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Article(id=1246046393165529357, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246046388547596731, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2024.033, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1721836800000, receivedDateStr=2024-07-25, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1775010210553, onlineDateStr=2026-04-01, pubDate=1735056000000, pubDateStr=2024-12-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1775010210553, onlineIssueDateStr=2026-04-01, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1775010210553, creator=13701087609, updateTime=1775010210553, updator=13701087609, issue=Issue{id=1246046388547596731, tenantId=1146029695717560320, journalId=1241755870837649424, year='2024', volume='45', issue='6', pageStart='709', pageEnd='856', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1775010209451, creator=13701087609, updateTime=1775010293319, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1246046740390986227, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246046388547596731, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1246046740390986228, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246046388547596731, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=846, endPage=856, ext={EN=ArticleExt(id=1246046393828229401, articleId=1246046393165529357, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=Traveling Wave Vibration Characteristics of a Coupled Double-Thin-Walled Cylindrical Shell Structure with Hard-Coating Damping at High Rotational Speeds, columnId=1244229834482757770, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Paper, runingTitle=null, highlight=null, articleAbstract=

The connection conditions and vibration suppression methods of coupled shell structures have received much attention, particularly as these structures play a crucial role in aero-engine components. In this paper, the vibration characteristics of a rotating hard coating damping double-thin-walled cylindrical shell coupled structure under bolt connection conditions are studied. First, a discontinuous arc connection is constructed to simulate actual bolt connection conditions by improving the artificial spring distribution method of the continuous entire circumference. And the artificial spring technique is used to define the boundary conditions of the shell structure. Next, the strain energy of the hard-coating shell structure is determined based on Sander's shell theory. The effect of rotational speed is considered, and the Rayleigh-Ritz method is used to derive the dynamic equations of the shell structure. In addition, the efficient state space method is used for calculation. The rationality and accuracy of the theoretical methods are validated through literature comparisons and finite element analysis. Additionally, the effects of rotational speed, connection stiffness, hard-coating thickness, and boundary conditions on the traveling wave vibration characteristics of the shell structure are analyzed. The results show that the traveling wave frequency increases significantly when the connection stiffness is in the range of 108～1010. Besides, the rotation leads to a separation phenomenon and an overall increasing trend in the traveling wave frequency. A greater hard coating thickness notably impacts the traveling wave frequency, exhibiting a maximum increase of 5.87% in the traveling wave frequency when the hard-coating thickness rises from 0 to 0.85 mm. These findings provide valuable theoretical insights and data support for the engineering design of hard-coating coupled double-thin-walled cylindrical shell structures.

, correspAuthors=Yu Wang, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, authorCompany=null, fund=null, authors=null, authorsList=Peng Wang, Yu Wang, Yufeng Chang, Liping Wang), CN=ArticleExt(id=1246046410618028616, articleId=1246046393165529357, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=高转速硬涂层阻尼双薄壁圆柱壳耦合结构的行波振动特性, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

对螺栓连接条件下旋转硬涂层阻尼双薄壁圆柱壳耦合结构的振动特性开展了研究. 首先，通过改进连续整周的人工弹簧分布方法，构造非连续弧连接来模拟实际螺栓连接条件. 其次，基于Sanders壳理论，确定硬涂层壳体结构的应变能. 最后，考虑转速的影响并采用Rayleigh-Ritz法推导动力学方程. 此外，利用状态空间法进行求解计算，通过文献和有限元方法验证理论方法的合理性. 结果表明：连接刚度在108~1010区间范围内行波频率显著增大；旋转导致行波频率出现分离现象且整体呈增大趋势；硬涂层厚度越大对行波频率的影响越明显，硬涂层厚度从0提高到0.85 mm时，行波频率最大增长程度可达5.87%.

