结构动力响应分析的李雅普诺夫法
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摘要
将结构的位移及速度响应作为状态变量,采用Lyapunov(李雅普诺夫)人工小参数法求解状态方程,导出状态方程一种新的离散形式的级数解。将秦九韶算法引入级数解的计算,提高了计算的效率和稳定性,同时给出了该文算法的计算格式和步骤。该算法无需对转换矩阵H求逆,也无须做指数矩阵eH运算,仅做矩阵向量相乘及向量求和运算,计算稳定而且效率高,精度仅由级数项数有关的几个参数控制,很容易达到任意精度要求,适合并行计算及压缩存储,可应用于工程结构抗震计算。最后通过典型算例进一步证实了该文算法的精度和效率。
In this paper, the displacement and velocity response of a structure are taken as the state variable. In order to solve the state equation, the Lyapunov’s artificial small parameter method is used, and a new series solution for the time-discrete form of the equation is presented. Hornor’s scheme is applied to the calculation of the series solution, and the computational efficiency and stability are therefore greatly improved. The corresponding computation scheme and procedure for the proposed algorithm are also presented. The algorithm is only composed of repetitious matrix-vector multiplication and vector summation and the calculations of the inverse matrix H -1 and exponential matrix eH are excluded, which results in high computational efficiency and stability. The accuracy of the algorithm is only controlled by the parameters related to the series, thusly the algorithm can easily achieve arbitrary-order accuracy in theory, and is suitable for parallel computing and compression storage. The proposed method can be used for seismic calculations of engineering structures. Two numerical examples are given to demonstrate the validity and efficiency of the method.
In this paper, the displacement and velocity response of a structure are taken as the state variable. In order to solve the state equation, the Lyapunov’s artificial small parameter method is used, and a new series solution for the time-discrete form of the equation is presented. Hornor’s scheme is applied to the calculation of the series solution, and the computational efficiency and stability are therefore greatly improved. The corresponding computation scheme and procedure for the proposed algorithm are also presented. The algorithm is only composed of repetitious matrix-vector multiplication and vector summation and the calculations of the inverse matrix H -1 and exponential matrix eH are excluded, which results in high computational efficiency and stability. The accuracy of the algorithm is only controlled by the parameters related to the series, thusly the algorithm can easily achieve arbitrary-order accuracy in theory, and is suitable for parallel computing and compression storage. The proposed method can be used for seismic calculations of engineering structures. Two numerical examples are given to demonstrate the validity and efficiency of the method.
引文
[1]钟万勰.结构动力方程的精细积分法[J].大连理工大学学报,1994,34(2):131―136.Zhong Wanxie.Precise integration method for structural dynamics equations[J].Journal of Dalian University of Technology,1994,34(2):131―136.(in Chinese)
[2]钟万勰.子域精细积分及偏微分方程数值解[J].计算结构力学及其应用,1995,12(3):253―260.Zhong Wanxie.Subdomain precise integration method and numerical solution of partial differential equations[J].Computational Structural Mechanics and Applica-tions,1995,12(3):253―260.(in Chinese)
[3]李金桥,于建华.非线性动力系统精细积分下的显示级数解[J].四川大学学报,2002,34(2):24―27.Li Jinqiao,Yu Jianhua.Explicit series solution under precise integration for nonlinear dynamics system[J].Journal of Sichuan University,2002,34(2):24―27.(in Chinese)
[4]李渊印,金先龙,张晓云.非线性动力方程精细积分级数解的并行算法[J].上海交通大学学报,2006,40(4):1809―1812.Li Yuanyin,Jin Xianlong,Zhang Xiaoyun.The parallel algorithms of series solution under precise integration for nonlinear dynamics system[J].Journal of Shanghai Jiaotong University,2006,40(4):1809―1812.(in Chinese)
[5]沈小璞,陈荣毅,沈鹏程.基于状态空间理论的结构动力响应解法[J].工程力学,1996(增刊):492―496.Sheng Xiaopu,Chen Rongyi,Sheng Pengcheng.State space iteration method for analysis of dynamic time history response of structures[J].Engineering Mechanics,1996(Supplement):492―496.(in Chinese)
[6]汪梦甫,周锡元.结构动力方程的高斯精细时程积分法[J].工程力学,2004,21(4):13―16.Wang Mengfu,Zhou Xiyuan.Gauss precise time-integration of structural dynamic analysis[J].Engineering Mechanics,2004,21(4):13―16.(in Chinese)
[7]张森文,曹开彬.计算结构动力响应的状态方程直接积分法[J].计算力学学报,2000,17(1):94―97.Zhang Senwen,Cao Kaibin.Direct integration of state equation method for dynamic response of structure[J].Chinese Journal of Computational Mechanics,2000,17(1):94―97.(in Chinese)
[8]王元丰,储德文.精细时程积分法在地震响应分析中的应用[J].土木工程学报,2004,37(1):20―23.Wang Yuanfeng,Chu Dewen.Application of precise time integration method for structural earthquake response[J].China Engineering Journal,2004,37(1):20―23.(in Chinese)
[9]Lyapunov A M.The general problem on stability of motion(English translation)[M].London:Taylor&Francis,1992.
