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基于近场动力学微分算子的变截面梁动力特性分析方法

李志远 黄丹 闫康昊

李志远, 黄丹, 闫康昊. 基于近场动力学微分算子的变截面梁动力特性分析方法[J]. 工程力学, 2022, 39(12): 23-30. doi: 10.6052/j.issn.1000-4750.2021.07.0579
引用本文: 李志远, 黄丹, 闫康昊. 基于近场动力学微分算子的变截面梁动力特性分析方法[J]. 工程力学, 2022, 39(12): 23-30. doi: 10.6052/j.issn.1000-4750.2021.07.0579
LI Zhi-yuan, HUANG Dan, YAN Kang-hao. METHOD FOR DYNAMIC CHARACTERISTIC ANALYSIS OF BEAMS WITH VARYING CROSS-SECTIONS BY USING PERIDYNAMIC DIFFERENTIAL OPERATOR[J]. Engineering Mechanics, 2022, 39(12): 23-30. doi: 10.6052/j.issn.1000-4750.2021.07.0579
Citation: LI Zhi-yuan, HUANG Dan, YAN Kang-hao. METHOD FOR DYNAMIC CHARACTERISTIC ANALYSIS OF BEAMS WITH VARYING CROSS-SECTIONS BY USING PERIDYNAMIC DIFFERENTIAL OPERATOR[J]. Engineering Mechanics, 2022, 39(12): 23-30. doi: 10.6052/j.issn.1000-4750.2021.07.0579

基于近场动力学微分算子的变截面梁动力特性分析方法

doi: 10.6052/j.issn.1000-4750.2021.07.0579
基金项目: 国家自然科学基金项目(12072104,51679077);中央高校基本科研业务费项目(B210203025);国家重点研发计划项目(2018YFC0406703)
详细信息
    作者简介:

    李志远(1994−),男,河南人,博士生,主要从事近场动力学研究(E-mail: lizhiyuan1007@163.com)

    闫康昊(1992−),男,河南人,博士生,主要从事计算力学方法研究(E-mail: yankanghao@foxmail.com)

    通讯作者:

    黄 丹(1979−),男,湖北人,教授,博士,博导,主要从事计算力学与工程安全研究(E-mail: danhuang@hhu.edu.cn)

  • 中图分类号: O326

METHOD FOR DYNAMIC CHARACTERISTIC ANALYSIS OF BEAMS WITH VARYING CROSS-SECTIONS BY USING PERIDYNAMIC DIFFERENTIAL OPERATOR

  • 摘要: 变截面梁式构件广泛应用于工程结构中,其动力特性更是结构设计和状态评估中的重要考虑因素之一。基于新兴的近场动力学微分算子(Peridynamic differential operator, PDDO),尝试提出了一种用于变截面梁动力特性分析的非局部方法。将变截面梁的动力学微分控制方程与边界条件通过PDDO由局部微分形式转化为对应的非局部积分形式,再结合拉格朗日乘数法与变分原理,将非局部积分形式的控制方程与边界条件转化为标准特征值问题表达形式,从而求得自振频率与振型。通过对等截面梁的自由振动分析并与解析解对比,验证了该方法良好的收敛性与准确性。进一步通过求解下边界一次、二次变化的连续变截面梁,证明了该方法对于任意变截面梁自由振动分析的适用性与可靠性。开展含孔变截面梁的自由振动分析,体现了该文的非局部方法在含缺陷构件振动分析和损伤识别问题方面的潜力,可为含缺陷变截面构件的动力分析问题提供新思路。
  • 图  1  任意变截面简支梁

    Figure  1.  Simply supported beam with arbitrarily and continuously varying cross-section

    图  2  物质点间的相互作用

    Figure  2.  Interaction of material points with arbitrary family

    图  3  均匀PD离散

    Figure  3.  Illustrations of uniform PD discretization

    图  4  等截面简支梁

    Figure  4.  The simply supported beam with constant cross-section

    图  5  各离散间距下前3阶无量纲自振频率的相对误差

    Figure  5.  Relative error of first 3 non-dimensional frequencies for different dense mesh

    图  6  下边界一次变化的简支梁

    Figure  6.  The beam with linearly varying lower surface

    图  7  下边界二次变化的简支梁

    Figure  7.  The beam with parabolic convex lower surface

    图  8  下边界一次、二次变化梁的前3阶振型(Ha/H = Hb/H = 2, y = 0.4H)

    Figure  8.  First 3 mode shapes in x and y directions at y = 0.4H for the beam with linearly varying and parabolic convex lower surface (Ha/H = Hb/H = 2, y = 0.4H)

    图  9  下边界二次变化的含孔梁

    Figure  9.  The beam with a hole and parabolic convex lower surface

    图  10  下边界二次变化含孔梁的前4阶轴向应力模态 (r/H = 0.6)

    Figure  10.  First 4 stress modes in x direction for the beam with a hole and parabolic convex lower surface (r/H = 0.6)

    表  1  等截面梁的无量纲自振频率

    Table  1.   Non-dimensional frequencies of the beam with constant cross-section

    阶次本文解解析解[10]相对误差/(%)
    1 9.7332 9.6420 0.946
    2 37.240 36.893 0.940
    3 78.588 77.860 0.935
    4 108.77 108.00 0.706
    5 129.62 128.42 0.929
    6 187.02 185.32 0.919
    7 217.26 215.76 0.699
    8 248.53 246.26 0.921
    下载: 导出CSV

    表  2  下边界一次变化梁的无量纲自振频率

    Table  2.   Non-dimensional frequencies of the beam with linearly varying lower surface

    阶次Ha/H=1.5Ha/H=2.0
    本文解解析解[10]本文解解析解[10]
    1 9.5882 9.6044 9.3260 9.3346
    2 37.898 37.937 37.010 37.012
    3 82.861 82.898 80.011 79.970
    4 142.08 142.03 135.53 135.34
    5 175.16 173.88 147.61 146.48
    6 212.93 212.68 200.70 200.29
    7 293.00 292.41 273.08 272.29
    8 348.52 346.02 291.10 289.03
    下载: 导出CSV

    表  3  下边界二次变化梁的无量纲自振频率

    Table  3.   Non-dimensional frequencies of the beam with parabolic convex lower surface

    阶次
    Hb/H = 1.6

    Hb/H = 2.0
    本文解解析解[10]本文解解析解[10]
    1 10.391 10.418 10.538 10.551
    2 38.115 38.092 37.175 37.143
    3 82.195 82.276 79.508 79.425
    4 139.93 139.56 133.39 133.02
    5 169.66 168.23 147.58 146.29
    6 207.24 207.10 195.33 194.89
    7 283.71 282.26 263.25 261.81
    8 319.66 317.26 272.84 270.87
    下载: 导出CSV

    表  4  下边界二次变化含孔梁的无量纲自振频率

    Table  4.   Non-dimensional frequencies of the beam with a hole and parabolic convex lower surface

    阶次r/H = 0.4r/H =0.6r/H =0.8
    1 10.235 10.335 10.153
    2 37.035 36.675 34.956
    3 79.879 80.177 79.080
    4 130.74 124.14 100.50
    5 141.13 129.60 108.26
    6 197.15 198.78 197.61
    7 254.23 233.76 199.19
    8 275.65 279.85 285.51
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-07-28
  • 录用日期:  2021-11-18
  • 修回日期:  2021-11-07
  • 网络出版日期:  2021-11-18
  • 刊出日期:  2022-12-01

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