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Geometric parameters

, figureFileSmall=null, figureFileBig=null, tableContent=
壳体长度（m）内表面半径（m）基体厚度（m）涂层厚度（m）
圆柱壳Ⅰ0.40.10.0020.0003
圆柱壳Ⅱ0.40.10.0020.0003
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几何参数

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壳体长度（m）内表面半径（m）基体厚度（m）涂层厚度（m）
圆柱壳Ⅰ0.40.10.0020.0003
圆柱壳Ⅱ0.40.10.0020.0003
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Material parameters

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材料层弹性模量（GPa）泊松比密度（kg/m3）
基体2060.37850
硬涂层54.4940.35560
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材料参数

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材料层弹性模量（GPa）泊松比密度（kg/m3）
基体2060.37850
硬涂层54.4940.35560
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Effect of truncation number on traveling wave frequency of coupling structure

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（m，n）前行波频率（Hz）后行波频率（Hz）
N=4N=8N=12N=16N=20N=4N=8N=12N=16N=20
（1，6）1719.81718.61718.51718.51718.51750.91749.61749.51749.51749.5
（1，7）2362.42361.02360.92360.92360.92389.32387.82387.82387.82387.8
（1，8）3103.73102.13102.03102.03102.03127.43125.73125.63125.63125.6
（1，9）3943.53941.73941.63941.63941.63964.63962.83962.73962.73962.7
（2，7）2402.02389.12388.82388.82388.82429.02416.02415.72415.72415.7
（2，8）3139.83128.03127.73127.73127.73163.53151.73151.43151.43151.4
), ArticleFig(id=1246046418675286779, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246046393165529357, language=CN, label=表3, caption=

截断数对耦合结构行波频率的影响

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（m，n）前行波频率（Hz）后行波频率（Hz）
N=4N=8N=12N=16N=20N=4N=8N=12N=16N=20
（1，6）1719.81718.61718.51718.51718.51750.91749.61749.51749.51749.5
（1，7）2362.42361.02360.92360.92360.92389.32387.82387.82387.82387.8
（1，8）3103.73102.13102.03102.03102.03127.43125.73125.63125.63125.6
（1，9）3943.53941.73941.63941.63941.63964.63962.83962.73962.73962.7
（2，7）2402.02389.12388.82388.82388.82429.02416.02415.72415.72415.7
（2，8）3139.83128.03127.73127.73127.73163.53151.73151.43151.43151.4
), ArticleFig(id=1246046418788532991, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246046393165529357, language=EN, label=Table 4, caption=

Comparison of dimensionless frequency of double thin-walled cylindrical shells

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阶次文献[21]本文De（%）
10.20940.20890.24
20.21560.21500.28
30.24280.24240.17
40.26940.26870.26
50.30400.30370.10
), ArticleFig(id=1246046418859836163, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246046393165529357, language=CN, label=表4, caption=

双薄壁圆柱壳无量纲频率对比

, figureFileSmall=null, figureFileBig=null, tableContent=
阶次文献[21]本文De（%）
10.20940.20890.24
20.21560.21500.28
30.24280.24240.17
40.26940.26870.26
50.30400.30370.10
), ArticleFig(id=1246046418977276679, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246046393165529357, language=EN, label=Table 5, caption=

Comparison of traveling wave frequencies of rotating hard coating thin-walled cylindrical shell

, figureFileSmall=null, figureFileBig=null, tableContent=
（m，n）前行波频率/Hz后行波频率/Hz
文献[14]本文De（%）文献[14]本文De（%）
1，21424.61424.20.0281500.11499.60.033
1，3946.1945.80.0321003.41003.00.040
1，4942.0941.80.021987.1986.90.020
1，51272.21272.60.0321309.61309.50.008
2，22925.42923.50.0652997.02995.10.063
2，31969.01967.70.0662025.12023.80.064
2，41570.71569.70.0191615.61614.60.062
2，51620.01619.30.0431657.01656.30.042
), ArticleFig(id=1246046419077939980, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246046393165529357, language=CN, label=表5, caption=

旋转硬涂层薄壁圆柱壳行波频率对比

, figureFileSmall=null, figureFileBig=null, tableContent=
（m，n）前行波频率/Hz后行波频率/Hz
文献[14]本文De（%）文献[14]本文De（%）
1，21424.61424.20.0281500.11499.60.033
1，3946.1945.80.0321003.41003.00.040
1，4942.0941.80.021987.1986.90.020
1，51272.21272.60.0321309.61309.50.008
2，22925.42923.50.0652997.02995.10.063
2，31969.01967.70.0662025.12023.80.064
2，41570.71569.70.0191615.61614.60.062
2，51620.01619.30.0431657.01656.30.042
), ArticleFig(id=1246046419170214670, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246046393165529357, language=EN, label=Table 6, caption=

Comparison of natural frequencies of hard coating double thin-walled cylindrical shells coupled structure

