Abstract:

The transverse parametric vibration of the axially moving structure is always one of the hot topics in the field of nonlinear dynamics. At present, most of the studies are considering the time-varying speed of dynamic model. The parametric excitation comes from harmonic fluctuations of the axial speed. However, the fluctuation of the axial tension in an axially moving structure is more extensive in the engineering application. There are few researches considering the time-varying tension. The bifurcation and the chaotic behavior of axially accelerating viscoelastic beams under time-varying tension are studied in this paper. A nonlinear integropartia-differential governing equation of the moving beam is established. The linear viscous damping and the Kelvin model in the viscoelastic constitution relation are introduced. The axial tension is assumed as a harmonic variation with time. The fourth-order Galerkin truncation is employed to discretize the governing equation. The dynamic behavior of axially accelerating viscoelastic beams is determined by applying the fourth-order Runge-Kutta algorithm. The influences of material's viscoelastic coefficients, the mean axial speeds, the axial tension fluctuation amplitudes, and the axial tension fluctuation frequencies on the bifurcation diagrams are demonstrated by some numerical results of the displacement and velocity at the midpoint of the beam. The maximum Lyapunov exponent diagram of the system is used to identify the period motion and chaos motion. The results show that the smaller mean axial speed leads to the periodic motion. The period-doubling bifurcation and chaotic behavior are easy to occur near the critical speed. The larger axial tension fluctuation amplitude results in the larger chaos interval. The less viscoelastic coefficient and axial tension fluctuation frequencies lead to the chaotic behavior of the axially moving beam. Furthermore, chaos motions are confirmed using different factors, such as the time history, the fast Fourier transforms, the phase-plane portrait and the Poincaré map.

Key words: axially moving beam, time-varying tension, Galerkin method, Runge-Kutta method, bifurcation, chaos