 Solution

For the normal modes of a double pendulum with a two-to-one length ratio, seek $\mathbf{\theta }=\mathbf{a}\cos (\Omega t+\phi )$. The generalized eigenvalue problem is $(\mathbf{V}-{\Omega }^2\mathbf{T})\mathbf{a}=0$. Put $s={\Omega }^{2}l/g$. A nonzero mode requiresThus the two angular frequencies areBoth squared frequencies are positive. The slower mode has ${\theta }_2/{\theta }_1=\sqrt{5}-1$ and is in phase; the faster mode has ${\theta }_2/{\theta }_1=-1-\sqrt{5}$ and is out of phase. These mode ratios also check the off-diagonal terms in the mass matrix.