, figureFileSmall=null, figureFileBig=null, tableContent=
阶次固支-固支固支-自由
有限元法本文De（%）有限元法本文De（%）
1468.0466.30.36164.6164.20.24
2529.8529.30.09279.8280.10.11
3748.0743.50.60379.9378.10.39
4785.1783.50.20476.4474.90.31
5861.8857.50.50551.5551.30.04
), ArticleFig(id=1246046419287655185, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246046393165529357, language=CN, label=表6, caption=

硬涂层双薄壁圆柱壳耦合结构固有频率对比

, figureFileSmall=null, figureFileBig=null, tableContent=
阶次固支-固支固支-自由
有限元法本文De（%）有限元法本文De（%）
1468.0466.30.36164.6164.20.24
2529.8529.30.09279.8280.10.11
3748.0743.50.60379.9378.10.39
4785.1783.50.20476.4474.90.31
5861.8857.50.50551.5551.30.04
), ArticleFig(id=1246046419409290006, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246046393165529357, language=EN, label=Table 7, caption=

Natural frequencies of rotating hard coating double thin-walled cylindrical shells coupled structure

, figureFileSmall=null, figureFileBig=null, tableContent=
m固有频率（Hz）
n=1n=2n=3n=4n=5n=6n=7n=8n=9n=10
11094505458742117417132353309239304867
222591188780857122517472381311839554891
33488190912021064132718112431316339974932
), ArticleFig(id=1246046419509953305, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246046393165529357, language=CN, label=表7, caption=

旋转硬涂层双薄壁圆柱壳耦合结构的固有频率

, figureFileSmall=null, figureFileBig=null, tableContent=
m固有频率（Hz）
n=1n=2n=3n=4n=5n=6n=7n=8n=9n=10
11094505458742117417132353309239304867
222591188780857122517472381311839554891
33488190912021064132718112431316339974932
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高转速硬涂层阻尼双薄壁圆柱壳耦合结构的行波振动特性
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王鹏 , 王宇 ** , 常宇枫 , 王立萍
固体力学学报 | 研究论文 2024,45(6): 846-856
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固体力学学报 | 研究论文 2024, 45(6): 846-856
高转速硬涂层阻尼双薄壁圆柱壳耦合结构的行波振动特性
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王鹏, 王宇** , 常宇枫, 王立萍
作者信息
  • 辽宁科技大学机械工程与自动化学院，鞍山，114051

通讯作者:

** E-mail：.
Traveling Wave Vibration Characteristics of a Coupled Double-Thin-Walled Cylindrical Shell Structure with Hard-Coating Damping at High Rotational Speeds
Peng Wang, Yu Wang** , Yufeng Chang, Liping Wang
Affiliations
  • School of Mechanical Engineering and Automation, University of Science and Technology Liaoning, Anshan, 114051
出版时间: 2024-12-25 doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.033
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摘要
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对螺栓连接条件下旋转硬涂层阻尼双薄壁圆柱壳耦合结构的振动特性开展了研究. 首先，通过改进连续整周的人工弹簧分布方法，构造非连续弧连接来模拟实际螺栓连接条件. 其次，基于Sanders壳理论，确定硬涂层壳体结构的应变能. 最后，考虑转速的影响并采用Rayleigh-Ritz法推导动力学方程. 此外，利用状态空间法进行求解计算，通过文献和有限元方法验证理论方法的合理性. 结果表明：连接刚度在108~1010区间范围内行波频率显著增大；旋转导致行波频率出现分离现象且整体呈增大趋势；硬涂层厚度越大对行波频率的影响越明显，硬涂层厚度从0提高到0.85 mm时，行波频率最大增长程度可达5.87%.

