Mathematical Methods for Physics and Engineering : A Comprehensive Guide

9.3 RAYLEIGH–RITZ METHOD

and that this mode has the same frequency as three of the other modes. The

general topic of the degeneracy of normal modes is discussed in chapter 29. The

movements associated with the final two modes are shown in diagrams (g) and

(h) of figure 9.5; this figure summarises all eight normal modes and frequencies.

Although this example has been lengthy to write out, we have seen that the

actual calculations are quite simple and provide the full solution to what is

formally a matrix eigenvalue equation involving 8×8 matrices. It should be

noted that our exploitation of the intrinsic symmetries of the system played a

crucial part in finding the correct eigenvectors for the various normal modes.

9.3 Rayleigh–Ritz method

We conclude this chapter with a discussion of the Rayleigh–Ritz method for

estimating the eigenfrequencies of an oscillating system. We recall from the

introduction to the chapter that for a system undergoing small oscillations the

potential and kinetic energy are given by

V=qTBq and T= ̇qTA ̇q,

where the components ofqare the coordinates chosen to represent the configura-

tion of the system andAandBare symmetric matrices (or may be chosen to be

such). We also recall from (9.9) that the normal modesxiand the eigenfrequencies

ωiare given by

(B−ω^2 iA)xi= 0. (9.14)

It may be shown that the eigenvectorsxicorresponding to different normal modes

are linearly independent and so form a complete set. Thus, any coordinate vector

qcan be writtenq=

∑

jcjx

j. We now consider the value of the generalised

quadratic form

λ(x)=

xTBx

xTAx

=

∑

m(x

m)Tc∗

mB

∑

icix

i

∑

j(x

j)Tc∗

jA

∑

kckx

k,

which, since both numerator and denominator are positive definite, is itself non-

negative. Equation (9.14) can be used to replaceBxi, with the result that

λ(x)=

∑

m(x

m)Tc∗

mA

∑

iω

2

icix

i

∑

j(x

j)Tc∗

jA

∑

kckx

k

=

∑

m(x

m)Tc∗

m

∑

iω

2

iciAx

i

∑

j(x

j)Tc∗

jA

∑

kckx

k. (9.15)

Now the eigenvectorsxiobtained by solving (B−ω^2 A)x= 0 are not mutually

orthogonal unless eitherAorBis a multiple of the unit matrix. However, it may