BIFURCATIONS AND CHAOS IN A QUASI-PERIODICALLY FORCED BEAM - THEORY, SIMULATION AND EXPERIMENT

Non-linear vibrations of a straight beam clamped at both ends and forc ed with two frequencies near the first mode frequency are theoreticall y and experimentally investigated. In an earlier paper, the occurrence of chaos in the forced beam was proved by using the Galerkin approxim ation, the averaging method and Melnikov's technique. First, the singl e mode Galerkin approximation for the beam is further analyzed here. T he existence of invariant tori corresponding to periodic orbits in the averaged system is established and their stability is determined. The occurrence of saddle-node and doubling bifurcations of tori, which co rrespond to saddle-node and period doubling bifurcations of periodic o rbits in the averaged system, respectively, is also detected. Second, numerical simulation results for a single mode equation and experiment al results for the beam are given. The existence of invariant tori and sustained chaotic motions is confirmed, and saddle-node and doubling bifurcations of tori are observed. The bifurcation sets and conditions for the existence of chaos are also obtained. These observations in n umerical simulations and experiments are compared with the theoretical predictions.