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圆锥角对功能梯度壳非线性振动响应的影响

张宇航 刘文光 刘超 吕志鹏

张宇航,刘文光,刘超,吕志鹏. 圆锥角对功能梯度壳非线性振动响应的影响 [J]. 应用数学和力学,2022,43(8):857-868 doi: 10.21656/1000-0887.420273
引用本文: 张宇航,刘文光,刘超,吕志鹏. 圆锥角对功能梯度壳非线性振动响应的影响 [J]. 应用数学和力学,2022,43(8):857-868 doi: 10.21656/1000-0887.420273
ZHANG Yuhang, LIU Wenguang, LIU Chao, LÜ Zhipeng. Effects of Cone Angles on Nonlinear Vibration Responses of Functionally Graded Shells[J]. Applied Mathematics and Mechanics, 2022, 43(8): 857-868. doi: 10.21656/1000-0887.420273
Citation: ZHANG Yuhang, LIU Wenguang, LIU Chao, LÜ Zhipeng. Effects of Cone Angles on Nonlinear Vibration Responses of Functionally Graded Shells[J]. Applied Mathematics and Mechanics, 2022, 43(8): 857-868. doi: 10.21656/1000-0887.420273

圆锥角对功能梯度壳非线性振动响应的影响

doi: 10.21656/1000-0887.420273
基金项目: 国家自然科学基金(51965042)
详细信息
    作者简介:

    张宇航(1998—),男,硕士生(E-mail:1103517976@qq.com

    刘文光(1978—), 男, 副教授,博士(通讯作者. E-mail:liuwg14@nchu.edu.cn

  • 中图分类号: O327

Effects of Cone Angles on Nonlinear Vibration Responses of Functionally Graded Shells

  • 摘要:

    研究了受外载荷下圆锥角对功能梯度壳的非线性振动的影响。首先,根据Voigt模型,在圆锥壳的厚度方向上建立功能梯度材料属性模型。然后,考虑一阶剪切变形理论和von Kármán非线性,利用Hamilton原理推导了功能梯度圆锥壳的非线性运动方程。之后,应用Galerkin法对运动方程进行离散化处理,再根据Volmir假设将方程简化为单自由度的非线性微分方程。最后,采用谐波平衡法和Runge-Kutta法对方程进行求解,分析了功能梯度圆锥壳的幅频响应特性曲线,讨论了不同材料分布函数以及陶瓷体积分数指数对圆锥壳幅频响应的影响,描述了不同锥角下圆锥壳的分岔图和不同激励幅值下圆锥壳的时间历程和相图,进一步通过Poincaré图反映了圆锥壳运动状态。结果表明:功能梯度圆锥壳呈现“渐硬”弹簧非线性特性;锥角增大,功能梯度圆锥壳混沌运动的现象被抑制,不易产生运动不稳定性;激励幅值增大,功能梯度圆锥壳运动呈现周期到多周期再到混沌的过程。

  • 图  1  几何模型

    Figure  1.  The geometric model

    图  2  陶瓷体积分数指数沿厚度方向的变化

    Figure  2.  Variations of ceramic volume fraction exponents along the thickness direction

    图  3  陶瓷体积分数指数对FGMs圆锥壳幅频响应特性曲线的影响

    Figure  3.  Effects of ceramic volume fraction exponents on nonlinear amplitude frequency responses of FGMs conical shells

    图  4  不同锥度下FGMs圆锥壳随激励幅值变化的分岔图

    Figure  4.  The bifurcation diagrams of FGMs conical shells for different excitation amplitudes under different cone angles

    5  不同激励幅值下FGMs圆锥壳时间历程曲线、速度位移相图以及Poincaré截面图(α0=15°)

    5.  The time history diagrams, phase plots and Poincaré maps of FGMs conical shells under different excitation amplitudes (α0=15°)

    表  1  FGMs圆柱壳的材料参数

    Table  1.   Material parameters of FGMs cylindrical shells

    materialelastic module E/ GPaPoisson’s ratio νdensity ρ/ (kg/m3)
    stainless steel207.7880.317 78 166
    Ni205.0980.38 900
    下载: 导出CSV

    表  2  FGMs圆柱壳频率对比(R1=1, R1/h=500, L/R=20, m=1)

    Table  2.   Comparison of frequencies of FGMs cylindrical shells (R1=1, R1/h=500, L/R=20, m=1)

    nf /HZ
    ref. [1]present
    616.45516.655
    722.63522.827
    829.77129.952
    937.86238.029
    1046.90547.056
    下载: 导出CSV

    表  3  纯金属圆锥壳频率对比(R2=1, R2/h=100, Lsin(γ0)/R2=0.25, α0=30°, m=1)

    Table  3.   Comparison of frequencies of pure metal conical shells (R2=1, R2/h=100, Lsin(γ0)/R2=0.25, α0=30°, m=1)

    nΩ=ω0R2((1-ν2)ρ/E)1/2
    ref. [16]ref. [17]present
    20.794 30.790 40.840 2
    30.708 50.727 40.741 2
    40.619 90.633 90.642 1
    50.543 70.551 40.558 6
    60.48960.49300.4993
    70.462 30.463 20.468 2
    80.462 70.462 30.465 4
    90.488 20.487 00.488 3
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-09-09
  • 录用日期:  2021-11-03
  • 修回日期:  2021-10-26
  • 网络出版日期:  2022-07-01
  • 刊出日期:  2022-08-01

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