Magneto-Elastic Bucking Bifurcation And Chaos Analysis About Bar And Plate

It is very common that the rods, plates and shells work in an electromagnetic environment as structural components with the development of modern advanced technology. One of the basic characters when an electromagnetic field and a mechanical field coupled is nonlinear. The mechanical behavior due to the nonlinear feature is very complex. It will affect the safety and reliability of systems significantly. So it attracted lots of researching interesting.The study of the magneto-elastic buckling problem was focus on the following two aspects previously: Finding the theoretical model of the buckling problem of soft ferromagnetic thin plate and its amendment; The stability study about the current-carrying coil and rods in the magnetic field of Tokamak fusion reactor as examples. However, it is on less of researching on the buckling of current-carrying components, like thin current-carrying plates, working in a strong magnetic field. It is almost no references for buckling Bifurcation and Chaos about current-carrying components working in electromagnetic filed.Considering the situation talking above, this paper showed some studying work using theoretical analysis and numerical computation about buckling, bifurcation and chaotic motion of current-carrying rods and thin plates. Research summarized in the following parts.(1)The analysis of the magnetic-elasticity buckling problem has been extended from coils and rods to non-ferromagnetic thin current-carrying plate here. Based on the non-linear magnetic-elasticity equations of motion, physical equations, geometric equations, expressions of Lorenz forces and electro-dynamic equations, the magnetic-elasticity dynamic buckling equation of a current plate under the action of a mechanical load in a magnetic field is derived. Then the buckling equation is changed into a standard form of the Mathieu equation using the Galerkin method. Thus, to solve the buckling problem is to solve the Mathieu equation. According to the eigenvalue relation of the coefficientsλandηin the Mathieu equation, the criterion equation for the buckling problem is also obtained here. As examples, the magnetic-elasticity buckling equation of a thin current-carrying plate applied four kinds of boundaries, simply supported, simply supported at three edges, simply supported and fixed opposite and fixed are obtained. The relation curves of the instability state and with variations in some parameters are also shown in this paper. The calculation results and the effects of the relative parameters are also discussed.(2)The bifurcation characters of a long slender bar in an electromagnetic field were discussed applying Lagrange description from the perspective of nonlinear dynamics. The bifurcation characters were discussed at the situation state mechanical load, linear and nonlinear dynamic load compression model individual. The buckling bifurcation conditions, bifurcation point and the type of bifurcation were gotten when the long slender bar was under the action of mechanical load in a magnetic field. A hinged aluminum beam and a non-ferromagnetic steel beam were taken as computational example. Their critical load and magnetic intensity value of causing bifurcation were carried out. The relationship curves between these critical values and the length of the beam were shown in the paper.(3)The bifurcation conditions, bifurcation point and bifurcation type of a thin current-carrying plate simply supported or fixed at all edges were obtained through discussed the stability of equilibrium when the plate applied the mechanical load in a electromagnetic field. The dynamic bifurcation conditions, bifurcation point and bifurcation type of the same plate were also obtained using LS method. At the same time, the critical load was carried out and the relation curves and the variation rules between critical load and the current density, the intensity of magnetic field and the geometry length of the plate were shown in the paper.(4)The nonlinear equations of motion of a plate simply supported or fixed at all edges were established when the plate applied mechanical load in a transverse magnetic field. The balance points of the nonlinear dynamic system were gotten in the case of non-disturbance. The critical condition of chaos in the sense of Smale horseshoe of the system was also obtained using Melnikov function method when a disturbance happened. (5)For a strip plate in a transverse electromagnetic field, the chaotic motion under the single-mode displacement was analyzed and obtained its critical chaotic conditions using Melnikov function method. More, the bifurcation point of the strip plate under the dual-mode displacement was obtained and the stability of bifurcation point was discussed using the averaging method. The differences of the simulation to the nonlinear behavior of a system using single-, dual-mode displacement mode were analyzed theoretically.To sum up, it contains very rich and complex dynamic behavior when structural components are in electromagnetic, mechanical and many other physical fields at the same time. Therefore, whether it is through theoretical research or experimental study, it has the theoretical and practical application value. At the same time, the results obtained here can also be used as references to the reliability design for related electromagnetic structure.