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Article(id=1228653355141099830, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228653350485422347, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2024.10.011, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1713283200000, receivedDateStr=2024-04-17, revisedDate=1720972800000, revisedDateStr=2024-07-15, acceptedDate=null, acceptedDateStr=null, onlineDate=1770863387141, onlineDateStr=2026-02-12, pubDate=1730044800000, pubDateStr=2024-10-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770863387141, onlineIssueDateStr=2026-02-12, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770863387141, creator=13701087609, updateTime=1770863387141, updator=13701087609, issue=Issue{id=1228653350485422347, tenantId=1146029695717560320, journalId=1225147924628267009, year='2024', volume='37', issue='10', pageStart='1625', pageEnd='1802', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770863386031, creator=13701087609, updateTime=1770863862999, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1228655351092936954, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228653350485422347, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1228655351092936955, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228653350485422347, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=1731, endPage=1738, ext={EN=ArticleExt(id=1228653355405340995, articleId=1228653355141099830, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Vibration characteristics of spinning cylindrical shells under various boundary conditions, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

Spinning cylindrical shells are critical components in practical engineering structures. The boundary conditions at the shell ends are diverse and significantly influence the vibration characteristics of the shell. To study these characteristics under various boundary conditions，a dynamic model of the spinning cylindrical shell is established using Lagrange equations and Novozhilov’s shell theory. The mathematical description of the boundary conditions for the cylindrical shell is combined with the discretized displacement functions，which are constructed based on a linear combination of Chebyshev polynomials. These functions satisfy the boundary conditions and are independent of the cylindrical shell's parameters. The vibration characteristics of stationary cylindrical shells are determined by solving the eigenvalue problems，revealing the influence of rotary inertia on the vibration characteristics. The applicability of different shell theories with respect to various geometrical parameters of the shell is discussed. Additionally，circumferential wave-dependent mode functions are identified and used to compute the natural frequencies of shell modes with the zero circumferential waves，as well as the travelling waves of the spinning cylindrical shell under different boundary conditions. The impact of structural parameters on the natural frequencies of the travelling waves is also analyzed.

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自旋圆柱壳作为实际工程结构中的重要部件，其壳体端部边界条件复杂多样，对部件的振动特性有着重要影响。为研究多种边界条件下自旋圆柱壳的振动特性，基于Lagrange方程和Novozhilov壳体理论建立自旋圆柱壳的结构动力学模型；根据圆柱壳边界条件的数学描述，采用Chebyshev多项式构建满足边界条件且与壳体结构参数无关的位移场离散函数；通过求解与振动特性相关的特征方程得到静止圆柱壳的固有频率和模态，分析壳体转动惯性对固有频率的影响，讨论壳体理论在不同壳体几何尺寸下的适用性；此外，发现了环向波数相关的模态函数，并将其用于圆柱壳零阶环向模态固有频率的求解及多种边界条件下自旋圆柱壳行波固有频率的求解，讨论了自旋圆柱壳行波固有频率随结构参数变化的情况。

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李映辉（1964―），男，博士，教授。E-mail：。
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董有恒（1992—），男，博士，副研究员。E-mail：。

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董有恒（1992—），男，博士，副研究员。E-mail：。

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董有恒（1992—），男，博士，副研究员。E-mail：。

