Page 108 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 4.2 Arbitrary Excitation 95

where hit - is the response to a unit impulse started at t = ^. Because the
system we are considering is linear, the principle of superposition holds. Thus, by
combining all such contributions, the response to the arbitrary excitation fit) is
represented by the integral
^ (0 = (4.2-1)
•'n
This integral is called the convolution integral and is sometimes referred to as the
superposition integral.
Example 4.2-1
Determine the response of a single-DOF system to the step excitation shown in Fig.
4.2-2.
Solution: Considering the undamped system, we have

hit) = —^— sin (o„t
^ ^ mco^ "
By substituting into Eq. (4.2-1), the response of the undamped system is
To

= - / ( l - cos a)„t) (4.2-2)
This result indicates that the peak response to the step excitation of magnitude Fq is

equal to twice the statical deflection.
For a damped system, the procedure can be repeated with
h(t) = sin o)J

or, alternatively, we can simply consider the differential equation
X + 2^(o^x -\- co^x =
m
whose solution is the sum of the solutions to the homogeneous equation and that of
the particular solution, which for this case is Fo/mw^. Thus, the equation
p
jc(^) = sin (v^l - - </>) H---- ^
-
’ mo)^
fitted to the initial conditions of x(0) = i( 0) = 0 will result in the solution, which is
fit)

Figure 4.2-2. Step function excita­
tion.