一种数值阻尼耗散可控的结构动力方程积分方法

刘伟; 童小龙; 金蓉

doi:10.21656/1000-0887.440292

一种数值阻尼耗散可控的结构动力方程积分方法

doi: 10.21656/1000-0887.440292

刘伟^{1, 2,},
童小龙^1, ,,
金蓉²

1.
湖南理工学院土木建筑工程学院，湖南岳阳 414006
2.
湖南理工学院南湖学院，湖南岳阳 414006

基金项目:

湖南省教育厅科学研究项目 22B0681

湖南省自然科学基金 2023JJ50281

2022年国家级大学生创新创业计划 S202212658006

详细信息

作者简介:
刘伟(1989―)，男，讲师，博士(E-mail: lwei_work@163.com)

通讯作者:
童小龙(1983―)，男，教授，博士(通讯作者. E-mail: tongxiaolong@hnist.edu.cn)

中图分类号: O39
计量
- 文章访问数: 100
- HTML全文浏览量: 50
- PDF下载量: 18
- 被引次数: 0
出版历程
- 收稿日期: 2023-09-23
- 修回日期: 2023-12-13
- 刊出日期: 2024-07-01

An Integration Method With Controllable Numerical Damping Dissipation for Structural Dynamic Equations

LIU Wei^{1, 2
,},
TONG Xiaolong^{1
, ,},
JIN Rong²

1.
College of Civil Engineering & Architecture, Hunan Institute of Science and Technology, Yueyang, Hunan 414006, P.R.China
2.
Nanhu College, Hunan Institute of Science and Technology, Yueyang, Hunan 414006, P.R.China

摘要

摘要: 数值耗散是数值积分方法的重要特性，直接影响数值仿真结果的准确性. 对于含有虚假高频振动的动力系统，数值耗散能够改善数值仿真结果，但是对于具有真实高频振动的动力系统，数值耗散则会导致计算结果失真. 该研究针对结构动力系统的求解，提出了一种数值耗散可控的两子步隐式数值积分方法. 基于理论推导，详细介绍了新积分方法的谱半径、稳定性、振幅衰减和周期延长等数值特性. 新隐式积分方法通过算法参数α能够对高频虚假振动数值耗散完全可控，相应的耗散比例为1-|α|，其中-1≤α≤1. 通过单自由度动力系统、高频虚假振动系统和多自由度非线性弹簧质量系统三个典型算例，分别证明了新隐式积分方法在计算精确性、高频数值耗散和非线性求解能力方面的优越性.
- 结构动力方程 /
- 隐式积分方法 /
- 数值阻尼 /
- 放大矩阵 /
- 谱半径
Abstract: Numerical dissipation is an important characteristic of numerical integration methods, which directly affects the accuracy of numerical simulation results. Numerical dissipation can improve numerical simulation results for dynamic systems with spurious high-frequency vibrations, but it can also cause distorted calculation results for dynamic systems with real high-frequency vibrations. A 2-sub-step implicit numerical integration method was proposed with controllable numerical damping dissipation to solve structural dynamic systems. Through theoretical derivations, the numerical properties of the new integration method, including the spectral radii, stability, amplitude decay, and period elongation, were introduced in detail. The new implicit integration method can utilize algorithm parameter α to control the numerical dissipation of spurious high-frequency vibration, with a corresponding dissipation ratio of 1-|α|, where -1≤α≤1. The advantages of the new method in terms of the computational accuracy, the high-frequency numerical dissipation, and the nonlinear solving ability were demonstrated through 3 typical examples of a 1-DOF dynamic system, a high-frequency spurious vibration system, and a multi-DOF nonlinear spring-mass system.
- structural dynamic equation /
- implicit integration method /
- numerical damping /
- amplification matrix /
- spectral radius

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图 1 算法的谱半径

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 1. Spectral radii of the proposed method

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图 2 本文方法和典型算法在振幅衰减和周期延长的对比

Figure 2. Comparison between the proposed method and typical algorithms for the percentage amplitude decay and the period elongation for ξ=0

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图 3 典型隐式算法绝对误差

Figure 3. Absolute errors of typical implicit methods for various steps

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图 4 本文隐式算法绝对误差

Figure 4. Absolute errors of proposed implicit methods for various steps

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图 5 三自由度弹簧系统模型

Figure 5. The model for the 3-DOF spring system

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图 6 典型隐式算法求解的节点2加速度时程曲线

Figure 6. Acceleration history curves of node 2 for typical implicit methods

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图 7 本文隐式算法和Wilson方法求解的节点2加速度时程曲线

Figure 7. Acceleration history curves of node 2 for the Wilson method and the proposed implicit method

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图 8 节点2的速度时程曲线

Figure 8. Velocity time histories of node 2

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图 9 节点2和节点3的位移

Figure 9. Displacements of node 2 and node 3

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图 10 节点3的加速度时程曲线

Figure 10. Acceleration time histories of node 3

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图 11 节点3的速度时程曲线

Figure 11. Velocity time histories of node 3

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图 12 多自由度弹簧动力系统

Figure 12. The multi-DOF nonlinear spring-mass system

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图 13 第2 000个弹簧质量块的位移响应

Figure 13. Displacement responses of the 2 000th mass

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图 14 第2 000个弹簧质量块的速度响应

Figure 14. Velocity responses of the 2 000th mass

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