Page 343 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

330 Introduction to the Finite Element Method Chap. 10

Case 2: A distributed force p{x) dx parallel to the beam will do virtual work
p(x)dx • 8u{x), where u(x) is the horizontal displacement due to deflection y(jc).
Displacement u{x) is equal to the diiTerence between the horizontal projection of
the deflected beam and x:

i{x) = / {ds —dx) = / dx f b ' " dr
where r is a dummy variable for x, and y' = dy/dr. The virtual displacement at x
is then equal to
8u{x) = f j 8y'^ dr
h)
where the integrand is interpreted as follows:

= i[(y' + 8 y ' f - y'^] = y' 8y'
Thus, the virtual work for the distributed force becomes

8W = - f p(x) f y'8y'drdx (10.8-4)
dn df) •'()
•'()
Substituting for y' in terms of the beam functions, we have

8^V = H f'pix) f drdx (10.8-5)
dn d(\
I J
8 W ^ rl
'
'
Qi= j ^ = - 4>,4>j drdx ( 10.8-6)
Example 10.8-2
An example of interest here is the helicopter blade whirling with angular speed il, as
shown in Fig. 10.8-2. For the first beam element, the loading is iCxmdx and Eq.
(10.8-6) applies without change. For subsequent elements, coordinate x must be
measured from the beginning of the new element to conform to that of the beam
functions. The load for the element is simply + x)mdx, where /, is the distance
from the rotation axis to the beginning of the new element. Presented here is the

il (/; -^X)mdx

Figure 10.8-2.