Research On Complex Modes Of Axially Moving Strings And Beams

The axially moving string and beam are the kind of important apparatus,such as conveyor belt,cableway,train,rocket,elevator,which are applied in engineering.Because of eccentric mass and external perturbation,inevitably,the system vibrations when the strings and beams move axially.The research on vibration and control is very important in engineering application.The Assumed Modes Method and the Galerkin Method are employed in the research of complex modes of the transverse vibration of axially moving strings and beams analytically.The Method of Varying Amplitudes is used to study the band gap of the axially moving belt with the periodic foundation.The content of the dissertation is as follows:Chapter 1 is the introduction,which mainly introduces the background and significance of the study of axial moving continua,and summarizes the research status at home and abroad,and finally presents the research content and innovation of the dissertation.In chapter 2,the modeling of a viscoelastic periodic foundation axially moving beam with load is studied,and a nonlinear governing equation is established based on the generalized Hamilton principle.In the modeling,the viscous coefficient,the periodic change of the stiffness of the elastic foundation and the geometric nonlinearity of the foundation are considered.The result is confirmed by the existed literature.In chapter 3,Fourier Galerkin Spectral method is applied to analyze the wellposedness of the solution of axially moving string,and the stability and convergence of the method for the solution of axially moving string is proved.Essentially,the spectral method is a modal analysis method,and its application on the modal analysis of the axially moving string is the development of the modal analysis method.In chapter 4,the modes of the transverse vibration of the axially moving string and beam are studied.Assumed Modes Method is applied to the system,found no axially moving tensioned string transverse vibration modes are real modes whose modes exhibit stand wave modes.The transverse vibration of the axially moving string is complex modes and take on the characteristics of the travelling wave and static nodes coupling together.The modes of axially moving strings have nodes,and there is backward travelling wave mode between two adjacent nodes.When the transport speed under the instability critical velocity,string and beam model are both shown the backward travelling wave modes.The travelling wave mode of beam model changes with transport speed,when the axial velocity exceeds the buckling instability speed,the first-order mode of the beam model transforms into forward travelling wave mode.In chapter 5,Assumed Modes Method is applied to study the time-varying support axial moving strings and the resonant response of complex mode.The explicit expression of system response with the time-varying support and the moving excitation is obtained.The number of nodes,the position of nodes and the relationship of system parameters are also obtained.By analyzing the explicit solution,the condition of the existence of nodes of system is obtained.The position of the node changes with the different excitation frequency,excited the position of the node can cause the resonance of the system.The resonance can be excited by external excitation when its frequency is equal to the natural frequency and when its position is on the node.In chapter 6,the Method of Varying Amplitudes is applied to a homogeneous axially moving belt with periodic elastic foundation.The dispersion relation and the band gap structure are obtained from the governing equation of the system.It is concluded that the band gaps appear when the divergence of the eigenvalue occurs and the veering phenomenon of mode shape begins.The dependence of the band gap on the parameters can be obtained by varying the parameters.The tunable band gap varying with the parameters of the system and the relation of band gap with the foundation stiffness,the stiffness modulation and the axial velocity are revealed.Hence,the foundation stiffness modulation can be designed to manipulate the band gap where the vibration is suppressed.The most important research of the dissertation is to propose the travelling wave modes of complex modes in axially moving strings and the change direction of travelling wave mode of beam model,and the band gap of periodic axially moving strings and beams caused by the abrupt changes of complex modes in periodic structures which the vibration is isolated.