基于加速度泰勒展开的动力学方程显式积分方法

文颖; 陶蕤

doi:10.6052/j.issn.1000-4750.2017.08.0661

基于加速度泰勒展开的动力学方程显式积分方法

文颖^1,2,,
陶蕤¹

1.
中南大学土木工程学院, 长沙 410075;
2.
重载铁路工程结构教育部重点实验室(中南大学), 长沙 410075

基金项目: 国家自然科学基金高铁联合基金重点项目（U1534206）；湖南省科技计划项目（2014FJ6036）

详细信息

作者简介:
陶蕤(1993-),男,湖北人,硕士生,从事车-桥系统垂向振动研究(E-mail:taorui_2014@163.com).

通讯作者:
文颖(1981-),男,湖南人,副教授,博士,重载铁路工程结构教育部重点实验室副主任,从事桥梁稳定极限承载力及车桥系统振动稳定性研究(E-mail:ywen_ce@csu.edu.cn).

中图分类号: TU311.3
计量
- 文章访问数: 641
- HTML全文浏览量: 33
- PDF下载量: 139
出版历程
- 收稿日期: 2017-08-29
- 修回日期: 2018-03-18
- 刊出日期: 2018-11-28

AN EXPLICIT TIME-DOMAIN INTEGRATION SCHEME FOR SOLVING EQUATIONS OF MOTION IN STRUCTURAL DYNAMICS BASED ON A TRUNCATED TAYLOR EXPANSION OF ACCELERATION

WEN Ying^1,2,,
TAO Rui¹

1.
School of Civil Engineering, Central South University, Changsha 410075, China;
2.
The Key Laboratory of Engineering Structures of Heavy Haul Railway(Central South University), Ministry of Education, Changsha 410075, China

摘要

摘要: 该文旨在提出兼顾适用性、可靠性与高效性的结构振动时域积分算法。基于加速度的泰勒展开式，引入截断系数考虑高阶项的影响，提出了具有4阶精度的加速度公式；通过积分并考虑典型时间步初始时刻系统动力平衡条件，建立了位移和速度的单步递推公式，运用终止时刻系统运动方程修正加速度。与多步积分法相比，单步积分法无需记录当前时间步以外时刻响应。稳定性分析表明，临界步长相比中心差分法增加40%。通过线性系统振动响应计算发现，当步长-系统固有周期（荷载周期）比达到0.2时，该文方法的振幅衰减率和周期延长率均小于5%；对于非线性系统，为降低算法阻尼和周期误差的影响，需控制步长周期比小于0.1。
- 显式时域积分 /
- 泰勒展开 /
- 稳定性分析 /
- 算法阻尼 /
- 周期延长
Abstract: The aim of this paper is to present a novel time integration algorithm with a high level of balance among applicability and reliability and computational efficiency for the dynamic analysis of structures. A formula for approximating acceleration with a forth-order degree of accuracy has been developed, based on the Taylor expansion approach. In applying the Taylor expansion method, a truncation parameter is defined to consider the contributions of high-order terms upon the accuracy of predicted results. Through an integration of the obtained acceleration and considering the dynamic equilibrium condition at the initial state of a typical time step, a single-step equation for computing displacement and velocity at the end state is correspondingly developed. A revised acceleration can be obtained from the calculated displacement and velocity through the equations of motion at the end state. In this regard, as compared with the multiple-step integration scheme, it is not required for the present method to temporarily record the state variables of previous steps. From the results of stability analysis, the maximum step length to period ratio within which the obtained responses remain bounded has been increased by 40% in comparison to the central difference method. By carrying out a series of numerical analyses for the purpose of demonstration, it is generally observed from the natural and forced vibration investigations for linear systems that the computational amplitude decay and period elongation were less than 5% even if the ratio between the time step length and system inherent period/load period mounts to 0.2. However, to reduce the effects of amplitude decay and period distortion for the time integration of nonlinear systems, the magnitude of the above mentioned ratio should generally be restricted below 0.1.
- explicit time integration /
- Taylor expansion /
- stability analysis /
- algorithm damping /
- period elongation

HTML全文

参考文献(22)

