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# Motion-Law Coefficients

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# Motion-Law Coefficients

## MOTION-LAWS COEFFICIENTS

Use Motion-Law Coefficients DE: Kennwert to compare motions that you design with the Traditional Motion-Laws.

### Motion-Law Coefficients

 Velocity Motion Coefficient Acceleration Motion Coefficient Jerk Motion Coefficient

The Motion-Law Coefficients are the maximum motion-values for the motion-derivative when the motion has a:

Motion Period ( ) = 1 second

AND

Output Displacement ( ) = 1 Linear or Angular Unit

### Actual Velocity and Actual Acceleration

You can use the Motion-Law Coefficients to calculate the actual motion-values for each motion-derivative if you know the Actual Displacement ( ) and the Actual Period ( ) of the Motion:

 Actual Maximum Velocity = Actual Maximum Acceleration = Actual Maximum Jerk =

### Torque Coefficients

 Output Torque Coefficient - E.g. the torque rotating the swinging arm of a Cam-Follower Input Torque Coefficient - E.g. the torque at the output of a gearbox connected to the Cam-Shaft

Note on Input Torque Coefficient

The maximum torque of a motion-law is important. Just as important is the rate-of-change of torque at crossover from acceleration to deceleration.

A positive input torque on the cam-shaft winds-up (twists) the cam-shaft. Similarly, a negative torque on the cam-shaft winds-down (untwists) the cam-shaft. When the rate-of-change of torque is rapid, the winding and unwinding of the cam-shaft is also rapid.

When the Torque changes from a positive to a negative value - at the crossover - backlash is traversed. The speed of the drive-motor may increase rapidly as the torque is released from it and then, after the Backlash has been traversed, becomes driven by the load. If the speed of the motor does increase, then the motion-law is also distorted. The maximum deceleration increases when the driving-shaft momentarily increases its speed.

### Constant Power

 Power - constant Torque and Angular Velocity Power - constant Force and Linear-Velocity

#### Variable Power

Of course, the Torque and the Angular Velocity of the output shaft continuously change throughout the  motion. Thus, the Power at the output shaft also changes continuously.

Use the suffix to indicate an instant in the motion, then the Instantaneous Power, when calculated at the output is:

 Instantaneous Power - varying Torque and Angular Velocity Instantaneous Power - varying Force and Linear-Velocity

Total Load Torque or Load Force are found from values of inertia, mass, and acceleration.

However, the:

Acceleration continually changes throughout the motion - of course.

Load Inertia and Mass, referred to the driven-shaft, can be constant (e.g. Dial-Plate) or can continually change (e.g. Toggle mechanism).

In the general case, the Load Inertia and Mass that is reflected to the Cam-Follower varies throughout the motion.

Use the suffix 'i' to indicate any instant in the motion, the instantaneous Load Torque and Load Force are:

Also, the instantaneous Load Power is:

When reflected Load Inertia is not a function of the motion, the Power-Coefficient is less complex.

### Power Coefficient

 Power Coefficient | (i = equal increments; from 1 to n)

### MOTION COEFFICIENTS OF THE TRADITIONAL MOTION-LAWS

Motion-Law Name

Velocity Coefficient

Cv

Acceleration Coefficient

Ca

Torque Coefficient

Cc

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 (in the Segment Editor) of the Sine-Constant-Cosine Acceleration (SCCA) Motion-Law to give many of the popular motion cam-laws for industrial cams.

Motion-Law Name

Coefficients

SCCA Parameters (Factors)

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

### 3-HARMONIC MOTION-LAWS

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

Motion-Law Name

Coefficients

Harmonic

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

0