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Mechanical Systems and Signal Processing， 2022， 169： 108730., articleTitle=A comprehensive study on the coupled multi-mode vibrations of cylindrical shells, refAbstract=null), Reference(id=1228653385331700187, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, doi=null, pmid=null, pmcid=null, year=2003, volume=41, issue=6, pageStart=1119, pageEnd=1130, url=null, language=null, rfNumber=［19］, rfOrder=23, authorNames=Amabili M, journalName=AIAA Journal, refType=null, unstructuredReference=Amabili M. Nonlinear vibrations of circular cylindrical shells with different boundary conditions［J］. AIAA Journal， 2003， 41（6）： 1119-1130., articleTitle=Nonlinear vibrations of circular cylindrical shells with different boundary conditions, refAbstract=null)], funds=[Fund(id=1228653383200993659, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, awardId=12102175, language=CN, fundingSource=国家自然科学基金资助项目(12102175), fundOrder=null, country=null), Fund(id=1228653383293268351, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, awardId=B240201128, language=CN, fundingSource=中央高校基本科研业务费专项资金资助项目(B240201128), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1228653379438702769, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, xref=1, ext=[AuthorCompanyExt(id=1228653379447091379, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, companyId=1228653379438702769, language=EN, country=null, province=null, city=null, 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tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, language=EN, label=Fig.2, caption=Analysis flow chart of the travelling wave characteristics of the spinning cylindrical shell under various boundary conditions, figureFileSmall=FH52+ZjI3FdYuY6kKt/nEA==, figureFileBig=eBwHxIvXjU3t5MCvDSN4GA==, tableContent=null), ArticleFig(id=1228653381770735925, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, language=CN, label=图2, caption=多种边界条件下自旋圆柱壳的行波振动特性分析流程图, figureFileSmall=FH52+ZjI3FdYuY6kKt/nEA==, figureFileBig=eBwHxIvXjU3t5MCvDSN4GA==, tableContent=null), ArticleFig(id=1228653381896565049, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, language=EN, label=Fig.3, caption=The relative errors between the natural frequencies calculated results based on Novozhilov shell theory and Donnell shell theory and the finite element results coming from Tab.2, figureFileSmall=dpGkIvlnW42LOwrAUT35BQ==, figureFileBig=OYRNAcpCZITwhvbKOm6YZg==, tableContent=null), ArticleFig(id=1228653381967868219, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, language=CN, label=图3, caption=表2中基于Novozhilov壳体理论和Donnell壳体理论的固有频率计算结果与有限元结果的相对误差, figureFileSmall=dpGkIvlnW42LOwrAUT35BQ==, figureFileBig=OYRNAcpCZITwhvbKOm6YZg==, tableContent=null), ArticleFig(id=1228653382039171391, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, language=EN, label=Fig.4, caption=Circumferential wave-number dependent mode configurations of the cylindrical shell under the C-C boundary condition, figureFileSmall=k8KelJkD+nvmX85Tjh4ZOg==, figureFileBig=l4mZ9mKwdaLP/l/+Ay5EcA==, tableContent=null), ArticleFig(id=1228653382106280260, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, language=CN, label=图4, caption=C-C边界条件下圆柱壳的环向波数相关模态构型, 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（（m，n）=（1，5））, figureFileSmall=76z73TF52db2qU3ywkKL7g==, figureFileBig=WcPdoGVeg++MgomaD3hGCQ==, tableContent=null), ArticleFig(id=1228653382429241684, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, language=CN, label=图6, caption=不同边界条件下自旋圆柱壳行波频率随自旋速度的变化（（m，n）=（1，5））, figureFileSmall=76z73TF52db2qU3ywkKL7g==, figureFileBig=WcPdoGVeg++MgomaD3hGCQ==, tableContent=null), ArticleFig(id=1228653382513127769, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, language=EN, label=Tab.1, caption=

Comparison of natural frequencies of the cylindrical shell under various boundary conditions （Unit： Hz） （m=1，E=70 GPa，=0.3，=2720 kg/m3，r=0.1 m，L=0.2 m， h=r/50，=0）

, figureFileSmall=null, figureFileBig=null, tableContent=
边界条件n = 0n = 1n = 2n = 3n = 4n = 5
有限元结果（8节点单元）Sm-Sm7857.264848.812778.581706.981325.651445.09
Sd-Sd7863.244861.682807.831759.281394.341508.38
Si-Si7865.344937.093184.352230.491772.671725.57
C-C7865.764955.063221.292278.941822.981768.49
Amabili-Reddy三阶剪切变形壳体理论结果[2]Sm-Sm7857.254848.772778.711707.581326.861446.60
Sd-Sd7862.134861.602808.451759.961394.591508.22
Si-Si7872.034937.173184.832231.431774.011727.04
C-C7872.784954.213219.952277.401821.561767.41
Novozhilov壳体理论结果Sm-Sm7856.764848.682778.781707.871327.671448.52
Sd-Sd7862.994861.442808.041760.361396.561511.86
Si-Si7865.504936.933184.432231.001773.951728.15
C-C7866.534954.973221.822280.051824.741771.31
), ArticleFig(id=1228653382601208159, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, language=CN, label=表1, caption=

多种边界条件下圆柱壳的固有频率对比（单位：Hz） （m=1，E=70 GPa，=0.3，=2720 kg/m3，r=0.1 m，L=0.2 m，h=r/50，=0）

