The Magneto-elastic Parameter Vibration Analysis Of A Rotating Conductive Circular Plate

An annular or discoid rotating plate member application is wide in Aerospace,Large Generating Units and Mechanical Engineering.The change of the rotation speed in complicated circumstance which includes electric,magnetic,force,etc.will cause large amplitude vibration,and lead to the loss of stability,even damage the rotatable structures.Hence,it is important to analyze the dynamic response of varying parameters in rotatable system.Excitation causes the parametric vibration through periodic change of parameters in the system when the system has parametric excitation.When the frequency of the small parameter excitation is far from the natural frequency of the system,The large response will occur if the frequency of the small parameter is twice the natural frequency of the system.In this paper,the magnetoelastic parametric vibration and stability of the rotating circular and annular plate in transverse magnetic field are investigated.According to the theory of large deflection and the influence of geometric nonlinearity,the kinetic energy,potential energy,external force work of rotating circular thin plate are given.Based on the Maxwell equations and electromagnetic constitutive relation,the expressions of electromagnetic force and the moment of the rotating circular plate in magnetic field are derived.The nonlinear magneto-elastic vibration differential equations of the spinning circular plate in the magnetic field are derived according to Hamilton princile.The stability of parametric vibration in transversal magnetic field of the rotating circular plate with varying velocity is studied.The Axisymmetric parameter vibration differential equations of the rotating circular plate are obtained through the Galerkin integral method.Through the coordinate transformation,the Mathieu equation is obtained.According to the equation of parametric vibration solved by using the averaging method and the Floquet theory,the stability conditions with different periods are given.Through the numerical calculation,the stability,vibration response and phase trajectory curves of parametric vibration system are obtained.The stability of different regions and the influence of different parameters on system stability are studied.Nonlinear principal parametric resonance and stability are investigated for rotating circular plate from the time-varying speed in the magnetic field.The method of multiple scales is applied to derive the nonlinear principal parametric resonance amplitudefrequency equation.The stability of the stationary solution is discussed and the critical conditions of stability of the plate are given.The amplitude-frequency characteristic curves and the relation curves of amplitude changing with other system parameters are obtained.The influences of detuning parameter,rotation rate,and magnetic induction intensity are investigated on the principal parametric resonance behavior.Magneto-elastic combination resonance is investigated for a rotating circular plate with variable speed under alternated loading.The method of multiple scales is applied to solve the nonlinear vibration defferential equation.Four different resonance conditions and corresponding amplitude-frequency equations are given by analyzing the secular term when combination resonance occur.The stability of the plate is discussed,and necessary and sufficient conditions of the system stability are given.Numerical computation obtains the amplitude-frequency characteristic curves,bifurcation diagrams of amplitude and the variety law of amplitude with magnetic induction intensity,radial force,etc.The influence of parameters variation on stability of the system and the different resonance conditions are investigated.The nonlinear magneto-elastic combination resonances of principal parameterprincipal and principal parameter-harmonic resonance for rotating annular plate in twoway magnetic field are investigated.The equations of non-axisymmetric transverse and longitudinal vibration of circular plates are given.The method of multiple scales is applied to derive the amplitude-frequency response equations.The relation curves of transverse and longitudinal vibration amplitudes changing with frequency detuning parameter are obtained.The influence of magnetic induction intensity,forced excitation and other parameters on resonance amplitude is discussed.The parametric,principal parametric,principal parameter-principal resonance and principal parameter-harmonic combination resonances of rotating annular and circular plates in coupled field are studied in this paper.The amplitude response equations are derived and the influence of different parameters on vibration performance is analyzed.This study provide theoretical basis for engineering analysis of rotating system.