Magnetism-solid Coupling Multiple Joint Resonances Of An Axially Moving Elastic Beam Excited By Current-carrying Wires

Axially moving conductive beams are widely used in engineering fields of machinery,electronics,aerospace,transportation,new energy,military manufacturing and so on,such as electromagnetic drive,electromagnetic track transmitter and magnetic levitation transportation.Axially moving beam structures working in the magnetic field environment show complex coupling dynamic behavior under the influence of various factors such as electricity,magnetism and force.In addition,changes of axial velocity and external excitation,or other nonlinear factors inevitably generate magnetoelastic multiple joint resonances,which may lead to reduced work efficiency,impact on product quality,structural instability,will have important effects on the safety and stability of structures.Therefore,studying the vibration mechanism of such structures and the influence of system parameters on dynamic characteristics can effectively control vibration and provide theoretical basis for structural analysis and design.Based on the basic theory of electromagnetic field,the distribution characteristics of the induced magnetic field excited by long straight current-carrying wires are analyzed,and the superimposed magnetic induction intensity,the electromagnetic forces and torques acting on the conductive beam are calculated.The expressions for the total kinetic energy,total potential energy,and external forces virtual work are derived.Based on the Hamiltonian principle,the magnetoelastic nonlinear vibration equations of axially uniform-speed and variable-speed moving beam are derived respectively.The principle-internal joint resonance of an axially uniform-speed moving beam is investigated.For a clamped-hinged beam,when the first and second order principle-internal joint resonance occurred,the state equations of the first-two order coupled modes and the amplitude-frequency response equations characterizing steady-state motion are derived respectively.The results show that the first and the second-order response are both excited,and different multi-solution regions are existed;The number of first and second order steady-state solutions varies simultaneously in several multi-solution regions,and the number depending on the external excitation force,moving velocity and current intensity.The superharmonic-internal joint resonance of an axially moving beam is researched.The nonlinear vibration equation is discretized into a group of ordinary differential equations by using displacement function and the Galerkin method.The modulation equations of amplitude and phase are obtained analytically,and the Jacobian matrix is also derived using Lyapunov stability theory.The variation law of amplitudes with different parameters is analyzed using numerical calculation software.The results show that the second order amplitude is obviously smaller than the first order;Moreover,the range of multi-solution regions decreases obviously with the increase of external excitation;As current intensity increases,the vibration amplitude decrease.The superharmonic-principal parametric joint resonance of an axially variable-speed moving beam is studied.Based on electromagnetic theory and Hamilton variational principle,the nonlinear forced vibration equation of the beam is derived.The approximate analytical solution is obtained and the stability of the steady-state solutions is analyzed.Through an example,the curves of amplitude versus frequency tuning value,external excitation,axial velocity,current intensity,and the position are exibited.The results show that the effect of axial velocity on resonance frequency is signifcantly larger than that of current intensity;As current intensity increases and the distance decreases,respectively,the system changes from quasi-periodic motion to single-frequency periodic motion.The principal-parametric joint resonance of an axially variable-speed moving beam with elastic constraints is explored.The vibration mode function is obtained according to the elastic boundary conditions,and then the equation is discretized by Galerkin integral method.The approximate analytical solution of the method of multiple scales is verified by numerical methods,and the judgment matrix for stability is calculated.Further more the effects of different parameters on the stability are analyzed.The results show that the stiffness of support spring and external excitation force mainly affect amplitude,while axial velocity,current intensity and axial force have effects on amplitude and frequency when the principal-parametric joint resonance occurs;The stability region increases with the increase of the stiffness of support spring.The principal-internal joint resonance of an axially uniform-speed moving beam with elastic constraints is considered.The displacement function is presented as a separate variable form of time and space,then equation is discretized into 2-DOF ordinary differential equations.The method of multiple scales is employed for obtaining the amplitude-frequency response equations.Numerical calculations are performed to obtain bifurcation diagrams,time history,phase,and Poincare maps.Results indicate that as current intensity increases,electromagnetic damping increases and amplitude decreases;As external excitation force increases and axial velocity decreases respectively,amplitude increase and the system changes from single periodic motion to quasi-periodic motion and then to single periodic motion.This thesis studies multiple resonances such as principal-internal,principalparametric,superharmonic-internal,superharmonic-principal parametric joint resonance,etc.of axially moving elastic beams with different boundary conditions,where a magnetic field is excited by long straight current-carrying wires.Analytical solutions under different joint resonances are obtained,and the influence of different parameters on resonance amplitude,frequency are analyzed.The bifurcation diagram,time history,phase,Poincare map,etc.varying with different control parameters are presented,and the dynamic response characteristics are explored when the main parameters changed.The research work can provide theoretical reference for analyzing complex resonance problems in practical systems under coupled fields.