Transverse Vibrations Of Axially Moving Beams And In-Plane Translating Plates: Modeling And Analysis

Moving structures are present in various industrial applications such as power transmission belts, tapes, paper tapes, textile fibers, band saws, aerialtramways, high-rise elevator cables, single cable aerial tramway, composite laminates, and the like. The moving structures suffer from the occurrence of large transverse vibrations due to variations in moving speed, so the study and control of the vibration of the moving structures are of great engineering, economic, and theoretical significances and have very broad application prospects. The modeling and analysis of moving structures are investigated in this dissertation. The dissertation is organized as following:Chapter 1 surveys the moving structures and the advance in the corresponding theories, and presents the significance, main contents, and main innovations of this dissertation.In Chapter 2, parametric stability of axially accelerating viscoelastic Euler beams and Timoshenko beams is revisited. The effects of the longitudinally varying tension due to the axial acceleration are highlighted. The dependence of the tension on the finite support rigidity is also considered. The generalized Hamilton principle and the Kelvin viscoelastic constitutive relation are applied to establish the governing equations and the associated boundary conditions for coupled planar motion of the beam. The method of multiple scales is employed to analyze parametric stability of transverse motion. As the basis of nonlinear vibration analysis, the mode function and the natural frequency of the linear free vibration are calculated. The stability boundaries are derived from the solvability conditions and the Routh-Hurwitz criterion for principal and sum resonances.In Chapter 3, nonlinear parametric vibration is investigated for axially accelerating viscoelastic beams subjected to parametric excitations resulting from longitudinally varying tensions and axial speed fluctuations are investigated. The dependence of the tension on the finite support rigidity is also modeled. A governing equation of coupled planar vibration and the associated boundary conditions are derived from the generalized Hamilton principle and the viscoelastic constitutive relation. The equation is simplified not a governing equation of transverse nonlinear vibration in small but finite stretching problems, which is a nonlinear integro-partial-differential equation with time-dependent and space-dependent coefficients. The method of multiple scales is employed to analyze the summation and the principal parametric resonance resonances. The amplitudes and the existence conditions of the steady-state responses are derived from the solvability conditions, and their stabilities are determined. In the studies of axially moving viscoelastic beams, the differential quadrature scheme is developed to solve numerically the governing equation, and the computational results confirm the outcomes of the method of multiple scales.In Chapter 4, the extended Hamilton principle and energy method are employed to obtain the governing equation and the associated boundary conditions. The plate's material obeys the Kelvin-Voigt model in which the material time derivative is used. The result of this chapter is the basis for subsequent applications chapters.In Chapter 5, dynamic stability in parametric resonance of in-plane translating viscoelastic plates is investigated. The in-plane translating speed is characterized as a simple harmonic variation about the constant mean speed. The method of multiple scales is applied directly to the governing equation to establish the solvability conditions in principal parametric resonances and combination parametric resonances of the additive type and the difference type.In Chapter 6, nonlinear free transverse vibrations of in-plane translating plates subjected to plane stresses are investigated. The method of multiple scales is employed to analyze the nonlinear partial differential equation. The solvability conditions are established in the cases without internal resonance and with 3:1 or 1:1 internal resonances. The relationship between the nonlinear frequencies and the initial amplitudes is analytically showed.In Chapter 7 and 8, nonlinear parametric vibration in internal resonances and forced vibration of in-plane translating viscoelastic plates is analytically and computationally investigated. The method of multiple scales is applied to establish the solvability conditions. The stability of the steady-state responses is determined based on the Routh-Hurvitz criterion. In the studies of in-plane translating viscoelastic plates, the differential quadrature schemes are developed for the two-dimensional full plate model and the one-dimensional reduced plate model to solve the nonlinear governing equations numerically. The numerical calculations confirm the approximate analytical results.