Modern Control Engineering

Example Problems and Solutions 87

A–3–2. Obtain the transfer function Y(s)/U(s)of the system shown in Figure 3–21. The input uis a

displacement input. (Like the system of Problem A–3–1, this is also a simplified version of an

automobile or motorcycle suspension system.)

Solution.Assume that displacements xandyare measured from respective steady-state

positions in the absence of the input u.Applying the Newton’s second law to this system, we

obtain

Hence, we have

Taking Laplace transforms of these two equations, assuming zero initial conditions, we obtain

EliminatingX(s)from the last two equations, we have

which yields

Y(s)

U(s)

=

k 1 Abs+k 2 B

m 1 m 2 s^4 +Am 1 +m 2 Bbs^3 + Ck 1 m 2 +Am 1 +m 2 Bk 2 Ds^2 +k 1 bs+k 1 k 2

Am 1 s^2 +bs+k 1 +k 2 B

m 2 s^2 +bs+k 2

bs+k 2

Y(s)=Abs+k 2 BY(s)+k 1 U(s)

Cm 2 s^2 +bs+k 2 DY(s)=Abs+k 2 BX(s)

Cm 1 s^2 +bs+Ak 1 +k 2 BDX(s)=Abs+k 2 BY(s)+k 1 U(s)

m 2 y$+by# +k 2 y=bx# +k 2 x

m 1 x$+bx# +Ak 1 +k 2 Bx=by# +k 2 y+k 1 u

m 2 y$=-k 2 (y-x)-b(y# -x#)

m 1 x$=k 2 (y-x)+b(y# -x#)+k 1 (u-x)

y

b

x

u

m 2

m 1

k 2

k 1

Figure 3–21
Suspension system.