Page 150 - Numerical methods for chemical engineering

136 3 Matrix eigenvalue analysis

to show that the trial form of the solution is valid. The angular vibrational frequency of the
normal mode is then
+
λ j
ω j = (3.188)
m eff
lattice 2D vib.m computes the normal modes of a 2-D lattice of point masses connected by
harmonic springs. animate 2D vib.m produces a movie of the oscillations for each mode. A
derivation of this lattice model and a discussion of the results is provided in the supplemental
material in the accompanying website.

Relaxing the assumption of equal masses

Above we have assumed that each degree of freedom has an equal effective mass. We now
relax this assumption, using Lagrange’s equation of motion,
d ∂L ∂L
= (3.189)
dt ∂ ˙ q j ∂q j
where the Lagrangian, the kinetic energy minus the potential energy, is

L(q, ˙q) = K(q, ˙q) − U(q) (3.190)
For a system of point masses, each described by Cartesian coordinates, Lagrange’s equa-
tion reduces to Newton’s second law of motion. For small departures about a minimum
energy state, Lagrange’s equations of motion typically generate a system of the linearized
form
d
2
M δ =−Hδ (3.191)
dt 2
in which the mass matrix M is symmetric, positive-definite, but not necessarily diagonal.
Since M is nonsingular, we can write
d 2 −1
δ =−M Hδ (3.192)
dt 2
−1
We thus must perform normal mode analysis on the matrix M H, where
M −1 Hw [ j] = λ j w [ j] ⇒ Hw [ j] = λ j Mw [ j] (3.193)

The generalized eigenvalue problem

A generalized eigenvalue problem, such as (3.193), is of the form
Aw = λBw (3.194)

where B is a nonsingular matrix. While there exist general techniques to convert (3.194)
into a standard eigenvalue problem using Schur decompositions, we here describe a sim-
pler approach that may be used when B is positive-definite and A is real symmetric, as
they are for the problem above with B = M and A = H. We compute the Cholesky