双薄壁圆柱壳  /  硬涂层阻尼  /  非连续弧连接  /  Rayleigh-Ritz法  /  状态空间法  /  行波振动

The connection conditions and vibration suppression methods of coupled shell structures have received much attention, particularly as these structures play a crucial role in aero-engine components. In this paper, the vibration characteristics of a rotating hard coating damping double-thin-walled cylindrical shell coupled structure under bolt connection conditions are studied. First, a discontinuous arc connection is constructed to simulate actual bolt connection conditions by improving the artificial spring distribution method of the continuous entire circumference. And the artificial spring technique is used to define the boundary conditions of the shell structure. Next, the strain energy of the hard-coating shell structure is determined based on Sander's shell theory. The effect of rotational speed is considered, and the Rayleigh-Ritz method is used to derive the dynamic equations of the shell structure. In addition, the efficient state space method is used for calculation. The rationality and accuracy of the theoretical methods are validated through literature comparisons and finite element analysis. Additionally, the effects of rotational speed, connection stiffness, hard-coating thickness, and boundary conditions on the traveling wave vibration characteristics of the shell structure are analyzed. The results show that the traveling wave frequency increases significantly when the connection stiffness is in the range of 108～1010. Besides, the rotation leads to a separation phenomenon and an overall increasing trend in the traveling wave frequency. A greater hard coating thickness notably impacts the traveling wave frequency, exhibiting a maximum increase of 5.87% in the traveling wave frequency when the hard-coating thickness rises from 0 to 0.85 mm. These findings provide valuable theoretical insights and data support for the engineering design of hard-coating coupled double-thin-walled cylindrical shell structures.

