Page 607 - Mechanical Engineers' Handbook (Volume 2)

598 Servoactuators for Closed-Loop Control

Motor equations:
• Electrical:
di
v K L a Ri (68)
a
a
m
E
dt
• Torque balance:
d
Ki J m m B T ƒm T m (69)
T
m
m
dt
• Gear equation:

n m m (70)
L L
• Lossless power transfer:
T T L (71)
m
L
m
• Load torque balance:
d
T J L L B T ƒL T e (72)
L
L
L
dt
If the gearing has backlash, the relationship between and is as shown in Fig. 41.
m L
Similarly, if there is a Coulomb friction load torque, the relationship between the torque and
the load velocity is as shown in Fig. 42. These nonlinear effects can be included in the
model, but the resulting set of algebraic and differential equations can be studied only
through digital simulation. Then the solution must be obtained for a specified input com-
mand; a general solution is not possible. It is often sufficient in analysis underlying prelim-
inary design to simplify the model by linearizing the equations and eliminating nonlinear
effects such as backlash and Coulomb friction. Such a linearized model can be very useful

Figure 41 Relationship between m and L due to backlash.