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浙江大学学报(工学版)
浙江大学学报(工学版)  2023, Vol. 57 Issue (9): 1794-1803    DOI: 10.3785/j.issn.1008-973X.2023.09.011
机械工程、能源工程     
基于梁函数-Ritz法的圆柱壳模态特性分析
徐港辉(),祝长生*()
浙江大学 电气工程学院，浙江 杭州 310027
Modal characteristics analysis of cylindrical shells based on beam functions-Ritz method
Gang-hui XU(),Chang-sheng ZHU*()
College of Electrical Engineering, Zhejiang University, Hangzhou 310027, China
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摘要：

单项梁函数作为圆柱壳轴向振型容许函数时计算精度不足，基于不同薄壳理论所得研究结果难以直接对比，为此将梁函数与Ritz法相结合，建立包含不同薄壳理论的统一方法，用于分析圆柱壳模态特性. 当梁函数项数取1时，梁函数-Ritz法退化为单项梁函数法，即单项梁函数法是梁函数-Ritz法的特例. 通过与有限元仿真及现有文献数据的对比，澄清现有文献在梁函数是否适用于模拟圆柱壳固支或自由边界条件上存在的分歧. 在不同边界条件下，总结长径比与厚径比对圆柱壳模态频率及理论计算精度的影响规律，得出不同薄壳理论的适用范围. 考虑的薄壳理论包括Donnell、Reissner、Sanders及Love理论. 结果表明，梁函数-Ritz法可以有效提升单项梁函数法的计算精度；Donnell理论的计算精度明显低于其他3种理论.
关键词： 圆柱壳;  模态;  梁函数;  容许函数;  Ritz法;  薄壳理论    
Abstract:

The calculation accuracy of a single beam function is insufficient, when the function is used as an admissible function of the axial modal shape of cylindrical shells, and the results obtained based on different thin shell theories are difficult to compare directly. The beam function was combined with the Ritz method to establish a unified method including different thin shell theories for analyzing the modal characteristics of cylindrical shells. When the number of the beam function terms was set to one, the beam functions-Ritz method degenerated to the single beam function method, i.e., the single beam function method was a special case of the beam functions-Ritz method. By comparing with the finite element simulation and the existing literature data, the disagreement in the existing literature on whether the beam function is suitable for simulating the clamped or free boundary conditions of cylindrical shells was clarified. The effects of length-to-radius ratio and thickness-to-radius ratio on the modal frequencies of cylindrical shells under different boundary conditions and the corresponding theoretical calculation accuracy were studied, and then the applicable scope of different thin shell theories was summarized. Four thin shell theories were considered, including the Donnell, Reissner, Sanders and Love theories. Results show that the beam functions-Ritz method can effectively improve the calculation accuracy of the single beam function method, and the calculation accuracy of Donnell theory was obviously lower than that of the other three theories.
Key words: cylindrical shell    mode    beam function    admissible function    Ritz method    thin shell theories
收稿日期: 2022-11-11 出版日期: 2023-10-16
CLC:  O 326  
基金资助: 国家自然科学基金资助项目(51975516)；重点基础研究项目(2020-ZD-232-00)
通讯作者: 祝长生     E-mail: gh_xu@zju.edu.cn;zhu_zhang@zju.edu.cn
作者简介: 徐港辉（1995—），男，博士生，从事电机振动分析与振动控制研究. orcid.org/0000-0002-8382-4798. E-mail： gh_xu@zju.edu.cn
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引用本文:

徐港辉,祝长生. 基于梁函数-Ritz法的圆柱壳模态特性分析[J]. 浙江大学学报(工学版), 2023, 57(9): 1794-1803.

Gang-hui XU,Chang-sheng ZHU. Modal characteristics analysis of cylindrical shells based on beam functions-Ritz method. Journal of ZheJiang University (Engineering Science), 2023, 57(9): 1794-1803.

链接本文:

https://www.zjujournals.com/eng/CN/10.3785/j.issn.1008-973X.2023.09.011    或    https://www.zjujournals.com/eng/CN/Y2023/V57/I9/1794

