Mathematical Methods for Physics and Engineering - Matematica.NET

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/MFigure 9.5 The displacements and frequencies of the eight normal modes ofthe 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 remainingfour, (e)–(h), of lower frequency, are degenerate.mode’. Expressing this motion in coordinate form gives as the fourth eigenvectorx (4) = 1 √2(−1 1 1 1 − 1 − 1 1 − 1) T .Evaluation of Bx (4) yieldsBx (4) =k4 √ 2 (−8 8 8 8 − 8 − 8 8 − 8)T =2kx (4) ,i.e. a multiple of x (4) , confirming that it is indeed an eigenvector. Further, sinceAx (4) = Mx (4) , it follows from (B − ω 2 A)x = 0 that ω 2 =2k/M forthisnormalmode. Diagram (d) of the figure illustrates the corresponding motions of the fourmasses.As the next step in exploiting the symmetry properties of the system we notethat, because of its reflection symmetry in the x-axis, the system is invariant underthe double interchange of y 1 with −y 3 and y 2 with −y 4 . This leads us to try aneigenvector of the formx (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-325