Motion-Law Coefficients

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Motion-Law Coefficients

MOTION-LAWS COEFFICIENTS

We can use Motion-Law Coefficients DE: Kennwert to compare Traditional Motion-Laws.

Motion-Law Coefficients

Velocity Motion-Coefficient

Acceleration Motion-Coefficient

Jerk Motion-Coefficient

Specifically, the Motion-Law Coefficients are the maximum values when the:

Motion Period = 1 second

AND

Output Displacement = 1 unit (Linear or Angular Unit)

Actual Velocity and Actual Acceleration

You can quickly 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 Output-Displacement  / Actual Period

Actual Maximum Acceleration =

× Actual Output-Displacement / Actual Period2

Actual Maximum Jerk =

× Actual Output-Displacement / Actual Period3

Torque Coefficients

Output Torque Coefficient

Input Torque Coefficient

--

 

Note on Input Torque Coefficient

Like acceleration, the maximum torque is important, and also how quickly the torque changes from a positive to a negative value - the 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 (untwist) in the other direction with a negative Torque on the cam-shaft. When the change in torque is rapid, the unwinding of the shaft is also rapid.

At the crossover, if there is backlash will be traversed. The Drive Torque becomes zero for a short while and the motor may accelerate rapidly.

Also, the load on the motor will tend to drive the motor as the load torque becomes negative. This will tend also  to increase the speed of the driving cam-shaft.

If the speed of the cam-shaft does increase, then the motion-law is also distorted. The maximum deceleration increases when the driving-shaft momentarily increases its speed.

Power

Constant Power  

Power when Torque and Angular Velocity are constant.

Power when Linear-Force and Linear-Velocity are constant.

Variable Power

The Torque and the Angular Velocity (or Force and Linear Velocity) continuously change throughout the  motion. Thus the instantaneous Power also changes continuously.

Thus, if we use the suffix 'i' to indicate any instant in the motion, then the Instantaneous Power, when calculated at the output is:

Power when load is rotating - e.g. a dial plate.

Power when load is sliding - e.g. a punch tool.

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 they can continually change (e.g. Toggle mechanism).

In the general case, the Load Inertia or Load Mass reflected to the Cam-Follower shaft varies throughout the motion.

Thus, if we use the suffix 'i' to indicate any instant in the motion, then the instantaneous Load Torque and Load Force are:

Load Torque with changing Load Inertia and Angular Acceleration.

Load Force with changing Load Mass and Linear Acceleration

Also, the instantaneous Load Power is:

Load Power with changing Load Inertia, Angular Acceleration, and Angular Velocity.

Load Power with changing Load Mass, Linear Acceleration, and Linear Velocity.

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

The instantaneous Load Power, with constant reflected Load Inertia or Load Mass is:

Load Power with constant Load Inertia, Angular Acceleration, and Angular Velocity.

Load Power with constant Load Mass, Linear Acceleration, and Linear Velocity.

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 Segment Parameters (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. The most common variations are symmetrical, with a zero-acceleration / constant-velocity in the middle-section of the motion.

 

Motion-Law Name

Coefficients

SCCA Parameters (Factors)

Velocity Coeff.

Cv

Acceleration Coeff.

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 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