Nonlinear dynamics of flexible beams undergoing a large linear motion of basement: Principal 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 the nonlinear differential equations and derive a set of nonlinear modulation equations for the principal parametric resonance of the first mode and 3:1 internal resonance between the first two modes. Then, the modulation equations are numerically solved to obtain the steady-state response and the stability assessment of the beam. The results show that the trivial, single, and two-mode solutions are possible. Supercritical and sub-critical fork bifurcation only occurs in single-mode equilibrium and the saddle-node bifurcation and Hopf bifurcation can be found in two-mode equilibrium. For a Hopf bifurcation, a limit cycle is found undergoing a series of period-doubling bifurcations and finally results in a jump of the response to either a single-mode or a two-mode stable equilibrium solution after a blue-sky catastrophe.