Nonlinear dynamics of flexible beams undergoing a large linear motion of basement: Combinational parametric and internal resonances

A set of nonlinear differential equations is established for the planar oscillation of flexible beams undergoing a large linear motion of the basement. The method of multiple scales combined with the Cartesian transformation is used to solve directly the nonlinear differential equations and derive a set of nonlinear modulation equations for the combinational parametric and 3:1 internal resonances between the first two modes. Then, the modulation equations are numerically solved to obtain the steady-state response and the stability of the beam. The results show that the trivial response and the two-mode non-trivial response of the system coexist and the responses arising from the 3:1 internal resonance belong to unstable saddles. Supercritical and sub-critical Hopf bifurcations in both trivial and non-trivial branches and the saddle-node bifurcations in non-trivial branches of the response curves are found. As usual, for the Hopf bifurcations, a limit cycle is found undergoing a series of period-doubling bifurcations and becomes a chaotic oscillation at last.