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微分求积法在结构动力分析中的应用

任争争 梅雨辰 李鸿晶

任争争, 梅雨辰, 李鸿晶. 微分求积法在结构动力分析中的应用[J]. 震灾防御技术, 2018, 13(4): 829-838. doi: 10.11899/zzfy20180410
引用本文: 任争争, 梅雨辰, 李鸿晶. 微分求积法在结构动力分析中的应用[J]. 震灾防御技术, 2018, 13(4): 829-838. doi: 10.11899/zzfy20180410
Ren Zhengzheng, Mei Yuchen, Li Hongjing. The Application of Differential Quadrature Method in Structural Dynamic Analysis[J]. Technology for Earthquake Disaster Prevention, 2018, 13(4): 829-838. doi: 10.11899/zzfy20180410
Citation: Ren Zhengzheng, Mei Yuchen, Li Hongjing. The Application of Differential Quadrature Method in Structural Dynamic Analysis[J]. Technology for Earthquake Disaster Prevention, 2018, 13(4): 829-838. doi: 10.11899/zzfy20180410

微分求积法在结构动力分析中的应用

doi: 10.11899/zzfy20180410
基金项目: 

国家自然科学基金 51478222

高等学校博士学科点专项科研基金 20123221110011

详细信息
    作者简介:

    任争争, 女, 生于1992年。硕士研究生。主要从事工程抗震研究。E-mail:876982248@qq.com

    通讯作者:

    李鸿晶, 男, 生于1966年。教授。主要从事地震工程学研究。E-mail:hjing@njtech.edu.cn

The Application of Differential Quadrature Method in Structural Dynamic Analysis

  • 摘要: 微分求积法(DQM)是1种求解微分方程初(边)值问题的数值方法,通常以较小的计算工作量即可获得较高的数值精度。这种方法应用于工程领域时多用来解决梁、板等结构的静力分析或结构特征值分析等问题,即对边值问题的微分方程的求解。结构动力分析属于初值问题,荷载和结构反应都具有特殊性,直接套用DQM求解边值问题并不能获得问题的解。本文尝试利用微分求积原理建立求解结构动力反应的具体方法。借鉴单元法的思想,将荷载持时划分为若干个时步,在每个时步内对动态荷载和结构反应进行离散,然后用DQM对时步逐个进行求解,得到体系在整个时域内的反应过程。通过对3种不同自振周期的线弹性单自由度体系在不同频率简谐激励下反应的计算,阐释了本文方法的可行性以及高精度、高效率的特点,通过数值试验确定了时步内相对较优的节点数,并为时步长度的选取提供了建议。
  • 图  1  简谐荷载激励下体系1的位移反应

    Figure  1.  Displacement response of system 1 under simple harmonic load

    图  2  简谐荷载激励下体系2的位移反应

    Figure  2.  Displacement response of system 2 under simple harmonic load

    图  3  简谐荷载激励下体系3的位移反应

    Figure  3.  Displacement response of system 3 under simple harmonic load

    表  1  体系的基本特性

    Table  1.   Basic characteristics of systems

    体系编号 自振周期Tn/s 阻尼比ξ
    1 0.68 0.05
    2 0.25 0.05
    3 0.08 0.05
    下载: 导出CSV

    表  2  简谐荷载的信息

    Table  2.   Information of simple harmonic load

    荷载函数/N 周期/s 振幅/m 持时/s
    sin2πt 1 1 40
    sin10πt 0.2 1 40
    sin20πt 0.1 1 40
    下载: 导出CSV

    表  3  荷载周期1s的平均相对误差

    Table  3.   Average relative error with load period of 1s

    m 体系编号
    1 2 3
    位移/% 速度/% 位移/% 速度/% 位移/% 速度/%
    2 98.734 98.734 98.734 98.734 98.734 98.734
    4 88.218 81.849 99.415 389.850 53.532 1293.500
    6 42.402 32.547 15.926 4.728 10.979 9.609
    8 6.698 8.020 1.543 7.814 2.042 2.741
    10 1.623 2.687 0.907 2.338 1.036 0.349
    12 0.121 0.138 0.897 3.341 0.819 4.156
    14 0.008 0.009 0.935 5.153 0.716 1.290
    16 0 0 0.930 3.899 0.635 0.825
    18 0 0 1.005 3.292 0.577 2.28
    20 0 0 1.268 4.715 0.527 0.899
    下载: 导出CSV

    表  4  荷载周期0.2s的平均相对误差

    Table  4.   Average relative error with load period of 0.2s

    m 体系编号
    1 2 3
    位移/% 速度/% 位移/% 速度/% 位移/% 速度/%
    2 99.749 99.749 99.749 99.749 99.749 99.749
    4 799.760 2460.400 403.870 411.780 108.540 225.480
    6 90.884 11.229 23.620 19.707 11.436 5.439
    8 6.052 15.762 2.102 2.135 1.068 2.520
    10 0.264 0.046 0.116 0.140 0.765 1.155
    12 0.009 0.022 0.005 0.004 0.807 3.325
    14 3681.600 256.240 251.760 2935.800 972.740 1334.400
    16 0 0 0 0 0.093 0.167
    18 0 0 0 0 0.010 0.013
    20 0 0 0 0 0.001 0.001
    下载: 导出CSV

    表  5  荷载周期为0.1s的平均相对误差

    Table  5.   Average relative error with load period of 0.1 seconds

    m 体系编号
    1 2 3
    位移/% 速度/% 位移/% 速度/% 位移/% 速度/%
    2 99.875 99.875 99.875 99.875 99.875 99.875
    4 1620.200 1370.200 578.690 459.540 51.868 52.298
    6 175.570 9.444 65.345 7.903 6.80×10282 8.63×10282
    8 11.571 9.569 4.413 3.254 1.951 2.655
    10 0.498 0.046 0.204 0.044 0.142 0.266
    12 0.016 0.013 0.007 0.005 0.006 0.007
    14 27683 306.960 1322.400 246.430 234.600 3888.600
    16 0 0 0 0 0 0
    18 0 0 0 0 0 0
    20 0 0 0 0 0 0
    下载: 导出CSV
  • 李鸿晶, 王通, 2011a.结构地震反应分析的逐步微分积分方法.力学学报, 43(2):430-435. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=CAS201303040000629782
    李鸿晶, 廖旭, 王通, 2011b.结构地震反应DQ解法的两种数值格式.应用基础与工程科学学报, 19(5):758-766. http://d.old.wanfangdata.com.cn/Periodical/yyjcygckxxb201105008
    廖旭, 李鸿晶, 孙广俊, 2013.基于DQ原理的结构弹塑性地震反应分析.工程力学, 30(7):161-166. http://www.cnki.com.cn/Article/CJFDTotal-GCLX201307026.htm
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出版历程
  • 收稿日期:  2017-11-25
  • 刊出日期:  2018-12-01

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