Page 615 - Mechanical Engineers' Handbook (Volume 2)

606 Servoactuators for Closed-Loop Control

˙
Y K K A (106)
a
1
E K B A 2
s steady state; F 0 2
L
The steady-state load sensitivity of the servoactuator is
Y ˙ K 2 (107)
F K B A 2
L steady state; E 0 2
s
Also observable from Eq. (105) are the natural frequency and the damping ratio associated
with the servomotor-load portion of the system. The natural frequency is
KB A 2
2
KM (108)
ns
3
and the damping ratio is
KM KB
2 3 (109)
s
2
2 KM(KB A )
3
2
The valve dynamics are often negligible compared to the servomotor-load dynamics. In
this case, Eq. (105) reduces to a second-order linear differential equation of the generalized
form:
1 s KKA K Ks
2
3
s
2
a
1
2
s
2 s 1 s( Y) KB A 2 E K B A 2 F L (110)
ns ns 2 2
2
In most cases, the term K B in Eqs. (105)–(110) is small compared to the term A . Simplified
2
forms of these equations result.
The dynamic behavior of the open-loop system may be viewed in terms of the speed
of response and the degree of damping. A fast system dynamic response requires a small
value of and a large value of . In order for to be large, K M must be small compared
ns
ns
3
2
to K B A . The value of M is usually fixed, but the designer often has some latitude in
2
varying K . The value of K can be minimized by making the volumes within the servomotor
3
3
chambers small. Increasing the ram area also provides for an increase in , but the effect
ns
is not as great as it first appears since K ƒ(V ) and V ƒ(A). Also, the actuator area is
3
i
i
often set by such practical considerations as maximum load or acceleration requirements.
The degree of damping in the open-loop system is governed by Eq. (109). In most
practical systems, 1, and K , M, and A are fixed by other considerations. Then the
s
3
damping may be increased by increasing the load damping B or the value of the effective
leakage coefficient, K . The value of K may be increased by increasing either C or C , that
1
L
L
2
is, increasing the valve underlap or the leakage across the actuator piston. Since normally
2
K B A , Eq. (107) shows that an increase in K results in an increase in sensitivity to load
2
2
disturbances. Likewise, an increase in the valve underlap (or the use of cross-port leakage)
results in an increase in quiescent power dissipation. Clearly, there is a trade-off between
degree of damping, steady-state load sensitivity, and quiescent power dissipation.
10.4 Electrohydraulic Servosystems
Figure 46 is a physical representation of a typical electrohydraulic servosystem intended to
accurately position an inertia load. The addition of position feedback converts the ‘‘rate-
type’’ open-loop system (servoactuator) into a position control system. The linearized model
given by Eq. (110) describes the open-loop portion of this position control system, as shown
in the block diagram of Fig. 47.