Page 342 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 10.8 Generalized Force Proportional to Displacement 329

distributed forces normal to the beam, and (2) for distributed forces parallel to the
beam.
Case (1): The term p(x) in the virtual work equation (10.7-2) is replaced by
f(x)yix), which results in the equation

ñW = j ‘f{ x )y { x ) 5y( j ) dx ( 10.8- 1)

With y(x) = where are the beam functions, and Qi are the element end
deflections as in Eq. (10.7-1), the virtual work is

dq, f'fi X) dx ( 10.8-2)

and the generalized force becomes

e , = (10.8-3)
which is proportional to the displacement.
Example 10.8-1
Figure 10.8-1 shows a cantilever beam with clastic foundation under the outer half of
the beam. The stiffness of the foundation is - k y Ibs/in., so that f i x ) ^ - k , a
constant. The equation of motion then takes the form
"I'l ' 0 , '
ml p é 8 E / r , T Ox Qz

’ + 7 T [^</] ‘ > = <
'■1 , ^'3 Qz
.04.
Evaluating the integral in Eq. (10.8-3) for an element of length /, we have
"0.3714 0.524/ 0.1286 -0.03095/ "
0.009524/' 0.03095/ -0.007143/'
{0,} = -kl 0.3714 -0.05238/ <
0.009524/'
When applied to this problem with 1 = 1/2 and transferred to the left side of the
equation, the stiffness of the beam is increased.

(D Figure 10.8-1.