= Normal modes of a rigid beam on two end springs
{title2=$\omega_s^2=2k/m,\quad\omega_a^2=6k/m$}
For a uniform rigid beam of <mass> $m$ and two identical vertical end <springs> of stiffness $k$, small vertical end <displacements> have <kinetic energy> $m(\dot x_1^2+\dot x_1\dot x_2+\dot x_2^2)/6$ and displacement-dependent <potential energy> $k(x_1^2+x_2^2)/2$. The <mass> and <stiffness matrices> are $\mathsf M=(m/6)\begin{pmatrix}2&1\\1&2\end{pmatrix}$ and $\mathsf K=k\mathsf1$. Their <generalized eigenvalue problem> gives symmetric translation $(1,1)$ and antisymmetric rocking $(1,-1)$ at the displayed <angular frequencies>. The centre coordinate contains only the symmetric mode. Neglect <spring> inertia, <damping> and finite-angle geometric corrections.
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