Based on the Maxwell equations, the nonlinear magneto-elastic vibration equations of a thin plate and the electrodynamic equations and expressions of electromagnetic forces are derived. In addition, the magneto-elasti...Based on the Maxwell equations, the nonlinear magneto-elastic vibration equations of a thin plate and the electrodynamic equations and expressions of electromagnetic forces are derived. In addition, the magneto-elastic combination resonances and stabilities of the thin beam-plate subjected to mechanical loadings in a constant transverse magnetic filed are studied. Using the Galerkin method, the corresponding nonlinear vibration differential equations are derived. The amplitude frequency response equation of the system in steady motion is obtained with the multiple scales method. The excitation condition of combination resonances is analyzed. Based on the Lyapunov stability theory, stabilities of steady solutions are analyzed, and critical conditions of stability are also obtained. By numerical calculation, curves of resonance-amplitudes changes with detuning parameters, excitation amplitudes and magnetic intensity in the first and the second order modality are obtained. Time history response plots, phase charts, the Poincare mapping charts and spectrum plots of vibrations are obtained. The effect of electro-magnetic and mechanical parameters for the stabilities of solutions and the bifurcation are further analyzed, Some complex dynamic performances such as perioddoubling motion and quasi-period motion are discussed.展开更多
The principle and 1:3 internal resonance of a rectangular thin plate in a transverse magnetic field is investigated.Based on the magneto-elastic vibration equation and electromagnetic force expressions of the thin pla...The principle and 1:3 internal resonance of a rectangular thin plate in a transverse magnetic field is investigated.Based on the magneto-elastic vibration equation and electromagnetic force expressions of the thin plates,the nonlinear magneto-elastic vibration differential equations of rectangular plates under external excitation in a transverse magnetic field are derived.For a rectangular plate with one side fixed and three other sides simply supported,the two-degree-offreedom nonlinear Duffing vibration differen-tial equations are proposed by the method of Galerkin.The method of multiple scales is adopted to solve the model equations and obtain four first-order ordinary differential equations governing modulation of the amplitudes and phase angles involved via the first-order or the second-order primary-internal reso-nances.With a numerical example,the amplitude frequency response curves,time history responses,phase portraits and Poincare maps of the first two order vibration modes via principle-internal resonance are respectively captured.And the effects of external excitation amplitudes,magnetic field intensities and thicknesses on the vibration of system are discussed.The results show that the response is dominated by the low mode when principle-internal resonance occurs.The internal resonance provides a mechanism for transferring energy from a high mode to a low mode.展开更多
文摘Based on the Maxwell equations, the nonlinear magneto-elastic vibration equations of a thin plate and the electrodynamic equations and expressions of electromagnetic forces are derived. In addition, the magneto-elastic combination resonances and stabilities of the thin beam-plate subjected to mechanical loadings in a constant transverse magnetic filed are studied. Using the Galerkin method, the corresponding nonlinear vibration differential equations are derived. The amplitude frequency response equation of the system in steady motion is obtained with the multiple scales method. The excitation condition of combination resonances is analyzed. Based on the Lyapunov stability theory, stabilities of steady solutions are analyzed, and critical conditions of stability are also obtained. By numerical calculation, curves of resonance-amplitudes changes with detuning parameters, excitation amplitudes and magnetic intensity in the first and the second order modality are obtained. Time history response plots, phase charts, the Poincare mapping charts and spectrum plots of vibrations are obtained. The effect of electro-magnetic and mechanical parameters for the stabilities of solutions and the bifurcation are further analyzed, Some complex dynamic performances such as perioddoubling motion and quasi-period motion are discussed.
基金supported by the National Natural Science Foundation of China (11472239)the Natural Science Foundation of Hebei Province of China (A2015203023)
文摘The principle and 1:3 internal resonance of a rectangular thin plate in a transverse magnetic field is investigated.Based on the magneto-elastic vibration equation and electromagnetic force expressions of the thin plates,the nonlinear magneto-elastic vibration differential equations of rectangular plates under external excitation in a transverse magnetic field are derived.For a rectangular plate with one side fixed and three other sides simply supported,the two-degree-offreedom nonlinear Duffing vibration differen-tial equations are proposed by the method of Galerkin.The method of multiple scales is adopted to solve the model equations and obtain four first-order ordinary differential equations governing modulation of the amplitudes and phase angles involved via the first-order or the second-order primary-internal reso-nances.With a numerical example,the amplitude frequency response curves,time history responses,phase portraits and Poincare maps of the first two order vibration modes via principle-internal resonance are respectively captured.And the effects of external excitation amplitudes,magnetic field intensities and thicknesses on the vibration of system are discussed.The results show that the response is dominated by the low mode when principle-internal resonance occurs.The internal resonance provides a mechanism for transferring energy from a high mode to a low mode.
