Page 308 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Chap. 9 Problems 295

9-9 A uniform bar of length / is fixed at one end and free at the other end. Show that the
frequencies of normal longitudinal vibrations are f = in \)c/2l, where c = yjE/p
is the velocity of longitudinal waves in the bar, and n = 0,1, 2,... .
9-10 A uniform rod of length / and cross-sectional area A is fixed at the upper end and is
loaded with a weight W on the other end. Show that the natural frequencies are
determined from the equation
/
ca/i/ -pr tan o)l\ -pr = — 777—
y E y E fr
9-11 Show that the fundamental frequency for the system of Prob. 9-10 can be expressed in
the form
(Oi = p^yJk/rM
where

k = 4K M = end mass
i ■
Reducing this system to a spring k and an end mass equal to M -r determine
an appropriate equation for the fundamental frequency. Show that the ratio of the
approximate frequency to the exact frequency found is
(l//?,)v^3r/(3 + /■)
9-12 The frequency of magnetostriction oscillators is determined by the length of the nickel
alloy rod, which generates an alternating voltage in the surrounding coils equal to
the frequency of longitudinal vibration of the rod, as shown in Fig. P9-12. Determine
the proper length of the rod clamped at the middle for a frequency of 20 kcps if the
modulus of elasticity and density are given as E = 30 X 10^ Ib/in.^ and p = 0.31
Ib/in.^, respectively.

Figure P9-12.
9-13 The equation for the longitudinal oscillations of a slender rod with viscous damping is
du H du Pi)
d?
where the loading per unit length is assumed to be separable. Letting u =
and p{x) = Hibj(l)i{x) show that
Po
■('f(i - sin ■dr

ml^l - i "y
bj = j i ‘p{x)4>j{x)dx
I Jn
Derive the equation for the stress at any point x.