Three-to-one mass double-pendulum modes (${\omega }_{-}^2=2g/(3\ell ),{\omega }_{+}^2=2g/\ell$)

An upper mass $3m$ and lower mass $m$ joined by equal light strings of length $\ell$ have small-angle mass and stiffness matrices $m{\ell }^2\left(\begin{array}{cc}4& 1\\ 1& 1\end{array}\right)$ and $mg\ell \left(\begin{array}{cc}4& 0\\ 0& 1\end{array}\right)$. The squared frequencies are $2g/(3\ell )$ and $2g/\ell$, with angular eigenvectors $(1,2)$ and $(1,-2)$ respectively. Equal-angle release decomposes with coefficients $3\alpha /4$ and $\alpha /4$. The angles are measured from vertical, not between strings.