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Article(id=1281323822972375305, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1281323788058996778, articleNumber=null, orderNo=null, doi=10.3969/j.issn.1007-7294.2026.03.013, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1720454400000, receivedDateStr=2024-07-09, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1783421005242, onlineDateStr=2026-07-07, pubDate=1773504000000, pubDateStr=2026-03-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1783421005242, onlineIssueDateStr=2026-07-07, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1783421005241, creator=13701087609, updateTime=1783421005241, updator=13701087609, issue=Issue{id=1281323788058996778, tenantId=1146029695717560320, journalId=1240685776644648972, year='2026', volume='30', issue='3', pageStart='341', pageEnd='506', issueExtLink='null', onlineDate='null', pubDate='1773504000000', pubDateStr='2026-03-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1783420996918, creator='13701087609', updateTime=1783422057887, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext={EN=IssueExt(id=1281328238156821342, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1281323788058996778, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1281328238156821343, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1281323788058996778, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null, downloadFileDto=null}, startPage=488, endPage=497, ext={EN=ArticleExt(id=1281323824733982987, articleId=1281323822972375305, tenantId=1146029695717560320, journalId=1240685776644648972, language=EN, title=Acoustic characteristics analysis of expansion muffler with flexible back cavity, columnId=1241023038381158513, journalTitle=Journal of Ship Mechanics, columnName=Hydro/Structural Acoustics, runingTitle=null, highlight=null, articleAbstract=

Compared with an expansion muffler with a rigid back cavity, the expansion muffler with a flexible back cavity has better low-frequency noise reduction performance. However, most of the current calculation methods for expansion mufflers with the flexible back cavity are based on transfer matrix methods under the plane wave assumption. The calculation error of such methods increases with acoustic-structure coupling, and it is difficult to obtain the modal function of the flexible structure under elastic boundary conditions with traditional calculation methods. Therefore, this paper proposed a calculation method based on the energy principle that does not rely on the plane wave assumption. The model of the muffler is decomposed into three sub-acoustic cavities, and these cavities are coupled to each other through coupling surfaces. Then, the sound pressure function of the sound field and the displacement function of the flexible structure are expanded into three-dimensional and two-dimensional Chebyshev series respectively, and the Rayleigh-Ritz method was used to solve the unknown coefficients in the Chebyshev series. The sound pressure and transmission loss of the muffler were obtained, and the correctness of the theoretical model was verified by comparing it with the FEM results. Finally, the coupling characteristics were analyzed, and the effects of boundary constraints and muffler parameters on transmission loss were studied. The results show that the expansion muffler with a flexible back cavity has a lower natural frequency and stronger low-frequency coupling effect than that with a rigid back cavity. The impact of boundary constraints on transmission loss is mainly reflected above 1000 Hz. When boundary constraints are released, the peak value of transmission loss moves to low frequencies and increases. This shift is conducive to improving sound attenuation performance. As the back cavity’s length or radius increases, the transmission loss curve moves to the low frequency, and the influence of back cavity’s radius on the acoustic performance of the muffler is more obvious. As the thickness of the flexible wall or Young's modulus decreases, the transmission loss curve will move further to the low frequency.

, authors=Jiu-xiao HOU, Su-wei YUAN, Hai-chao ZHU, authorsList=Jiu-xiao HOU, Su-wei YUAN, Hai-chao ZHU, authorCompany=null, correspAuthors=Jiu-xiao HOU, authorNote=null, correspAuthorsNote=null, copyrightStatement=Copyright ©2026 Journal of Ship Mechanics. All rights reserved., 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, fund=null), CN=ArticleExt(id=1281323831247737118, articleId=1281323822972375305, tenantId=1146029695717560320, journalId=1240685776644648972, language=CN, title=弹性背腔扩张式消声器声学特性分析, columnId=1241023038515376243, journalTitle=船舶力学, columnName=流体与结构声学, runingTitle=null, highlight=null, articleAbstract=

弹性背腔扩张式消声器相较于刚性背腔扩张式消声器具有更优良的低频消声性能。目前针对弹性背腔扩张式消声器的计算方法多为基于平面波假设的传递矩阵法，此类方法的计算误差随着声固耦合作用的增强而变大，且传统计算方法难以获得弹性边界条件下弹性结构的模态函数。因此，本文提出了一种基于能量原理、不依赖于平面波假设的计算方法。通过子结构分解法将弹性背腔扩张式消声器模型分解为三个子声腔，随后将声场的声压函数及弹性结构的位移函数分别展开为三维与二维切比雪夫级数形式，结合Rayleigh-Ritz法求解切比雪夫级数中的未知系数，从而得到消声器内部声压及传递损失，通过与有限元结果的对比验证了理论模型的正确性。最后，分析了其耦合特性并研究了边界约束和消声器参数对传递损失的影响规律。结果表明，弹性背腔扩张式消声器相较于刚性背腔具有更低的固有频率和更强的低频耦合作用。边界约束对传递损失的影响主要体现在1000 Hz以上，释放边界约束可使传递损失峰值向低频移动并增大峰值，从而提升消声性能。此外，背腔长度或半径的增大会使传递损失曲线向低频移动，其中背腔半径对消声器的声学性能影响更为显著；弹性壁厚度或杨氏模量的减小会使传递损失曲线进一步向低频移动。

