Normal modes of three freely ended coupled pendulums (${\omega }^2=g/L+(k/m)\lambda ,\lambda =0,1,3$)

For three equal simple pendulums in a straight open chain with springs only between neighbours, the spring stiffness matrix in displacement coordinates is $k\left(\begin{array}{ccc}1& -1& 0\\ -1& 2& -1\\ 0& -1& 1\end{array}\right)$. Its eigenvectors $(1,1,1)$, $(1,0,-1)$ and $(1,-2,1)$ have respective dimensionless eigenvalues $0,1,3$. Adding the gravitational restoring term yields the displayed squared eigenfrequencies. These are free spring ends: fixed wall springs would change the matrix and mode shapes.