﻿ The Motion-Law Coefficients

# The Motion-Law Coefficients

## MOTION LAWS COEFFICIENTS: Cv ,Ca, Cj,Cc ,Pc

We can compare Traditional Motion-Laws with their 'Motion Law Coefficients DE: Kennwert'.

To obtain a Motion Law Coefficient, we normalize the motions, so that the X-axis has a range from 0 to 1, and the Y-axis position graph also has a range of 0 to 1.  Then, the motion-derivatives of the functions automatically become normalised values.

The three Motion-Law Coefficients, Cv , Ca , and Cj , which are the maximum velocity, acceleration and jerk that a motion-law has when the range of the output [linear or angular unit] and input = '1' [second].

 • Velocity Coefficient, Cv = Maximum Velocity of a motion-law when the output has a stroke and period of '1'.
 • Acceleration Coefficient, Ca = Maximum Acceleration of a motion-law when the output has a stroke and period of '1'.
 • Jerk Coefficient, Cj = Maximum Jerk of a motion-law when the output has a stroke and period of '1'.
 • Dynamic Torque Coefficient, CT = Maximum of the Product of Velocity X Acceleration of a motion-law when the output has a stroke and period of '1'.

### Actual Velocity and Actual Acceleration

To calculate the actual velocity and acceleration when the stroke and period are not '1':

 • Actual Maximum Velocity = Cv × Actual Stroke ÷ Actual Period
 • Actual Maximum Acceleration = Ca × Actual Stroke ÷ Actual Period2
 • Actual Maximum Jerk = Cj × Actual Stroke ÷ Actual Period3

### Input Torque Coefficient, Cc

The Input Torque Coefficient, Cc [also given as 'Q' in indexer catalogues. However, we use Q as the 'inertia ratio' [See the MechDesigner manual].

 • Input Torque Coefficient, Cc = max(vi × ai)/Ca
 Note: Like acceleration, it is not only the absolute maximum of the Torque given by the 'Input Torque Coefficient', but also how quickly the torque changes from a 'positive' to a 'negative' torque - 'rate of change of Torque', particularly at crossover from Acceleration to Deceleration'. A positive Torque on the cam-shaft will tend to 'wind-up' [twist] the cam-shaft, and similarly, the cam-shaft will 'wind-down' in the other direction during negative Torque on the cam-shaft. When the change in torque at the 'cross-over' is rapid, then the winding and unwinding of the shaft in one direction and then the other is also rapid. The load on the motor that drives the shaft reduces to zero, and then, worse, the load then drives the motor [when the efficiency of the drive is high]. There is then a greater tendency for the input transmission to 'over-run', or increase its speed. If the cam-shaft increases its speed, then the load also moves more quickly, and the torque fluctuation also increases.

### Power Coefficient, Pc

When Torque and Angular Velocity are constant, then Power is P = τ × ω

Similarly, when Linear Force and Linear Velocity are constant, then Power is:

P = F × v

Clearly, when the Torque and Angular Velocity [or Force and Linear Velocity] both change continuously throughout the motion, the instantaneous Power also changes continuously.

Thus, if we use the suffix 'i' to indicate any instant in the motion, then the 'instantaneous' Power is:

Pi = τi × ωi,

or

Pi = Fi × vi

Torque and Force are found from instantaneous values of reflected inertia and acceleration.

 • Acceleration changes throughout the motion.
 • Reflected Inertia is the inertia referred back to the Power Source [usually the Drive Motor or Servomotor]. This would vary for any mechanism that has a variable transmission ratio between the drive and payload. Again, in the general case, the reflected inertia varies throughout the motion.

τi = Ii × αi,

or

Fi = mi × ai

When Reflected Inertia or Mass are not a function of the motion, the Power Coefficient is normalized by ignoring their values. We can use the instantaneous values of angular acceleration and linear acceleration. Thus:

### MOTION COEFFICIENTS OF THE TRADITIONAL MOTION-LAWS

Motion-Law Name

Non-dimensional Maximum Velocity

Cv

Non-dimensional Maximum Acceleration

Ca

Non-dimensional Input Torque Coefficient

Cc

Non-dimensional Power Coefficient

Pc

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 3-4-5

1.875

5.773503

1.159

6.6925

Polynomial 4-5-6-7

2.1875

7.5132

1.431

10.750

Modified Sine

1.759603

5.527957

0.987

5.4575

### SINE-CONSTANT-COSINE ACCELERATION (SCCA) with CONSTANT VELOCITY

Edit the Segment Parameters (Fractions) of the (SCCA) Motion Law to give many of the popular motion cam-laws for industrial cams. The most common variations are symmetrical, with a zero-acceleration / constant-velocity in the middle-section of the motion. This tends to r

Enter the Segment-Parameters, indicated below, in the boxes in the Segment Editor.

Motion Law Name

Coefficients

SCCA Parameters (Factors)

Velocity Coefficient

Cv

Acceleration Coefficients

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

### 3-HARMONIC MOTION-LAWS

Edit the Segment Parameters of the Triple Harmonic Motion Law to give alternatives to some of the popular motion-laws.

Enter the Segment-Parameters for each Harmonic in the boxes in the Segment Editor.

Motion Law Name

Coefficients

Harmonic

Velocity Coefficient

Cv

Acceleration Coefficients

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

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