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## Example

For the vibrating mass-spring system introduced in §2.1 and Figure 2.1, we assume that

1. all masses , so , and
2. 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