Quasi-periodic vibration of an axially moving beam under conveying harmonic varying mass

In this work, quasi-periodic vibration of an axially moving beam under conveying harmonic varying mass are studied. Coupled modelling equations of nonlinear vibration of an axially moving beam are established with the aid of the generalized Hamilton principle. A nonlinear partial-integro-differential equation as the transverse vibration modelling equation is deduce by decoupled method. After obtaining the equilibrium buckled state of the axially moving beam, a transverse vibration governing equation around the buckled state is presented. The Galerkin method is applied to truncate the governing equation into a set of nonlinear ordinary differential equations. The harmonic balance method with three time variables is applied to analyze quasi-periodic vibration of the nonlinear dynamic system with combined parametric and forced external excitation. Time traces and phase-plane portraits of quasi-periodic vibration are obtained, which are in excellent agreement with those of the direct time integration method. Frequency amplitude responses of the vibration are determined by the harmonic balance method with three time variables. Effects of the velocity of the axially moving beam, amplitude of the varying mass, and the external excitation frequency on the vibration amplitude of axially moving beam and the ratio of the two fundamental frequencies are investigated.