= End-driven two-bead string
{title2=$\omega_0^2=T/(2ML)$}
Place two equal <masses> $M$ on a massless <mechanical string> with spans $L,2L,L$ and uniform <tension> $T$. Fix one end and drive the other with displacement $h(t)$. Linear transverse force balance gives $\ddot y_1+\omega_0^2(3y_1-y_2)=0$ and $\ddot y_2+\omega_0^2(3y_2-y_1)=2\omega_0^2h$. The mass-normalized stiffness matrix has symmetric and antisymmetric <normal modes> with squared angular frequencies $2\omega_0^2$ and $4\omega_0^2$. The mode coordinates $(y_1+y_2)/2$ and $(y_1-y_2)/2$ separate the forced equations.
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