Page 295 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

282 Vibration of Continuous Systems Chap. 9

The first part of Eq. (9.5-3) states that the rate of change of shear along the
length of the beam is equal to the loading per unit length, and the second states
that the rate of change of the moment along the beam is equal to the shear.
From Eq. (9.5-3), we obtain the following:
dV
(9.5-4)
d\:^
The bending moment is related to the curvature by the flexure equation, which, for
the coordinates indicated in Fig. 9.5-1, is
d^y
M = El (9.5-5)
dx^
Substituting this relation into Eq, (7.4-4), we obtain
d^
(9.5-6)
dx'
For a beam vibrating about its static equilibrium position under its own
weight, the load per unit length is equal to the inertia load due to its mass and
acceleration. Because the inertia force is in the same direction as p{x), as shown
in Fig. 9.5-1, we have, by assuming harmonic motion.
p{x) = pw^y (9.5-7)
where p is the mass per unit length of the beam. By using this relation, the
equation for the lateral vibration of the beam reduces to

(9.5-8)
dx^
In the special case where the flexural rigidity El is a constant, the preceding
equation can be written as

E/ - pw y = 0 (9.5-9)
ax
On substituting
(9.5-10)

we obtain the fourth-order differential equation

« 4 0 (9.5-11)

for the vibration of a uniform beam.
The general solution of Eq (9.5-11) can be shown to be
y = A cosh (3x + B sinh /3x + C cos ¡3x D sin ^x (9.5-12)