Axially Accelerating Viscoelastic Beams: Asymptotic Perturbation Analysis And Numerical Validation

Axially moving beams can represent many engineering devices, such as band saws, power transmission belts, aerial cable tramways, crane hoist cables, flexible robotic manipulators, and spacecraft deploying appendages. However, vibrations associated with the devices have limited their applications. Therefore, understanding transverse vibrations of axially moving beams is important for the design of the devices. The investigations on vibrations of axially moving beams have theoretical as well, because an axially moving beam is a typical representative of distributed gyroscopic systems. The method of analyzing vibrations of an axially moving beam can be applied to other more complicated distributed gyroscopic systems. In this paper, an asymptotic perturbation method is proposed to and the differential quadrature scheme is developed to investigate stability and steady-state response of the parameter vibrations of an axially moving viscoelastic beam. Meanwhile, the numerical calculations validate the analytical results.Stability and steady-state response of an axially moving viscoelastic beam constituted by Kelvin model are investigated. The material time derivative is used in the viscoelastic constitutive relation. Asymptotic perturbation method is applied to analyze stability region of transverse parameter vibration of axially moving beam for the principle and summation resonance. The solvability condition and instability boundary of stability are obtained for the principle and summation resonance via asymptotic analysis. Nonlinear steady-state response is investigated in summation parameter resonance of axially moving viscoelastic beam. Nontrivial amplitude and existence condition of nontrivial solution for steady-state response of parameter vibrations are obtained. Base on the Routh-Hurvitz criterion, stability condition of trivial and nontrivial solutions in summation parameter resonance are obtained when the steady-state response occurs. The differential quadrature scheme is developed to study numerically stability region and steady-state response of axially moving viscoelastic beam. Finally, the numerical and the analytical results are compared.The standard linear solid model is used to describe viscoelastic material of axially moving beam. Linear and nonlinear governing equations of traverse parameter vibration of beam are created. Asymptotic perturbation method is developed to investigate analytically linear stability region and nonlinear steady-state response of transverse vibration of axially moving viscoelastic beam. Numerical results are validated by the analytical results via differential quadrature scheme. Numerical examples show asymptotic perturbation method is applied to analyze the transverse vibration of axially moving beam with high accuracy.The main innovations of this dissertation are as follows:1. For first time, the idea that stability and steady-state response of axially moving viscoelastic beam is investigated via asymptotic perturbation method is proposed.2. The differential quadrature method is used to solve numerically governing equations of transverse vibration of axially moving viscoelastic beam, and validate the analytical results of above-mentioned problems.3. Creating the linear and nonlinear equations of an axially moving viscoelastic beam with viscoelastic constitutive relation constituted by the standard linear solid model, then asymptotic perturbation method and differential quadrature method are applied to investigate stability and steady-state response of axially moving viscoelastic beam.4. Analytical and numerical methods of present dissertation are used to revisit linear stability and steady-state response of an axially moving viscoelastic beam with Kelvin model, and then the resualts is the same as those via method of multiple scales.