求解周期性分段线性系统动态响应的高效数值方法  

An Efficient Numerical Method for Computing Dynamic Responses of Periodic Piecewise Linear Systems

作　　者：何冬东 高强[1,2] 钟万勰[1,2] HE Dongdong;GAO Qiang;ZHONG Wanxie(State Key Laboratory of Structural Analysis for Industrial Equipment（Dalian University of Technology）,Department of Engineering Mechanics,Faculty of Vehicle Engineering and Mechanics,Dalian University of Technology,Dalian,Liaoning 115024,P.R.China)

机构地区：[1]工业装备结构分析国家重点实验室(大连理工大学) [2]大连理工大学运载工程与力学学部工程力学系,辽宁大连116024

出　　处：《应用数学和力学》2018年第7期737-749,共13页Applied Mathematics and Mechanics

基　　金：国家自然科学基金(11572076;914748203); 国家重点基础研究发展计划(973计划)(2014CB049000)

摘　　要：基于参变量变分原理,提出了一种求解具有大量间隙弹簧的周期性分段线性系统动态响应的高效率数值方法.通过参变量变分原理来描述间隙弹簧,将复杂的非线性动力问题转化为线性互补问题求解,避免了求解过程中的迭代和刚度阵更新,该算法能准确判断间隙弹簧的压缩和松弛状态.基于结构的周期性和能量传播速度的有限性,提出了一种求解系统动态响应的高效率精细积分方法.该算法指出周期结构的矩阵指数中存在大量的相同元素和零元素,从而不需要重复计算和存储这部分元素,节省了计算量并降低了计算机存储要求.分析了一个五自由度分段线性系统在简谐荷载作用下的动力学行为,包括稳定的周期运动、准周期运动和混沌运动.通过与Runge-Kutta方法的比较,该文方法的正确性和高效率得到了验证.An efficient method based on the parametric variational principle（ PVP） was proposed for computing the dynamic responses of periodic piecewise linear systems with multiple gap-activated springs.Through description of gap-activated springs with the PVP,the complex nonlinear dynamic problem was transformed to a standard linear complementary problem. This method can avoid iterations and updating the stiffness matrix in the computing process and can accurately determine the states of the gap-activated springs. Based on the periodicity of the system and the precise integration method（ PIM）,an efficient numerical time-integration method was developed to obtain the dynamic responses of the system. This method indicates that there are a large number of identical elements and zero elements in the matrix exponents of a periodic structure,and saves computation load and computer storage by avoiding repeated calculation and storage of these elements. Numerical results validate the proposed method. The dynamic behaviors of a 5-DOF piecewise linear system under harmonic excitations were analyzed,including the stable periodic motion,the quasi-periodic motion and the chaotic motion. In comparison with the Runge-Kutta method,the proposed method has satisfactory correctness and efficiency.

关 键 词：分段线性系统 动态响应 参变量变分原理 周期性 精细积分方法 

分 类 号：TB122[理学—工程力学] O322[理学—力学]