, figureFileSmall=null, figureFileBig=null, tableContent=
边界条件n = 0n = 1n = 2n = 3n = 4n = 5
有限元结果（8节点单元）Sm-Sm7857.264848.812778.581706.981325.651445.09
Sd-Sd7863.244861.682807.831759.281394.341508.38
Si-Si7865.344937.093184.352230.491772.671725.57
C-C7865.764955.063221.292278.941822.981768.49
Amabili-Reddy三阶剪切变形壳体理论结果[2]Sm-Sm7857.254848.772778.711707.581326.861446.60
Sd-Sd7862.134861.602808.451759.961394.591508.22
Si-Si7872.034937.173184.832231.431774.011727.04
C-C7872.784954.213219.952277.401821.561767.41
Novozhilov壳体理论结果Sm-Sm7856.764848.682778.781707.871327.671448.52
Sd-Sd7862.994861.442808.041760.361396.561511.86
Si-Si7865.504936.933184.432231.001773.951728.15
C-C7866.534954.973221.822280.051824.741771.31
), ArticleFig(id=1228653382672511331, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, language=EN, label=Tab.2, caption=

Comparison of natural frequencies of the Sm-Sm boundary cylindrical shell under different radius-to-thickness ratios： influence of the rotation inertia of the shell （Unit： Hz） （m=1，E=70 GPa，=0.3，=2720 kg/m3，r=0.1 m，L=0.2 m，=0）

, figureFileSmall=null, figureFileBig=null, tableContent=
r/hn有限元结果（8节点单元）Amabili-Reddy三阶剪切变形壳体理论结果考虑壳体转动惯性忽略壳体转动惯性
Novozhilov壳体理论结果Donnell壳体理论结果Novozhilov壳体理论结果Donnell壳体理论结果
5022778.582778.712778.782783.072778.912783.21
31706.981707.581707.871722.521708.071722.73
41325.651326.861327.671359.461327.971359.77
51445.091446.601448.521492.551449.071493.11
61866.461867.961871.821919.991872.871921.06
72457.722459.122466.122515.312468.032517.26
20041062.971063.041063.061065.571063.081065.59
5776.79776.90776.95782.14776.96782.16
6680.48680.62680.70689.04680.72689.07
7719.56719.72719.85730.42719.88730.46
8846.48846.63846.83858.46846.89858.52
91027.981028.121028.431040.471028.511040.56
), ArticleFig(id=1228653382777368937, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, language=CN, label=表2, caption=

不同半径-厚度比情况下Sm-Sm边界圆柱壳的固有频率对比：壳体转动惯性的影响（单位：Hz）（m=1，E=70 GPa，=0.3，=2720 kg/m3，r=0.1 m，L=0.2 m，=0）

, figureFileSmall=null, figureFileBig=null, tableContent=
r/hn有限元结果（8节点单元）Amabili-Reddy三阶剪切变形壳体理论结果考虑壳体转动惯性忽略壳体转动惯性
Novozhilov壳体理论结果Donnell壳体理论结果Novozhilov壳体理论结果Donnell壳体理论结果
5022778.582778.712778.782783.072778.912783.21
31706.981707.581707.871722.521708.071722.73
41325.651326.861327.671359.461327.971359.77
51445.091446.601448.521492.551449.071493.11
61866.461867.961871.821919.991872.871921.06
72457.722459.122466.122515.312468.032517.26
20041062.971063.041063.061065.571063.081065.59
5776.79776.90776.95782.14776.96782.16
6680.48680.62680.70689.04680.72689.07
7719.56719.72719.85730.42719.88730.46
8846.48846.63846.83858.46846.89858.52
91027.981028.121028.431040.471028.511040.56
), ArticleFig(id=1228653382857060715, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, language=EN, label=Tab.3, caption=

Natural frequencies solved by using the circumferential wave-number dependent mode functions （m=1，E=68.4 GPa，=0.3，=2740 kg/m3，r=0.1224 m，h=2.196 mm，L=0.264 m，=0）

, figureFileSmall=null, figureFileBig=null, tableContent=
计算（1，5）模态下固有频率时，式（15）中使用的离散函数固有频率/Hz相对误差/%
（1，5）模态下的离散函数1264.010.00
（1，4）模态下的离散函数1266.520.20
（1，3）模态下的离散函数1277.131.04
（1，2）模态下的离散函数1296.092.54
), ArticleFig(id=1228653382924169583, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, language=CN, label=表3, caption=

使用环向波数相关的模态函数求解的固有频率（m=1，E=68.4 GPa，=0.3，=2740 kg/m3，r=0.1224 m，h=2.196 mm，L=0.264 m，=0）

, figureFileSmall=null, figureFileBig=null, tableContent=
计算（1，5）模态下固有频率时，式（15）中使用的离散函数固有频率/Hz相对误差/%
（1，5）模态下的离散函数1264.010.00
（1，4）模态下的离散函数1266.520.20
（1，3）模态下的离散函数1277.131.04
（1，2）模态下的离散函数1296.092.54
), ArticleFig(id=1228653383008055667, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, language=EN, label=Tab.4, caption=

Comparison of travelling wave natural frequencies of the spinning cylindrical shell under various boundary conditions（Unit： Hz）

