Bifurcation And Chaos Of A Rotating Magneto-elastic Circular Plate

The rotating circular and annular plate exists widely in machinery, instrument, electrical and other fields, and the presence of magnetic field affects the vibration characteristics of circular and annular plate. The magneto elastic theory and Galerkin’s method are used to derive vibration equations. And with numerical example, Bifurcation and chaos problem of magnetic field in rotary motion of circular and annular plate are studied.Based on Maxwell Equations and the nonlinear theory of large deflection, the dynamics equations of nonlinear magneto elastic rotational motion are derived with Hamiltonian principle, and using the Bessel function as the modal function of Galerkin’s integrate, get the transverse vibration equation.The effects of bifurcation parameters on the bifurcation and chaotic motion of the system are studied under different boundary conditions such as fixed both, simply supported both, fixed on the one boundary and simply supported on the other boundary for annular plate and fixed supported boundary for circle plate. The Galerkin method and Bessel mode shape function was used to achieve the ordinary differential equations for axisymmetric transverse vibration. And the poincare map, waveform graph, spectrum graph and phase diagram are drawn, and the influence of the bifurcation parameters to the bifurcation and chaos of system is discussed. The results show that with the change of the bifurcation parameters, the system experiences a complicated process from chaos to quasi-periodic or multi-periodic motion to chaos, and the system present intermittent chaos. With increase of transverse force and decrease of magnetic induction, the probability of chaos in the system increases. The influence of boundary conditions is also obvious.With the singularity theory, the transition of rotating annular plate in magnetic field Primary resonance is obtained, and the conditions for the existence of multiple values are discussed. The chaotic analytic prediction by using Melnikov method is implemented under the boundary condition of fixed on.the outer boundary and free on the inner boundary. And the effects of parameters such as excitation frequency, amplitude, rotating speed, thickness and magnetic induction to chaos in the system are studied. The results show that the smaller the excitation frequency, the rotational speed and the magnetic induction intensity, the greater the excitation force amplitude, the more prone to rupture of the heteroclinic Orbits, the chaos or almost periodic motion.