double-thin-walled cylindrical shells  /  hard-coating damping  /  discontinuous arc connection  /  Rayleigh-Ritz method  /  state space approach  /  traveling wave vibration
王鹏, 王宇, 常宇枫, 王立萍. 高转速硬涂层阻尼双薄壁圆柱壳耦合结构的行波振动特性. 固体力学学报, 2024 , 45 (6) : 846 -856 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.033
Peng Wang, Yu Wang, Yufeng Chang, Liping Wang. Traveling Wave Vibration Characteristics of a Coupled Double-Thin-Walled Cylindrical Shell Structure with Hard-Coating Damping at High Rotational Speeds[J]. Chinese Journal of Solid Mechanics, 2024 , 45 (6) : 846 -856 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.033
正文
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0　引言
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薄壁圆柱壳由于具有重量轻、刚度高和适用性好等特点，已广泛应用于航空航天和船舶等领域[1-3]. 在工程应用中，双薄壁圆柱壳耦合结构经常通过螺栓连接方式进行组合，例如航空发动机中的鼓筒转子结构和薄壁机匣[4]，其中不同组合壳体之间的螺栓是非连续分布的，如何简单有效的模拟实际螺栓连接已成为关键问题. 同时，壳体结构在高速旋转状态下工作时，由于受气动载荷、流场和摩擦冲击的影响，容易过度振动而引起失稳和疲劳损伤等问题[5]，因此考虑有效的方法抑制过度振动显得至关重要.
近年来，通过在结构表面涂覆硬涂层阻尼材料进行振动控制的方法已成为研究热点，硬涂层阻尼减振技术广泛应用于板[6]、叶片[7]和圆柱壳[8]等结构的减振，例如：Han等[9]基于状态矢量法和传递矩阵法，对硬涂层薄板的振动特性开展了分析. Blackwell等[10]基于实验法，研究了钛板涂覆硬涂层阻尼材料的振动特性. Yang等[11]基于高阶剪切变形理论和哈密顿变分原理，研究了悬臂硬涂层板的自由振动. Mitra等[12]通过有限单元法和谐波平衡法，研究了硬涂层叶片的自由振动. Song等[13]基于传递矩阵法和Love壳理论，研究了旋转状态下硬涂层薄壁圆柱壳的行波振动特性. Du等[14]基于Sanders壳理论，对旋转硬涂层薄壁圆柱壳的行波振动进行了研究，并讨论了硬涂层厚度、边界条件、转速和周向波数对壳体固有频率的影响. 王宇等[15]采用Viogt-Reuss原理和传递矩阵法，研究了不同边界条件下硬涂层薄壁圆柱壳的行波振动特性. Li等[16]基于经典层合壳理论和Rayleigh-Ritz法，对涂覆NiCoCrAlY+YSZ硬涂层阻尼材料的纤维增强复合薄壁圆柱壳进行了振动特性研究. Du等[17]使用Kirchhoff理论和Sanders壳理论，研究了硬涂层鼓-盘结构的非线性振动特性. 可以看出，对硬涂层材料的研究主要集中在单一结构，而硬涂层材料在旋转耦合结构行波振动特性方面的研究较少.
对于壳体耦合结构，连接条件对动力学特性的影响得到了学者们的广泛重视，具有代表性的成果如下：Chai等[18]基于Donnell壳理论和人工弹簧技术，对刚性连接条件下，圆锥壳和圆柱壳耦合结构的行波振动特性进行了分析. Li等[19]利用一阶剪切变形理论，对非均匀厚度球-柱-球壳在弹性连接条件下的自由振动特性进行了研究. Zhao等[20]通过Rayleigh-Ritz法，对轴-盘-鼓耦合结构在弹性连接条件下的行波振动特性开展了研究. Li等[21]将复合薄壁圆柱-圆柱壳作为研究对象，通过改变连接弹簧刚度的方法，研究了部分螺栓松动下的自由振动和强迫振动. Tang等[22]基于Sanders壳体理论，利用非连续点连接模拟螺栓并研究了薄壁圆柱壳-圆柱壳结构的固有频率和响应. Ma等[23]利用Donnell壳体理论，研究了非连续点连接条件下粘弹性双薄壁圆柱壳耦合结构的自由振动. 目前对于耦合结构中连接条件的研究主要认为是刚性连接、弹性连接和非连续点连接. 在实际工程中，螺栓既不是类似于刚性和弹性连接的整周连接，也不是在圆周方向上没有作用范围的非连续点连接，而是具有一定影响区域的非连续弧连接[24]. 因此，开展非连续弧连接条件下，高速旋转硬涂层双薄壁圆柱壳耦合结构的行波振动特性分析具有重要意义.
综上所述，本文在现有双薄壁圆柱壳耦合结构动力学研究的基础上，通过建立非连续弧连接方式来模拟螺栓连接，并考虑硬涂层抑振手段和高转速变化，基于Sanders壳理论和Rayleigh-Ritz法推导动力学方程，结合高效率的状态空间法求解行波频率，验证本文方法的有效性，并讨论转速、连接刚度、硬涂层厚度和边界条件对壳体振动特性的影响.
1　理论方法
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1.1　力学模型
