Section 4–4 / Hydraulic Systems 133
+
(a)(b)
Spring Area=A
constant=k
Density
of oil =r
Oil
under
pressure
Resistance=R
e
x
a
b
y
z
E(s) b X(s) Y(s)
a+b
a
a+b
K
s
Ts
Ts+ 1
Z(s)
or
Taking the Laplace transforms of both sides of this last equation, assuming zero initial
conditions, we obtain
The transfer function of this system thus becomes
Let us define RA^2 rk=T.(Note that RA^2 rkhas the dimension of time.) Then
Clearly, the dashpot is a differentiating element. Figure 4–21(c) shows a block diagram
representation for this system.
Obtaining Hydraulic Proportional-Plus-Integral Control Action. Figure 4–22(a)
shows a schematic diagram of a hydraulic proportional-plus-integral controller. A block
diagram of this controller is shown in Figure 4–22(b). The transfer function Y(s)/E(s)
is given by
Y(s)
E(s)
=
b
a+b
K
s
1 +
Ka
a+b
T
Ts+ 1
Z(s)
Y(s)
=
Ts
Ts+ 1
=
1
1 +
1
Ts
Z(s)
Y(s)
=
s
s+
k
RA^2 r
sY(s)=sZ(s)+
k
RA^2 r
Z(s)
dy
dt
=
dz
dt
+
kz
RA^2 r
Figure 4–22
(a) Schematic diagram of a hydraulic proportional-plus-integral controller; (b) block diagram of the controller.