Page 614 - Mechanical Engineers' Handbook (Volume 2)

10 Steady-State and Dynamic Behavior of Servoactuators and Servosystems 605

6. The mass of the moving parts in the servomotor (M ) and the viscous damping in
a
the servomotor (B ) are small compared to the mass (M) and the viscous damping
m
(B) associated with the load.
7. The Coulomb friction and stiction forces in the servomotor and load are negligible.
Mathematical Model
The typical servoamplifier is governed by the equation
I KE (103)
as
Equation (61) describes the steady-state behavior of the electrohydraulic servovalve. This
equation can be used in the dynamic analysis of the servoactuator if the servovalve dynamics
are negligible compared to the servomotor-load dynamics. Otherwise, either Eq. (65) or (66)
should be used. Equations (38)–(44) describe the dynamic behavior of the servomotor. The
external load force in Eq. (38) may be expressed as follows:
dY dY
2
F F M B K(Y Y) (104)
L
e
p
dt 2 dt
The set of algebraic equations outlined above cannot be combined into a single dynamic
equation or transfer function because of nonlinearities. Also, since the set of equations is
rather complex, conclusions about the performance of the system cannot be drawn without
actually solving the equations for a variety of conditions. A computer-based analysis is the
only practical method of determining the steady-state and dynamic performance of the system
unless the equations are linearized in some fashion.
A linearized model which assumes small perturbations of all variables about an initial
steady-state operating condition can be very useful in quickly assessing system dynamic
behavior. Such a model is particularly useful for preliminary design and as a reference when
an analysis is made using the set of nonlinear describing equations.
It can be shown 45 that for the case when V V V (see Fig. 28) and under the
1i
2i
i
assumptions listed above, the linearized model may be expressed as follows:
KM s KM BK 3
2
3
2
KB A 2 K B A 2 s 1 s( Y)
2
2
KA G (s)( E ) K Ks
a
2
3
L
KB A 2 1 s K B A 2 ( F ) (105)
2 2
where K C C 2
1
2
C servovalve pressure–flow sensitivity [see Eq. (64)]
1
C internal leakage flow-rate coefficient
2
K A /K A(1/K 1/K 1/K)
s
3
t
0
2
2
K 2 A /V stiffness of the sealed chamber
i
0
K stiffness of the load drive
K stiffness of the structural mounting
s
(s) is the transfer function for the servovalve. When the servovalve dynamics are
and G i
negligible compared to the servomotor-load dynamics, G (s) K [see Eq. (63)]. When the
1
i
servovalve dynamics are of the same order as the servomotor-load dynamics, G (s)is given
i
by the right-hand side of either Eq. (65) or Eq. (66).
Four important measures of performance can be observed from Eq. (105) without ac-
tually solving the equation. The steady-state gain or sensitivity of the servoactuator is