一种Birkhoff形式下结构动响应问题的保辛中点格式

A symplectic midpoint scheme for structural dynamic response problems in Birkhoffian form

结构动响应预测是结构设计的基础,是结构振动控制、载荷识别的前提。本文在辛体系下针对结构动响应问题,提出了一种Birkhoff形式下的保辛中点格式。首先引入状态变量,并基于摄动方法将结构动响应方程转化为线性自治Birkhoff方程的形式,进一步利用中心差分推导出线性自治Birkhoff方程的中点格式,其证明是保辛的。该格式不要求Birkhoff方程系数矩阵非奇异,因此适用于奇数维系统。两个不同数值算例的结果充分验证了本文方法的卓越性,也凸显了相对于传统算法在计算精确度和稳定性方面的明显优势。

Structural dynamic response prediction is the foundation of structural design and serves as a prerequisite for structural vibration control and load identification.In this paper,we address structural dynamic response problems within the symplectic framework and propose a symplectic midpoint scheme in Birkhoffian form.The state variables are first introduced and the structural dynamic response equations are transformed into the form of linear autonomous Birkhoffian equations based on the perturbation method.The central difference is further used to derive the midpoint scheme of the linear autonomous Birkhoffian equation,which is proved to be symplectic.This scheme does not require the coefficient matrix of the Birkhoffian equation to be non-singular and is therefore suitable for odd-dimensional systems.The results from two distinct numerical test cases provide ample validation of the excellence of the method presented in this paper,highlighting the significant advantages it possesses over traditional algorithms in terms of computational accuracy and stability.