Stability and bifurcation for a flexible beam under a large linear motion with a combination parametric resonance

Stability and bifurcation behaviors for a model of a flexible beam undergoing a large linear motion with a combination parametric resonance are studied by means of a combination of analytical and numerical methods. Three types of critical points for the bifurcation equations near the combination resonance in the presence of internal resonance are considered, which are characterized by a double zero and two negative eigenvalues, a double zero and a pair of purely imaginary eigenvalues, and two pairs of purely imaginary eigenvalues in nonresonant case, respectively. The stability regions of the initial equilibrium solution and the critical bifurcation curves are obtained in terms of the system parameters. Especially, for the third case, the explicit expressions of the critical bifurcation curves leading to incipient and secondary bifurcations are obtained with the aid of normal form theory. Bifurcations leading to Hopf bifurcations and 2-D tori and their stability conditions are also investigated. Some new dynamical behaviors are presented for this system. A time integration scheme is used to find the numerical solutions for these bifurcation cases, and numerical results agree with the analytic ones.