MotionDesigner Reference & User Interface

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 normalized 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 catalogs. 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,

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

Fi = mi × ai

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 : Pc= max of (vi × ai)

RMS of Power Coefficient: PCrms = √(∑((vi × ai)2)/n) ; i = equal increments; from 1 to n.

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 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. This tends to r Enter the Segment-Parameters, indicated below, in the boxes in the Segment Editor.
Motion Law Name	Coefficients		SCCA Parameters (Factors)
Motion Law Name	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
Motion Law Name	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