Page 203 - Servo Motors and Industrial Control Theory -

200 Appendix B

26. This is a simple problem to guide to you the next problem which is consider-
ably more complicated. The figure below shows a simple mass-spring-damper
system which often is used for suspension of cars or in train stations to absorb
the impact of the trains.
x
K
F
M
C

Write the governing differential equation for this system and find the transfer
function which relates the displacement x to the applied force F. Determine the
natural frequency and damping ratio in terms of K and C. For K = 20000 (N/m)
and M = 500 kg, determine the value of C so that the damping ratio is 0.5. This
method gives you an idea on how to design the dampers for next problem.
27. Solve problem 25 before attempting this problem. The figure below shows a
simple representation of a train with three wagons. The force between the wag-
ons must be absorbed by spring damper systems.

y(t)
K1.C1 K2.C2
F
M1 M2 M3

Assume that y , y , y are the displacements of the three masses. Write the equa-
1
2
3
tion of motion for each mass by drawing the free body diagram of each mass
first. Then, transform the governing differential equations to state space form. In
practical situation the stiffness of each spring must be high enough to limit the
displacement of each mass and usually K2 must be larger than K1 because the
maximum dynamic force is applied to the front wagon. Assume the following
numerical values for the parameters involved in state space model are,
M1 M2= = M3 = 500 kg
K1 50000 N/m=
K2 100000 N/m=
Find the values of C1 and C2 so that the damping ratio of all eigenvalues are at
least 0.5. Your state model should be of six orders and there must be one or two
zero eigenvalues because the system is subjected to free motion.
Determine the eigenvector for each eigenvalue and discuss the meaning of each
eigenvector for the system.