 Solution

For a normal mode write $x_j=A_j{e}^{i\omega t}$. The generalized eigenvalue problem isThe three independent eigenvectors and corresponding squared frequencies areThe last two have zero mass-weighted displacement and are orthogonal in the mass inner product. The lowest frequency is zero: it represents a rigid translation with both springs unchanged. Its general motion includes uniform centre-of-mass velocity, not just a constant displacement. Superpositions of the two oscillatory normal modes and this translation give the general solution.