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For the vibrating mass-spring system introduced in §2.1 and
Figure 2.1, we assume that
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.
- all masses , so , and
- all damping constants , so .
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
Later sections deal with the cases of nonunit masses
and nonzero damping constants .