Bifurcation And Chaos Of Axially Accelerating Viscoelastic Beams

Many engineering devices can be modeled as axially moving structures in mechanical applications.Axially moving beams as the most typical mechanical model,including band saws,power transmission belts,robot manipulators and aerial tramways.Those structures have an extensive application in the engineering filed.Due to the axial speed,those structures generally produce large transverse vibration which affect the efficiency of engineering equipment.Since the parameter vibration caused by the external excitation,the axial moving structures possess complicated vibration characteristics.The axial speed and the axial tension act as the time-dependent characteristic parameters possess particular effects on axially moving beams.Therefore,the introduction of axially time-varying tension and time-varying speed are of great significance to the study of transverse vibration of axially moving structures.In this paper,the dynamic models of axially moving viscoelastic Euler beams and Timoshenko beams are established,respectively.The mechanical properties of axially moving systems are studied by combining with approximate analysis and numerical methods.The dissertation is organized as following:1.The transverse and longitudinal coupling model of axially moving Euler beams under varying tension is investigated.The eighth-order Galerkin method is used to solve the coupling model.Based on the results of direct multiscale method and differential quadrature method of the simplified model,and the results of the eighth-order Galerkin truncation of coupling model,the difference of the response amplitude at the midpoint of the beam between those two models are compared.The advantages and disadvantages of those two models under different methods are investigated via the operation time and the accuracy of the results.The motion morphology of coupling model is demonstrated by means of the time history,phase diagram,spectrum analysis and the Poincare? section.2.The nonlinear integro-partial-differential equations of axially moving Euler beams under time dependent tension is discussed.The dynamic equations are discretized by the fourth-order Galerkin truncation.The period-doubling bifurcation phenomena of the transverse vibration are exhibited by the numerical solution of the amplitude of the beam.The period motion and chaos are identified via the largest Lyapunov exponent quantitatively.The bifurcation and chaos characteristics on the subharmonic parameter resonance of axially moving beams are examined.The method of time-series analysis is employed to identify chaos motion by the time history,phase diagram,the frequency spectrum and the Poincare? section.3.The nonlinear model of axially moving Euler beams with speed dependent tension and tension dependent speed is studied.The results of approximate analysis and different numerical methods on the table steady state amplitude at the midpoint of the beam are compared.The tristable and bistable domains of attraction on the stable steady state solution with a three-to-one internal resonance are analyzed emphatically by the fourth-order Galerkin truncation and the differential quadrature method,respectively.The period-doubling bifurcation and chaos of the beam along the axial speed fluctuation amplitudes,the mean axial speeds,the viscoelastic coefficients,the axial tension fluctuation amplitudes and the fluctuation frequencies are simulated to exhibit the dynamic phenomena of axially moving beams.4.The linear vibration model of axially moving Timoshenko beams under time dependent tension is established.The dynamic stability of harmonic parameter resonance with time dependent tension is investigated by direct multiscale method.The effects of the shear deformation coefficients,viscoelastic coefficients,the mean axial speeds,the rotational inertia and the stiffness on the stability boundary are discussed,respectively.Based on the application of the different order Galerkin truncation,the numerical verification is simulated.Numerical results show that higher order Galerkin truncation contribute to accurate results of the linear dynamics of axially moving Timoshenko beams.