图 1  各向同性薄圆柱壳示意图
薄壳理论 $ {\varepsilon _x} $ $ {\varepsilon _\theta } $ $ {\gamma _{x\theta }} $
Donnell $ {\varepsilon _{x_ {\rm{D}}}} $ $ {\varepsilon _{\theta_ {\rm{D}}}} $ $ {\gamma _{x\theta_ {\rm{D}}}} $
Reissner $ {\varepsilon _{x_ {\rm{D}}}} $ ${\varepsilon _{\theta_ {\rm{D}}} }+z\dfrac{1}{ { {r^2} } }\dfrac{ {\partial v} }{ {\partial \theta } }$ ${\gamma _{x\theta_ {\rm{D}}} }+z\dfrac{1}{r}\dfrac{ {\partial v} }{ {\partial x} }$
Sanders $ {\varepsilon _{x_ {\rm{D}}}} $ ${\varepsilon _{\theta_ {\rm{D}}} }+z\dfrac{1}{ { {r^2} } }\dfrac{ {\partial v} }{ {\partial \theta } }$ ${\gamma _{x\theta_ {\rm{D}}} }+z\left(\dfrac{3}{ {2r} }\dfrac{ {\partial v} }{ {\partial x} } - \dfrac{1}{ {2{r^2} } }\dfrac{ {\partial u} }{ {\partial \theta } }\right)$
Love $ {\varepsilon _{x_ {\rm{D}}}} $ ${\varepsilon _{\theta_ {\rm{D}}} }+z\dfrac{1}{ { {r^2} } }\dfrac{ {\partial v} }{ {\partial \theta } }$ ${\gamma _{x\theta_ {\rm{D}}} }+z\dfrac{2}{r}\dfrac{ {\partial v} }{ {\partial x} }$
表 1  常用薄壳理论对比
图 2  圆柱壳模态频率的理论计算值随项数的变化关系
图 3  不同边界条件下不同项数理论计算的相对误差
n fC-C/Hz fC-F/Hz fS-S/Hz
文献[6] I=1 I=50 FEM 文献[6] I=1 I=50 FEM 文献[6] I=1 I=50 FEM
0 8300.4 8300.4 7829.2 7784.4 2295.4 5341.7 3903.4 3892.1 7784.2 7784.1 7784.1 7784.1
1 5314.9 5314.8 4916.8 4895.5 1325.2 2830.7 2309.0 2304.8 4811.1 4811.1 4811.1 4811.1
2 3452.8 3452.7 3164.2 3150.2 588.4 1317.3 1152.5 1151.4 2750.0 2750.0 2750.0 2750.0
3 2361.9 2361.9 2172.0 2163.7 303.3 700.9 637.9 637.3 1630.0 1630.0 1630.0 1629.8
4 1692.2 1692.1 1569.1 1564.4 159.8 432.2 402.5 401.0 1041.7 1041.7 1041.7 1041.1
5 1263.8 1263.8 1182.9 1179.9 80.9 317.2 301.8 298.6 723.5 723.5 723.5 722.1
6 986.3 986.3 931.9 929.3 194.2 293.7 286.2 281.2 555.9 555.9 555.9 553.3
7 814.1 814.1 777.2 774.2 283.5 330.3 326.8 320.5 488.4 488.4 488.4 484.5
8 724.9 724.9 700.1 696.5 377.6 402.5 400.9 393.9 494.5 494.5 494.5 489.5
9 705.1 705.1 688.8 684.4 481.5 496.8 496.0 488.5 551.8 551.7 551.7 546.2
表 2  3种边界条件下的圆柱壳模态频率对比(梁函数)
(m,n) fC-C/Hz
文献[5]-MF 文献[5]-OP 文献[5]-CP 本研究I=25
(1,9) 684.7 684.2 684.0 684.7
(1,8) 697.3 696.3 696.2 697.1
(1,10) 726.2 725.9 726.2 726.0
(1,7) 775.4 774.0 774.3 775.7
(1,11) 808.1 807.8 808.1 808.0
(1,12) 919.9 919.6 919.6 919.8
(1,6) 930.6 928.8 929.2 932.0
(1,13) 1054.0 1054.0 1054.0 1054.1
(2,11) 1141.0 1139.0 1139.0 1156.2
(2,10) 1167.0 1165.0 1165.0 1190.6
表 3  两端固支圆柱壳模态频率对比(不同类型容许函数)
(m,n) fS-S/Hz
文献[5]-MF 文献[5]-OP 文献[5]-CP 本研究I=25
(1,7) 484.6 484.6 484.6 484.6
(1,8) 489.6 489.6 489.6 489.6
(1,9) 546.2 546.2 546.2 546.2
(1,6) 553.3 553.3 553.3 553.4
(1,10) 636.8 636.8 636.8 636.9
(1,5) 722.1 722.1 722.1 722.2
(1,11) 750.7 750.7 750.7 750.7
(1,12) 882.2 882.2 882.2 882.3
(2,10) 968.1 968.1 968.1 968.2
(2,11) 983.4 983.4 983.4 983.5
表 4  两端简支圆柱壳模态频率对比(不同类型容许函数)
图 4  圆柱壳轴向振型(I=50, FEM)与梁弯曲振型(I=1)对比
图 5  不同长径比与厚径比时圆柱壳模态频率的分布
图 6  不同长径比与厚径比时理论计算结果的误差分布
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