Two vertically suspended springs

For two masses hung in series from springs of natural length $\ell$, the lower equilibrium spring tension is $m_{2}g$ and the upper tension is $(m_1+m_2)g$. Hence their equilibrium lengths are

For equal masses $m$ and spring constants $k$, vertical displacements from equilibrium have stiffness and mass matricesThe two squared frequencies and displacement ratios areGravity shifts the equilibrium but does not enter these vertical frequencies.