On a generalization of the concepts of self-adjointness and of Rayleigh's quotient

MECH. RES. COMM.

Vol.l, 67-72, 1974

ON A GENERALIZATION

OF THE C O N C E P T S

Pergamon Press.

Prlnted in USA.

OF S E L F - A D J O I N T N E S S

AND OF

QUOTIENT 1

RAYLEIGH'S

Leipholz Professor, D e p a r t m e n t of Civil Engineering, Division, U n i v e r s i t y of W a t e r l o o , Waterloo, H.H.E.

Solid M e c h a n i c s Ontario, C a n a d a

(Received 8 January 1974; accepted as ready for print 27 February 1974)

Introduction Let a b o u n d a r y - e i g e n v a l u e problem be given as f o l l o w s : -ISIY + S2Y = 0,

(1)

[ u y ] B = o.

(2)

In (1) I i s t h e e i g e n v a l u e , S. and S^ are l i n e a r d i f f e r e n t i a l o p e r a t o r s , • 1 i and y i s the s o l u t i o n o f the b o u n d a r y - e i g e n v a l u e problem. In (2), U i s again a l i n e a r d i f f e r e n t i a l o p e r a t o r • and s u b s c r i p t B i n d i c a t e s t h a t t h e o p e r a t i o n on y i s to be c a r r i e d out a t boundary B. The posed problem i s s a i d t o be s e l f - a d j o i n t • /[-XSlV + S2v ] udV = f[-ESlU + S2u ] vdV V V

if (3)

holds for any pair of admissible functions u and v• i.e. functions which are sufficiently often differentiable and satisfy boundary conditions (2). In (3) the integration is to be carried out over domain V bounded by boundary B. N o w • l e t R a y l e i g h ' s Q u o t i e n t be i n t r o d u c e d . t i o n y, t h e e x p r e s s i o n

Using t h e a d m i s s i b l e func-

f y S2YdV XR _ V

(4)

f Y SlYdV V i s formed, which i s R a y l e i g h ' s Q u o t i e n t .

As well known• I R i s an approxima-

IDedicated to Professor Heinz Parkus in honour of his sixty-fifth birthday. Scientific communication

67

68

H.H.E. LEIPHOLZ

VoI.I, No. 2

tion of the true eigenvalue ~, and = min %R in S,

(s)

S = space of all admissible functions, is the condition which allows to approximate ~ as close as desired by hR. It is essential to point out that condition (5) holds only in the case that problem (i), (2) is self-adjoint and that both functionals on the right hand side of (4), in the numerator and in the denominator, are positive definite. Hence, the usefulness of Rayleigh's Quotient ~_ as a means of • K approximating the true eigenvalue ~ depends heavily on the fact that problem (i), (2) is self-adjoint. It is assumed that a proof for this statement is not necessary as this fact is well known. Aim of the paper is to define a more general kind of self-adjointness than the classical one and to show that even for eigenvalue problems which are in the strict, classical sense non-self-adjoint, a generalized kind of Rayleigh's Quotient may be found for approximating the eigenvalue of the problem by virtue of a minimum principle inherent in the generalized Rayleigh's Quotient.

A Generalization of Self-Adjointness

Consider the same kind of boundary-eigenvalue problem as described by (i), (2). However, now

(6)

f[-ISlV + S2v ] TudV = f[-lSlU + S2u ] TvdV V V shall hold instead of (3), where T is an approximate linear differential operator, the existence of which depends certainly not only on the nature of S 1 and S 2 but probably more on the kind of boundary conditions (2). If an identity (6) holds for any pair of admissible functions u and v, the boundary-eigenvalue problem (i), (2) shall be called self-adjoint with

respect to operator T and boundary conditions (2). Obviously, the classical case of self-adjointness as described by (3) is a special case of (6) for T ~ E, where E is the unit operator.

A Generalization of Rayleigh's Quotient

Insert in (i) in place of y the admissible function y. (I) by Ty and integrate this product•

The result is

Then, multiply

Vol. I, No. 2

SELF-ADJOINTNESS AND RAYLEIGII' S QUOTIENT

-IRG vITY SlYdV + vITY S2YdV = 0,

69

(7)

G where in a d d i t i o n ~ has been r e p l a c e d by an a p p r o p r i a t e value IR in o r d e r to s u s t a i n t h e v a l u e zero on t h e r i g h t hand s i d e o f (7). Expression (7) can be G solved f o r tR thus y i e l d i n g the G e n e r a l i z e d R a y l e i g h ' s Quotient f Ty S2Y dV G

XR =

(s)

v

f Ty SlY dV V The proposition is that %RG is again a useful approximation of % because = min kRG in S

(9)

under the assumptions that (i) and (ii)

f Ty S2Y dV and f Ty SlY dV are positive definite, V V problem (i), (2) is self-adjoint with respect to operator T and

boundary conditions

(2).

The correctness of this proposition will now be proven: Since assumptions

(i) and (ii) underlying

(8) do hold specifically for

the true solution y of problem (i), (2), also

=

f Ty S2Y dV V

(lO)

f Ty SlY dV V is valid.

Now, t h e v a r i a t i o n o f I may be c a l c u l a t e d .

Obviously, we a r r i v e

first at

-~

f Ty SlY dV - hi T~y SlY dV - ~ f Ty S 1 6y dV V V V

+ f T6y S2Y dV + f Ty S 2 6y dV = 0. V V

(ll)

If the variation of y is assumed to be carried out in S, then ~y is an admissible function like y itself.

