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内共振条件下直线运动梁的动力稳定性
引用本文:冯志华,胡海岩.内共振条件下直线运动梁的动力稳定性[J].力学学报,2002,34(3):389-400.
作者姓名:冯志华  胡海岩
作者单位:南京航空航天大学振动工程研究所,南京,210016
基金项目:国防基础科研计划项目(10172005)资助.
摘    要:基于Kane方程,建立起了包含有耦合的三次几何及惯性非线性项大范围直线运动梁动力学控制方程.利用多尺度法并结合笛卡尔坐标变换,对所得方程进行一次近似展开,着重对满足一、二阶模态间3:1内共振现象的两端铰支梁参激振动平凡解稳定性进行了详尽的分析,得出了稳定性边界的解析表达式.采用中心流形定理对调制微分方程组进行降维处理,分析了相应Hopf分岔类型并通过数值计算发现了稳定的极限环存在.

关 键 词:  动力稳定性  参激振动  内共振  多尺度法  大范围直线运动
修稿时间:2001年4月27日

DYNAMIC STABILITY OF A SLENDER BEAM WITH INTERNAL RESONANCE UNDER A LARGE LINEAR MOTION
Feng Zhihua Hu HaiyanInstitute of Vibration Engineering Researeh,Nanjing University of Aeronautics and Astronautics,Nanjing ,China.DYNAMIC STABILITY OF A SLENDER BEAM WITH INTERNAL RESONANCE UNDER A LARGE LINEAR MOTION[J].chinese journal of theoretical and applied mechanics,2002,34(3):389-400.
Authors:Feng Zhihua Hu HaiyanInstitute of Vibration Engineering Researeh  Nanjing University of Aeronautics and Astronautics  Nanjing  China
Institution:Feng Zhihua Hu HaiyanInstitute of Vibration Engineering Researeh,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China
Abstract:Dynamic modeling of a flexible beam undergoing a large linear motion is presentedin this paper at first. The equations of motion for the beam are derived by using Kane's equationand then simplified through the Rayleigh-Ritz method. Different from the linear modeling methodwhere the generalized inertia forces and the generalized active forces are linearized in the model-ing process, the present model takes the coupled cubic non-linearities of geometrical and inertialtypes into consideration. In the case of a simply supported slender beam under certain averageacceleration of base, the second natural frequency of the beam may approximate to the tripled firstone so that the condition of 3:1 internal resonance of the beam holds true. The method of multi-ple scales is used to solve directly the nonlinear differential equations and to derive the nonlinearmodulation equation for either the principal parametric resonance or the combination parametricresonance with 3:1 internal resonance between the first two modes of the beam. The dynamicstability of the trivial state of the system is investigated by using Cartesian transformation indetail. The equations of approximate transition curves in the plane of dimensionless frequency andexcitation parameter that separate stable from unstable solution are derived. For the case of prin-cipal parametric resonance of the first mode, in addition to the principal instability region, thereexist several new narrow instability regions because of the presence of internal resonance. Thesenarrow regions move from the left of the principal instability region to the right of it when thefrequency detuning parameter of internal resonance increases from the negative to the positive. Incontrast with the case of principal parametric resonance of the first mode, single mode equilibriumsolution is possible for the principal parametric resonance of the second mode. Furthermore, thestability of the trivial solution for the third mode or the higher ones has not been affected by theinternal resonance between the first two modes. Finally,the modulation equations are reduced toa two-dimensional system and the type of a Hopf bifurcation is determined in the vicinity of thebifurcation via the center manifold theorem and a limit cvcle is found.
Keywords:beam  dynamic stability  parametric excitation  internal resonance  method ofmultiple scales  large linear motion
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