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

摘要

Non-linear vibrations of a straight beam clamped at both ends and forced with two frequencies near the first mode frequency are theoretically and experimentally investigated. In an earlier paper, the occurrence of chaos in the forced beam was proved by using the Galerkin approximation, the averaging method and Melnikov's technique. First, the single mode Galerkin approximation for the beam is further analyzed here. The 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 correspond to saddle-node and period doubling bifurcations of periodic orbits in the averaged system, respectively, is also detected. Second, numerical simulation results for a single mode equation and experimental 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 numerical simulations and experiments are compared with the theoretical predictions. [References: 46]