Mathematical Methods for Physics and Engineering : A Comprehensive Guide

9.4 EXERCISES

Estimate the eigenfrequencies of the oscillating rod of section 9.1.

Firstly we recall that

A=

Ml^2

12

(

63

32

)

and B=

Mlg

12

(

60

03

)

.

Physical intuition suggests that the slower mode will have a configuration approximating
that of a simple pendulum (figure 9.1), in whichθ 1 =θ 2 , and so we use this as atrial
vector.Takingx=(θθ)T,

λ(x)=

xTBx

xTAx

=

3 Mlgθ^2 / 4

7 Ml^2 θ^2 / 6

=

9 g

14 l

=0. 643

g

l

,

and we conclude from (9.19) that the lower (angular) frequency is≤(0. 643 g/l)^1 /^2 .We
have already seen on p. 319 that the true answer is (0. 641 g/l)^1 /^2 and so we have come
very close to it.
Next we turn to the higher frequency. Here, a typical pattern of oscillation is not so
obvious but, rather preempting the answer, we tryθ 2 =− 2 θ 1 ; we then obtainλ=9g/l
and so conclude that the higher eigenfrequency≥(9g/l)^1 /^2. We have already seen that the
exact answer is (9. 359 g/l)^1 /^2 and so again we have come close to it.

A simplified version of the Rayleigh–Ritz method may be used to estimate the

eigenvalues of a symmetric (or in general Hermitian) matrixB, the eigenvectors

of which will be mutually orthogonal. By repeating the calculations leading to

(9.18),Abeing replaced by the unit matrixI, it is easily verified that if

λ(x)=

xTBx

xTx

is evaluated foranyvectorxthen

λ 1 ≤λ(x)≤λm,

whereλ 1 ,λ 2 ...,λmare the eigenvalues ofBin order of increasing size. A similar

result holds for Hermitian matrices.

9.4 Exercises

9.1 Three coupled pendulums swing perpendicularly to the horizontal line containing
their points of suspension, and the following equations of motion are satisfied:

−m ̈x 1 =cmx 1 +d(x 1 −x 2 ),

−M ̈x 2 =cMx 2 +d(x 2 −x 1 )+d(x 2 −x 3 ),

−m ̈x 3 =cmx 3 +d(x 3 −x 2 ),

wherex 1 ,x 2 andx 3 are measured from the equilibrium points;m,Mandm
are the masses of the pendulum bobs; andcanddare positive constants. Find
the normal frequencies of the system and sketch the corresponding patterns of
oscillation. What happens asd→0ord→∞?
9.2 A double pendulum, smoothly pivoted atA, consists of two light rigid rods,AB
andBC, each of lengthl, which are smoothly jointed atBand carry massesmand
αmatBandCrespectively. The pendulum makes small oscillations in one plane