, figureFileSmall=null, figureFileBig=null, tableContent=
n行波Sm-Sm边界Sd-Sd边界
=1000 r/min=2000 r/min=1000 r/min=2000 r/min
文献[2]本文解文献[2]本文解文献[2]本文解文献[2]本文解
4前行波931.07931.52928.64929.07973.37974.37970.70971.69
后行波946.94947.39960.37960.81989.22990.231002.411003.42
5前行波1023.551024.631025.631026.691061.631063.631063.401065.38
后行波1036.531037.611051.581052.651074.591076.601089.341091.33
6前行波1332.441334.631336.961339.131361.301364.411365.611368.70
后行波1343.361345.561358.801360.991372.211375.341387.451390.55
7前行波1762.081766.081767.991771.951783.701788.521789.481794.26
后行波1771.491775.501786.801790.791793.101797.941808.291813.10
8前行波2277.842284.582284.662291.362294.642302.112301.382308.81
后行波2286.102292.852301.172307.902302.892310.382317.892325.35
), ArticleFig(id=1228653383096136053, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228653355141099830, language=CN, label=表4, caption=

不同边界条件下自旋圆柱壳的行波固有频率对比（单位：Hz）

, figureFileSmall=null, figureFileBig=null, tableContent=
n行波Sm-Sm边界Sd-Sd边界
=1000 r/min=2000 r/min=1000 r/min=2000 r/min
文献[2]本文解文献[2]本文解文献[2]本文解文献[2]本文解
4前行波931.07931.52928.64929.07973.37974.37970.70971.69
后行波946.94947.39960.37960.81989.22990.231002.411003.42
5前行波1023.551024.631025.631026.691061.631063.631063.401065.38
后行波1036.531037.611051.581052.651074.591076.601089.341091.33
6前行波1332.441334.631336.961339.131361.301364.411365.611368.70
后行波1343.361345.561358.801360.991372.211375.341387.451390.55
7前行波1762.081766.081767.991771.951783.701788.521789.481794.26
后行波1771.491775.501786.801790.791793.101797.941808.291813.10
8前行波2277.842284.582284.662291.362294.642302.112301.382308.81
后行波2286.102292.852301.172307.902302.892310.382317.892325.35
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多种边界条件下自旋圆柱壳振动特性研究
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董有恒 1 , 李映辉 2 , 李翔宇 2 , 茅晓晨 1
振动工程学报 | 2024,37(10): 1731-1738
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振动工程学报 | 2024, 37(10): 1731-1738
多种边界条件下自旋圆柱壳振动特性研究
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董有恒1 , 李映辉2 , 李翔宇2, 茅晓晨1
作者信息
  • 1河海大学力学与工程科学学院，江苏　南京　211100
  • 2西南交通大学力学与航空航天学院，四川　成都　611756

    • 董有恒（1992—），男，博士，副研究员。E-mail：。

通讯作者:

李映辉（1964―），男，博士，教授。E-mail：。
Vibration characteristics of spinning cylindrical shells under various boundary conditions
You-heng DONG1 , Ying-hui LI2 , Xiang-yu LI2, Xiao-chen MAO1
Affiliations
  • 1College of Mechanics and Engineering Science, Hohai University, Nanjing 211100, China
  • 2School of Mechanics and Aerospace Engineering, Southwest Jiaotong University, Chengdu 611756, China
出版时间: 2024-10-28 doi: 10.16385/j.cnki.issn.1004-4523.2024.10.011
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摘要
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自旋圆柱壳作为实际工程结构中的重要部件，其壳体端部边界条件复杂多样，对部件的振动特性有着重要影响。为研究多种边界条件下自旋圆柱壳的振动特性，基于Lagrange方程和Novozhilov壳体理论建立自旋圆柱壳的结构动力学模型；根据圆柱壳边界条件的数学描述，采用Chebyshev多项式构建满足边界条件且与壳体结构参数无关的位移场离散函数；通过求解与振动特性相关的特征方程得到静止圆柱壳的固有频率和模态，分析壳体转动惯性对固有频率的影响，讨论壳体理论在不同壳体几何尺寸下的适用性；此外，发现了环向波数相关的模态函数，并将其用于圆柱壳零阶环向模态固有频率的求解及多种边界条件下自旋圆柱壳行波固有频率的求解，讨论了自旋圆柱壳行波固有频率随结构参数变化的情况。

