Page 339 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

326 Introduction to the Finite Element Method Chap. 10

The equation indieates that the system is decoupled for l\ = = in which
case the equation reduces to
\ /
1
kl^ \
- p (l92 + ^ ) V2
El
I
P 1 , kP \
M ^02
)
The deflection at the center is then
~{PP/EI) MP/EI
So =
192 + kP/EI 16P + KE/EI
Example 10.6-2
Determine the natural frequencies of the constrained beam of Example 10.6-1 when
/, = /^ = 1/2.
Solution: For this determination, we need the mass matrix, which can be assembled from
Eq. (10.2-10) as

/ m ■ 156(/, +/2) - 2 2 { i j - q y 156 0 '
- i "’M
- 2 2 ( / ? - / | ) 4(/? + / 0 \420j 0
The equation of motion then becomes

156 0 0
(o^ml El
420
0 0
( “ ' ■ - " I r )
Again, coordinates V2 and 62 are decoupled. By letting A = oj^mP/420EI, the

equation for U2 gives

1 / icE \
^ ^ 156 ^ E7 r " ^ 0.00641 El
and the natural frequency for this mode is

El I ( LrP'
'(S)
= 22.73 1/1 0.005211
Similarly, the equation for 62 results in

A = 16 -l-

El
and
^2 = 81.98W— T 1/1 + 0.0625
ml m