Transverse Vibration Of An Axially Moving Beam

In this paper, the transverse vibration of an axially moving beam and the related works are studied.Firstly, the free vibration with and without nonlinear effects is investigated by wave propagation. When the linear vibration is analysed, the differential equation of transverse motion is derived using the Hamilton's Principle. The frequency equation of such a beam is derived with the phase-closure principle. The modal functions are also obtained. With the nonlinear effects, the nonlinear normal modes and natural frequencies for a stationary beam are derived using the phase-closure principle. Then the nonlinear normal modes and natural frequencies for a travelling beam are obtained. Unlike the linear ones, the nonlinear modes and the associated natural frequencies are amplitude-dependent. The intercoupling between the different linear normal modes is clearly shown to exist in the nonlinear modal vibration.Secondly, a combination of harmonic balance and perturbation techniques is used to analyse the forced vibration of the axially moving beam with and without nonlinear effects. The governing equation and the boundary conditions are written in the standard state-space form in terms of the nonlinear normal modes. The steady-state response to a harmonic excitation is also obtained. As expected, the nonlinear response has multiple solutions in some range of excitation frequency. The stability analysis is carried out to confirm which motion may actually come true. The stable and unstable regions are obtained by perturbing the periodic solutions.Thirdly, the stability in parametric resonance of axially moving viscoelastic beam with time-dependent speed is investigated. The viscoelastic material of the beam obeys the differential Kelvin model. The equation of motion is derived from Newton's second law. The method of multiple scales is applied directly to the partial differential equation governing the transverse vibration. The stability boundary is derived from the solvability condition. Numercial results demonstrate that summation parametric resonance occurs if the axial speed fluctuation frequency is close to the sum of any two natural frequencies of unperturbed system. Principal parametric resonance occurs when the axial speed fluctuation frequency is close to two times of a natural frequency of unperturbed system. With the speed fluctuation amplitude increasing, the instability region becomes larger. While the viscosity cofficient can decrease the instability region. In addition , the viscosity cofficient influents more on the stability boundary in higher order principal parametric resonance.Finally , the results of the thesis are summarized and the further work is suggested.