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基于应变的几何非线性梁建模与分析

许秋怡 孟杨 李书

许秋怡,孟杨,李书. 基于应变的几何非线性梁建模与分析[J]. 北京航空航天大学学报,2023,49(8):2039-2049 doi: 10.13700/j.bh.1001-5965.2021.0627
引用本文: 许秋怡,孟杨,李书. 基于应变的几何非线性梁建模与分析[J]. 北京航空航天大学学报,2023,49(8):2039-2049 doi: 10.13700/j.bh.1001-5965.2021.0627
XU Q Y,MENG Y,LI S. Strain-based geometrically nonlinear beam modeling and analysis[J]. Journal of Beijing University of Aeronautics and Astronautics,2023,49(8):2039-2049 (in Chinese) doi: 10.13700/j.bh.1001-5965.2021.0627
Citation: XU Q Y,MENG Y,LI S. Strain-based geometrically nonlinear beam modeling and analysis[J]. Journal of Beijing University of Aeronautics and Astronautics,2023,49(8):2039-2049 (in Chinese) doi: 10.13700/j.bh.1001-5965.2021.0627

基于应变的几何非线性梁建模与分析

doi: 10.13700/j.bh.1001-5965.2021.0627
详细信息
    通讯作者:

    E-mail:summy@buaa.edu.cn

  • 中图分类号: V214.3

Strain-based geometrically nonlinear beam modeling and analysis

More Information
  • 摘要:

    柔性机翼在气动载荷作用下产生较大变形,几何非线性因素不容忽视。利用机翼柔性特点,通常采用梁模型进行结构建模。从几何精确梁理论出发,结合Hamilton原理推导了几何非线性梁的动力学平衡方程。不同于经典的位移基有限元,采用梁广义应变作为插值变量,得到广义质量阵、广义阻尼阵、刚度阵及载荷列向量,建立非线性应变梁模型。结合Newmark数值算法和牛顿-拉夫森(Newdon-Raphson)迭代法建立了动力学方程求解算法。针对典型算例,开展静、动力学分析,并分别与有限元软件的仿真结果进行对比。结果表明:在相当计算精度下,所建模型收敛特性更好。为进一步验证所建模型在工程实际应用中的精度和有效性,开展针对典型大展弦比机翼的地面静力试验。试验表明:仿真结果与激光位移计和光纤传感设备测得的变形值具有较高的一致性,验证了所建模型具有较高精度。

     

  • 图 1  梁空间构形示意图

    Figure 1.  Kinematic dagram of beam spaxial configuration

    图 2  单元划分示意图

    Figure 2.  Illustration of elements

    图 3  悬臂梁模型示意图

    Figure 3.  Illustration of cantilevered beam

    图 4  悬臂梁受端部弯矩作用变形图

    Figure 4.  Deformation of cantilever beam subjected to tip moment

    图 5  梁端部位移随载荷大小变化

    Figure 5.  Tip displacements with respect to force magnitudes

    图 6  Nastran非线性静力学迭代

    Figure 6.  Nonlinear static iterations in Nastran

    图 7  本文模型非线性静力学求解迭代收敛曲线

    Figure 7.  Iterative curve for solving nonlinear statics of proposed model

    图 8  悬臂梁在定向载荷和随动载荷作用下的变形

    Figure 8.  Deformation of cantilevered beam subjected to dead force and follower force

    图 9  端部定向正弦载荷作用下位移响应

    Figure 9.  Responses of tip displacement subject to sinusoidal dead force

    图 10  端部随动正弦载荷作用下位移响应

    Figure 10.  Responses of tip displacement subject to sinusoidal follower force

    图 11  用于地面试验的机翼模型图

    Figure 11.  Wing model used in ground test

    图 12  地面静力试验示意图

    Figure 12.  Diagram of ground static tests

    图 13  光纤光栅传感器黏贴示意图

    Figure 13.  Installation of fiber grating sensors

    图 14  机翼主梁表面弯曲应变

    Figure 14.  Bending strains on beam surface at different positions

    图 15  自重作用下机翼垂向变形

    Figure 15.  Vertical displacements of wing under gravity

    表  1  悬臂梁模型物理属性

    Table  1.   Properties of cantilevered beam

    参数数值
    长度L/m1.00
    弹性模量E/GPa210
    剪切模量G/GPa84.2
    横截面宽度b/mm35
    横截面高度h/mm1.5
    密度$\rho $/(kg·m−3)$7.75 \times {10^3}$
    扭转惯量${I_x}$/(kg·m)$5.37 \times {10^{ - 9}}$
    面外弯曲惯量${I_y}$/(kg·m)$9.84 \times {10^{ - 12}}$
    面内弯曲惯量${I_{\textit{z}}}$/(kg·m)$5.36 \times {10^{ - 9}}$
    下载: 导出CSV

    表  2  不同端部载荷作用下端部垂向位移

    Table  2.   Tip vertical displacements with respect to different tip forces

    端部载荷/N端部垂向位移/m相对误差/%
    Nastran仿真本文模型
    0.60.095840.09578−0.063
    1.20.186550.18643−0.064
    1.80.268520.26836−0.060
    2.40.340200.33999−0.062
    3.00.401650.40140−0.062
    下载: 导出CSV

    表  3  不同端部载荷作用下端部轴向位移

    Table  3.   Tip axial displacements with respect to different tip forces

    端部载荷/N端部轴向位移/m相对误差/%
    Nastran仿真本文模型
    0.6−0.005525−0.005523−0.040
    1.2−0.021120−0.021110−0.047
    1.8−0.044360−0.044340−0.045
    2.4−0.072380−0.072350−0.041
    3.0−0.102720−0.102670−0.049
    下载: 导出CSV

    表  4  本文模型各迭代步端部垂向位移

    Table  4.   Tip vertical displacements of each iteration step in proposed model

    迭代次数端部垂向位移/m
    10.447575
    20.386251
    30.405881
    40.400023
    50.401817
    60.401272
    70.401438
    80.401388
    90.401403
    100.401398
    下载: 导出CSV

    表  5  机翼模型主要参数

    Table  5.   Main parameters of wing model

    机翼展长/m 机翼弦长/m 机翼总质量/kg
    1.00.010.672
    下载: 导出CSV

    表  6  不同展向位置机翼垂向变形

    Table  6.   Vertical displacements at different positions

    展向位置/m试验值/m垂向变形${{\textit{z}}_2}$/m${{\textit{z}}_2} - {{\textit{z}}_1} $的
    相对误差/%
    激光
    位移计
    光纤
    传感
    平均值${{\textit{z}}_1}$
    0.2−0.01851−0.02457−0.02154−0.02364−0.21
    0.4−0.07833−0.08559−0.08196−0.08243−0.05
    0.6−0.15509−0.16753−0.16131−0.16149−0.02
    0.8−0.24355−0.25978−0.25167−0.250580.11
    1.0−0.34−0.35587−0.34794−0.343230.47
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-10-22
  • 录用日期:  2021-11-26
  • 网络出版日期:  2021-12-30
  • 整期出版日期:  2023-08-31

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