Modern Control Engineering

Example Problems and Solutions 365

A–6–11. Obtain the transfer function of the mechanical system shown in Figure 6–74. Assume that the

displacementxiis the input and displacement xois the output of the system.

Solution.From the diagram we obtain the following equations of motion:

Taking the Laplace transforms of these two equations, assuming zero initial conditions, and then

eliminatingY(s),we obtain

This is the transfer function between and By defining

we obtain

This mechanical system is a mechanical lead network.

Xo(s)

Xi(s)

=a

Ts+ 1

aTs+ 1

=

s+

1

T

s+

1

aT

b 1

k

=T,

b 2

b 1 +b 2

=a 61

Xo(s) Xi(s).

Xo(s)

Xi(s)

=

b 2

b 1 +b 2

b 1

k

s+ 1

b 2

b 1 +b 2

b 1

k

s+ 1

b 1 Ax#o-y#B=ky

b 2 Ax#i-x#oB=b 1 Ax#o-y#B

− 8 − 7 − 6 − 5 − 4 − 3 − 2 −10 1 2

0

1

3

2

4

5

− 2

− 1

− 5

− 4

− 3

Real Axis

Imag Axis

Root-Locus Plot of G(s) = K(s + 1)/[(s^2 + 2s + 2)(s^2 + 2s + 5)]

Figure 6–73
Plot of root loci.

b 2

b 1

k y

xi

xo

Figure 6–74
Mechanical system.