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变截面变曲率梁振型的有限元超收敛拼片恢复解和网格自适应分析

王永亮

王永亮. 变截面变曲率梁振型的有限元超收敛拼片恢复解和网格自适应分析[J]. 工程力学, 2020, 37(12): 1-8. doi: 10.6052/j.issn.1000-4750.2020.02.0065
引用本文: 王永亮. 变截面变曲率梁振型的有限元超收敛拼片恢复解和网格自适应分析[J]. 工程力学, 2020, 37(12): 1-8. doi: 10.6052/j.issn.1000-4750.2020.02.0065
WANG Yong-liang. SUPERCONVERGENT PATCH RECOVERY SOLUTIONS AND ADAPTIVE MESH REFINEMENT ANALYSIS OF FINITE ELEMENT METHOD FOR THE VIBRATION MODES OF NON-UNIFORM AND VARIABLE CURVATURE BEAMS[J]. Engineering Mechanics, 2020, 37(12): 1-8. doi: 10.6052/j.issn.1000-4750.2020.02.0065
Citation: WANG Yong-liang. SUPERCONVERGENT PATCH RECOVERY SOLUTIONS AND ADAPTIVE MESH REFINEMENT ANALYSIS OF FINITE ELEMENT METHOD FOR THE VIBRATION MODES OF NON-UNIFORM AND VARIABLE CURVATURE BEAMS[J]. Engineering Mechanics, 2020, 37(12): 1-8. doi: 10.6052/j.issn.1000-4750.2020.02.0065

变截面变曲率梁振型的有限元超收敛拼片恢复解和网格自适应分析

doi: 10.6052/j.issn.1000-4750.2020.02.0065
基金项目: 国家自然科学基金项目(41877275,51608301);中央高校基本科研业务费专项资金项目(2019QL02);天津市重点实验室开放课题基金项目(2017SCEEKL003);中国矿业大学(北京)越崎青年学者计划项目(2019QN14)
详细信息
    通讯作者:

    王永亮 (1985−),男,河北唐山人,副教授,博士,硕导,主要从事矿山岩体计算力学的研究 (E-mail: wangyl@cumtb.edu.cn)

  • 中图分类号: O242.21

SUPERCONVERGENT PATCH RECOVERY SOLUTIONS AND ADAPTIVE MESH REFINEMENT ANALYSIS OF FINITE ELEMENT METHOD FOR THE VIBRATION MODES OF NON-UNIFORM AND VARIABLE CURVATURE BEAMS

  • 摘要: 该文提出变截面变曲率梁振型的有限元后处理超收敛拼片恢复方法,建立各阶振型的超收敛解,并基于振型超收敛解进行变截面曲梁面内和面外自由振动的自适应分析。在位移型有限元后处理阶段,引入超收敛拼片恢复方法和高阶形函数插值技术,得到振型(位移)的超收敛解。利用振型超收敛解估计当前网格下振型有限元解的能量模形式下的误差,并指导网格进行自适应细分加密分析,获得优化的网格和满足预设误差限的高精度解答。数值算例表明该算法适于求解不同曲线形态、多类边界条件、变截面、变曲率形式的曲梁面内和面外自由振动连续阶频率和振型,解答精确、分析过程高效可靠。
  • 图  1  平面变截面变曲率梁坐标系和符号

    Figure  1.  Coordinate systems and symbols of planar non-uniform and variable curvature curved beam

    图  2  抛物线曲梁几何模型

    Figure  2.  Geometric model of parabolic curved beam

    图  3  本文方法解答与密集网格高精度解的误差

    Figure  3.  Errors of results of proposed method and high-precision solutions under dense mesh

    图  4  第1阶振型$(u_1^h,\;v_1^h,\;\psi _{{\textit{z}}1}^h)$

    Figure  4.  First order vibration mode $(u_1^h,\;v_1^h,\;\psi _{{\textit{z}}1}^h)$

    图  5  第2阶振型$(u_2^h,\;v_2^h,\;\psi _{{\textit{z}}2}^h)$

    Figure  5.  Second order vibration mode $(u_2^h,\;v_2^h,\;\psi _{{\textit{z}}2}^h)$

