Page 287 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

274 Vibration of Continuous Systems Chap. 9

d^TTT
dT dx

Figure 9.3-1. Torque acting on an
element dx.
mass per unit volume, the differential equation of motion becomes
d^e d^e G \ d^e
plpdx ^t2 Ip(j ^ dt ^ dx^ 7 (9.3-3)
^ . 7
n
This equation is of the same form as that of longitudinal vibration of rods, where 0
and G/p replace u and E/p, respectively. The general solution hence can be
written immediately by comparison as

6 = I A sin (oJ ^ X B cos coJ x\(C sin cot + D cos cot) (9.3-4)

Example 9.3-1
Determine the equation for the natural frequencies of a uniform rod in torsional
oscillation with one end fixed and the other end free, as in Fig. 9.3-2.
Solution: Starting with equation
+ B COS c o y j p / G x) sin CO /

6 = (^A sin ÎÜ ^ J p / G x
apply the boundary conditions, which are
(!) when jc = 0, 0 = 0,
(2) when x = I, torque = 0, or
= 0
dx
Boundary condition (!) results in 5 = 0.
Boundary condition (2) results in the equation

cos co^Jp/G I = 0
which is satisfied by the following angles
P , _ 77 3 t7 577
(-a

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Figure 9.3-2.