Page 306 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Chap. 9 Problems 293

If N is very large, the angle tt/AN is small and
. ( 2 / — 1)77 ( 2 / — 1 ) 77-
sin ^ ^ ~ ^ ^ —
AN ~ AN
for the lower frequeneies. The previous equation can then be approximated by
AE
= (2' - 1)t ì / Â/T ’ / « N (b)
which is the exact equation for the longitudinal vibration of the uniform rod. For
higher frequencies, the assumption And = 0 will not be valid (see Fig. 9.7-2), and Eq.
(a) for the discrete mass system must be used.

P R O B L E MS

9-1 Find the wave velocity along a rope whose mass is 0.372 kg/m when stretched to a
tension of 444 N.
9-2 Derive the equation for the natural frequencies of a uniform cord of length / fixed at
the two ends. The cord is stretched to a tension T and its mass per unit length is p.
9-3 A cord of length / and mass per unit length p is under tension T with the left end
fixed and the right end attached to a spring-mass system, as shown in Fig. P9-3.
Determine the equation for the natural frequencies.

Figure P9-3.

9-4 A harmonic vibration has an amplitude that varies as a cosine function along the
x-dircction such that
y = u cos kx • sin a)/

Show that if another harmonic vibration of same frequency and equal amplitude
displaced in space phase and time phase by a quarter wavelength is added to the first
vibration, the resultant vibration will represent a traveling wave with a propagation
velocity equal to c = co/k.
9-5 Find the velocity of longitudinal waves along a thin steel bar. The modulus of elasticity
and mass per unit volume of steel are 200 X 10'^ N /m “ and 7810 kg/m \ respectively.
9-6 Shown in Fig. P9-6 is a flexible cable supported at the upper end and free
to oscillate under the influence of gravity. Show that the equation of lateral