The dynamics of co- and counter rotating coupled spherical pendula

The dynamics of co- and counter-rotating coupled spherical pendula (two lower pendula are mounted at the end of the upper pendulum) is considered. Linear mode analysis shows the existence of three rotating modes. The linear modes allow us to understand the nonlinear normal modes, which are visualized in frequency-energy plots. With the increase of energy in one mode we observe a symmetry breaking pitchfork bifurcation. In the second part of the paper we consider energy transfer between pendula having different energies. The results for co-rotating (all pendula rotate in the same direction) and counter-rotating motion (one of lower pendula rotates in the opposite direction) are presented. In general, the energy fluctuations in counter-rotating pendula are found to be higher than in the co-rotating case.