磁场中旋转运动圆环板主共振分岔及混沌研究

Principal Resonance Bifurcation and Chaos of Rotating Annular Plates in Magnetic Fields

研究了磁场中旋转运动圆环板的磁弹性主共振及分岔、混沌问题.通过Hamilton(哈密顿)原理推得磁场中旋转运动圆环板的横向振动方程,并采用Bessel(贝塞尔)函数作为振型函数进行Galerkin(伽辽金)积分,得到磁场中旋转运动圆环板的无量纲非线性振动常微分方程.利用多尺度法展开,得到静态分岔方程、对应的转迁集与分岔图,以及物理参数作为分岔控制参数时的分岔图.利用Mel’nikov(梅利尼科夫)方法,对系统混沌特性进行研究,得到外边夹支内边自由边界条件下异宿轨破裂的条件;通过数值计算,得到外激振力幅值作为分岔控制参数时系统的分岔图与指定参数条件下系统响应图.结果表明,磁场扼制多值现象的产生;激振频率、转速、磁感应强度越小,激振力幅值越大,系统的异宿轨越容易发生破裂,从而引发混沌或概周期运动.

The magneto-elastic principal resonance bifurcation and chaos of rotating annular plates in magnetic fields were studied. Based on the expressions of kinetic energy,strain energy and virtual work done by external forces and electromagnetic forces,the nonlinear vibration equations of a rotating annular plate in magnetic field were deduced with the Hamiltonian principle. The Galerkin method with the Bessel mode shape functions was used to achieve the ordinary differential vibration equations. The static bifurcation equations and corresponding transition sets with the physical parameters as the bifurcation control parameters were achieved by means of the method of multiple scales. Finally,the critical conditions for the break of the heteroclinic orbits were obtained under the conditions of fixed outer boundary and free inner boundary with the Melnikov method. Moreover,the global bifurcation diagrams under the external forces as the control parameters and other response diagrams with specified control parameters were drawn. The results show that the magnetic field deters the occurence of multi-value phenomena. With the decreasing of the external force frequency,the rotating speed and the magnetic induction,and with the increasing of the external force,the systems heteroclinic orbits break more easily,meanwhile chaos or almost periodic motion of the system is induced.