Quasi-periodic motions of high-dimensional nonlinear models of a translating beam with a stationary load subsystem under harmonic boundary excitation

Abstract

Bifurcations and quasi-periodic motions of high-dimensional nonlinear models of a translating beam with a stationary load subsystem under harmonic boundary excitation, where there are combined parametric and forcing excitations, are investigated. It is demonstrated that by adjusting the frequency of the boundary excitation beyond bifurcation points, the nonlinear system exhibits quasi-periodic motion rather than the periodic response reported in an earlier publication. Particular attention is paid to the nonlinear dynamics of models with five and six included trial functions, where quantitative and qualitative results of frequency responses and quasi-periodic motions are significantly different from each other. The nonlinear governing equations of motion of the translating beam are established by using the Newton's second law. The Galerkin method is used to truncate the governing partial differential equation into a set of nonlinear ordinary differential equations. The incremental harmonic balance (IHB) method is used to solve for periodic responses of the high-dimensional models of the translating beam. The Floquet theory along with the precise Hsu's method is used to investigate stability of the periodic responses. The IHB method with two time scales developed earlier is extended to analyze quasi-periodic motion of the nonlinear system with combined parametric and forcing excitations whose spectrum contains uniformly spaced sideband frequencies. Quasi-periodic motion obtained from the IHB method with two time scales is in excellent agreement with that from numerical integration using the fourth-order Runge-Kutta method.