Three identical simple pendula (of mass m and equilibrium string length l) are attached together by springs of spring constant k, as shown in the figure.
The frequencies of small oscillations are given by \(\sqrt{\frac{g}{l}}, \sqrt{\frac{k}{m}+\frac{g}{l}}, \sqrt{\frac{3 k}{m}+\frac{g}{l}}\). The normal modes (without normalisation) corresponding to these frequencies respectively are
1
(1, 1, 1), (1, 0, 1), (1, -2, 1)
2
(1, 1, 1), (1, 0, -1), (1, 2, 1)
3
(1, 1, 1), (1, 0, -1), (1, -2, 1)
4
(1, 2, 1), (1, 0, -1), (1, 1, 1)