Middle-mass-independent mode of a fixed three-mass spring chain ($q=(1,0,-1{)}^T,{\omega }^2=2k/m$)

Three consecutive masses $m,M,m$ joined by four identical springs of constant $k$ between fixed walls have an antisymmetric normal mode with zero central displacement. The two adjacent spring forces cancel on the central mass, while each outer mass experiences restoring force $-2kx$. This proves that the displayed angular frequency is independent of $M$ for every positive central mass. The two symmetric modes satisfy $(2k-m{\omega }^2)(2k-M{\omega }^2)-2k^2=0$ and generally do depend on $M$. For $M=m$, all three mode shapes are $(1,\sqrt{2},1)$, $(1,0,-1)$ and $(1,-\sqrt{2},1)$, with squared frequencies $(2-\sqrt{2})k/m$, $2k/m$, $(2+\sqrt{2})k/m$.