Page 122 - Servo Motors and Industrial Control Theory -

7.2 A Simple Mechanically Controlled Servo System 117

(a+b)
x: = e (7.2)
b
Then it is assumed that the point B is fixed and point C moves by amount y. From
the similarity of the two resulting triangles, the relation between x and y can be
found as

x : = a (7.3)
y b

which can be written as,
a
x: = y (7.4)
b

From the principle of superposition, the relation between x and e and y for any in-
stant of time can be written as
(a +b)·e a
x: = − y (7.5)
b b

The second term is taken as negative because when y is moved, x moves in the op-
posite direction of x positive.
Equation (7.5) shows that there is a mechanical feedback. It means that when
an input of e is applied, the mass moves with the amount y until x becomes zero.
In fact, the displacement e can be calibrated in terms of displacement y. The point
B can be moved between points A and C. The location of point B in fact changes,
as will be shown later, the gain of the system. It also changes the calibration of the
displacement y. If the point B is moved above point A, then the connection between
the spool valve and hydraulic jack must be changed so as to have a position control
system.
Designing such a system, the stability and accuracy of the system must be stud-
ied. This is achieved by deriving a mathematical model for each part of the system
and deriving the overall transfer function which relates the displacement y to the
input displacement e.
When the spool valve is displaced by x, the flow equation becomes

q: = C ·x· (P − P) (7.6)
S
d
where C is the valve coefficient which is function of valve parameters such as
d
cross sectional area, oil density, and the gravitational acceleration and the shape of
the valve. The manufacturer of valves must provide these parameters. P , p are the
s
supply pressure and the pressure on the side of the jack. Equation (7.6) is nonlinear
and it is proportional to displacement x and square root of the jack pressure. Using
the linearization technique Eq. (7.6) may be written as