Transition scenario to turbulence in thin vibrating plates

Abstract

A thin plate, excited by a harmonic external forcing of increasing amplitude, shows transitions from a periodic response to a chaotic state of wave turbulence. By analogy with the transition to turbulence observed in fluid mechanics as the Reynolds number is increased, a generic transition scenario for thin vibrating plates, first experimentally observed, is here numerically studied. The von Kármán equations for thin plates, which include geometric non-linear effects, are used to model large amplitude vibrations, and an energy-conserving finite difference scheme is employed for discretisation. The transition scenario involves two bifurcations separating three distinct regimes. The first regime is the periodic, weakly non-linear response. The second is a quasiperiodic state where energy is exchanged between internally resonant modes. It is observed only when specific internal resonance relationships are fulfilled between the eigenfrequencies of the structure and the forcing frequency; otherwise a direct transition to the last turbulent state is observed. This third, or turbulent, regime is characterized by a broadband Fourier spectrum and a cascade of energy from large to small wavelengths. For perfect plates including cubic non-linearity, only third-order internal resonances are likely to exist. For imperfect plates displaying quadratic nonlinearity, the energy exchanges and the quasiperiodic states are favored and thus are more easily obtained. Finally, the turbulent regime is characterized in the light of available theoretical results from wave turbulence theory.