Page 141 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

128 Transient Vibration Chap. 4

what is the maximum peak acceleration of the ejection pulse applied to the pilot?
Assume as in Example 4.3-3 that the seated pilot of 160 lb can be modeled with a
spinal spring stiffness of k = 450 Ib/in.
4-31 A spring-mass system with viscous damping is initially at rest with zero displacement.
If the system is activated by a harmonic force of frequency to = ca,, = deter­
mine the equation for its motion.
4-32 In Prob. 4-31, show that with sraall damping, the amplitude will build up to a value
(1 - e *) times the steady-state v^alue in time t = l // i ^ (5 = logarithmic decrement).
4-33 Assume that a lightly damped system is driven by a force F„sin ca,,/, where ia,, is the
natural frequency of the system. Determine the equation if the force is suddenly

removed. Show that the amplitude decays to a value e * times the initial value in the
time i = l//„ 5 .
4-34 Set up a computer program for Example 4.7-1.
4-35 Draw a general flow diagram for the damped system with zero initial conditions
excited by a force with zero initial value.
4-36 Draw a flow diagram for the damped system excited by base motion y(t) with initial
conditions x(0) = A, and i(0 ) == K,.
4-37 Write a Fortran program for Prob. 4-36 in which the base motion is a half-sine wave.
4-38 Determine the response of an undamped spring-mass system to the alternating square
wave of force shown in Fig. P4-38 by superimposing the solution to the step function
and matching the displacement and velocity at each transition time. Plot the result and
show that the peaks of the response will increase as straight lines from the origin.

rr 'pTT
CÛ 3 ^
Figure P4-38.

4-39 For the central diflference method, supply the first higher-order term left out in the
recurrence formula for T,, and verify that its error is 0(/?“).
4-40 Consider a curve x = and determine x /a t t = 0.8, 0.9, 1.0, 1.1, and 1.2. Calculate
Xjo by using x, = + i ^ ^ ^ ^ 0.10, and show that the
error is approximately 0(/z^).

4-41 Repeat Prob. 4-40 with x, = 1//z(x, - x,_ ,) and show that the error is approximately
m i
4-42 Verify the correctness of the superimposed exact solution in Example 4.7-1, Figure
4.7-4.
4-43 Calculate the problem in Example 4.7-2 by using the R ungc-K utta computer program
RUNGA (see Chapter 8).