Page 340 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 10.7 Generalized Force for Distributed Load 327

Thus, both natural frequencies are increased by the constraining springs. If
k = K = {), the exact natural frequencies for the beam with fixed ends are
ia, = 22.37i ca, = 61.67i

so that the error in the finite element approach is 1.61 percent for the first mode and
33.9 percent for the second mode. Dividing the beam into shorter elements reduces
these errors.

10.7 GENERALIZED FORCE FOR DISTRIBUTED LOAD

As discussed in Chapters 7 and 8, the generalized force is found from the virtual
work of the applied forces. With the displacement expressed as
y(x) = (10.7-1)
the virtual work of the applied distributed force p(x) is

8W = [ p{x) 5y(x) dx
h)

= 3v^ f p(x)(f)i(x) dx + i p{x)(f)2{x) dx (10.7-2)
•'o •'()

+ i ^ ■*" ^^2 i p{x)(f)^{x) dx
•^0 ' •'()
The integrals in Eq. (10.7-2) are the generalized forces.
If the same procedure is applied to the end forces, Ej, M,, F2, and M2, the
virtual work is

8W = F^8v^ + M i 86 ^ F F2 8 V2 (10.7-3)
Equating the virtual work in the previous two cases, we obtain the following
relationships:

F^= ^ P{x)4>¡(x) dx F2
'0
(10.7-4)
M, = i^p(x)(f)2(x) dx M2 f^p(x)(t)4(x) dx

Thus, for the distributed load, the equivalent finite element loads are the general­
ized forces just given.
Example 10.7-1
Figure 10.7.1 shows a cantilever beam of length ¡¡ with a uniform load p(x) =
p Ib/in. over the outer half of the beam. Determine the deflection and slope at the
free end using the method of this section.