Page 255 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

242 Computational Methods Chap. 8

the /th iterated vector X. From the preceding equation, we obtain
( 1) ( 1)
a:, = - a:. -
X. - X,

where the last two equations have been introduced as identities. Expressed in
matrix form, this equation is
'I ( 1) ( 1)
mn
-ïl 0 - m. - — m. —
•«2 0 (8.5-5)
0
= [S]{X}
This is the constraint equation for removing the first mode, and [S]is the sweeping
matrix. By replacing X on the left side of Eq. (8.3-1) by this constraint equation, it
becomes
ASX = k X (8.5-6)
Iteration of this equation now sweeps out the undesired c/>| component in each
iteration step and converges to the second mode cfyj-
For the third and higher modes, the sweeping procedure is repeated with the
normal modes already found. This reduces the order of the matrix equation by 1
each time. Thus, the matrix [AS] is referred to as the deflated matrix.
It is well to mention that the convergence for higher modes becomes more
critical if impurities are introduced through the sweeping matrix, i.e., the lower
modes used for the sweeping matrix must be accurately found. The highest mode
can be checked by the inversion of the original equation, which is the equation
formulated in terms of the stiffness matrix.
Computer notes. To program the iteration procedure for the digital
computer, it is convenient to develop another form of the sweeping matrix S based
on the Gram-Schmidt orthogonalization.^ Rewriting Eq. (8.4-2) as
X^ 2 X j (X j (f) j C 2 2 e (f) 3 • • • -|- (8.5-7)
where a, is the unwanted (/>, component, we again premultiply this equation by
(f)^M to obtain
0[M (2i, - acf)^) = 0

Wilson E. Klus-Jurgen Bathe, Numerical Methods in Finite Element Analysis (Englewood Cliffs,
NJ: Prentice-Hall, 1976), p. 440.