自旋圆柱壳  /  多种边界条件  /  转动惯性  /  位移场离散

Spinning cylindrical shells are critical components in practical engineering structures. The boundary conditions at the shell ends are diverse and significantly influence the vibration characteristics of the shell. To study these characteristics under various boundary conditions，a dynamic model of the spinning cylindrical shell is established using Lagrange equations and Novozhilov’s shell theory. The mathematical description of the boundary conditions for the cylindrical shell is combined with the discretized displacement functions，which are constructed based on a linear combination of Chebyshev polynomials. These functions satisfy the boundary conditions and are independent of the cylindrical shell's parameters. The vibration characteristics of stationary cylindrical shells are determined by solving the eigenvalue problems，revealing the influence of rotary inertia on the vibration characteristics. The applicability of different shell theories with respect to various geometrical parameters of the shell is discussed. Additionally，circumferential wave-dependent mode functions are identified and used to compute the natural frequencies of shell modes with the zero circumferential waves，as well as the travelling waves of the spinning cylindrical shell under different boundary conditions. The impact of structural parameters on the natural frequencies of the travelling waves is also analyzed.

spinning cylindrical shell  /  various boundary conditions  /  rotary inertia  /  displacement field discretization
董有恒, 李映辉, 李翔宇, 茅晓晨. 多种边界条件下自旋圆柱壳振动特性研究. 振动工程学报, 2024 , 37 (10) : 1731 -1738 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.10.011
You-heng DONG, Ying-hui LI, Xiang-yu LI, Xiao-chen MAO. Vibration characteristics of spinning cylindrical shells under various boundary conditions[J]. Journal of Vibration Engineering, 2024 , 37 (10) : 1731 -1738 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.10.011
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引言
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凭借其中心对称特性和较高的比刚度，圆柱壳结构作为重要零部件在机械领域和航空航天领域得到了广泛应用。高速离心机的内胆和外壳、风电机组联轴器的合金钢中间管、航空发动机的叶片轮毂等零部件均可简化为自旋圆柱壳力学模型。不同于静止圆柱壳的驻波特性，由自旋运动引起的科氏效应会导致自旋圆柱壳结构出现行波振动［1-2］。同时，实际工程中的圆柱壳结构边界条件种类较多，常见的简支边界条件难以满足振动特性求解精度的要求。因此，针对自旋圆柱壳结构，发展除简支边界条件外的其他多种边界条件描述方法，并准确分析自旋圆柱壳的行波振动特性，能够为实际工程结构中多种边界自旋圆柱壳的动力学分析提供研究基础。
