基于二次特征值法的矩形薄板的动力稳定性研究  

Dynamic stability research on rectangular thin plate based on the quadratic eigenvalue method

作　　者：钟子林[1] 刘爱荣[1] 卢汉文[1] 黄友钦[1] 

机构地区：[1]广州大学-淡江大学工程结构灾害与控制联合研究中心,广东广州510006

出　　处：《中山大学学报（自然科学版）》2016年第3期68-76,共9页Acta Scientiarum Naturalium Universitatis Sunyatseni

基　　金：国家自然科学基金资助项目(51578166);广州市羊城学者资助项目(1201541551);广东省教育厅高层次人才资助项目

摘　　要：基于Von-Karman薄板大挠度理论,利用Galerkin法得到面内周期荷载作用下四边简支矩形薄板的二阶常微Mathieu-Hill型参数振动方程;运用二次特征值法分别求出矩形薄板线性参数振动方程周期为2T和T时的主要与次要动力不稳定域,并用有限元数值分析方法验证了二次特征值法的精确性,同时定性地分析了主要参数共振下非线性弹性对系统定态振幅的影响。分析结果表明:1当激发力频率近薄板两倍自振频率时,薄板发生强烈的横向参数共振;2二次特征值法可精确计算矩形薄板发生动力不稳定时对应的频率和激发系数;3随着薄板振幅的增长,非线性的存在抑制了定态振动幅值的无限增长,牵引系统向大频率方向振动,导致振幅稳定增加或迅速增大的复杂振动状态。Based on the Von-Karman large deflection theory of thin plates,the second-order ordinary differential Mathieu-hill Parametric vibration equation was established for of simply supported rectangular thin plate under in-plane periodic loading using the Galerkin method. The quadratic eigenvalue method was adopted to obtain the main and secondary instability domains of the rectangular thin plate with the periodic solutions of 2T and T,then the accuracy of the method was verified by finite element method,meanwhile,the influence of nonlinear elasticity on stationary amplitude caused by the main Parametric resonance was qualitatively analyzed. The analytic results showed that strong transverse parametric resonance will occur when the frequency of excitation force is about twice as large as the natural vibration frequency of the thin plate. The quadratic eigenvalue method can be used to calculate accurately those parameters,such as the frequency and the excitation coefficient of the dynamic instability region of the rectangular thin plate. As the increasing of the amplitude,the infinite increase of vibration amplitude was controlled owing to the nonlinear factor which tows system to the direction of large vibration frequency.Furthermore,the nonlinear factor can bring the system into a complex vibration state in which the vibration amplitude increased stably or rapidly.

关 键 词：矩形薄板 二次特征值法 参数共振 动力不稳定域 周期荷载 

分 类 号：TU317[建筑科学—结构工程]