Page 107 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

94 Transient Vibration Chap. 4

Substituting for the initial condition i(0) = F/m , we arrive at the equation
F
-e - i“«' sin ^J\ - (4.1-6)
mo)
The response to the unit impulse is of importance to the problems of
transients and is identified by the special designation h{t). Thus, in either the
damped or undamped case, the equation for the impulsive response can be
expressed in the form
X = Fh{t) (4.1-7)
where the right side of the equation is given by either Eq. (4.1-4) or (4.1-6).

4.2 ARBITRARY EXCITATION

By having the response /i(r) to a unit impulse excitation, it is possible to establish
the equation for the response of the system excited by an arbitrary force fit). For
this development, we consider the arbitrary force to be a series of impulses, as
shown in Fig. 4.2-1. If we examine one of the impulses (shown crosshatched) at
time i its strength is

and its contribution to the response at time t is dependent upon the elapsed time
it - 0 , or
f i 0 A ^ h i t - 0

Figure 4.2-1.