Page 105 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

4

Transient

Vibration

When a dynamical system is excited by a suddenly applied nonperiodic excitation
F(t), the response to such excitation is called transient response, since steady-state
oscillations are generally not produced. Such oscillations take place at the natural
frequencies of the system with the amplitude varying in a manner dependent on
the type of excitation.
We first study the response of a spring-mass system to an impulse excitation
because this case is important in the understanding of the more general problem of
transients.

4.1 IMPULSE EXCITATION

Impulse is the time integral of the force, and we designate it by the notation F\
¡F {t)d t ( 4 . 1 - 1 )
We frequently encounter a force of very large magnitude that acts for a very short
time but with a time integral that is finite. Such forces are called impulsive.
Figure 4.1-1 shows an impulsive force of magnitude F/e with a time duration
of 6. As 6 approaches zero, such forces tend to become infinite; however, the
impulse defined by its time integral is F, which is considered to be finite. When F
is equal to unity, such a force in the limiting case 6 0 is called the unit impulse,
or the delta function. A delta function ai t = ^ is identified by the symbol 8(t - ^)
and has the following properties:
~ ^ 0 for all i ^ ^
= greater than any assumed value for t = ^ M 1 21
-00 '
d(t - dt = 1.0 0 < i <oo
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