Page 253 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

240 Computational Methods Chap. 8

Thus, the frequency of the lovs/cst mode is
= 0.457-1/ —
^ V it : 32m V ^
with the mode shape
0.250)
c/>, == {0.790

1.000 j
It should be mentioned here that if the equation of motion was forriiulated in
terms of the stiffness matrix, the iteration equation would be
AX = AX
[M~'K]X = co"X
Because the iteration procedure always converges to the largest eigenvalue, the
stiffness equation would converge to the highest mode. In vibration analysis, the lower
modes arc generally of greater interest than the higher modes, so that the matrix
iteration procedure will find its use mainly for equations formulated in terms of
flexibility where the eigenvalues are proportional to the reciprocal of co^.

8.4 CONVERGENCE OF THE ITERATION PROCEDURE

To show that the iteration proeedure eonverges to the largest eigenvalue, whieh for
the equation formulated in terms of flexibility is the lowest fundamental mode, the
assumed trial veetor X^ is expressed in terms of the normal modes (/>, by the
expansion theorem:

X^ = + r*2^2 ^3^3 T ■■‘ T (8.4-1)

where are constants. Multiplying this equation by the dynamic matrix A, we
have
A X ^ — X 2 — c^Acf)^ + C2v4</>2 T c^Acf)^ + * * ■ + (8.4-2)

Because each normal mode satisfies the following equation

(8.4-3)

the right side of Eq. (8.4-2) becomes
4^2 = C| — <(>| + C2 — <I>2 + — ■ +
CU 1 CO 2 OJ 2
which is the new displacement vector X 2. Again, premultiplying X 2 by the
dynamic matrix and using Eq. (8.4-3), the result is

AX^ = X 2 = Cj — -f C2 — 4>2 + <^ 3 ~4 ^ 3 T * • ' T
O), CO 2