Application Of The Stability Of Mathieu Equation's Solution To The Analysis Of Magneto-Elastic Stability In The Thin Plate

It is very common that the plates and shells work in the electromagnetic environment as structural components with the development of modern advanced technology.When there is no machinery to restrain ,the system has an unstable movement mode at least,and while exists restrain,after up to a certain critical value in electric current or the magnetic fielding,buckling must take place in the system.So the magnetoelastic stability analysis of thin plates and thin shells is an important theory and question of using.It will affect the safety and reliability of systems significantly.Based on the analysis and summary of the predecessors has studied the stability of current-carrying coils and poles,using special function to study the magnetoelastic stability of thin current-carryig plates under the effect of alternating magnetic field and mechanical load.The main work of our research is summarized in the following parts:Firstly,the magnetoelastic stability,the domestic and foreign research situation and primary content of our research are introduced,the theory of stability analysis and stability criteria are derived and basic knowledge related to mathematics are described.Secondly,the text derives and provides the non-linear magnetoelastic movement equation of thin current-carrying plates,the geometry equations and the physics equations,the expression formula of Lorent's force,electrodynamics equation through the theory.On this basis,draw magnetoelastic movement stability equation of thin current-carrying plates under the effect of alternating magnetic field and mechanical load.Using the Galerkin principle to change the stability equation for the special function the Mathieu standard form,and utilizing the boundary of the Mathieu solves'steady area and unsteady area,according to the eigenvalue relation in the Mathieu equation to draw the criterion equation of losing the magnetoelastic steady critical state.Finally,through regarding as the example concretely,the magneticelastic stability equation of a thin current-carrying plate applied four kinds of boundaries,simply supported at three edges,simply supported,simply fixed and simply supported and fixed opposite are obtained.The relation curves of losing the magnetoelastic steady critical state and variations in some parameters are shown in this paper.The calculation results and the effects of the relative parameters are also discussed.