基于加速度泰勒展开的动力学方程显式积分方法

doi:10.6052/j.issn.1000-4750.2017.08.0661

工程力学 ›› 2018, Vol. 35 ›› Issue (11): 26-34.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

收稿日期:2017-08-30 修回日期:2018-03-19 出版日期:2018-11-07 发布日期:2018-11-07
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). E-mail:ywen_ce@csu.edu.cn
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).
基金资助:
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）

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

Received:2017-08-30 Revised:2018-03-19 Online:2018-11-07 Published:2018-11-07

摘要/Abstract

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。

关键词: 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.

Key words: explicit time integration, Taylor expansion, stability analysis, algorithm damping, period elongation

中图分类号:

TU311.3

文颖, 陶蕤. 基于加速度泰勒展开的动力学方程显式积分方法[J]. 工程力学, 2018, 35(11): 26-34.

WEN Ying, TAO Rui. AN EXPLICIT TIME-DOMAIN INTEGRATION SCHEME FOR SOLVING EQUATIONS OF MOTION IN STRUCTURAL DYNAMICS BASED ON A TRUNCATED TAYLOR EXPANSION OF ACCELERATION[J]. Engineering Mechanics, 2018, 35(11): 26-34.

参考文献

[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.

Metrics

Viewed

Full text

Abstract

Cited

Shared

Discussed

基于加速度泰勒展开的动力学方程显式积分方法

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

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 10

[1]	杨杰, 马萌璠, 王旭. 随机结构动力可靠度计算的条件概率方法[J]. 工程力学, 2018, 35(S1): 17-21.
[2]	马会环, 余凌伟, 王伟, 范峰. 铝合金半刚性椭圆抛物面网壳静力稳定性分析[J]. 工程力学, 2017, 34(11): 158-166.
[3]	孙玉进, 宋二祥, 杨军. 基于非线性强度准则的土工结构安全系数有限元计算[J]. 工程力学, 2016, 33(7): 84-91.
[4]	陆念力, 都亮. 多级阶梯柱侧向刚度与轴压临界力的精确分析及其实用算式[J]. 工程力学, 2015, 32(8): 217-222.
[5]	樊星,袁奇,高进,余沛坰. 超临界机组给水泵汽轮机挠性支承结构稳定性计算与实验研究[J]. 工程力学, 2013, 30(4): 410-416.
[6]	蔡建国, 涂展麒, 冯健, 张晋. 初始缺陷对三向张弦梁结构整体稳定性影响研究[J]. 工程力学, 2012, 29(8): 220-226.
[7]	陆念力孟丽霞. 基于二阶理论的弹性约束变截面悬臂梁刚度与稳定性分析[J]. 工程力学, 2012, 29(12): 365-369.
[8]	李增志;别社安;任增金. 抛石防波堤稳定性的离散单元法分析[J]. 工程力学, 2009, 26(增刊 I): 111-114.
[9]	陈永辉;王新泉;刘汉龙;. 基于尖点突变理论的Y型桩屈曲临界荷载分析[J]. 工程力学, 2009, 26(4): 119-127.
[10]	陆念力;张宏生. 计及二阶效应的一种变截面梁精确单元刚度阵[J]. 工程力学, 2008, 25(12): 60-064,.
[11]	郭泽英;李青宁;张守军. 结构地震反应分析的一种新精细积分法[J]. 工程力学, 2007, 24(4): 0-040.
[12]	魏东;刘应华;朱建明;岑章志. 压弯钢构件火灾下的极限稳定性分析[J]. 工程力学, 2007, 24(2): 0-079.
[13]	乔广;王丽萍;郑铁生. 可倾瓦滑动轴承系统线性稳定性分析[J]. 工程力学, 2007, 24(11): 0-185.
[14]	杨维好;宋雷. 顶部自由、底部嵌固桩的稳定性分析[J]. 工程力学, 2000, 17(5): 63-66,3.
[15]	周建春;寿楠椿;刘光栋. 钢筋混凝土梯形截面桁粱桥侧倾稳定性分析[J]. 工程力学, 1995, 12(3): 55-62.