Magneto-Aeroelastic Nonlinear Dynamics Of A Circular Plate Rotating In Magnetic Field

At present,the new type of conductive and magnetic materials are used to the hightech equipments.Many investigators have focused on the dynamics of conductive and magnetic elastic structures with high-speed rotation in multi-physical fields.However,due to complex structures and physical environment,the corresponding basic principle or theory are still in a continuous development and improvement,e.g.,magnetoelastic mechanics,magnetoelastic vibration and coupling dynamics.Even the nonlinear dynamics behaviors and laws of the most basic structural elements,such as beams,plates,shells,etc.,which work in multi-physical fields,need further extension and betterment.In this paper,the magneto-aeroelastic dynamic model of a circular plate rotating in the air-magnetic fields is established to investigate the internal mechanism and laws of the complex nonlinear dynamic behaviors generated by the magneto-aero-mechanical coupling effects.Based on the large deflection theory of thin plate,the aeroelasticity theory,the rotary damping theory,electromagnetic field principles and electrodynamics,considering the influence of centrifugal forces and rotation effects,the expressions of kinetic energy,potential energy and virtual work of external force are obtained,and the magneto-aeroelastic dynamics equation of a conductive rotating circular plate is derived by Hamilton variational principle.The magneto-aeroelastic linear vibrations of a rotating conducting circular plate are presented.Based on the Kirchhoff-Love assumation and the mode shape functions in the form of series is used to derive the magneto-aeroelastic linear vibration differential equation is deduced.The influence of magnetic field,rotating speed,aerodynamic parameters,geometric parameters and forced excitation on the natural frequency,amplitude frequency characteristics and stability of the system under different boundary conditions is analyzed.The magneto-aeroelastic primary resonances,bifurcations and chaos of a rotating conducting circular plate are studied.Considering the geometrical nonlinearity of the large deflection of the plate,the magneto aeroelastic axisymmetric nonlinear forced vibration differential equation of the plate is given.When the excitation frequency is close to the first natural frequency of the circular plate,the primary resonances are excited.The multi-scale method and the average method are used to solve the differential equations,respectively.The steady state equation of the system under the corresponding boundary conditions and the stability criterion of the solution are obtained.The numerical results of amplitude frequency response curve and the amplitude varying with different parameters are exhibited to expound the characteristics and laws of the nonlinear vibration and analyze the stability of the system.In addition,under the simple-edge conditions,the bifurcation diagrams,maximum Lyapunov exponent spectrums and system responses with different control paramters of different control parameters are plotted to reveal the mechanism of the complex nonlinear dynamics behaviors e.g.,bifurcation and chaos.The axisymmetric modal interaction and Hopf bifurcation of a rotating circular plate are investigated.Considering the combined relationship between the modes of the circular plate and the influence of the centrifugal force,the magneto-aeroelastic modal interaction differential equations of of a rotating circular plate are developed.The multi-scale method with the polar coordinate transformation are employed to solve the differential equations and achieve the modal interaction modulation equations.The conditions of the single mode response,the multi-modal response and the corresponding stability criteria of the steadystate solution are discussed.The influence of magnetic induction intensity,aerodynamic parameters,rotation speed and excitation force on the single-mode response and modal interaction characteristics of the system are analyzed.Moreover,a Hopf bifurcation can be found in three-mode equilibrium by choosing appropriate parameters,where a limit cycle occurs and evolves into chaos undergoing a series of period-doubling bifurcations and finally results in a jump of the response to a single-mode stable equilibrium.The magneto-aeroelastic primary-internal resonances of a rotating circular plate is analyzed.Considering the influence of gyroscopic effect,the differential equations of nonlinear 2-DOF gyroscopic system are given.By using the multi-scale method and solvable condition of gyroscopic system,the 3:1 internal resonance modulation equations he system are deduced.Furthermore,the stability of the solution is determined according to the characteristic roots of Jacobian matrix of coefficient equations.The result of the curves calculated illustrates the influence of magnetic induction intensity and rotation speed on the eigenfrequencies of the two degenerated modes corresponding to the same nodal-diameter vibration of the system.In addition,under primary and 3:1 internal resonances,the curves that the resonance amplitude varying with detuning parameters and excitation amplitudes are plotted reveal the mechanism of Hopf bifurcation instability and the intermittent response induced by gyroscopic effect-magneto-mechanical coupling.