Mathematical Methods for Physics and Engineering - Matematica.NET

9.3 RAYLEIGH–RITZ METHODand that this mode has the same frequency as three of the other modes. Thegeneral topic of the degeneracy of normal modes is discussed in chapter 29. Themovements 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 theactual calculations are quite simple and provide the full solution to what isformally a matrix eigenvalue equation involving 8 × 8 matrices. It should benoted that our exploitation of the intrinsic symmetries of the system played acrucial part in finding the correct eigenvectors for the various normal modes.9.3 Rayleigh–Ritz methodWe conclude this chapter with a discussion of the Rayleigh–Ritz method forestimating the eigenfrequencies of an oscillating system. We recall from theintroduction to the chapter that for a system undergoing small oscillations thepotential and kinetic energy are given byV = q T Bq and T = ˙q T A˙q,where the components of q are the coordinates chosen to represent the configurationof the system and A and B are symmetric matrices (or may be chosen to besuch). We also recall from (9.9) that the normal modes x i and the eigenfrequenciesω i are given by(B − ω 2 i A)x i = 0. (9.14)It may be shown that the eigenvectors x i corresponding to different normal modesare linearly independent and so form a complete set. Thus, any coordinate vectorq can be written q = ∑ j c jx j . We now consider the value of the generalisedquadratic form∑λ(x) = xT Bxx T Ax = ∑ m(xm ) T c ∗ mB ∑ i c ix ij (xj ) T c ∗ j A ∑ k c kx k ,which, since both numerator and denominator are positive definite, is itself nonnegative.Equation (9.14) can be used to replace Bx i , with the result thatλ(x) ==∑m (xm ) T c ∗ mA ∑ ∑i ω2 i c ix ij (xj ) T c ∗ j A ∑ k c kx k∑m (xm ) T c ∗ m∑i ω2 i c iAx i∑j (xj ) T c ∗ j A ∑ k c kx k . (9.15)Now the eigenvectors x i obtained by solving (B − ω 2 A)x = 0 are not mutuallyorthogonal unless either A or B is a multiple of the unit matrix. However, it may327