[1]	Argyris J H, Mlenjnek H P. Dynamics of structures[M]. New York:North Holland Publishing House, 1991.
[2]	Thompson W T, Dahleh M D. Theory of vibration with applications, fifth edition[M]. New York:Pearson Education Press, 1997.
[3]	Chang S Y. An explicit method with improved stability[J]. International Journal for Numerical Methods in Engineering, 2009, 77(8):1100-1120.
[4]	Chen C, Ricles J M. Development of direct integration algorithms for structural dynamics using discrete control theory[J]. Journal of Engineering Mechanics, 2008, 134(8):676-683.
[5]	Bathe K J. Finite element procedures[M]. New York:Prentice Hall Publishing Company, 1996.
[6]	Hilbert H M, Hughes T J R, Taylor R L. Improved numerical dissipation for time integration algorithms in structural dynamics[J]. Earthquake Engineering & Structural Dynamics, 1977, 5(3):283-292.
[7]	Bathe K J, Noh G. Insight into an implicit time integration scheme for structural dynamics[J]. Computers & Structures, 2012, 98(1):1-6.
[8]	Chopra A K. Dynamics of structures[M]. 5th ed. New York:Prentice Hall Publishing Company, 2016.
[9]	宋佳, 许成顺, 杜修力, 等. 基于精细时程积分的u-p格式饱和两相介质动力问题的显-显式时域算法[J]. 工程力学, 2017, 34(11):9-17. Song Jia, Xu Chengshun, Du Xiuli, et al. A time-domain explicit-explicit algorithm based on the precise time-integration for solving the dynamic problems of fluid-saturated porous media in u-p form[J]. Engineering Mechanics, 2017, 34(11):9-17. (in Chinese)
[10]	吕和祥, 蔡志勤, 裘春航. 非线性动力学问题的一个显式精细积分算法[J]. 应用力学学报, 2001, 18(2):34-40. Lu Hexiang, Cai Zhiqin, Qiu Chunhang. An explicit precise integration algorithm for nonlinear dynamics problems[J]. Chines Journal of Applied Mechanics, 2001, 18(2):34-40. (in Chinese)
[11]	王进廷, 杜修力. 有阻尼体系动力分析的一种显式差分法[J]. 工程力学, 2002, 19(3):109-111. Wang Jinting, Du Xiuli. An explicit difference method for dynamic analysis of a structure system with damping[J]. Engineering Mechanics, 2002, 19(3):109-111. (in Chinese)
[12]	冉田苒, 王涛, 周惠蒙, 等. 一种无条件稳定的显式数值积分算法[J]. 防灾减灾工程学报, 2016, 36(1):50-54. Ran Tianran, Wang Tao, Zhou Huimeng, et al. An explicit algorithm with unconditional stability for pseudo dynamic testing[J]. Journal of Disaster Prevention and Mitigation Engineering, 2016, 36(1):50-54. (in Chinese)
[13]	陈学良, 袁一凡. 求解振动方程的一种显式积分格式及其精度与稳定性[J]. 地震工程与工程振动, 2002, 22(3):9-14. Chen Xueliang, Yuan Yifan. Explicit integration formula for vibration equation and its accuracy and stability[J]. Earthquake Engineering and Engineering Vibration, 2002, 22(3):9-14. (in Chinese)
[14]	杜晓琼, 杨迪维, 赵永亮. 一种无条件稳定的结构动力学显式算法[J]. 力学学报, 2015, 47(2):310-319. Du Xiaoqiong, Yang Diwei, Zhao Yongliang. An unconditional stable explicit algorithm for structural dynamics[J]. Chinese Journal of Theoretical and Applied Mechanics, 2015, 47(2):310-319. (in Chinese)
[15]	Noh G, Bathe K J. An explicit time integration scheme for the analysis of wave propagations[J]. Computers & Structures, 2013, 129(1):178-193.
[16]	袁驷, 袁全, 闫维明, 等. 运动方程自适应步长求解的一个新进展——基于EEP超收敛计算的线性有限元法[J]. 工程力学, 2018, 35(2):13-20. Yuan Si, Yuan Quan, Yan Weiming, et al. New development of solution of equations of motion with adaptive time-step size-Linear FEM based on EEP super convergence technique[J]. Engineering Mechanics, 2018, 35(2):13-20. (in Chinese)
[17]	Rossi D F, Ferreira W G, Mansur W J, et al. A review of automatic time-stepping strategies on numerical time integration for structural dynamic analysis[J]. Engineering Structures, 2014, 80(1):118-136.
[18]	曾庆元, 周智辉, 文颖. 结构动力学讲义[M]. 北京:人民交通出版社, 2015. Zeng Qinyuan, Zhou Zhihui, Wen Ying. Lectures on dynamics of structures[M]. Beijing:China Communication Press, 2015. (in Chinese)
[19]	刘祥庆, 刘晶波, 丁桦. 时域逐步积分算法稳定性与精度的对比分析[J]. 岩石力学与工程学报, 2007, 26增刊1):3001-3008. Liu Xiangqing, Liu Jingbo, Ding Hua. Comparative analysis of stabilization and accuracy of step-by-step time integration methods[J]. Chinese Journal of Rock Mechanics and Engineering, 2007, 26(Suppl1):3001-3008. (in Chinese)
[20]	王进廷, 张楚汉, 金峰. 有阻尼动力方程显示积分方法的精度研究[J]. 工程力学, 2006, 23(3):1-5. Wang Jinting, Zhang Chuhan, Jin Feng. On the accuracy of several explicit integration schemes for dynamic equation with damping[J]. Engineering Mechanics, 2006, 23(3):1-5. (in Chinese)
[21]	杨超, 肖守讷, 阳光武, 等. 一类非耗散的显式时间积分方法[J]. 振动工程学报, 2015, 28(3):441-449. Yang Chao, Xiao Shoune, Yang Guangwu, et al. Non-dissipative explicit time integration methods of the same class[J]. Journal of Vibration Engineering, 2015, 28(3):441-449. (in Chinese)
[22]	Yang Y B, Wu Y S. A versatile element for analyzing vehicle-bridge interaction response[J]. Engineering Structures, 2001, 23(5):452-469.

施引文献(11)

期刊类型引用(6)

1.	孙劲文，夏峰，齐文浩，陈赛斌，吴亚强，景立平. 自适应最小二乘支持向量机加速度积分新方法. 防灾科技学院学报. 2025(01): 18-30 . 百度学术
2.	赵建锋，赵旭. 一类带高频耗散的动力学显式算法. 应用力学学报. 2023(04): 939-946 . 百度学术
3.	胡林烽，雷家艳，吴嘉伟，高婧. 基于计算扰动修正的Wilson-θ法稳定性和精度分析. 厦门大学学报(自然科学版). 2022(04): 563-571 . 百度学术
4.	余才志，李岳，王鹏，孙长库. 三轴微振动实时测量技术研究. 传感技术学报. 2022(12): 1635-1642 . 百度学术
5.	赵旭，赵建锋. 基于修正四次样条插值法的结构动力学显式算法. 青岛理工大学学报. 2021(05): 123-130 . 百度学术
6.	李述涛，刘晶波，宝鑫，陶西贵，陈一村，肖兰，贾艺凡. 采用粘弹性人工边界单元时显式算法稳定性分析. 工程力学. 2020(11): 1-11+46 . 本站查看