Comment on: Transfer matrix relationships for repeated structures

Int. J. mech. Sci. P e r g a m o n P r e s s . 1 9 7 2 . V o l . 14, p p . 7 0 9 - 7 1 0 . P r i n t e d i n G r e a t B r i t a i n

L E T T E R S TO T H E E D I T O R C o m m e n t o n : T r a n s f e r m a t r i x relationships for repeated structures b y W . T. T~O~SON, I n t . J . mech. Sci. 13, 737 (1971) (Received 18 F e b r u a r y 1972)

THE ~E~HOD p r o p o s e d b y t h e a u t h o r is a n efficient m e t h o d for c a l c u l a t i n g t h e t r a n s f e r m a t r i x o f s y s t e m s c o n s i s t i n g of r e p e a t e d i d e n t i c a l sections. F o r s o m e i n v e s t i g a t i o n s h o w e v e r it is d e s i r a b l e t o k n o w t h e v a l u e of [T]" for n = 1, 2 . . . . , m w h e r e m > n ; in s u c h cases it m a y b e m o r e efficient t o use t h e C a y l e y - H a m i l t o n t h e o r e m , t F o r t h e case considered b y t h e a u t h o r , t h i s s t a t e s t h a t [T] ~ = a~[T] +a~[1], where a~ = t r a c e [ T ]

=A+D=fh+fq,

a2 = - d e t [T] = - ( A D - B C ) = - g t t~,. T h u s successive p o w e r s of [T] c a n b e c a l c u l a t e d f r o m t h e r e c u r r e n c e r e l a t i o n s h i p [T]~+ ~ = a~[T]~+~ + a,[T] ~. This m e t h o d m a y ~Iso be used to generate powers of n x ~ matrices. A m e t h o d which

is f a i r l y efficient n u m e r i c a l l y is as follows: Step 1

Calculate t h e m a t r i c e s [ T `"~] = [T]"

b y m e a n s of t h e r e c u r r e n c e r e l a t i o n ,

[ T (i+1)] = [ T ] x [ T (el

fori=

1. . . . . n - 1 .

Step 2

T h e C a y l e y - H a m i l t o n i a n t h e o r e m a s s e r t s t h a t t h e r e exists a r e l a t i o n of t h e f o r m [ T (n)] = a l [ T ("-~] + a2[T 'n-~)] + . . . + a,[1]. T h e v a l u e s of al, a2 . . . . . an c a n n o w b e o b t a i n e d b y e q u a t i n g coefficients in t h e a b o v e m a t r i x e q u a t i o n a n d s o l v i n g n u m e r i c a l l y , t h e r e s u l t i n g set of l i n e a r e q u a t i o n s . T h i s set is of course o v e r - d e t e r m i n e d , a c o n v e n i e n t s u b s e t for s o l u t i o n consists of t h e d i a g o n a l terms : ~11

'

~11

,

""",

~(n-1) ~$2 '

,,(n-2) ~$2 ~

<~-~',

< V ~, . . . . .

" " "'

tn, t22)

t..,

w h e r e t h e c o m p o n e n t s o f [T] are t~¢, etc. 709

] 1

~

0~1 ~2

~

=

t {n)

<-~

~

710

Letters to the E d i t o r

Step 3 Once t h e n u m e r i c a l values of a 1. . . . . a n are knowrt t h e r e m a i n i n g powers of IT] m a y be d e t e r m i n e d using t h e recttrrence relation IT ('+~] = ax[T ('+~-x)] + . . . + a ~ [ T ~)]

for p = 1 . . . . . m - n .

This m e t h o d avoids t h e p r o b l e m of d e t e r m i n i n g t h e eigenvalues a n d eigenvectors of t h e m a t r i x [T] which can be quite tedious p a r t i c u l a r l y w h e n t h e eigenvalues are complex.

REFERENCE I. S. PERLIS, Theory of Matrices, p. 136. Addison-Wesley,

School of Civil Engineering University of Sydney Sydney, 1V.S. W. 2006 Australia

Massachusetts.

(1958).

J . R . BOOKEI~

Reply to J. R. Bool~er's comment: (Received 17 April 1972) DR. B o o x ~ R ' s suggestion to use t h e C a y l e y - H a m i l t o n t h e o r e m for the e v a l u a t i o n of IT] n has considerable m e r i t in t h a t t h e necessity of c o m p u t i n g t h e eigenvalues a n d the eigenvectors of IT] is avoided. Since t h e d e t e r m i n a n t of [T] is u n i t y for these problems a n d t h e trace [TJ is available b y inspection, the use of t h e recurrence formula appears to be quite a t t r a c t i v e . The p r o b l e m of d e t e r m i n i n g t h e eigenvalues and the eigenvectors of [TJ is t o d a y a routine operation on the c o m p u t e r , e v e n for complex quantities. Since these c o m p u t a t i o n s are r e q u i r e d for e v e r y f r e q u e n c y chosen, the chances are t h a t a c o m p u t e r p r o g r a m would be w r i t t e n to c a r r y o u t t h e entire c o m p u t a t i o n .

University of California Santa Barbara California, U.S.A.

W . T . T~oMso~