微分求积法的改进及在蜂窝夹层板非线性动力学分析中的应用

Nonlinear dynamics analysis is an international frontier topics in the field of engineering science , the dynamics model of engineering structures are mostly described by complex non-linear partial differential equations (group) , and its solution can only rely on approximate methods. Therefore, research on the numerical approximation method of nonlinear partial differential equations(group) is critial for the analysis of nonlinear dynamics of engineering structures. The project intends to combine the numerical and theoretical analysis to study the application of differential quadrature method in nonlinear dynamic systems , focus on how to improve the differential quadrature method to further enhance its solution accuracy and efficiency , to study the effect of the complex boundary conditions , the nonlinear term bring by material and structure on solution accuracy and efficiency, to improve radial basis meshless differential quadrature method to provide a new simple and effective calculation method for the study of multivarible high dimensional nonlinear dynamical systems. Improved differential quadrature method was used to study complex nonlinear dynamics properties of honeycomb sandwich cantilever plate, so that the valuable theoretical guidance is put forward for the optimal design of engineering structure and vibration control . Based on differential quadrature method in the application of the honeycomb sandwich cantilever plate, a set of theory of nonlinear meshless differential quadrature method that is suitable for nonlinear dynamic analysis of plate type structure will be put forward .

非线性动力学分析是工程科学领域的国际前沿课题，而描述工程结构的动力学模型大多为复杂的非线性偏微分方程（组），其求解只能依赖于近似的方法。因此研究非线性动力学偏微分方程（组）的数值近似方法对于工程结构的非线性动力学分析至关重要。本项目拟采用数值和理论分析相结合的方法，研究微分求积法在非线性动力系统中的应用，重点研究如何改进微分求积法以进一步提高其求解精度和效率，研究复杂边界条件的处理，由材料、结构等带来的非线性项对求解的影响，改进径向基无网格微分求积法为研究多元高维非线性动力系统提供一种新的简单高效的计算方法。利用改进的微分求积方法研究蜂窝夹层悬臂板的复杂非线性动力学性质，为航空工程结构的优化设计和振动控制提出有价值的理论指导。基于微分求积法在蜂窝夹层悬臂板中的应用，提出一整套适合板类结构的非线性无网格微分求积方法分析理论。

微分求积法相比于非线性动力学分析常用近似方法具有原理简单(不依赖变分原理)、计算量小、精度高、易于操作等优点，然而将微分求积法用于非线性动力学偏微分方程求解进行长时间复杂非线性动力学性质分析的研究工作还很少。改进和完善微分求积法使之更为高效，可为解决实际工程中非线性动力学问题提供一种简单高效的分析方法，具有十分重要的意义。本项目采用数值和理论分析相结合的方法，研究了微分求积法在非线性动力系统中的应用。从理论上将微分求积法推广到一维、二维非线性动力学偏微分方程的求解，给出了微分求积法分析非线性动力学系统的方法步骤；针对复杂边界条件，提出了一种新的处理方法，进一步提高了微分求积法的求解精度和效率；改进了径向基微分求积法，建立了多元高维非线性动力系统的无网格微分求积方法分析理论，为研究多元高维非线性动力系统提供了一种新的简单高效的计算方法。利用改进的微分求积方法研究了粘弹性梁、蜂窝夹层板的复杂非线性动力学性质，为航空航天等机械领域工程结构的优化设计和振动控制提供有价值的理论指导。本课题发表学术论文5篇，其中SCI期刊1篇，EI收录2篇，中文核心1篇。