由于单个三角函数能准确描述简支边界条件下圆柱壳关于轴向坐标的模态构型，现有研究中关于圆柱壳振动特性的研究大多在两端简支边界条件下开展［3-9］。徐港辉等［3］通过正、余弦模态函数研究了两端简支圆柱壳动力学响应中的模态参与问题。Bich等［4］研究了简支边界功能梯度材料圆柱壳在外加载荷作用下的非线性振动，首先将壳体位移分量在轴向和环向离散为单模态下的三角函数形式，然后在非线性运动控制方程的基础上忽略壳体面内惯性项，得到关于壳体径向位移的非线性微分方程，最后得到了圆柱壳的频响函数和非线性动力学响应的数值解。Wang［5］运用Donnell非线性壳体理论建立了复合材料自旋圆柱壳关于径向位移的非线性运动控制方程，将壳体径向位移用三角函数表示，运用谐波平衡法研究了简支边界条件下圆柱壳的强迫振动，研究发现较高的自旋速度会导致复杂的频响曲线并产生新的稳态。基于一阶剪切变形壳体理论，Sofiyev等［6］研究了弹性地基上功能梯度材料圆柱壳的非线性振动，考虑壳体端部简支边界，假设壳体单模态振动，将壳体位移场离散为三角函数形式，通过数值算例讨论了系统参数对圆柱壳频响曲线的影响规律。
为研究多种边界条件下圆柱壳的振动特性，学者们用三角函数和双曲函数组合形式的梁模态函数来描述圆柱壳关于轴向坐标的振型［10-12］。孙述鹏等［10］采用固支边界梁的模态函数描述圆柱壳模态函数在轴向的构型，分别研究了横向简谐力和恒力作用下两端固支自旋圆柱壳的行波振动响应。此外，Jin等［13］和Lin等［14］通过在圆柱壳端部添加虚拟弹簧来模拟弹性边界，其中不同的壳体边界条件由不同形式的弹簧刚度系数组合决定，此特定的刚度系数组合随壳体几何尺寸和材料性质而发生变化。石先杰等［15］采用边界虚拟弹簧模拟功能梯度圆柱壳的任意经典边界，研究了壳体自由振动和瞬态振动特性。庞福振等［16］研究了边界虚拟弹簧约束下半球壳的受迫振动响应。
为建立多种边界条件下与圆柱壳结构参数无关的壳体模态函数，Kurylov等［17］将圆柱壳位移场离散为正交多项式或三角函数的多项求和形式，在此基础上对比分析了简支和固支边界圆柱壳的固有频率。Dong等［18］使用Chebyshev多项式分别给出了满足圆柱壳8种边界条件的位移场离散函数，得到了较高精度的壳体固有频率。
综上所述，学者们对多种边界条件下圆柱壳的振动特性分析已开展广泛研究，本文将进一步研究多种边界条件下自旋圆柱壳的行波振动特性。给出含有自旋运动的壳体结构模态分析流程，同时分析壳体转动惯性对固有频率求解精度的影响，并通过数值算例对比不同壳体理论的适用范围，研究不同边界条件对行波振动特性的影响。
1　自旋圆柱壳动力学模型
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自旋圆柱壳力学模型及其横截面微元如图1所示，其中红色虚线圆所在的环面为圆柱壳中面，圆柱壳以自旋速度（r/min）绕其中心轴线作自旋运动。圆柱壳长度、壁厚、中面半径分别为L，h，r；圆柱壳材料的杨氏模量、泊松比、密度分别为E，，。为建立自旋圆柱壳动力学模型，并使用Chebyshev多项式表示位移场离散函数，需将圆柱壳轴向坐标转移到多项式的定义区间［-1，1］。将惯性坐标系（，，）的轴置于圆柱壳中心轴线，柱坐标系（，，z）固结于壳体中面，其中，［-1，1］，［0，2］，［-0.5h，0.5h］。在固结于中面的柱坐标系下，圆柱壳中面任意点沿，，z三个方向的位移分别为，，。
考虑圆柱壳横截面转动，则圆柱壳任意点沿，，z三个方向的位移，，可表示为关于圆柱壳中面位移场的函数：
式中　变量下标中的“，”表示对下标“，”后面变量的微分。
本文使用薄壁壳体理论中较为精确的Novozhilov壳体理论［19］，其应变场表示为：
其中，壳体中面应变和曲率变化分量分别表示为：
圆柱壳应力-应变关系则表示为：
式中　 （i，j=1，2，4）为与杨氏模量和泊松比有关的弹性常数［2］。
为方便对圆柱壳边界条件进行描述，接着给出圆柱壳横截面处轴力和弯矩的定义式：
根据图1所示的壳体横截面微元，圆柱壳应变能可表示为：
同时，自旋圆柱壳的动能K包含结构的振动动能和自旋运动引起的动能，具体表示为：
式中　变量上方加‘·’表示对时间变量t的微分。
由离心惯性力产生的环向拉伸应变能为：
结合式（1）和（5），可给出圆柱壳在端部可移动简支（Sm）、端部不可移动简支（Si）、滑移（Sd）、固支（C）、自由（F）等多种边界条件下，关于壳体边界处（=-1或1）的位移场、轴力和弯矩的数学描述：
2　振动特性
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为了研究多种边界条件下自旋圆柱壳的行波振动特性，需要对位移场进行离散，再通过求解特征值问题来得到壳体结构固有频率。
由于圆柱壳位移场关于环向坐标的模态构型呈花瓣状，且相互正交，故可通过单一的正、余弦函数准确描述。而圆柱壳位移场关于轴向坐标的离散函数与壳体边界条件有关，文献［18］给出的数值结果证明，若使用梁模态函数（三角函数和双曲函数的组合形式）近似描述多种边界条件下壳位移场关于轴向坐标的离散函数，得到的固有频率存在较大误差。
