### 6.2 Undamped dynamic problem

\[ \mathbf K \mathbf u + \mathbf M \ddot {\mathbf u} = \mathbf f \]

where \(\ddot {\mathbf u}\) is defined as
\(\text d ^2 \mathbf u / \text d t^2\).
\[\begin{align}
\frac {\text d} { \text d t } \sin {\omega t} & = \omega \cos \omega t \\
\frac {\text d} { \text d t } \cos {\omega t} & = - \omega \sin \omega t
\end{align}\]

If we consider harmonic vibrations, \(\mathbf M \ddot {\mathbf u}\)
becomes \(- \omega^2 \mathbf M \mathbf u\). Because of the lack
of damping (or energy dissipation),
forcing functions at certain frequencies will lead to infinite
displacements, so we consider the *unforced* problem:
\[\mathbf K \mathbf u - \omega^2 \mathbf M \mathbf u = 0\]

R. Funnell
Last modified: 2018-11-07 12:52:06
Slide show generated from fem.html by Weasel 2018 Nov 7