工程力学 ›› 2018, Vol. 35 ›› Issue (11): 26-34.doi: 10.6052/j.issn.1000-4750.2017.08.0661

• 基本方法 • 上一篇    下一篇

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. 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 Ying1,2, TAO Rui1   

  1. 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: 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
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