Page 135 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

122 Transient Vibration Chap. 4

[3] J a c o b s e n , L. S. a n d A y r e , R. S. Engineering Vibrations. New York: McGraw-Hill,
1958.
[4] N e e s o n , F. C. Shock & Vibration Isolation: Breaking the Academic Paradigm, Proceed­
ings of the 61st Shock & Vibration Symposium, Vol. 1, October 1990.
[5] S a c z a e s k i , K. J. Vibration Analysis Methods Applied to Forensic Engineering Problems,
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ing Division, Vol. 34, pp. 197-206.

P R OB L E MS

4-1 Show that the time t^ corresponding to the peak response for the impulsively excited
spring-mass system is given by the equation
tan Vi = Vi - C a
4-2 Determine the peak displacement for the impulsively excited spring-mass system, and
show that it can be expressed in the form
4km
^ peak - ,
= exp - -tan
Plot this result as a function of C
4-3 Show that the time t^ corresponding to the peak response of the damped spring-mass
system excited by a step force F,) is = tt/ \ - .
4-4 For the system of Prob. 4-3, show that the peak response is equal to
I = 1 -r exp I ------p
/ max \/l
4-5 For the rectangular pulse of time duration t^, derive the response equation for t > t^
using the free-vibration equation with initial conditions xit^) and Compare with
Eq. (4.4-3b).
4-6 If an arbitrary force f(t) is applied to an undamped oscillator that has initial
conditions other than zero, show that the solution must be of the form
x(t) = X() cos co^t H--- - sin oj
-
4-7 Show that the response to a unit step function, designated by g(t), is related to the
impulsive response h(t) by the equation h(t) = g(t).
4-8 Show that the convolution integral can also be written in terms of g(t) as

x(o = /(o)g(o + j ' f ( O g ( t - n d ^
-'n
where git) is the response to a unit step function.
4-9 In Sec. 4.3, the subsidiary equation for the viscously damped spring-mass system was
given by Eq. 4.3-(a). Evaluate the second term due to initial conditions by the inverse
transforms.