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 conc ertina mode and a chequerboard mode) whose non-linear interaction breaks th e in-out symmetry of the linear vibration theory. The two-mode interaction leads to preferred vibration patterns with larger deflection inwards than o utwards, and at internal resonance, significant energy transfer occurs betw een the modes. This has regular and chaotic features. Here, direct numerica l integration is employed to examine chaotic motions. Using a set of 2-D Po incare sections, each valid for a fixed level of the Hamiltonian, H, the in stability under increasing H appears, as a supercritical period-doubling pi tchfork bifurcation. Chaotic motions near a homoclinic separatrix appear im mediately after the bifurcation, giving an irregular exchange of energy. Th is chaos occurs at arbitrarily low amplitude as perfect tuning is approache d. The instability manifests itself as repeating excursions around the sepa ratrix, and a number of practical predictions can be made. These include th e magnitude of the excursion, the time taken to reach this magnitude and th e 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, gi ving a response that resembles, for a while, the lower-energy quasi-periodi c orbits of the underlying Hamiltonian system. (C) 2001 Academic Press.