磁场中旋转运动圆板的分叉与混沌研究

应用数值模拟方法研究磁场中旋转运动圆板的分叉与混沌问题。首先,基于薄板理论和麦克斯韦电磁场方程组,给出了动能、应变势能、外力虚功以及电磁力的表达式,再利用哈密顿原理,得到磁场中旋转运动圆板横向振动的非轴对称非线性磁弹性振动微分方程组。其次,采用贝塞尔函数作为圆板的振型函数进行伽辽金积分,得到了轴对称情况下横向振动的常微分方程组表达式。最后,针对主共振,取周边夹支边界条件的圆板作为算例,得到了当振型函数取一阶时,将磁感应强度、外激励振幅和激励频率作为控制参数的分叉图及庞加莱映射图等计算结果,并讨论了分叉参数对系统的分叉与混沌的影响。数值计算结果表明,这些控制参数的变化影响系统稳定性,在分叉参数逐渐变化的过程中,系统经历从混沌到多倍周期运动再到混沌的往复过程。

Bifurcation and chaos of a rotating circular plate in magnetic field

Numerical simulation was used to research the bifurcation and the chaos of rotating circular plate in magnetic field.Firstly,based on the thin plate theory and Maxwell Equations,the expressions of kinetic energy,potential energy,virtual work by external force and electromagnetic force were derived.Using Hamilton''s principle,the nonlinear non-axisymmetric magnetoelastic vibration differential equations for vibration of the rotating circular plate in the magnetic field were investigated.Secondly,the Galerkin method and Bessel mode shape function were used to derive the ordinary differential equations for axisymmetric transverse vibration.Finally,the primary resonance of the circular plat fixed boundary was studied.Considering the first order mode shape function,the results such as bifurcation diagrams and Poincare map were obtained under the control parameters of magnetic induction,transverse force magnitude and frequency respectively.The influence of bifurcation parameters on the bifurcation and chaos of system was discussed.The results show that the bifurcation parameters affect the stability of the system,and with the system experiences a complicated process from chaos to multi periodic motion to chaos as these parameters vary.