam-operated indexing mechanisms.

Additionally, you can look at the Motion-Law Coefficients of the common cam motion-laws. 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

#### Explanatory Notes

 Peak Acceleration This merit rating applies to the nominal maximum output acceleration during the motion period, calculated by the motion-law equation. Output Vibration Output 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 Velocity Peak Velocity is the nominal maximum output velocity during the motion period, calculated by the motion-law equation. Its value is also increased by superimposed vibration. Impact / Backlash Impact 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 Torque The 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. Motion-Laws with good, that is low, acceleration do not necessarily have good input torque. Input Vibration The elasticity and backlash of the input transmission can cause serious 'over-run' and 'input-vibration'. 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 Vibration Residual 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 motion-law is vital in some cases.

### Motion Laws Compared - another table.

This table includes 'Period Ratio' as a parameter with which you can select a Motion-Law. It is from an ESDU* item.

* Engineering Science Data Unit, published in the UK. Tutorial and Reference Help Files for MechDesigner and MotionDesigner 13.2 + © Machine, Mechanism, Motion and Cam Design Software by PSMotion Ltd