Page 294 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 9.5 Euler Equation for Beams 281

With the boundary conditions, 0(0, t) = Oil, t) = 0, the natural frequencies are
found from
sin a>-\/ 17 / = 0
K

— I = TT,2tt,. .., nir
K
For a node at midspan, n = 2:

■^2 / y 7

T 39 400
2 - ^ / t = 2 8 0 0 ^ T W
- 3.94 s
which agrees closely with the observed period of 4 s.

9.5 EULER EQUATION FOR BEAMS

To determine the differential equation for the lateral vibration of beams, consider
the forces and moments acting on an element of the beam shown in Fig. 9.5-1.
V and M are shear and bending moments, respectively, and p{x) represents
the loading per unit length of the beam.
By summing forces in the y-direction,
d V - p { x ) dx ^ 0 (9.5-1)
By summing moments about any point on the right face of the element,
dM - Vdx - {p(x)(dx f = 0 (9.5-2)

In the limiting process, these equations result in the following important relation­
ships:
dV , , dM
(9.5-3)

p{x) dx

dM
< O l> '
^ 1/ j. wi/
V ^ dV
^dx
Figure 9.5-1