Page 388 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 12.1 Rayleigh Method 375

The additional strain energy in element dx is
dU = {aAedx = {EAe^ dx
where A is the cross-sectional area, a is the stress due to tension, and e = \idy/dxY
is the unit strain.
Equating the kinetic energy to the total strain energy of bending and tension,
we obtain
= + dx

The preceding equation then leads to the frequency equation:

d^y
fEI dx -y
dx^
jy^ dm

which contains an additional term due to tension.

Accuracy of the integral method over differentiation. In using
Rayleigh’s method of determining the fundamental frequency, we must choose an
assumed curve. Although the deviation of this assumed deflection curve compared
to the exact curve may be slight, its derivative could be in error by a large amount
and hence the strain energy computed from the equation

u - I / e ; dx
dx^
may be inaccurate. To avoid this difficulty, the following integral method for
evaluating U is recommended for some beam problems.
We first recognize that the shear V is the integral of the inertia loading
ma)^y from the free end of the beam, as indicated by both Figs. 12.1-3 and 12.1-4.

V{^) = c o ^fm ie )y(^)d ^ (12.1-13)

Because bending moment is related to the shear by the equation
dM
dx = E (12.1-14)

Figure 12.1-3.