Page 307 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

294 Vibration of Continuous Systems Chap. 9

motion IS
d^y I d^y dy
dt^ dx^ dx

I T ^ d T

J^pgdx

Figure P9-6.

9-7 In Prob. 9-6, assume a solution of the form y = Y{x) cos o)t and show that Y(x) can
be reduced to a Bessel’s differential equation

d^Y(z) 1 dY(x)
+ + y (z) = 0
dz^ z dz
with solution
y(z)=Jo(2) or

y (x ) =7o 2

by a change in variable 2^ = 4co’x/g.
9-8 A particular satellite consists of iiwo equal masses of m each, connected by a cable of
length 2/ and mass density p, as shown in Fig. P9-8. The assembly rotates in space with
angular speed coq. Show that if the variation in the cable tension is neglected, the
differential equation of lateral motion of the cable is
^ y d^y 2
dx^ mcog/ \ dt^
and that its fundamental frequency of oscillation is

Figure P9-8.