= Vertical modes of a horizontally constrained suspended bar
{title2=$M\omega^4-[\kappa_0+2\kappa_1+M\kappa_1/m]\omega^2+\kappa_0\kappa_1/m=0$}
A horizontal bar of <mass> $M$ hangs from a <spring> of incremental stiffness $\kappa_0>0$, and two equal masses $m$ hang separately from the bar through springs of stiffness $\kappa_1>0$. The bar remains horizontal and all motion is vertical. For physical displacements $q=(q_1,q_2,q_3)$ from the loaded <equilibrium>, the <kinetic energy> and quadratic <potential energy> are
$$
T=\frac12[M\dot q_1^2+m\dot q_2^2+m\dot q_3^2],\qquad V=\frac12\kappa_0q_1^2+\frac12\kappa_1[(q_2-q_1)^2+(q_3-q_1)^2].
$$
<Gravity> determines the static extensions; its linear terms cancel at equilibrium. Exchange of the lower masses splits the <normal modes> into the antisymmetric mode $(0,1,-1)$ with $\omega^2=\kappa_1/m$, and two symmetric modes $(A,B,B)$. The latter obey $(\kappa_0+2\kappa_1-M\omega^2)A-2\kappa_1B=0$ and $-\kappa_1A+(\kappa_1-m\omega^2)B=0$. Their determinant gives the title quadratic in $\omega^2$, and $B/A=\kappa_1/(\kappa_1-m\omega^2)$. For $M=2,m=1,\kappa_0=4,\kappa_1=1$, their squared <angular frequencies> are $2\pm\sqrt2$, with displacement ratios $(1,1\mp\sqrt2,1\mp\sqrt2)$ respectively. The lower symmetric mode moves all masses in phase; the higher one moves the lower pair against the bar.
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