Modern Control Engineering

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204 Chapter 5 / Transient and Steady-State Response Analyses

EXAMPLE 5–8 Consider the mechanical system shown in Figure 5–30, where m=1kg,b=3N-secm, and

k=2Nm. Assume that at t=0the mass mis pulled downward such that x(0)=0.1m and

(0)=0.05msec. The displacement x(t)is measured from the equilibrium position before the

mass is pulled down. Obtain the motion of the mass subjected to the initial condition. (Assume

no external forcing function.)

The system equation is

with the initial conditions x(0)=0.1m and (xis measured from the equilib-

rium position.) The Laplace transform of the system equation gives

or

Solving this last equation for X(s)and substituting the given numerical values, we obtain

This equation can be written as

Hence the motion of the mass mmay be obtained as the unit-step response of the following

system:

MATLAB Program 5–14 will give a plot of the motion of the mass. The plot is shown in Figure 5–31.

G(s)=

0.1s^2 +0.35s

s^2 +3s+ 2

X(s)=

0.1s^2 +0.35s

s^2 + 3 s+ 2

1

s

=

0.1s+0.35

s^2 +3s+ 2

X(s)=

mx(0)s+mx#(0)+bx(0)

ms^2 +bs+k

Ams^2 +bs+kBX(s)=mx(0)s+mx#(0)+bx(0)

mCs^2 X(s)-sx(0)-x#(0)D+bCsX(s)-x(0)D+kX(s)= 0

x

(0)=0.05 msec.

mx

$

+bx

+kx= 0

x

MATLAB Program 5–14

% --------------- Response to initial condition ---------------

% ***** System response to initial condition is converted to

% a unit-step response by modifying the numerator polynomial *****

% ***** Enter the numerator and denominator of the transfer

% function G(s) *****

num = [0.1 0.35 0];

den = [1 3 2];

% Enter the following step-response command

step(num,den)

% Enter grid and title of the plot

grid

title('Response of Spring-Mass-Damper System to Initial Condition')

m

k

b x

Figure 5–30

Mechanical system.

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