Rockwell Automation Publication 750-AT006D-EN-P - January 2022
9
Chapter 1 Background
Compliance
Compliance is the elastic deformation of mechanical components. In rotating machinery, it is a twisting of these components that stores
and releases energy in the form of torque. More flexing or twisting means more energy storage. Compliance is common in couplings, shafts,
linkages, and belts. The effect is modeled as a two mass system that is coupled by a spring.
Figure 2 - Compliant Two Mass System
The following definitions are given.
T
M
= Motor torque
V
M
= Motor velocity, typically produced by encoder feedback
T
R
= Reaction torque on the motor shaft from the load
k = Spring constant
b = Viscous friction
V
L
= Load velocity
T
D
= Load torque disturbance
When motor torque is applied, the coupling can twist when transferring torque to the load. As the coupling twists, it stores energy and does
not immediately rotate the load. The energy is then released when the motor torque is removed, which causes continued acceleration of the
load. A dynamic model is given where PI is a proportional-integral controller and LPF is a low pass filter.
Figure 3 - Compliant Load Model
Here, the spring constant k is in units of [N•m/rad], which can be typically found in rotary coupling specifications. For example, a shaft with
a spring constant of 1.0 deflects or twists by 1 radian when a steady 1 N•m torque is applied to one end while the other end is anchored to a
stationary rigid body.
The effect of compliance is frequency dependent, which creates tuning problems in the form of resonances. A compliant coupling creates a
resonance and an anti-resonance in signals that are measured by the motor side encoder. Furthermore, another resonance is present at the
end of the load or end effector. This load side resonance has a resonant frequency equal to the motor side anti-resonant frequency.
s
J
L
1
T
M
s
J
M
1
V
M
V
L
b + k/s
T
R
T
D
V
M
V
CMD
1
Power
Converter
T
m
PI
LPF
+
/s
V
L
Velocity Loop
1
T
D
T
R