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Example
For the vibrating mass-spring system introduced in §2.1 and
Figure 2.1, we assume that
- all masses
, so
, and
- all damping constants
, so
.
This simplies the equations of motion to
.
We solve them by substituting
, where
is
a constant vector and
is a constant scalar to be determined.
This yields
Thus
is an eigenvector
and
is an eigenvalue of the symmetric positive
definite tridiagonal matrix
. Thus
is pure imaginary
and we get that
is periodic with period
.
Symmetric tridiagonal matrices have particularly fast and efficient
eigenvalue algorithms.
Later sections deal with the cases of nonunit masses
and nonzero damping constants
.
Susan Blackford
2000-11-20