Transverse Vibrations Of Axially Moving Viscoelastic Beams: Modeling, Analysis, And Simulation

A variety of engineering systems with axially moving strings and beams involve power transmission chains, band saw blades, aerial cableways and paper sheets during processing. The study of the transverse vibration of the axially moving strings and beams is of great significance. The research of the transverse vibration in that case may pay contribution to the context of continuous gyroscopic systems.The modeling, analysis and simulation of axially moving viscoelastic strings and beams are investigated in this dissertation. The governing equations of coupled planar of axially moving strings and beams are modeled, and nonlinear models of transverse motion of axially moving strings and beams are computationally investigated. The method of multiple scales is applied to mathematical models to calculate the stability and the steady-state response. The finite difference schemes and the differential quadrature schemes are respectively developed to numerically solve the equations of axially accelerating viscoelastic beams. And the numerical results confirm the results derived from the method of multiple scales. By analyzing the time series based on the numerical solutions of the differential equations, the nonlinear dynamical behaviors like bifurcation and chaos are identified. The main points of the concrete content are as follows:Nonlinear models of transverse motion of axially moving strings and beams are computationally investigated. The governing equations of coupled planar is reduced to the partial differential equation of transverse motion by neglecting longitudinal terms partial differential equations. The integro-partial-differential equation of transverse motion is derived from the partial differential equation by averaging of the string disturbed tension. The finite difference schemes are developed to numerically solve the coupled equations, the the partial differential equation, and the integro-partial-differential equation. Free vibration of the strings and beams, forced vibration of the axially moving viscoelastic beams, and transverse parametric vibration of axially accelerating viscoelastic beams from the partial differential equations and the integro-partial-differential equations are respectively compared with the transverse component calculated from the coupled equation from the angles of transverse responses, the stability of steady-state responses and the transient transverse responses.Modeling transverse vibration of nonlinear beams are investigated via numerical solutions of partial differential equations and an integro-partial-differential equation. The governing equation is derived from the viscoelastic constitution relation by using material derivative, not simply by the partial time derivative. The method of multiple scales is applied directly to the governing equations without discretization to calculate the instability regions and the steady-state response for axially accelerating viscoelastic linear beam, nonlinear forced vibration and nonlinear parametric resonance respectively. Neighboring mode has not affected primary resonance, and subharmonic resonance is theoretically proved. Results are compared to previous work in which the partial time derivative was used in the viscoelastic constitutive relation. The finite difference schemes and the differential quadrature schemes are respectively developed to numerically solve the equations of axially accelerating viscoelastic beams for numerical simulations the instability regions of linear subharmonic resonance, the steady-state response of nonlinear primary resonance and nonlinear subharmonic resonance. And the numerical results confirm the results derived from the method of multiple scales.By analysis of the time series based on the numerical solutions to the partial differential equations and the integro-partial-differential equations calculated by the differential quadrature method, the nonlinear dynamical behaviors like bifurcation and chaos are identified. The bifurcation diagrams are presented in the case that the amplitude of speed fluctuation is varied while other parameters are fixed. The phase plane, the Poincare map, the Lyapunov exponent, and initial value sensitivity are used to identify the periodic motions or chaotic motions occurring in the transverse vibrations of the axially accelerating viscoelastic beam.The main innovation of this dissertation are as follows:1. For the first time it compares the coupled equations, the the partial differential equation, and the integro-partial-differential equation from many aspects based on the numerical solutions of the differential equations calculated by the finite difference method;2. For the first time it theoretically proves that neighboring mode has not affected primary resonance in the transverse forced vibration and subharmonic resonance in the transverse parametric vibration of the axially accelerating viscoelastic beam;3. For the first time it respectively develops the finite difference schemes and the differential quadrature schemes to numerically solve the equations of axially accelerating viscoelastic beams for confirming the results derived from the method of multiple scales.