导电薄板的磁弹性组合共振分析

Magneto-Elastic Combination Resonances Analysis of Current-Conducting Thin Plate

基于Mexwell方程,给出了导电薄板的非线性磁弹性振动方程、电动力学方程和电磁力表达式.在此基础上,研究了横向磁场中梁式导电薄板的磁弹性组合共振问题,应用Galerkin法导出了相应的非线性振动微分方程组.利用多尺度法进行求解,得到了系统稳态运动下的幅频响应方程,分析了组合共振激发的条件.根据Liapunov近似稳定性理论,对稳态解的稳定性进行了分析,得到了稳定性的判定条件.通过数值计算,给出了一、二阶模态下共振振幅随调谐参数、激励幅值和磁场强度的变化规律曲线图,以及系统振动的时程响应图、相图、Poincaré映射图和频谱图,进一步分析了电磁、机械等参量对解的稳定性及分岔特性的影响,并讨论了系统的倍周期和概周期等复杂动力学行为.

Based on Maxwell equations, the nonlinear magneto-elastic vibration equations of a thin plate were derived. The electrodynamic equations and expressions of electromagnetic forces were also derived. In addition, the magneto-elastic combination resonances and stabilities of the thin beam-plate subjected to mechanical loadings in a constant transverse magnetic flied were studied. By means of the Galerldn Method, the corresponding nonlinear vibration differential equations were derived. The amplitude frequency response equation of the system in steady motion was obtained by the method of multiple scales. The excitation condition of combinadon resonances was analyzed. Based on the Lia- punov stability theory, the stabilities of steady solutions were analyzed and the critical conditions of stability were also obtained. Through the numerical calculation, the curves which resonance-amplitudes changing with dettming parameters, excitation amplitudes and magnetic intensity in the first and the second order modality were obtained respectively. The time history response plots, the phase charts, the Poincaré mapping charts and the spectrum plots of vibrations were also 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 the period-doubling motion and the quasi-period motion were discussed.