1∶2内共振情况下点阵夹芯板动力学的奇异性分析  

A Singularity Analysis on Dynamics of Symmetric Cross-Ply Composite Sandwich Plates Under 1 ∶ 2 Resonance

作　　者：郭宇红 张伟[1] 杨晓东[1] GUO Yuhong;ZHANG Wei;YANG Xiaodong(College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, P.R. China)

机构地区：[1]北京工业大学机电学院,北京100124

出　　处：《应用数学和力学》2018年第5期506-528,共23页Applied Mathematics and Mechanics

基　　金：国家自然科学基金(11290152;11072008;11272016)~~

摘　　要：内共振是一种典型的非线性动力学行为,点阵夹芯板在航空航天领域中有着广泛的应用背景.研究点阵夹芯板的内共振问题具有重要的理论及工程意义.在横向激励与面内激励联合作用下,基于四边简支点阵夹芯板的动力学方程,利用多尺度法得到极坐标形式的平均方程,进而化简成稳态形式的代数方程,研究其在1∶2内共振情况下的非线性动力学行为.该文利用推广的奇异性理论研究分叉问题,基于稳态形式的代数方程,计算出含有两个调谐参数、一个横向激励和一个面内激励这4个参数的限制切空间;在强等价的条件下,简化了稳态形式的代数方程;在非退化的情况下,计算出简化的代数方程的正规形;对于含有两个状态变量和4个分叉参数的一般非线性动力学方程的奇异性理论进行了推广;利用推广的奇异性理论得到正规形余维4的18个普适开折的表达式;计算出普适开折转迁集的表达式;这样清楚了点阵夹芯板受到小扰动,当分叉、滞后和双极限点产生时,调谐参数和激励参数之间的关系,数值仿真了转迁集和分叉图,结果表明在不同的分叉区域有不同的振动形式.Inner resonance is a typical nonlinear dynamic behavior, and the symmetric cross-ply composite sandwich plates have been widely used in aerospace. The studies about inner res-onance of such sandwich plates have both theoretical and engineering significances. Based on the dynamic equations for the sandwich plates, of which the boundary conditions were simply supported on 4 sides, the transverse and in-plane excitations were both considered. The average equations in the polar form were obtained with the multiscale method, and the algebraic equa-tions in the steady state form were derived through the average equations. The singularity theory was utilized to investigate I ： 2 resonant bifurcations of the symmetric cross-ply sandwich plates. Based on the algebraic equations in the steady state form, the restricted tangent space was obtained for the bifurcation equations with 2 tuning parameters, an in-plane excitation and a transverse excitation. Then the algebraic equations were simplified under strong equivalence, and the normal form of the algebraic equations were obtained in non-degenerate cases. The sin-gularity theory were generalized for the general nonlinear dynamic equations with 2 state varia-bles and 4 bifurcation parameters, and the 18 universal unfoldings of bifurcation equations with codimension 4 were obtained in the case of 1 ： 2 internal resonance. The transition sets in the parameter plane and the bifurcation diagrams were depicted. The relationships between the tun-ing parameters and the exciting parameters were determined when bifurcation, hysteresis, and double limit points happened. The numerical results indicate that the vibration modes in differ-ent bifurcation re^ions are different.

关 键 词：分叉方程 奇异性理论 普适开折 转迁集 

分 类 号：O322[理学—一般力学与力学基础]