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We continue to use the example introduced in §2.1 and Figure 2.1. We now consider the fully general case of nonzero masses $m_i$ and damping constants $b_i$. This leads to the equations of motion $M \ddot{x}(t) = -B \dot{x} (t) -K x(t)$. We solve them by changing variables to

y(t) = \bmat{c} \dot{x}(t) \\ x(t) \emat,


\bmat{cc} M & 0 \\ 0 & I \emat \dot{y}(t)
= \bmat{c} M \dd...
...ot{x}(t) \emat
= \bmat{cc} -B & -K \\ I & 0 \emat \cdot y(t)

or $A \dot{y}(t) = C y(t)$. We solve this by substituting $y(t) = e^{\lambda t} y$, where $y$ is a constant vector and $\lambda$ is a constant scalar to be determined. This yields

\begin{displaymath}Cy = \lambda Ay.\end{displaymath}

Thus $y$ is an eigenvector and $\lambda$ is an eigenvalue of the generalized nonsymmetric eigenvalue problem.

Susan Blackford 2000-11-20