Nonlinear Dynamics and Bifurcation Analysis of a Hinged-Clamped Beam Under Parametric and Internal Resonances

Abstract

The present study examines the stability and bifurcation of axially moving hinged-clamped beams subjected to parametric excitation originating due to speed alterations. Attention is paid to the principal parametric resonance amid 3:1 internal resonance in the subcritical speed regime. The direct method of multiple scales is applied to predict the nonlinear behavior of the beam modeled in the form of the nonlinear integro-partial differential equation. A continuation algorithm is used to solve numerical examples to illustrate the stability and bifurcations of periodic solutions for the given set of system parameters. The fixed point solution is different for different modes. Three kinds of equilibrium solution curves like trivial, two-mode, and single-mode solutions are obtained from numerical computations. The first two kinds are available for both first and second modes while the third one is available for the second mode only. The introduction of material damping converts the isolated two-mode solution into an isolated closed-loop form. The system displays saddle-node and Hopf bifurcation points in both modes while super and subcritical pitchfork bifurcation points in the second mode only. Decreasing viscous damping strengthens the effect of internal resonance. The numerically simulated results are unique, interesting, and are not available in the existing literature.