Magnetoelastic Internal Resonance Characteristics Of Conductive Thin Plates

Thin-plate structures are widely used in space stations and aerospace field,so it is necessary to study the complex dynamic problems of thin plates under large deformation.With the rapid development of modern science and technology fields such as aerospace,nuclear industry,magnetic levitation transportation,electromechanical power system and large-scale water conservancy and hydropower project,the research on electromagnetic elastic mechanics theory and its application has attracted people's attention,The study of structural magneto-elastic dynamics in electromagnetic field is of theoretical and practical significance.Internal resonance is one of the unique phenomena distinguishing the nonlinear vibrations from the linear ones.For multi-dof systems,mutual interference between different modes causes the system energy to exchange between modes that interfere with each other to cause internal resonance,when the parameters of the system change to make some natural frequencies of the system reducible or approximate reducible.In this paper,for the conductive material thin plate in magnetic field,the magneto-elastic nonlinear vibration problems such as free vibration,main resonance and super-harmonic resonance are studied when the first two modes of the system satisfy the approximation relationship,that is,internal resonance occurs in the system.Based on Kirchhoff's basic hypothesis and considering geometric nonlinearity,the virtual work of deformation potential energy and external force on virtual displacement of thin plate under virtual displacement is calculated.The expressions of electromagnetic force and moment are obtained according to electrodynamics equations.The nonlinear magneto-elastic vibration differential equations of conductive thin plate are derived by using the principle of virtual work.The internal resonance characteristics of conductive strip under free vibration are studied.By setting the displacement function and separating the time and space variables,the differential equation of magneto-elastic vibration of strip plate is obtained by Galerkin integral method.Using the multi-scale method,the modulation equation of amplitude and phase under the condition of 1:3 internal resonance is obtained.The time history and phase diagrams of the first two coupled modes are plotted.The effects of initial values and magnetic field intensity on the internal resonance are discussed.The 1:1 internal resonance and 1:3 internal resonance characteristics of conductive rectangular thin plate in transverse magnetic field under different size ratios are studied.The multi-scale method is used to solve the vibration differential equations,and the amplitude and phase modulation equations are obtained.By numerical calculation,the response diagrams of the system under 1:1 and 1:3 internal resonance are obtained,and the internal resonance characteristics of the system under free vibration are analyzed.The principal and internal resonance and also the super-harmonic and internal resonance of rectangular conductive thin plates subjected to lateral external excitation in a transverse magnetic field is studied.Set up the displacement function of rectangular conductive thin plates with one side fixed and three sides simply supported.According to Galerkin integral method and multi-scale method,the amplitude-frequency response equations under combined resonance of 1:3 internal resonance and external excitation are obtained.Through numerical examples,the amplitude-frequency characteristic curves and dynamic response diagrams of the system are given,and the effects of tuning parameters,magnetic field strength and external excitation amplitude on the vibration are discussed.The 1:3 internal resonance under parametric resonance of a rectangular thin plate with in-plane alternating tension in transverse uniform magnetic field is studied.The multi-scale method is employed to solve the derived magneto-elastic vibration differential equation.Under 1:3 internal resonance,the amplitude-frequency response equations of the first two order principal parameter resonance and the combined-parameter resonance are obtained.The curves of amplitude varying with frequency,magnetic intensity and amplitude of tension disturbance are obtained,respectively.The effects of magnetic intensity,tuning parameters and the amplitude of tension disturbance on the vibration characteristics of the system are discussed.