Page 106 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 4.1 Impulse Excitation 93

Figure 4.1-1.

If 8(t - is multiplied by any time function f(t), as shown in Fig. 4.1-2, the
product will be zero everywhere except at t = and its time integral will be
r f i t ) 8 { t - ^ ) d t = f u ) o < ^ < o c (4.1-3)

Because Fdt = mdv, the impulse F acting on the mass will result in a
sudden change in its velocity equal to F/m without an appreciable change in its
displacement. Under free vibration, we found that the undamped spring-mass
system with initial conditions x(0) and i(0) behaved according to the equation
i(0) .
jc == ^.—sm (oj + x(0) cos (oj

Hence, the response of a spring-mass system initially at rest and excited by an
impulse F is
X = sin coj = Fh{t) (4.1-4)
mo)^
where
1
h{t) = sm ù)^t (4.1-5)
mo)^
is the response to a unit impulse.
When damping is present, we can start with the free-vibration equation, Eq.
(2.6-16), with x(0) = 0:
i(0 )^ -W r------ y
----^ = = s i n V^l -

Figure 4.1-2.