Page 258 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 8.5 Convergence to Higher Modes 245

which can be expressed by the matrix equation
0 0 0.25
0 0 -0 .7 9
0 0 1.00
This matrix is devoid of the first two modes and can be used as a sweeping matrix for
the third mode. Applying this to the original equation, we obtain
■ 4 2 1 ■ '0 0 0.25'
4 8 4 0 0 -0 .7 9
4 8 7 0 0 1.00

This equation results immediately in the third mode, which is
0.25 \
1.68 -0.79 I = 3k
1.00 I ^3

The natural frequency of the third mode is then found to be

^ = 1 34,/ A
V IT .68w V m
These natural frequencies were checked by solving the stiffness equation, which

”4 0 0“ 4 - 1

m 0 2 0 < ^2 > + A -1 2 = 0
V ^ j
0 0 1 0 -1

With A = mco^/k, the determinant of this equation set equal to zero gives
8(1 - A)’ - 5(1 - A) = (1 - A )[8(l - A)- - 5 ] = 0
Its solutions are
0.250
A, = 0.2094 ca, = 0.4576 0 , = { 0.791
1.000

Ao = 1.0000 = 1.0000-1/ —
; i i
1 [ 0.250
A ^= 1.7906 ia. = 1.3381-i/ — -0.791
1 1 1.000
Figure 8.5-1 is a printout from the computer program ITERA TE that solves for
A = k/m o)^ instead of A = mco^/k. Thus, for mode 1, we have for comparison,
A = 1 /A = 1 /4.775 = 0.2094. The mode shapes, however, are not altered.