Identifying internal resonance regimes in free-vibrating systems with multiple autoparametric couplings

Abstract

This paper introduces a method for identifying internal resonance regimes in free-vibrating systems with multiple nonlinear couplings, illustrated using a chain of two Rott’s pendula. The method based on decomposing Hamiltonian and subsequent treatment of the coupling energies allows for a comprehensive understanding of autoparametric system behaviour. A novel signal-processing-based coupling energy evaluation significantly enhances the effectiveness of the method, revealing internal resonances with slow energy exchange. In the theoretical part, a general step-by-step procedure applicable to any conservative coupled oscillators is introduced. We demonstrate that concerning such free-vibrating systems, the proposed method effectively uncovers internal resonances influenced by system parameters, initial conditions, and the initial energy of the system. Through an illustrative example of the system with multiple quadratic couplings, we also show that slow energy exchange occurs for sum and difference internal resonances.