Local bifurcation analysis of an accelerating beam subjected to two frequency parametric excitations

Stability and bifurcation behaviors of an axially moving beam subjected to two frequency excitations were investigated both analytically and numerically. Simultaneous principal parametric resonance as well as combination parametric resonance with 3:1 internal resonance was considered. The critical points are classified into the following categories: a double zero accompanied by two negative eigenvalues, a simple zero coexisting with a pair of pure imaginary eigenvalues, and two pairs of pure imaginary eigenvalues in the non-resonant case. Based on the normal form theory, the stability regions of the initial equilibrium point were studied and the explicit expressions of the critical bifurcation curves for the occurrence of static bifurcation and Hopf bifurcation were obtained. The critical line from which a two-dimensional torus can arise was also discussed. Moreover, numerical simulations were given, which exhibit strong concordance with the analytical predictions. The analytical results obtained here contribute to a better comprehension of the dynamical behaviors of the system and may be helpful to researchers attempting to design the system parameters of related engineering structures.