Free three-mass spring chain (${\omega }^2=0,k/m,(1+2/\alpha )k/m$)

For free-end masses $m,\alpha m,m$ joined by two identical springs, the displayed squared frequencies have displacement vectors $(1,1,1)$, $(1,0,-1)$ and $(1,-2/\alpha ,1)$. Distinct eigenvectors are orthogonal in the mass inner product. The zero mode carries translation, while the middle mass does not move in the antisymmetric internal mode. Fixed end springs would give a different spectrum.