Page 261 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

248 Computational Methods Chap. 8

8.8 SYSTEMS WITH DISCRETE MASS MATRIX
For the lumped-mass system in which the coordinates are chosen at each of the
masses, the mass matrix is diagonal and U is simply equal to the square root of
each diagonal term. The inverse of U is then equal to the reciprocal of each term
in U, so that we have

m,
M = m 22 U = =

= u - T

Thus, the dynamic matrix A = U ^KU *of Eq. (8.7-3) is simply determined.

Example 8.8-1
Consider the system of Example 6.8-1, which is shown again in Fig. 8.8-1. The mass
and stiffness matrices for the problem are
2 0 3 -1
M = m 0 1 K = k - 1 1

We first decompose the mass matrix to M = U^U = Because M is
diagonal, the matrix U is simply found from the square root of the diagonal terms. Its
inverse is also found from the inverse of the diagonal terms, and its transpose is
identical to the matrix itself.

u = = vW v/2 0 u^' = u~^ = l/v/2 0 1 0.7071 0
0 1 4m 0 1 '/m . 0 1.
Thus, the terms of the standard equation become
U-^MU~' = U -^ U ^U U -' = I
k 0.707 01 [ 3 -1 ] 0.707 0 k 1.50 -0.707
m .0 l] 1 -1 l) 0 1 m -0.707 1.0

2m
Zk
Figure 8.8-1.