[10]董付国,厉玉蓉.秦九韶算法思想在RSA密码算法中的应用研究[J].计算机工程与应用,2008,44(28):65―78.Dong Fuguo,Li Yurong.Study on Qinjiushao algorithm and its application in RSA[J].Computer Engineering and Applications,2008,44(28):65―78.(in Chinese)
[2]钟万勰.子域精细积分及偏微分方程数值解[J].计算结构力学及其应用,1995,12(3):253―260.Zhong Wanxie.Subdomain precise integration method and numerical solution of partial differential equations[J].Computational Structural Mechanics and Applica-tions,1995,12(3):253―260.(in Chinese)
[3]李金桥,于建华.非线性动力系统精细积分下的显示级数解[J].四川大学学报,2002,34(2):24―27.Li Jinqiao,Yu Jianhua.Explicit series solution under precise integration for nonlinear dynamics system[J].Journal of Sichuan University,2002,34(2):24―27.(in Chinese)
[4]李渊印,金先龙,张晓云.非线性动力方程精细积分级数解的并行算法[J].上海交通大学学报,2006,40(4):1809―1812.Li Yuanyin,Jin Xianlong,Zhang Xiaoyun.The parallel algorithms of series solution under precise integration for nonlinear dynamics system[J].Journal of Shanghai Jiaotong University,2006,40(4):1809―1812.(in Chinese)
[5]沈小璞,陈荣毅,沈鹏程.基于状态空间理论的结构动力响应解法[J].工程力学,1996(增刊):492―496.Sheng Xiaopu,Chen Rongyi,Sheng Pengcheng.State space iteration method for analysis of dynamic time history response of structures[J].Engineering Mechanics,1996(Supplement):492―496.(in Chinese)
[6]汪梦甫,周锡元.结构动力方程的高斯精细时程积分法[J].工程力学,2004,21(4):13―16.Wang Mengfu,Zhou Xiyuan.Gauss precise time-integration of structural dynamic analysis[J].Engineering Mechanics,2004,21(4):13―16.(in Chinese)
[7]张森文,曹开彬.计算结构动力响应的状态方程直接积分法[J].计算力学学报,2000,17(1):94―97.Zhang Senwen,Cao Kaibin.Direct integration of state equation method for dynamic response of structure[J].Chinese Journal of Computational Mechanics,2000,17(1):94―97.(in Chinese)
[8]王元丰,储德文.精细时程积分法在地震响应分析中的应用[J].土木工程学报,2004,37(1):20―23.Wang Yuanfeng,Chu Dewen.Application of precise time integration method for structural earthquake response[J].China Engineering Journal,2004,37(1):20―23.(in Chinese)
[9]Lyapunov A M.The general problem on stability of motion(English translation)[M].London:Taylor&Francis,1992.
[10]董付国,厉玉蓉.秦九韶算法思想在RSA密码算法中的应用研究[J].计算机工程与应用,2008,44(28):65―78.Dong Fuguo,Li Yurong.Study on Qinjiushao algorithm and its application in RSA[J].Computer Engineering and Applications,2008,44(28):65―78.(in Chinese)
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