二次非线性圆板自由振动分岔解  

BIFURCATION SOLUTION FOR FREE VIBRATION OF CIRCULAR PLATE WITH STRONG QUADRATIC NONLINEARITY

作　　者：李银山[1] 刘波[1] 张明路[1] 段国林[1] 

机构地区：[1]河北工业大学机械工程学院,天津300130

出　　处：《机械强度》2011年第4期505-510,共6页Journal of Mechanical Strength

基　　金：国家自然科学基金(10472076);山西省自然科学基金(20001007)资助项目~~

摘　　要：计及材料的非线性弹性,建立圆板自由振动的非线性动力学方程。采用Galerkin法,将圆板的非线性动力学偏微分方程简化成四种标准类型的二次非线性微分方程。提出一类强非线性动力系统的初值变换法,将描述动力系统的二阶常微分方程,化为以角频率、振幅和偏心距为独立变量的不完备非线性代数方程组,考虑初始条件补充约束方程,构成频率、振幅和偏心距为变量的完备非线性代数方程组。利用Maple程序可以方便地求解。结果表明,初值变换法不仅适合于对称振动问题,而且适合于非对称振动问题。首次给出二次非线性自由振动的偏—频曲线。The nonlinear dynamic equation of free vibration of a circular plate is derived with nonlinear theory of elasticity.By using Galerkin＇s method,the governing partial differential equation was reduced to four standard types of quadratic nonlinear ordinary ones.A method of initial-value transformation is presented for a class strongly nonlinear dynamic system.By using Ritz-Galerkin＇s method,an oscillation system governed by a set of second order ordinary differential equations,can be transformed into a set of incomplete non-linear algebraic equations with angular frequencies,amplitudes and central offsets as independent variables.Then,by supplement of some initial-value restrictions,the incomplete equations can be made complete,obtaining a set of non-linear algebraic equations with angular frequencies,amplitudes and central offsets,which can be solved conveniently by Maple program.The results show that the method of initial-value transformation can solve not only symmetry problems,but also asymmetry problems for free vibration.The offset-angular frequency backbone curves of quadratic nonlinear equations were presented for the first time.

关 键 词：初值变换法 分岔 二次强非线性 非对称振动 偏—频曲线 

分 类 号：O322[理学—一般力学与力学基础]