Page 186 - Servo Motors and Industrial Control Theory -

182 Appendix A

With back pressure of 50 bar and with valve opening 0.5 the flow rate is 5 lit/
min. Determine the coefficients of the linearized equation. Also assume that the
leakage coefficient of the piston is one lit/min when the pressure is 150 bars.
The diameter of the jack in the cylinder is 0.2 m.
By writing the governing differential equation for each part derive the block
diagram of the system. Assume that the compressibility of hydraulic oil for this
particular application is negligible and that the friction on the load is negligible.
Any parameters that have not been specified use some engineering value for
them. Determine the transfer function of the system which relates the output
displacement (x) to the command signal (xi). Find the appropriate gain (K) so
that the damping ratio of the dominant root is 0.7. Study the steady state errors
of the system for step and ramp inputs.
Repeat the analysis when an external force of F is applied to the mass. Deter-
mine the steady state error of the system for step and ramp input of force on
the mass.
Repeat the calculation for when the hydraulic oil compressibility is not negli-
gible and it has a bulk modulus of compressibility is 100,000 (N/(m ).
2
Determine the maximum steady state speed of the piston when it is operated in
open loop with maximum pressure and maximum valve opening.
56. The figure below shows a mass-spring-damper system in closed loop control
form. The force applied to the mass determines the position of the mass. In this
case it is required to operate the system in closed loop form so that the position
of the mass is kept at a certain value determined by the command signal and in
the face of disturbance that may be applied to the mass.
The force applied to the mass and it is relationship from output voltage of con-
troller can be approximated by first order lag as,

F := 10
V . 05 s 1+
The mechanism which generates the force is not shown. It may be a hydraulic
jack or a servo motor. But it should be noted that because for this particular
application the loading mechanism can be approximated by a gain and a first
order lag with time constant of 0.5 s.
The diagram also shows a PID controller for this particular application. Deter-
mine the gains of the controller so the system with dominant roots of the char-
acteristic equation has at least a damping ratio of 0.7 with natural frequency of
20 rad/sec. Use the following numerical values to calculate the transfer func-
tion. Use both root locus and Bode diagram to study the stability of the system.

M 100 kg=
K = 10000 N/m
C10N/(m/s)=