Page 239 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

226 Lagrange’s Equation Chap. 7

By letting A = co^ml^/EI, the characteristic determinant
( 1 - | A ) ( 1 - i A ) ^
( 1 - l A ) ( l - l A )

reduces to the following polynomial equation for the eigenvalues:
A^ - 34.666A + 79.999 = 0

Solving for A, we have
I 2.486
A = 17.333 ± 14.847 =

\ 32.180
and the natural frequencies are
ioi = 1.577i

ia2 = 5.672i

The exact values for this problem are
EA EA
= 1.5708
ml^ ml^
Stt EA EA
^ 4.7124
~2 ml^
which indicates good agreement for the first mode. The second mode frequency is
20.4 percent high, which is to be expected with only two modes.
From the first equation, the ratio of the amplitudes is
( i - i q
1 - iA
By substituting Aj = 2.486, the first mode ratio is
-0.378 -1.0
Q2 ~ 0.171 “ 0.453
For the second mode, we substitute A2 = 32.18 and obtain
_ -7.05 _ -1.0
Q2 ~ 9.73 “ 1.38
The displacement equation for each mode can now be written as
«.(^) = - ( j ) + 0.453(y)'

Uiix) = - ( 7 ) + l-38( j )