RayleighRitz method

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 = qT Bq
and
T = q̇T Aq̇,
where the components of q are the coordinates chosen to represent the configuration of the system and A and B are symmetric matrices (or may be chosen to be
such). We also recall from (9.9) that the normal modes xi and the eigenfrequencies
ωi are given by
(B − ωi2 A)xi = 0.
(9.14)
It may be shown that the eigenvectors xi corresponding to different normal modes
are linearly independent and so form a complete set. Thus, any coordinate vector
q can be written q = j cj xj . We now consider the value of the generalised
quadratic form
mT ∗ (x ) c B
ci xi
xT Bx
= m j T ∗m i k ,
λ(x) = T
x Ax
j (x ) cj A
k ck x
which, since both numerator and denominator are positive definite, is itself nonnegative. Equation (9.14) can be used to replace Bxi , with the result that
mT ∗ 2 i
m (x ) cm A i ωi ci x
λ(x) = j T ∗
k
j (x ) cj A
k ck x
mT ∗ 2
i
m (x ) cm i ωi ci Ax
= .
(9.15)
∗
j
T
k
j (x ) cj A
k ck x
Now the eigenvectors xi obtained by solving (B − ω 2 A)x = 0 are not mutually
orthogonal unless either A or B is a multiple of the unit matrix. However, it may
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NORMAL MODES
be shown that they do possess the desirable properties
(xj )T Axi = 0
and
(xj )T Bxi = 0
if i = j.
(9.16)
This result is proved as follows. From (9.14) it is clear that, for general i and j,
(xj )T (B − ωi2 A)xi = 0.
(9.17)
But, by taking the transpose of (9.14) with i replaced by j and recalling that A
and B are real and symmetric, we obtain
(xj )T (B − ωj2 A) = 0.
Forming the scalar product of this with xi and subtracting the result from (9.17)
gives
(ωj2 − ωi2 )(xj )T Axi = 0.
Thus, for i = j and non-degenerate eigenvalues ωi2 and ωj2 , we have that
(xj )T Axi = 0, and substituting this into (9.17) immediately establishes the corresponding result for (xj )T Bxi . Clearly, if either A or B is a multiple of the unit
matrix then the eigenvectors are mutually orthogonal in the normal sense. The
orthogonality relations (9.16) are derived again, and extended, in exercise 9.6.
Using the first of the relationships (9.16) to simplify (9.15), we find that
|ci |2 ωi2 (xi )T Axi
.
λ(x) = i
2 k T
k
k |ck | (x ) Ax
(9.18)
Now, if ω02 is the lowest eigenfrequency then ωi2 ≥ ω02 for all i and, further, since
(xi )T Axi ≥ 0 for all i the numerator of (9.18) is ≥ ω02 i |ci |2 (xi )T Axi . Hence
λ(x) ≡
xT Bx
≥ ω02 ,
xT Ax
(9.19)
for any x whatsoever (whether x is an eigenvector or not). Thus we are able to
estimate the lowest eigenfrequency of the system by evaluating λ for a variety
of vectors x, the components of which, it will be recalled, give the ratios of the
coordinate amplitudes. This is sometimes a useful approach if many coordinates
are involved and direct solution for the eigenvalues is not possible.
2
may also be
An additional result is that the maximum eigenfrequency ωm
2
estimated. It is obvious that if we replace the statement ‘ωi ≥ ω02 for all i’ by
2
2
‘ωi2 ≤ ωm
for all i’, then λ(x) ≤ ωm
for any x. Thus λ(x) always lies between
the lowest and highest eigenfrequencies of the system. Furthermore, λ(x) has a
stationary value, equal to ωk2 , when x is the kth eigenvector (see subsection 8.17.1).
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