Vertical normal modes of two equal suspended masses
= Vertical normal modes of two equal suspended masses
For equal masses $m$ and spring constants $k$, vertical displacements from equilibrium have stiffness and mass matrices
$$
K=k\begin{pmatrix}2&-1\\-1&1\end{pmatrix},
\qquad M=mI.
$$
The two squared frequencies and displacement ratios are
$$
\omega_\pm^2=\frac{k}{2m}(3\pm\sqrt5),
\qquad
\frac{\eta_2}{\eta_1}=\frac{1\mp\sqrt5}{2}.
$$
Gravity shifts the equilibrium but does not enter these vertical frequencies.