Page 380 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Chap. 11 Problems 367

M t t t A
Figure PI 1-8.
response yix, t) in terms of the normal modes of the beam. Indicate what modes are
absent and write down the first two existing modes.
11-9 A slender rod of length /, free at x = 0 and fixed at x = /, is struck longitudinally by
a time-varying force concentrated at the end x = 0. Show that all modes are equally
excited (i.e., that the mode-participation factor is independent of the mode number),
the complete solution being
i 7T X \ I 3 tT X \
2FqI
u(x, t) -h • • •
AE
(?)
11-10 If the force of Prob. 11-9 is concentrated at x ==1/3, determine which modes will be

absent in the solution.
11-11 In Prob. 11-10, determine the participation factor of the modes present and obtain a
complete solution for an arbitrary time variation of the applied force.
11-12 Consider a uniform beam of mass M and length / supported on equal springs of total
stiffness k, as shown in Fig. PI 1-12a. Assume the deflection to be
y(x,t) = <p|(jr)<?|(0 + <p2{x)q2{t)

Figure Pll-12.