Chaotic dynamics in the forced nonlinear vibration of an axially accelerating viscoelastic beam

In this paper, the chaotic behaviour in the transverse vibration of an axially moving viscoelastic tensioned beam under the external harmonic excitation is studied. The parametric excitation comes from harmonic fluctuations of the moving speed. A nonlinear integro-partial-differential governing equation is established to include the material derivative in the viscoelastic constitution relation and the finite axial support rigidity. Moreover, the longitudinally varying tension due to the axial acceleration is also considered. The nonlinear dynamics of axially moving beam is investigated under incommensurable relationships between the forcing frequency and the parametric frequency. Based on the Galerkin truncation and the Runge-Kutta time discretization, the numerical solutions of the nonlinear governing equation are obtained. The time history of the center of the axially moving viscoelastic beam is chosen to represent the motion of the beam. Based on the time history of the axially moving beam, the Poincaré map is constructed by sampling the displacement and the velocity of the center. The bifurcation diagram of the axially moving beam is used to show the influence of the external excitation. Furthermore, quasi-periodic motions are identified using different methods including the Poincaré map, the phase-plane portrait, and the fast Fourier transforms.