基于此，本文选取Chebyshev多项式描述圆柱壳位移场关于轴向坐标的离散函数。当存在自旋运动时，多项式描述的陀螺矩阵具有奇异性和稀疏性，无法直接求解相对应的特征值问题。因此，首先通过对静止圆柱壳的模态分析，得到多种边界条件下各阶模态的离散函数描述形式，在此基础上求解多种边界条件下自旋圆柱壳的行波振动特性，具体流程图如图2所示。
首先给出多种边界条件下圆柱壳单模态振动的壳体中面位移场离散形式：
式中　 U（），V（）和W（）表示位移场关于轴向和环向坐标的离散函数；f（t）为与时间有关的简谐函数。
考虑第一类Chebyshev多项式，其前j阶多项式依次表示为：，，。式（10）中的离散函数可表示为的线性组合形式：
式中　 n表示圆柱壳各阶模态的环向波数；表示Chebyshev多项式的项数；，，表示Chebyshev多项式的系数。
将式（10）和（11）代入式（9），通过运算即可得到满足圆柱壳不同边界条件的Chebyshev多项式组合形式。此时，满足边界条件的圆柱壳离散函数为：
式中　fun表示满足边界条件的第j个关于Chebyshev多项式的线性组合函数；M表示U（），V（）和W（）关于轴向坐标的离散函数的求和项数；，，表示Chebyshev多项式的修正系数。
本文使用如下Lagrange方程求解圆柱壳固有振动特性：
式中　 q为向量=｛，…，，，…，，，…，｝中的元素。
将式（10）和（12）代入能量表达式（6）~（8），再结合式（13），并假设圆柱壳的简谐振动f（t）=，其中表示圆柱壳的固有频率。当自旋速度=0时，可得到静止圆柱壳的关于固有频率和固有振型的特征方程：
式中　 ，分别表示静止圆柱壳的质量矩阵和刚度矩阵。
当自旋速度0时，自旋圆柱壳出现行波振动，固有频率分为前行波频率和后行波频率。对于环向波数为n的行波模态，此时圆柱壳关于轴向坐标和环向坐标的离散函数为：
式中　 ，和为满足壳体边界条件的模态函数，与对应的静止圆柱壳模态函数一致，由式（12）和（14）得到，如图2流程图所示。此时，q为向量q=｛，，，，，｝中的广义变量，根据Lagrange方程得到：
式中　 M，G和K分别表示自旋圆柱壳的质量矩阵、陀螺矩阵和刚度矩阵。值得注意的是，由自旋运动引起的科式效应和离心效应，分别在矩阵G和K中体现。
假设简谐振动形式q=，其中和分别为待定标量和向量，将此式代入式（16），即可得到二次特征值问题。为方便求解行波固有频率，使用状态空间描述，令，广义坐标q在状态空间中的运动变为，得到关于自旋圆柱壳行波固有频率的特征值方程：
式（17）与（16）具有相同的特征值。求解式（17）可得到6个频率值，其中最小的两个频率对应该边界条件下自旋圆柱壳的前行波频率和后行波频率。
2.1　数值验证与壳体转动惯性的影响
为验证本文中基于Chebyshev多项式的模态离散函数的有效性，及多种边界条件下圆柱壳振动特性的理论推导与数值模拟的正确性，首先将本文中基于Novozhilov壳体理论的圆柱壳固有频率与有限元结果、文献［2］中基于Amabili-Reddy三阶剪切变形壳体的理论结果进行对比，如表1所示。通过数值解对比发现，当式（12）中多项式项数M=20时，固有频率值收敛，故在本文数值解求解过程中M均取为20。
值得注意的是，在C-C和Si-Si边界条件下，表1中针对圆柱壳零阶环向模态（n=0）的固有频率计算，在C-C和Si-Si边界条件下，使用式（12）和（14）无法直接得到其固有频率。此时，采用（1，1）模态下求得的离散函数作为（1，0）模态频率计算的离散函数，可得到较高精度的固有频率。由表1可知，本文解与有限元解、文献［18］的解有较好的对照，验证了Chebyshev多项式在模拟圆柱壳多种边界条件时的有效性。
现有文献在关于圆柱壳振动特性和非线性振动响应的分析中，为了便于计算分析，一般忽略壳体转动惯性［4，6，10-11，17］，在式（7）中仅考虑中面位移场相关的动能。基于Novozhilov壳体理论和Donnell壳体理论，表2分别给出了考虑壳体转动惯性和忽略壳体转动惯性情况下圆柱壳的固有频率，同时对比了圆柱壳在不同半径-厚度比（r/h）情况下，上述两种壳体理论的适用性。为了更直观地进行分析，将表2中基于上述两种壳体理论的固有频率计算结果与有限元结果对比的相对误差展示在图3中。
从表2和图3中可以看出，针对两种不同的壳体理论，忽略壳体转动惯性造成的固有频率计算误差较小，这是因为壳体厚度与壳体长度相比较小，且转动惯性是比壳体中面惯性更高阶的量，故在较大半径-厚度比情况下，忽略壳体转动惯性的假设是合理的。此外，对比Novozhilov壳体理论和Donnell壳体理论的适用性，Novozhilov壳体理论比Donnell壳体理论具有更高的固有频率计算精度。在薄壁壳情况下（r/h200），与有限元结果对比发现，基于Donnell壳体理论的计算结果最大误差不超过1.6%。在中厚壳情况下（），基于Novozhilov壳体理论的计算结果仍具有较高的精度，与Amabili-Reddy三阶剪切变形壳体理论的结果相差较小，而基于Donnell壳体理论的计算结果存在较大误差。故Novozhilov壳体理论在中厚壳描述中也是一个较好的选择。
2.2　环向波数相关的模态函数
