Normal modes of two equal hinged uniform rods (${\omega }_{\pm }^2=(g/a)(3\pm 6/\sqrt{7})$)

For two rods of length $a$ and mass $m$, one hinged to a fixed point and the other to its free end, use the absolute rod angles from the downward vertical. The small-oscillation mass matrix is $m{a}^2\left(\begin{array}{cc}4/3& 1/2\\ 1/2& 1/3\end{array}\right)$ and the stiffness matrix is $(mga/2)\mathrm{diag}(3,1)$. Their generalized eigenvalue equation gives $7{\kappa }^2-42\kappa +27=0$ for $\kappa =a{\omega }^2/g$. The low-frequency mode has angles of the same sign and the high-frequency mode has opposite signs.