### 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