Energy balance of two synchronized self-excited pendulums with different masses

We consider the synchronization of two self-excited pendulums with different masses. We show that such pendulums hanging on the same beam can show almost-complete (in-phase) and almost-antiphase synchronizations in which the difference of the pendulums displacements is small. Our approximate analytical analysis allows one to derive the synchronization conditions and explains the observed types of synchronizations as well as gives an approximate formula for amplitudes of both the pendulums and the phase shift between them. We consider the energy balance in the system and show how the energy is transferred between the pendulums via the oscillating beam allowing synchronization of the pendulums.