旋转硬涂层双薄壁圆柱壳耦合结构的模型如图1所示，左右两部分为圆柱壳Ⅰ和圆柱壳Ⅱ，壳体以角速度Ω转动，NiCrAlCoY+YSZ硬涂层均匀地附着在圆柱壳Ⅰ和Ⅱ的外表面. 在圆柱壳Ⅰ和Ⅱ的底部分别建立正交曲线坐标系O1-x1θ1z1和O2-x2θ2z2，中面上任意点pi的轴向、周向和径向位移分别为ui、vi和wi（i=1，2表示圆柱壳Ⅰ和圆柱壳Ⅱ），Ri、Li、hi、Rs，i、Ts，i和Tc，i分别为壳体中面半径、轴向长度、内壁到中面的距离、基体内表面半径、基体和硬涂层厚度.
利用人工弹簧技术模拟连接条件，非连续弧连接方式如图2（a）所示，在两个硬涂层薄壁圆柱壳之间有着非连续均匀分布的弧，连接弹簧只分布在每个弧上，ku，p、kv，p、kw，p和kθ，p分别表示轴向、周向和径向的连接弹簧刚度系数，θr为第r段弧的起始角度，Δθ为连接弧的有效角度. 同时壳体耦合结构两端定义三组平动弹簧和一组旋转弹簧模拟边界条件，边界条件如图2（b）所示，ku，i、kv，i、kw，i和kθ，i分别为轴向、周向和径向的边界弹簧刚度系数.
1.2　动力学方程
基于Sanders壳体理论，两个硬涂层薄壁圆柱壳在任意点的应变与位移关系可表示为
式中：、和分别为壳体中面应变；kx，i、kθ，i和kxθ，i为壳体中面曲率，具体表达式为
式中：ηi是由xi/Li得到的无量纲长度.
根据广义胡克定律，应力和应变关系表示为
式中：k=s和c分别表示基体层和硬涂层；σx，i、σθ，i与σxθ，i分别表示轴向应力、切向应力以及剪应力；（a，b=1，2，6）为第k层包含弹性模量和泊松比的缩减刚度系数，具体表达式为
式中：Ek，i和νk，i分别为弹性模量和泊松比.
中面上力和力矩的表达式为
式中：Nx，i、Nθ，i和Nxθ，i为内力；Mx，i、Mθ，i和Mxθ，i为力矩.
本构方程的表达式为
其中
式中：Aab，i、Bab，i和Dab，i分别为拉伸刚度、耦合刚度和弯曲刚度，具体表达式为
式中：hi为内壁到中面的距离，表达式为
式中：Es，i和Ec，i为基体和硬涂层的弹性模量.
旋转硬涂层薄壁圆柱壳的动能为
式中：ρi为密度，表达示为
式中：ρs，i和ρc，i分别为基体和硬涂层密度.
旋转引起的初应变能为
式中：为环向初应力，表达式为
旋转态壳体的应变能为
圆柱壳Ⅰ和Ⅱ储存在边界弹簧的势能为
人工弹簧在连接处储存的势能为
式中：NS为均匀分布弧的总数.
旋转硬涂层双薄壁圆柱壳耦合结构在中面上任意一点的位移表达式为
式中：n为周向波数；ω为固有角频率；Ui（ηi）、Vi（ηi）和Wi（ηi）为轴向振型函数，表达式为
式中：m为轴向半波数；N为截断数；am，i、bm，i和cm，i为未知系数；Pm，i（ηi）为位移容许函数，通过格拉姆-施密特正交化方法可以得到[25]
式中：ψm，i（ηi）为正交多项式，表达式为
其中
取能量表达式的最大值，利用Rayleigh-Ritz法得到拉格朗日函数为
对式（27）未知系数求偏导，表达式为
式中：X为里兹向量.
通过式（28）得到旋转态非连续弧连接硬涂层双薄壁圆柱壳耦合结构的动力学方程为
式中：M和M*为质量矩阵；K为总刚度矩阵，表达式为
式中：Kε、Kspr和Kcoup分别为势能刚度矩阵、边界刚度矩阵和非连续弧连接刚度矩阵.
为了提高计算效率，利用状态空间法求解式（29），表达式为
式中：B*和A*为12N×12N的矩阵；q为12N×1的向量，表达式为
等式（31）可以简化为
其中
将运动方程的固有频率求解问题转化为矩阵-（B*）-1A*特征值计算问题，可以提高固有频率的计算效率. 由于旋转的影响，各阶模态得到数值不同的两个频率，其中较小的为前行波，较大的为后行波.
2　收敛性和有效性验证
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采用Rayleigh-Ritz法对模型进行验证和分析时，硬涂层薄壁圆柱壳的几何和材料参数[14]如表1和表2所示，连接处的弹簧刚度为1010 N/m（Nm/rad）（后文刚度单位与此一致），均匀分布弧的总数为16，每段弧的有效角度为π/18，第一段弧的起始角度为0°.
为了保证计算精度，需要进行收敛性分析来确定截断数，当转速为300 rad/s时，固支-固支边界条件下截断数对行波频率的影响如表3所示. 可以看出，当截断数从4增加到8时，行波频率持续下降，当截断数为12时，行波频率已经收敛，因此在后文的计算中截断数取12.
首先通过文献[21]对非连续弧连接双薄壁圆柱壳耦合结构的有效性进行验证，其中几何和材料参数为：ρs，1=ρs，2=7850 kg/m3，Ts，1=Ts，2=0.002 m，L1=L1=0.1 m，Es，1=Es，2=206 GPa，R1=R2=0.1 m，νs，1=νs，2=0.3. 当边界弹簧和连接弹簧刚度系数为1012，计算结果如表4所示；计算结果的误差值可通过式（37）得到；无量纲角频率定义为. 从表4可以看出，本文方法和文献[21]中固有频率的最大误差值为0.28%，具有较好一致性，表明此方法满足所需精度要求.
式中和表示本文结果和文献结果.
其次对旋转态硬涂层薄壁圆柱壳的有效性进行验证，几何参数如表1所示，材料参数为：Es，1=212 GPa，Ec，1=54.494 GPa，ρs，1=7850 kg/m3，ρc，1=5560 kg/m3，νs，1=νs，2=0.3. 当转速为300 rad/s时，行波频率如表5所示. 可以发现，在固支-固支边界条件下，本文结果与文献[14]中行波频率的最大误差值为0.066%，具有良好一致性. 因此，表明本文方法适用于旋转硬涂层双薄壁圆柱壳耦合结构的振动特性分析.
