Supercritical Nonlinear Vibration Of Axially Moving Viscoelastic Beams:Modeling And Simulation

Moving structures can be used as mechanical models for a variety of common engineering components,such as power conveyor belt,band saw,strip steel,textile fiber,elevator cable,and so on.In order to improve production efficiency,these engineering systems have a very high demand for moving speed.At the same time,the existence of the motion speed in most cases also makes the system produce violent transverse vibration,and thus seriously affects the production and processing.On the other hand,the axially moving beam as a typical gyro continuum,its theoretical research can be further developed to other gyro continua,such as axially moving plates,shells,pipes conveying fluid and so on.Therefore,the study of the transverse vibration of axially moving structures has a wide range of engineering application prospect and important theoretical research value.In the past,the research of the transverse vibration of axially moving structures in the supercritical regime usually focused on the study of Euler-Bernoulli beam which is mostly applied for slender structures by small deformation theory.But there are a lot of axially moving short and thick structures in the engineering projects.Therefore,a more precise dynamical model of the supercritical transverse vibration of axially moving structures is established by the finite deformation theory.And then the nonlinear vibration characteristics under parametric excitation and forced excitation are studied.The specific research contents are shown as follows.In the first chapter,the research background and current situation of axially moving structures are introduced in detail,especially focused on the research progress of supercritical moving structures.In particular,the research progress of supercritical axially moving structures is illustrated.And then the purpose and significance of this thesis are expounded.In the end of this chapter,the main contents and innovation points are introduced in detail.In Chapter 2,based on the finite deformation theory and generalized Hamilton principle,the integro-partial-differential model to describe the nonlinear transverse vibration of axially moving viscoelastic beams is established by considering the Kelvin viscoelastic constitutive relation and taking the material time derivative.Since the Galerkin truncation method is used to discretize the governing equations,the fourth order Runge-Kutta method is applied to solve the ordinary differential equations.Based on the numerical solution,the effects of different system parameters on the nonlinear vibration characteristics of axially moving beams,such as the axial mean moving speed and the amplitude of the axial speed variation,are investigated under the parametric excitation through the observation of Poincaré section.In addition,the natural frequencies of the axially moving beam,modeled by the finite deformation principle and the small deformation principle,respectively,are calculated and compared.The third chapter shows the transverse vibration model of viscoelastic axially moving beams under the parametric excitation and external harmonic excitation.And then its nonlinear dynamic behaviors are studied.Considering whether the relation of the frequencies is commensurable or incommensurable,the chaotic characteristics of the moving system are discussed by the numerical method.Furthermore,the influence of different system parameters on the nonlinear vibration of the vibration model is investigated.In the fourth part,the nonlinear dynamic characteristics of the axially moving structures subjected to forced excitation in the supercritical regime are studied.Also employing the finite deformation theory and using the Kelvin model and material time derivative,the new supercritical model in the transverse vibration of axially moving beams is investigated after the critical velocity and the nontrivial configuration solution are derived.Through Galerkin truncation method,the nonlinear dynamics of the system under different axial velocity and external excitation amplitude is studied.Furthermore,the influence on the nonlinear dynamical behaviors of two axially moving the beam models,established by finite deformation theory and small deformation principle,respectively,are compared.The fifth chapter analyzed the parametric vibration of the supercritical axially moving beam established by the finite deformation theory.Since the numerical methods are used to solve the governing equation,the nonlinear dynamical behaviors are studied from a variety of points,including the period-doubling bifurcation diagrams with the system parameters changing,the time history,the frequency spectrum,the phase diagram,the Poincaré map and the initial sensitivity.In Section 6,the responses of the transverse nonlinear vibration of axially moving viscoelastic structures modeled by the Timoshenko beam principle are investigated under parametric excitation and external excitation.Introducing the varying axial tension caused by the axial acceleration along the longitudinal direction and the finite support's rigidity,the forced vibration model of the axially moving Timoshenko beam is established.Moreover,the bifurcation and chaotic dynamical behaviors of the system are numerically illustrated under two cases,whether the parametric excitation frequency can be commensurable by the external harmonic excitation frequency or not.Chapter 7 gives the summary of the full thesis presents the conclusion of this study and its related research work which is to be carried out in further depth.Through the comparison and discussion of the small deformation theory,the application range of the finite deformation theory and the Timoshenko theory extended.And the influence of the finite deformation theory on the nonlinear vibration of the structure axially moving at a high speed is clarified,which plays an important role in the engineering application and practical development.