Modern Control Engineering

132 Chapter 4 / Mathematical Modeling of Fluid Systems and Thermal Systems

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(a) (b) (c)

q R

P 2 P 1

A

k

y

y

z

z

t

t

Y(s) Z(s)

1

Ts

T=RA

(^2) r
k

The transfer function between yandebecomes a constant. Thus, the hydraulic controller

shown in Figure 4–20(a) acts as a proportional controller, the gain of which is Kp. This gain

can be adjusted by effectively changing the lever ratio b/a. (The adjusting mechanism is

not shown in the diagram.)

We have thus seen that the addition of a feedback link will cause the hydraulic

servomotor to act as a proportional controller.

Dashpots. The dashpot (also called a damper) shown in Figure 4–21(a) acts as a

differentiating element. Suppose that we introduce a step displacement to the piston po-

sitiony. Then the displacement zbecomes equal to ymomentarily. Because of the spring

force, however, the oil will flow through the resistance Rand the cylinder will come back

to the original position. The curves yversustandzversustare shown in Figure 4–21(b).

Let us derive the transfer function between the displacement zand displacement y.

Define the pressures existing on the right and left sides of the piston as and

respectively. Suppose that the inertia force involved is negligible. Then the

force acting on the piston must balance the spring force. Thus

whereA=piston area, in.^2

k=spring constant, lbfin.

The flow rate qis given by

where q=flow rate through the restriction, lbsec

R=resistance to flow at the restriction, lbf-secin.^2 -lb

Since the flow through the restriction during dtseconds must equal the change in the

mass of oil to the left of the piston during the same dtseconds, we obtain

wherer=density, lbin.^3. (We assume that the fluid is incompressible or r=constant.)

This last equation can be rewritten as

dy

dt

-

dz

dt

=

q

Ar

=

P 1 - P 2

RAr

=

kz

RA^2 r

qdt=Ar(dy-dz)

q=

P 1 - P 2

R

AAP 1 - P 2 B=kz

P 2 (lbfin.^2 ),

P 1 (lbfin.^2 )

Figure 4–21

(a) Dashpot; (b) step change in yand the corresponding change in zplotted versus t; (c) block

diagram of the dashpot.

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