    图  6  第3阶振型$(u_3^h,\;v_3^h,\;\psi _{{\textit{z}}3}^h)$

    Figure  6.  Third order vibration mode $(u_3^h,\;v_3^h,\;\psi _{{\textit{z}}3}^h)$

    图  7  变截面变曲率梁几何模型

    Figure  7.  Geometric model of non-uniform and variable curvature curved beam

    图  8  椭圆弧曲梁几何模型

    Figure  8.  Geometric model of of elliptic curved beam

    图  9  第7阶振型$(w_7^h,\;\psi _{y7}^h,\;\varphi _{x7}^h)$

    Figure  9.  Seventh order vibration mode $(w_7^h,\;\psi _{y7}^h,\;\varphi _{x7}^h)$

    图  10  圆弧曲梁几何模型

    Figure  10.  Geometric model of circular curved beam

    表  1  抛物线曲梁面内自由振动无量纲频率值

    Table  1.   Non-dimensional frequencies of in-plane vibration of parabolic curved beam

    高跨比${h / L}$阶次k频率值$\bar \omega $
    两端固支一端固支一端简支两端简支
    $\bar \omega _k^h$$\bar \omega _k^h$[25]$\bar \omega _k^h$$\bar \omega _k^h$[25]$\bar \omega _k^h$$\bar \omega _k^h$[25]
    0.2146.024546.025236.573336.603728.758728.7644
    287.126587.272978.059278.158968.243368.3075
    3126.287126.280123.343123.3630123.282123.2460
    0.8110.935310.93598.341758.343406.379256.38010
    225.775625.795821.708321.719917.963717.9707
    345.696145.744140.325240.351435.355235.3749
    下载: 导出CSV

    表  2  变截面变曲率梁面内自由振动频率值

    Table  2.   Frequencies of in-plane vibration of non-uniform and variable curvature curved beam

    阶次k频率值$\omega /{\rm{Hz}}$
    $\omega _k^h$$\omega _k^h$[26]
    172.1035 72.05
    2150.842150.78
    3267.482267.34
    4407.912407.77
    下载: 导出CSV

    表  3  椭圆弧曲梁面内自由振动无量纲频率值

    Table  3.   Non-dimensional frequencies of in-plane vibration of elliptic curved beam

    阶次k频率值$\bar \omega $阶次k频率值$\bar \omega $
    $\bar \omega _k^h$$\bar \omega _k^h$[25]$\bar \omega _k^h$$\bar \omega _k^h$[25]
    167.435267.55344211.537212.414
    284.093184.21855325.811327.516
    3160.403160.6476358.439358.960
    下载: 导出CSV

    表  4  抛物线曲梁面外自由振动无量纲频率值

    Table  4.   Non-dimensional frequencies of out-of-plane vibration of parabolic curved beam

    阶次k频率值$\bar \omega $
    两端固支一端固支一端简支两端简支
    $\bar \omega _k^h$$\bar \omega _k^h$[27]$\bar \omega _k^h$[28]$\bar \omega _k^h$$\bar \omega _k^h$[27]$\bar \omega _k^h$[28]$\bar \omega _k^h$$\bar \omega _k^h$[27]$\bar \omega _k^h$[28]
    117.044217.0317.1211.127611.1211.156.082746.0796.090
    248.396548.3748.7738.965738.9439.1030.403730.3830.40
    395.021394.9796.0682.193582.1482.6170.033869.9970.03
    4109.942109.9109.9109.828109.8109.8109.839109.7109.8
    5156.526156.4158.7140.486140.4141.4125.037125.0125.0
    6203.793203.7203.8203.788203.7203.8193.982193.8194.0
    7230.935230.8234.7212.172212.1213.8203.764203.7203.8
    下载: 导出CSV

    表  5  圆弧曲梁面外自由振动无量纲频率值

    Table  5.   Non-dimensional frequencies of out-of-plane vibration of circular curved beam

    阶次k频率值$\bar \omega $
    $\bar \omega _k^h$$\bar \omega _k^h$[29]$\bar \omega _k^h$[30]
    116.8851616.8816.885
    239.7004339.7039.700
    340.9346340.9040.934
    470.5807570.5170.581
    下载: 导出CSV
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出版历程
  • 收稿日期:  2020-02-10
  • 修回日期:  2020-04-20
  • 网络出版日期:  2020-12-10
  • 刊出日期:  2020-12-10

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