为进一步验证本文方法的有效性，对硬涂层双薄壁圆柱壳耦合结构的固有频率和有限元法进行对比，几何和材料参数如表1和表2所示. 当连接刚度为1012时固有频率如表6所示. 从中看出，本文结果与有限元结果的最大误差值为0.60%，进一步表明本文方法的有效性.
3　数值分析和讨论
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3.1　转速的影响
当转速为0时，固支-固支边界条件下硬涂层双薄壁圆柱壳耦合结构的固有频率如表7所示. 可以看到，固有频率最低阶模态为m=1，n=3；同时壳体固有频率随周向波数增加呈现出先减小后增大的趋势. 在复杂的工作环境下，壳体的最低阶模态容易出现共振现象，因此若未特殊说明，后文选取（1，3）阶模态进行振动特性分析.
旋转对硬涂层双薄壁圆柱壳耦合结构振动特性的影响如图3所示. 从图3中可以看出，在转速增加初始阶段，由于旋转的影响，壳体频率会出现前、后行波频率分离现象，且后行波频率始终大于前行波频率；同时，后行波频率随着转速的增加持续增大，前行波频率随着转速的增加先减小后增大，整体呈增大趋势；此外，随着周向波数增加，各阶模态前后行波分离程度逐渐减小，且m=2相较于m=1时，前后行波分离程度较大.
3.2　连接刚度的影响
在两端固支边界条件下，连接刚度对壳体结构行波频率的影响如图4所示. 可以看出，当连接刚度从106变化到1014时，前后行波在不同转速下的频率变化趋势相似，连接刚度从106变化到108时，行波频率增加趋势不明显，但是当连接刚度从108变化到1010时，行波频率显著增加，此后随着连接刚度的增大，行波频率基本保持不变，这是因为当连接刚度足够大时连接状态已经接近于刚性连接；同时在不同连接刚度下，后行波频率随转速增加而增加，前行波频率随转速增加先减小后增大.
3.3　硬涂层厚度的影响
在两端固支边界条件下，硬涂层厚度和转速变化时对壳体频率的影响如图5所示. 可以看到，不同硬涂层厚度的行波曲线随着转速的增加，前、后行波分别呈现出先减小后增大和持续增大的趋势；同时可以发现，行波频率随硬涂层厚度增加而逐渐增大；当硬涂层厚度从0.00 mm提高到0.35 mm时，行波频率增加程度不明显，当硬涂层厚度从0.60 mm提高到0.85 mm时，行波频率变化程度增大，与涂层厚度为0 mm相比硬涂层厚度为0.85 mm时，行波频率的提升幅度最大可达5.87%. 表明硬涂层厚度越大对结构行波频率的影响越显著. 工程应用中可以通过改变硬涂层的厚度，根据实际需求对结构的振动抑制进行处理.
3.4　边界条件的影响
在固支-自由、简支-简支、固支-固支和固支-简支边界条件下，旋转硬涂层双薄壁圆柱壳耦合结构的振动特性如图6所示. 由于边界弹簧刚度的影响，固支-自由边界条件下的固有频率最小，固支-固支边界条件下的固有频率最大；当转速增加时前后行波曲线出现先减小后增大和持续增大的现象，并且前后行波频率从大到小的边界顺序为固支-固支、固支-简支、简支-简支和固支-自由，前后行波频率分别在3000 rad/s时达到最大值.
4　结论
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本文以高速旋转硬涂层阻尼双薄壁圆柱壳耦合结构为研究对象，基于Sanders壳理论、Rayleigh-Ritz法和状态空间法，研究了壳体结构的行波振动特性，主要结论如下：
（1）验证了本文方法的收敛性和正确性，当截断数为12时具有良好的计算精度和计算效率，适用于旋转硬涂层双薄壁圆柱壳耦合结构的振动特性分析；旋转导致行波曲线出现分离现象，且行波曲线随转速的增加整体呈增大趋势.
（2）连接刚度在一定取值区间内对行波振动特性的影响显著，表现为连接刚度从106变化到108时，行波频率基本不变，连接刚度从108变化到1010时，行波频率明显增加，连接刚度大于1010时，行波频率再次稳定不变，这种现象是因为连接刚度足够大时，连接条件接近刚性连接状态.
（3）硬涂层厚度越大对壳体行波频率的影响越明显，与无涂层相比硬涂层厚度为0.85 mm时，行波频率的最大增长幅度可达5.87%；行波频率从大到小的边界条件为固支-固支、固支-简支、简支-简支和固支-自由. 本文研究可为此类薄壁耦合结构的工程设计及振动特性分析提供一定的理论参考.
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2024年第45卷第6期
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.033
  • 接收时间：2024-07-25
  • 首发时间：2026-04-01
  • 出版时间：2024-12-25
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    辽宁科技大学机械工程与自动化学院，鞍山，114051

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2种不同金属材料的力学参数
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Genus		 种数
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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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