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二维功能梯度壁板热颤振本征问题的精确解

代林桐 邢誉峰

代林桐, 邢誉峰. 二维功能梯度壁板热颤振本征问题的精确解[J]. 北京航空航天大学学报, 2021, 47(10): 2097-2104. doi: 10.13700/j.bh.1001-5965.2020.0351
引用本文: 代林桐, 邢誉峰. 二维功能梯度壁板热颤振本征问题的精确解[J]. 北京航空航天大学学报, 2021, 47(10): 2097-2104. doi: 10.13700/j.bh.1001-5965.2020.0351
DAI Lintong, XING Yufeng. Exact solutions of thermal flutter of two-dimensional functionally graded panel[J]. Journal of Beijing University of Aeronautics and Astronautics, 2021, 47(10): 2097-2104. doi: 10.13700/j.bh.1001-5965.2020.0351(in Chinese)
Citation: DAI Lintong, XING Yufeng. Exact solutions of thermal flutter of two-dimensional functionally graded panel[J]. Journal of Beijing University of Aeronautics and Astronautics, 2021, 47(10): 2097-2104. doi: 10.13700/j.bh.1001-5965.2020.0351(in Chinese)

二维功能梯度壁板热颤振本征问题的精确解

doi: 10.13700/j.bh.1001-5965.2020.0351
基金项目: 

国家自然科学基金 11672019

详细信息
    通讯作者:

    邢誉峰, E-mail: xingyf@buaa.edu.cn

  • 中图分类号: V221+.3;TB553

Exact solutions of thermal flutter of two-dimensional functionally graded panel

Funds: 

National Natural Science Foundation of China 11672019

More Information
  • 摘要:

    为了得到二维功能梯度壁板热颤振的精确解并揭示颤振机理,根据经典薄板理论及一阶活塞理论,建立了超声速气流下二维功能梯度壁板的本征控制微分方程并求得了精确解,根据得到的本征根对颤振机理进行了分析。针对功能梯度材料(FGM)的不同体积分数,分别研究了壁板在恒温场及非线性温度场下的颤振边界随马赫数的变化规律,并比较了2种温度场下的结果。通过分析简支、固支及其组合边界情况下的壁板颤振特性,从数学角度发现颤振现象的发生是由于挠度的一阶导数导致刚度非对称,且功能梯度材料能够有效提高热环境下壁板的颤振边界,同时利用ABAQUS软件对功能梯度壁板的振动特性进行了模拟,进一步验证了所提方法的有效性。

     

  • 图 1  两端简支壁板几何模型

    Figure 1.  Geometric model of simply supported panel at both ends

    图 2  临界动压值随梯度指数变化曲线

    Figure 2.  Critical dynamic pressure versus gradient index

    图 3  临界颤振频率随温度的变化曲线

    Figure 3.  Critical flutter frequency versus temperature

    图 4  两种热环境下频率随马赫数的变化曲线

    Figure 4.  Frequency versus Mach number in two thermal environments

    表  1  简支和固支边界条件

    Table  1.   Boundary conditions for simple support and clamp

    BCs(边界条件) x=0或x=a
    简支(S) w=0,
    固支(C) w=0,
    下载: 导出CSV

    表  2  二维壁板颤振频率方程和本征函数

    Table  2.   Eigensolutions of two-dimensional panel flutters

    BCs 频率方程 本征函数的系数
    简支-简支(SS)
    ϕ(0)=0
    ϕ(1)=0
    ϕ″(0)=0
    ϕ″(1)=0
    固支-固支(CC)
    ϕ(0)=0
    ϕ(1)=0
    ϕ′(0)=0
    ϕ′(1)=0
    简支-固支(SC)
    ϕ(0)=0
    ϕ″(0)=0
    ϕ(1)=0
    ϕ′(1)=0
    4ϑα1β1sinh 2ϑ-β1(4ϑ2-α12-β12)sin α1cosh β1-α2(4ϑ2+α12+β12)cos α1sinh β1=0
    下载: 导出CSV

    表  3  功能梯度材料弹性常数

    Table  3.   Elastic constants of FGM

    材料名称 组成成分 E/GPa υ ρ/(kg·m-3)
    陶瓷金属 Si3N4(氮化硅陶瓷) 322 0.24 2 370
    SUS304(不锈钢) 207 0.32 8 166
    下载: 导出CSV

    表  4  分层为5层时密度和弹性模量

    Table  4.   Density and modulus of elasticity for 5 layers

    坐标方向 厚度方向坐标/(10-4m) 密度/(kg·m-3) 弹性模量/(1011Pa)
    z -8.000 8 165.942 2.070
    z -4.000 8 151.916 2.073
    z 0 7 984.875 2.106
    z 4.000 7 191.866 2.263
    z 8.000 4 743.520 2.749
    下载: 导出CSV

    表  5  分层为10层时密度和弹性模量

    Table  5.   Density and modulus of elasticity for 10 layers

    坐标方向 厚度方向坐标/(10-4m) 密度/(kg·m-3) 弹性模量/(1011Pa)
    z -9.000 8 165.998 2.070
    z -7.000 8 165.560 2.070
    z -5.000 8 160.340 2.071
    z -3.000 8 135.558 2.076
    z -1.000 8 059.047 2.091
    z 1.000 7 874.296 2.127
    z 3.000 7 493.496 2.203
    z 5.000 6 790.582 2.343
    z 7.000 5 594.284 2.580
    z 9.000 3 681.166 2.959
    下载: 导出CSV

    表  6  不同边界条件下FGM板频率

    Table  6.   Frequency of FGM plate under different boundary conditions

    边界条件 模态阶数 频率/Hz
    精确解 ABAQUS(5层) ABAQUS(10层)
    SS 1 131.679 3 129.345 7 131.010 7
    2 523.997 7 517.596 2 521.925 4
    CC 1 296.864 6 293.355 6 295.806 1
    2 818.792 2 808.520 3 815.306 1
    SC 1 204.444 7 202.048 4 203.851 7
    2 663.293 8 654.707 9 660.551 3
    下载: 导出CSV

    表  7  本文解与Galerkin方法结果的对比

    Table  7.   Comparison of result between present method and Galerkin method

    方法 固有频率(Ma=2) 颤振参数
    ω1/Hz ω2 /Hz ωf /Hz Maf
    Galerkin 650.725 8 2 130.318 9 1 773.443 7 7.177 4
    本文 651.975 4 2 130.663 1 1 759.928 3 6.989 6
    下载: 导出CSV

    表  8  不同边界下FGM板的临界颤振频率和临界动压的比较

    Table  8.   Comparison of flutter frequency and critical dynamic pressure of FGM plate under different boundary conditions

    T/K n SS CC SC
    λ* ω* λ* ω* λ* ω*
    Tt=300
    Tb=300
    1 434.22 26.0 788.45 41.7 594.19 33.3
    5 388.51 21.1 719.89 34.1 548.49 27.3
    50 354.23 19.0 651.33 30.6 491.35 24.5
    Tt=500
    Tb=300
    1 399.94 24.8 754.17 40.7 571.34 32.4
    5 365.66 20.2 685.61 33.2 514.21 26.3
    50 319.95 17.9 617.05 29.7 457.07 23.4
    Tt=500
    Tb=500
    1 365.66 23.4 708.46 39.3 525.63 30.9
    5 331.38 18.9 639.90 31.9 479.93 25.1
    50 285.67 16.6 571.34 28.5 422.79 22.3
    下载: 导出CSV
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出版历程
  • 收稿日期:  2020-07-21
  • 录用日期:  2020-08-21
  • 网络出版日期:  2021-10-20

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