Angular expression for the Rayleigh quotient

Journal of Sound and Vibration (1987) 113(1), 215-216

ANGULAR EXPRESSION FOR THE RAYLEIGH QUOTIENT

1.

INTRODUCTION

The Rayleigh quotient proposed by Lord Rayleigh [1] as a method to estimate the fundamental frequency of elastic systems is now an essential topic in vibration analysis. In this note, an expression for this quotient based on the angle between a trial vector and the system eigenvectors is presented. This expression provides a new way to demonstrate the properties of the Rayleigh quotient and can be helpful in lecturing use. 2.

DEFINITION OF RAYLEIGH QUOTIENT

Consider a positive definite system and write the eigenvalue problem in the form [k]{u} = A[m]{u}, where [k] and Em] are symmetric matrices, {u} is a vector and A is a

number. When A is an eigenvalue Ai, the solution of this equation is the corresponding eigenvector {u(il}. For an arbitrary vector u the expression R(u) = {u}T[k]{u}j{u}T[m]{u} depending on the vector {u} is known as Rayleigh quotient. R(u) has a stationary value when the vector u is in the neighborhood of an eigenvector. (Throughout this discussion, of course, the dimension n of the vectors concerned is the same as that of the eigenvalue.) 3.

ANGULAR EXPRESSION OF THE RAYLEIGH QUOTIENT

An expression for the Rayleigh quotient in terms of an angle in vector space can be provided in the following way. First one defines a "kinetic energy scalar product", denoted by ( h., as follows: (u, V)k. ={u}T[m]{v}; it has the corresponding norm lIullk. = (u, U)k.)1/2.

N ow the "kinetic energy" scalar product of two vectors can also be written as (u, v h. = where (J is the "angle" between the two "kinetic energy vectors". One can assume that an arbitrary vector {u} is a superposition of the normalized eigenvectors {u(i)} of the system: i.e.,

I u Ilk.11 vllke cos (0),

{u}=£

and

Ci{U(l)}

Ilu(i)llke=l,

(U ( i ) ~ uU»

ke

= 8..y.

i=1

It is then easy to show that, for such an arbitrary vector {u}, n

R(u)=

I

2

Ai cos (8;).

i=l

Here Aj is the eigenvalue associated with the eigenvector {u(i)}, (}j is the angle between the arbitrary vector {u} and {u(i)}, and Ai = w~, where Wi is the corresponding natural frequency of vibration of the system. One can now see that Cj = (u, u (i» ke =" I u I ke cos ( 8;). Then _ {u}T[k]{u} R(u) IIul11e

(i)

_1_

T

Ilull~. L I CjCj{u } [k]{u

W}'

.

But {u(i)V[k]{u(j)} = Al'y, and so

R(U)=(£ A

ie7)/IIU

11

(=1

ie="

£ A1coS

2

( Oi)'

i=1

215 0022-460X/81/040215 +02 $03.00/0

© 1987 Acadsmic Press Inc. (London) Limited

216

LEITERS TO THE EDITOR

Also one has that

II u"~. == (U, U)ke = 2: L CjCj(u(i), uU)h. = L c; == II uIIi. L cos'' (8J. j

j

j

j

Then

L cos" (8 = 1 j )

and

R( u)

=L (Ai -

Ar ) cos 6j + A" 2

i

where Ar is the eigenvalue associated with the eigenvector {u(r)}. Thus when {u} is near {u(r)} then Or ~ e, is very small, OJ == e, + 71'/2, where e, is a small number, and R(u)=Ar+I (Aj-Ar)d. j

From these expressions one can see that (i) the Rayleigh quotient has a stationary value in the neighborhood of an eigenvector; (ii) the stationary value is a minimum in the neighborhood of the fundamental eigenvector. Other properties (see reference [2]) also can be deduced. The idea used here, of course, is simply that a vector is in the neighborhood of another vector if the angle between them, defined through the "kinetic energy scalar product", is small. COPPEI VFRJ c» 68503, 21941·Rio de Janeiro, Brasil Now at Laboratoire de Mecanique et d' Acoustique, Centre National de la Recherche Scientifique, B.P. 71, 13277 Marseille Cedex 9, France

J. G.

SLAMA

(Received 30 October 1986) REFERENCES

1. LORD RAYLEIGH 1877 Theory of Sound, Volume 1. New York: Dover Publications, second edition, 1945 re-issue. 2. L. MEIROVITCH 1967 Analytical Methods in Vibrations. New York: MacMillan.