各种板边条件下大挠度圆板自由振动的分岔解  

BIFURCATE SOLUTIONS OF FREE VIBRATION TO A CIRCULAR PLATE FOR VARIOUS BOUNDARY CONDITIONS

作　　者：李银山[1] 张善元[2] 刘波[1] 董青田[1] 

机构地区：[1]河北工业大学力学系,天津300130 [2]太原理工大学应用力学研究所,太原030024

出　　处：《机械强度》2007年第1期30-35,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：计及几何非线性,在各种板边条件下,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三种标准类型的Duffing方程。提出一类强非线性动力系统的两项谐波法,将描述动力系统的二阶常微分方程,化为以频率、振幅为变量的非线性代数方程组,考虑初始条件补充约束方程,构成频率、振幅为变量的封闭非线性代数方程组。利用Maple程序可以方便地求解。结果表明,两项谐波法不仅适合于对称振动问题,而且适合于非对称振动问题。The nonlinear dynamic equation of free vibration to a circular plate for various boundary conditions, with the geometric nonlinear effects taken into account. By using Galerkin＇ s method, the governing partial differential equation was reduced to three standard types of Duffing＇ s equation. Two harmonics method is presented for strongly nonlinear dynamic-system. An oscillation system which is described as a second order ordinary differential equation, is expressed as a set of non-linear algebraic equations with a frequency and amplitudes as the independent variables. Considering binding equation of initial conditions, they constitutes a complete set of non-linear algebraic equations with a frequency and amplitudes as the independent variables. Using Maple program, the algebraic equations can be solved conveniently. The results show that two harmonics method can solve beth a symmetry problem of free vibration and a non-symmetry problem of free vibration. The backbone curve and bifurcation diagram about amphtude-frequency of Dufling-Holmes equation were gained. Via contrast it can be clearly seen that the results are beheved to be rather general and complete.

关 键 词：圆板 强非线性动力系统 非对称振动 两项谐波法 

分 类 号：O322[理学—一般力学与力学基础]