Page 104 - Electric Drives and Electromechanical Systems

Chapter 3 Power transmission and sizing 97

transmitted torque will be proportional to this difference. If K is the shaft stiffness (Nm
1
rad ), and B is the damping constant (Nm rad 1 s) then for the motor end of the shaft,
2
T m ¼ I m s q m þ Bsðq m q L Þþ Kðq m q L Þ (3.35)
and at the load end the torque will turn the load in the same direction as the motor,
hence,
2
Bsðq m q L Þþ Kðq m q L Þ¼ I L s q L þ T L (3.36)
where s is the differential operator, d =dt. If these equations are solved it can be shown
that the undamped natural frequency of the system is given by,
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K K
u o ¼ þ (3.37)
I m I L
The damped oscillation frequency is given by, where z is the damping ratio,
ffiffiffiffiffiffiffiffiffiffiffiffiffi
q
u n ¼ u o 1 z 2 (3.38)
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
B 1 1
x ¼ p ffiffiffiffi þ (3.39)
2 K I m I L
which gives,
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
B 2 1 1
u n ¼ 1 þ (3.40)
4K I m I L
In order to produce a stable system, the damped oscillation frequency must be
significantly different from any torque pulsation frequencies produced by the drive or
driven system.

3.7 Linear drive considerations

For many high-performance linear applications, including robotic or similar high-
performance applications, the use of leadscrews, timing belts or rack and pinions
driven by rotary motors, are not acceptable due to constrains imposed by backlash and
limited acceleration. The use of a linear three phase brushless motor (Section 6.3) or the
Piezoelectric motor (Section 9.3), provides a highly satisfactory solution to many motion
control problems. If the required application requires only a small high-speed
displacement, the voice coil (Section 9.1) can be considered.
The following advantages are apparent when a linear actuator is compared to
conventional system based on a driving a belt or leadscrew:

When compared to a belt and pulley system, a linear motor removes the problems
associated with the compliance in the belt. In many cases the compliance will
cause vibration when the load comes to rest, and hence limits the peak speed and
acceleration of a belt drive. However, it should be noted that a high-performance
belt drive can have a repeatability error in excess of 50 mm.