Mathematical Methods for Physics and Engineering : A Comprehensive Guide

9.2 SYMMETRY AND NORMAL MODES

(a)ω^2 =0 (b)ω^2 =0 (c)ω^2 =0 (d)ω^2 =2k/M

(e)ω^2 =k/M (f)ω^2 =k/M (g)ω^2 =k/M (h)ω^2 =k/M

Figure 9.5 The displacements and frequencies of the eight normal modes of

the system shown in figure 9.4. Modes (a), (b) and (c) are not true oscillations:

(a) and (b) are purely translational whilst (c) is a mode of bodily rotation.

Mode (d), the ‘breathing mode’, has the highest frequency and the remaining

four, (e)–(h), of lower frequency, are degenerate.

mode’. Expressing this motion in coordinate form gives as the fourth eigenvector

x(4)=

1

√

2

(− 1111 − 1 − 11 −1)T.

Evaluation ofBx(4)yields

Bx(4)=

k

4

√

2

(− 8888 − 8 − 88 −8)T=2kx(4),

i.e. a multiple ofx(4), confirming that it is indeed an eigenvector. Further, since

Ax(4)=Mx(4), it follows from (B−ω^2 A)x= 0 thatω^2 =2k/Mforthisnormal

mode. Diagram (d) of the figure illustrates the corresponding motions of the four

masses.

As the next step in exploiting the symmetry properties of the system we note

that, because of its reflection symmetry in thex-axis, the system is invariant under

the double interchange ofy 1 with−y 3 andy 2 with−y 4. This leads us to try an

eigenvector of the form

x(5)=(0 α 0 β 0 −α 0 −β)T.

Substituting this trial vector into (B−ω^2 A)x= 0 gives, of course, eight simulta-