Page 123 - Servo Motors and Industrial Control Theory -

118 7 Electrohydraulic Servo Motors

q: = K x − C p (7.7)
p
1
where,

d d
K: = q C : = q
1
dx p dp
p: = con x: = con
The reader must be aware that the lower case variables represent variation from an
operating point. This has not been implemented in Eq. (7.6) because it is clear that
the equation represents the absolute values of the flow rate.
The pressure forces the piston to move according to the equation of motion of

Ap: = Ms y + Csy (7.8)
2
In Eq. (7.8), M is the total mass of moving parts of the jack, A is the cross sectional
area of the piston, and C represents the viscous friction that the numerical values
must be given by manufacturers. The reader must know that s is the Laplace Trans-
form operator and is the same as differentiation with respect to time assuming that
all the initial conditions are zero.
On the other hand, the flow equation from the spool valve assuming that the oil
is incompressible moves through the jack according to the equation,

q: = Asy + C · p (7.9)
1
where, C is the leakage coefficient and it is proportional to pressure difference p
1
across the piston and p is the pressure variation. The first term in Eq. (7.9) is the
flow rate required to move the piston by the speed (sy).
Eliminating the variables p, q, and x and with some algebraic manipulation, the
overall transfer function becomes

 (a +b)
 a 
y: =  
⋅
⋅
⋅
2
 C M (C M) s   C (C C) b 
p
s1
b⋅    1  +  ⋅  +   C ⋅  + A + 1  ⋅  ⋅+
⋅
p
   A  A  K 1a ⋅     A A  AK 1 
(7.10)
Equation (7.10) represents a second-order transfer function which was discussed
in detail in previous chapters. The natural frequency which represents the speed of
response is given by the coefficient of s in the denominator and the damping ratio
2
is given by the coefficient of s in the denominator. As was discussed in the previous
chapters, the important behavior of second-order transfer functions is stable but the
damping ratio might become very small resulting in excessive oscillations. From
the following equations, the natural frequency and damping ratio can be obtained.