依据图2所示的流程求解特征方程式（14），可得到圆柱壳在不同边界条件下的固有频率和对应的模态构型。图4给出了C-C边界条件下，圆柱壳的（1，2）、（1，3）、（1，4）、（1，5）、（1，6）、（1，7）模态构型沿轴向的变化情况。
在相同轴向半波数条件下，分别对比图4中，和在不同环向波数情况下随圆柱壳轴向坐标的变化，发现圆柱壳模态构型在不同环向波数中存在明显差异，称此现象为环向波数相关的模态函数。
为进一步验证圆柱壳环向波数相关的模态函数，表3分别给出了采用（1，2）、（1，3）、（1，4）、（1，5）模态下已知的关于轴向坐标的离散函数作为模态构型时，求解得到的固有频率。由表3的计算结果及其与准确值间的相对误差可知，通过自身模态去求解固有频率可以得到准确值，当用于求解固有频率的模态函数所对应的环向波数与待求模态波数相差较大时，会造成固有频率求解误差较大。
值得注意的是，与圆柱壳的环向波数相关的模态函数在除简支-简支边界外的其他边界条件中均存在，此现象将在多种边界条件下，圆柱壳零阶环向模态的固有频率求解、自旋圆柱壳行波振动特性求解、圆柱壳的多模态耦合振动分析中有重要应用。
图5对比了C-C，Sm-Sm边界条件下，圆柱壳固有频率随环向波数的变化。从图5中可以看出，不同于梁、板结构，圆柱壳的最低固有频率一般不在最小模态波数处取得。相同模态波数情况下，固支边界的固有频率远高于端部可移动简支边界的固有频率。
2.3　自旋圆柱壳行波固有频率
依据图2所示的流程，考虑Sm-Sm，Sd-Sd边界条件，表4给出了自旋圆柱壳行波振动的固有频率，并与现有文献［2］的结果进行了对比。从表4中可以看出，本文采用Chebyshev多项式模拟圆柱壳多种边界条件下的位移场离散函数，并使用环向波数相关的模态函数求解多种边界条件下的自旋圆柱壳行波振动特性是可行的，具有较高的精度。图6则给出了Sm-Sm，C-C边界条件下，自旋圆柱壳（1，5）行波模态的固有频率随自旋速度的变化。
3　结论
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本文基于Novozhilov壳体理论建立了圆柱壳动力学模型，运用Chebyshev多项式组合形式的位移场离散函数研究了多种边界条件下自旋圆柱壳的行波振动特性。主要结论如下：
（1）Novozhilov壳体理论可用于中厚壳的理论分析，在圆柱壳振动特性分析中能达到与高阶剪切变形壳体理论相近的准确度，且具有较简便的形式（包括非线性项），在圆柱壳多模态耦合振动分析中，Novozhilov壳体理论是一个较好的选择。
（2）在较大半径-厚度比情况下，壳体转动惯性可忽略不计，以简化振动特性求解过程。圆柱壳的环向波数相关的模态函数在除简支-简支边界外的其他边界条件中均存在。
（3）在多种边界条件下，由Chebyshev多项式构成的模态离散函数可以精确满足圆柱壳的多种边界条件。环向波数相关的模态函数在圆柱壳零阶环向模态的固有频率求解、自旋圆柱壳行波振动特性求解、圆柱壳的多模态耦合振动分析具有重要作用。
在后续的研究中，进一步将环向波数相关的模态函数应用于自旋圆柱壳的非线性多模态耦合振动，研究多种边界条件下自旋圆柱壳的非线性动力学响应。
基金
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  • 国家自然科学基金资助项目(12102175)
  • 中央高校基本科研业务费专项资金资助项目(B240201128)
文献
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2024年第37卷第10期
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doi: 10.16385/j.cnki.issn.1004-4523.2024.10.011
  • 接收时间：2024-04-17
  • 首发时间：2026-02-12
  • 出版时间：2024-10-28
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  • 收稿日期：2024-04-17
  • 修回日期：2024-07-15
基金
国家自然科学基金资助项目(12102175)
中央高校基本科研业务费专项资金资助项目(B240201128)
作者信息
    1河海大学力学与工程科学学院，江苏　南京　211100
    2西南交通大学力学与航空航天学院，四川　成都　611756

通讯作者:

李映辉（1964―），男，博士，教授。E-mail：。
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https://castjournals.cast.org.cn/joweb/zdgcxb/CN/10.16385/j.cnki.issn.1004-4523.2024.10.011
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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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