Bifurcation And Chaotic Responses In The Nonlinear Vibration Of Axially Accelerating Viscoelastic Beams

Axially moving beams can model many engineering devices, such as powertransmission belts, tapes, paper tapes, aerial tramways, high-rise elevator cables, andthe like. One important problem in these axially moving systems is the occurrence oflarge unwanted transverse vibrations. Since axially moving beam are typical gyrocontinua, the research of the moving beams can be extended to more complicatedgyro continua. Therefore, the study of the transverse vibration of the movingstructures is of great engineering and theoretical significances and has very broadapplication prospects.In this dissertation, the nonlinear dynamic behaviors of the axially movingviscoelastic Euler beams and Timoshenko beams are investigated:(1) The effects of the longitudinally varying tension due to the axial acceleration areconsidered to establish the integro-partial-differential governing equations todescribe the transverse vibration of the beams. The Galerkin truncation isapplied to discretize the governing equations into a set of nonlinear ordinarydifferential equations. Based on the solutions obtained by the four-orderRunge-Kutta algorithm, the nonlinear dynamical behaviors like chaos wereidentified by the observation of the courses of displacement and velocitychanging with time at the midpoint of the beam;(2) Furthermore, the dependence of the tension on the finite support rigidity isintroduced to establish the dynamical equations. The bifurcation and chaos ofthe axially moving Euler beams are studied via the high-order Galerkintruncation as well as the differential and integral quadrature method in thesupercritical regime. Meanwhile, the bifurcation diagram, the phase plane, andthe Poincaré map are used to identify the dynamical behaviors by differentnumerical methods and various terms Galerkin truncations. Numerical resultsshow the discernible differences between the motion form predictions from the Galerkin truncation and the differential quadrature method. At the same time,there is qualitative disagreement among the different term truncated system;(3) Another dynamic model is established to include two frequency excitations,namely, the external harmonic excitation and the parametric excitation fromharmonic fluctuations of the moving speed. The effects of the external harmonicexcitation on the bifurcation of the moving beams are examined. At the sametime, the results of different terms Galerkin truncations on predicting nonlineardynamic behaviors are compared. Numerical simulations reveal that the forcedresonance excited by the external excitation will shift the motion form in thenonlinear vibration of the axially accelerating viscoelastic Euler beam, and thenthere is an agreement for the motion form predictions of quasiperiodic motionby different terms Galerkin truncations;(4) Since the effects of shear deformation and rotary inertia are neglected in theEuler–Bernoulli model, which should be considered if a beam is thick, theassumption of Timoshenko beam theory is carried out to establish the governingequations. Based on the application of the various terms Galerkin truncations,the steady-state response and nonlinear dynamic behaviors of an axially movingviscoelastic Timoshenko beam are studied. Moreover, the effects of variousterms Galerkin truncation on the amplitude-frequency responses as well asbifurcation diagrams are discussed. Numerical examples indicate that the4-termor higher terms Galerkin truncation is needed to study the nonlinear vibration ofthe axially accelerating viscoelastic Timoshenko beam.The exploration of the above-mentioned problems plays an important role todevelop the theory of the gyro continua and improve the engineering application ofthe axially moving beams.