Page 109 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

96 Transient Vibration Chap. 4

given as
f cos(]/\ - - i//) (4.2-3)
V I '
where
tan ijj =

Figure 4.2-3 shows a plot of xk/F^) versus with as a parameter, and it is evident

that the peak response is less than 2Fq/ / c when damping is present.
Base excitation. Often, the support of the dynamical system is subjected
to a sudden movement specified by its displacement, velocity, or acceleration. The
equation of motion can then be expressed in terms of the relative displacement
z = X - y as follows:
z + ~ (4.2-4)
and, hence, all of the results for the force-excited system apply to the base-excited

system for z when the term F/w, is replaced by —y or the negative of the base
acceleration.
For an undamped system initially at rest, the solution for the relative
displacement becomes

^ ^ f ÿ { è ) sin io„{t - (4.2-5)

Example 4.2-2
Consider an undamped spring-mass system where the motion of the base is specified
by a velocity pulse of the form
y(0 = v„e-‘/'»ii{t)
where u{t) is a unit step function. The velocity together with its time rate of change is
shown in Fig. 4.2-4.