Synchronization of two flow excited pendula

Two aerodynamically excited pendula are considered as a simple example of two linearly coupled, self sustained mechanical oscillators, modelled by two coupled Van-der-Pol equations. The considered mechanical application admits of a systematic survey of synchronized regimes within the framework of standard nonlinear stability analysis. Using normal form theory and the prevailing direct averaging approach the occurring Hopf bifurcation with two distinct pairs of purely imaginary eigenvalues is studied in the non-resonant case and in the 1:1-resonance corresponding, respectively, to strong and weak coupling. In particular, for the resonant case a graphical approach permits a comprehensive interpretation relating the stable stationary solutions of the averaged system with synchronized regimes and allows an analytical computation of the oscillation amplitudes and the synchronous frequency.