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

The Motion-Law Coefficients are the maximum values of each motion-derivative when the:

Input-period = 1 second

AND

Displacement = 1 unit (Linear or Angular Unit).

Velocity Motion-Coefficient

Acceleration Motion-Coefficient

Jerk Motion-Coefficient

Dynamic Torque-Coefficient

Actual Velocity and Actual Acceleration

Actual Maximum Velocity =

× Actual Output-Displacement  / Actual Period

Actual Maximum Acceleration =

× Actual Output-Displacement / Actual Period2

Actual Maximum Jerk =

× Actual Output-Displacement / Actual Period3


Input Torque Coefficient

Input Torque Coefficient

Note on Input Torque Coefficient

Like acceleration, it is not only the maximum of the torque that is important, 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 , worse, the load then drives the motor to increase its speed.

If the cam-shaft increases its speed, then the load also moves more quickly, and the torque fluctuation also increases.

Constant Power  

Power when Torque and Angular Velocity are constant.

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

Variable Power

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:

Power when Torque and Angular Velocity are varying.

Power when Linear-Force and Linear-Velocity are varying

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

Acceleration changes throughout the motion.

Reflected Inertia and Mass is referred back to the Power Source (usually the Drive Motor or Servomotor). In the general case, the reflected inertia varies throughout the motion.

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

Torque with varying Reflected-Inertia and Angular-Acceleration.

Torque with varying Reflected-Mass and Linear-Acceleration

When Reflected Inertia is 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:

Power-Coefficient

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

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