Hence, according to (6), and because

assumption (ii) is supposed to hold, -~f Ty S 1 6y dV + f Ty S 2 6y dV = -El T6y SlY dV + f TSy S2Y dV. V V V V

(12)

Using (12) in (11) y i e l d s -~

f Ty SlY dV = -2 f [-~SlY ÷ S2Y] T~y dV. V V

(13)

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H.H.E. LEIPHOLZ

Vol. l, No. 2

However, a c c o r d i n g t o ( 1 ) , and due t o assumption ( i ) , which s t a t e s t h a t f Ty SlY dV i s p o s i t i v e d e f i n i t e , V

(14)

6~ = O. G

The conclusion is t h a t ~R assumes a stationary value for y : y. MoreG G over, the stationary value of %R has to be even a minimum since %R is always positive as the right hand side of (8) is supposed to be positive definite according to assumption (i).

Ex,~ple

C o n s i d e r P f l f i g e r ' s rod (Fig. 1). l i n e a r mass d i s t r i b u t i o n tributed,

That i s a p i n n e d - r o d o f l e n g t h £,

~, f l e x u r a l r i g i d i t y

c o m p r e s s i v e f o l l o w e r f o r c e s g.

t i o n s o f t h e rod about i t s ~

+ ~w

IV

a, s u b j e c t e d t o u n i f o r m l y d i s -

M a t h e m a t i c a l l y , t h e small v i b r a -

e q u i l i b r i u m p o s i t i o n a r e d e s c r i b e d by

+ g(~-x)w" = 0,

(IS)

w ( o , t ) : w " ( o , t ) : w ( ~ , t ) = w " ( ~ , t ) : O.

i~t

Assuming a s o l u t i o n i n the form w = e

(16)

y ( x ) , where m i s the frequency o f

vibration and y the mode shape, (15) and (16) can be transformed into the boundary eigenvalue problem _~y + ~y,V + g ( £ - x ) y " = 0, y(o)

= y,,(o)

= y(~)

= y',(~)

(17) = 0,

(18)

= ~2,

(19)

which i s j u s t o f t h e kind ( 1 ) , (18) i s n o t s e l f - a d j o i n t

(2).

As can e a s i l y be checked, problem (17),

in t h e c l a s s i c a l

sense.

However, i t

a d j o i n t in t h e g e n e r a l s ens e w i t h r e s p e c t t o t h e o p e r a t o r T ~

is selfd2 .../dx 2

Hence,

f [ - ~ v + a v ' V + g ( £ - x ) v " ] u '' dx = f [-~u+au'V+g(E-X)U"]v '' dx o

holds.

(20)

o

The c o r r e c t n e s s o f (20) can be shown by means o f i n t e g r a t i o n by p a r t s

and t a k i n g i n t o a c c o u n t t h a t c o n d i t i o n s (18) do h o l d f o r t h e two a d m i s s i b l e f u n c t i o n s u and v. The g e n e r a l i z e d R a y l e i g h ' s Q u o t i e n t f o r P f l ~ g e r ' s rod r e a d s

Vol. i, No. 2

71

SELF-ADJOINTNESS AND RAYLEIGH' S QUOTIENT £

£

f[ag'v÷gfz-x)9,,]9

,' dx

G = 0 ~R

I( 9 '''2 - g f Z - x ) 9 " 2 ) d x (21)

5 0

/[Y]9" dx o

f o

dx

This quotient can be used for determining approximately the eigenvalue curve F(A,g) = 0 (Fig. 2).

As can be seen in the figure, the buckling loads

i = 1,2,3 .... , of Pfluger's rod are given by the intersection of gi crit the eigenvalue curves with the positive load axis. Hence, buckling occurs for I = 0.

Using this fact in {21),

£ f ~ , , , 2 dx g o gR = £ / ( £ - x ) y ''2 dx o

{22)

r e s u l t s as a g e n e r a l i z e d R a y l e i g h ' s Q u o t i e n t f o r t h e f i r s t This s t a t e m e n t i s j u s t i f i e d

by t h e f a c t t h a t {22) corresponds t o R a y l e i g h ' s

Q u o t i e n t (8) i f t h i s e x p r e s s i o n i s w r i t t e n down f o r {17), and f o r ~ = 0.

b u c k l i n g load. (18), Ty ~ y " ,

Consequently,

g = min g~ in S.

(23)

The fact that g is a minimum follows from the fact that the right hand side of (22) is always positive. for

approximating the

Thus, it has been proved that (22) can be used

lowest buckling load gl crit of Pfluger's rod from

above. Using as an admissible function y = a sin ~x/£ in {22) yields G = 22/£3 = 19.7394~/~5, and using y = a(x4/12-xS£/6 + £5x/12) in {22) gR yields g~ = 20e/£ 5. These values have to be compared with gl crit 18"95~/~3 which is the exact value as calculated by A. Pfl~ger himself [I]. The conG clusion is that the approximate values gR do approach the exact value g indeed from above as predicted, and that the approximations are fairly good ones in spite of the fact

that only

one term expansion of y in terms of

admissible functions has been used for {22) in the numerical examples.

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H.H.E. LEIPROLZ

VoI.I, No. 2

Acknowledgement

This research was carried out under the support of the National Research Council of Canada under Grant No. A 7297.

Reference I.

A. Pfl~ger, StabilitEtsprobleme der Elastostatik, p. 193, SpringerVerlag, Berlin, G~ttingen, Heidelberg, (1950).

9 ~x

g2crl Q1crit w(x,t) FIG. 1

Pfl[iger' s rod

~UjoI

/~2 FIG. 2

Eigenvalue curve of Pfl~ger's rod moi' i = 1,2,3,.... are the frequencies of free vibration g"i crit' i = 1,2,3 ..... are the buckling loads