CHAOTIC ENERGY EXCHANGE THROUGH AUTO-PARAMETRIC RESONANCE IN CYLINDRICAL SHELLS

Internal auto-parametric instabilities in the free non-linear vibrations of a cylindrical shell are studied numerically, focusing on two modes (a concertina mode and a chequerboard mode) whose non-linear interaction breaks the in–out symmetry of the linear vibration theory. The two-mode interaction leads to preferred vibration patterns with larger deflection inwards than outwards, and at internal resonance, significant energy transfer occurs between the modes. This has regular and chaotic features. Here, direct numerical integration is employed to examine chaotic motions. Using a set of 2-D Poincaré sections, each valid for a fixed level of the Hamiltonian, H, the instability under increasing H appears, as a supercritical period-doubling pitchfork bifurcation. Chaotic motions near a homoclinic separatrix appear immediately after the bifurcation, giving an irregular exchange of energy. This chaos occurs at arbitrarily low amplitude as perfect tuning is approached. The instability manifests itself as repeating excursions around the separatrix, and a number of practical predictions can be made. These include the magnitude of the excursion, the time taken to reach this magnitude and the degree of chaos and unpredictability in the outcome. The effect of small damping is to pull the motion away from what was the chaotic separatrix, giving a response that resembles, for a while, the lower-energy quasi-periodic orbits of the underlying Hamiltonian system.