, authors=侯九霄, 袁苏伟, 朱海潮, authorsList=侯九霄, 袁苏伟, 朱海潮, authorCompany=null, correspAuthors=侯九霄, authorNote=

袁苏伟（1997–），男，博士研究生，E-mail：

, correspAuthorsNote=
侯九霄（1990–），男，讲师，通讯作者，E-mail：
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袁苏伟（1997–），男，博士研究生，E-mail：

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袁苏伟（1997–），男，博士研究生，E-mail：

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International Journal of Mechanical Sciences, 2017, 133: 91‒99., articleTitle=null, refAbstract=null)], funds=null, companyList=[AuthorCompany(id=1281323833122591008, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, xref=null, ext=[AuthorCompanyExt(id=1281323833135173921, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, companyId=1281323833122591008, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=National Key Laboratory on Ship Vibration and Noise, Naval University of Engineering, Wuhan 430033, China), AuthorCompanyExt(id=1281323833147756834, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, companyId=1281323833122591008, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=海军工程大学 船舶振动噪声重点实验室，武汉 430033)])], 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tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=CN, label=图2, caption=简支条件下理论结果与有限元结果的对比, figureFileSmall=5EMWpgjk5DGkGFxdpoSePA==, figureFileBig=KvGQLIW34zX/3mbmPGsE5Q==, tableContent=null), ArticleFig(id=1281323838021538112, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=EN, label=Fig.3, caption=Comparison of theoretical and FEM results under fixed supported conditions, figureFileSmall=pv+APUoirIVLUYd1qz97rQ==, figureFileBig=dh9uqfoofoxBoZoY9YRmPg==, tableContent=null), ArticleFig(id=1281323838080258369, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=CN, label=图3, caption=固支条件下理论结果与有限元结果的对比, figureFileSmall=pv+APUoirIVLUYd1qz97rQ==, figureFileBig=dh9uqfoofoxBoZoY9YRmPg==, tableContent=null), ArticleFig(id=1281323838151561538, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, 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tableContent=null), ArticleFig(id=1281323838659072329, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=CN, label=图7, caption=背腔半径对传递损失的影响, figureFileSmall=cvl8qTeqbuyybgs9jJ0dLw==, figureFileBig=tMyBgZrPbXhdVF5H/ES4Vw==, tableContent=null), ArticleFig(id=1281323838805872970, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=EN, label=Fig.8, caption=Influence of flexible wall thickness on transmission loss, figureFileSmall=kKdDqLXnEyQ6mYOMGS5dvQ==, figureFileBig=hxPVlGerUBtT4XYZE2kB1w==, tableContent=null), ArticleFig(id=1281323838889759051, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=CN, label=图8, caption=弹性壁厚度对传递损失的影响, figureFileSmall=kKdDqLXnEyQ6mYOMGS5dvQ==, figureFileBig=hxPVlGerUBtT4XYZE2kB1w==, tableContent=null), ArticleFig(id=1281323838965256524, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=EN, label=Fig.9, caption=Influence of flexible wall Young's modulus on transmission loss, figureFileSmall=LE8z8wW4NL/ZJwkm4OsZ0A==, figureFileBig=RdOxlvjImLZhzWuEo5t8FA==, tableContent=null), ArticleFig(id=1281323839044948301, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=CN, label=图9, caption=弹性壁杨氏模量对传递损失的影响, figureFileSmall=LE8z8wW4NL/ZJwkm4OsZ0A==, figureFileBig=RdOxlvjImLZhzWuEo5t8FA==, tableContent=null), ArticleFig(id=1281323839216914766, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=EN, label=Tab.1, caption=

Geometric and material parameters of the muffler with a flexible back cavity

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L1 /mmL2 /mmL3 /mmR1 /mmR2 /mmR3 /mmEs /GPahs /mm $ \mu $ $ {\rho }_{\mathrm{s}} $ /（kg·m−3）
20060020050150503060.481200
), ArticleFig(id=1281323839288217935, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=CN, label=表1, caption=

弹性背腔扩张式消声器的几何参数与材料参数

, figureFileSmall=null, figureFileBig=null, tableContent=
L1 /mmL2 /mmL3 /mmR1 /mmR2 /mmR3 /mmEs /GPahs /mm $ \mu $ $ {\rho }_{\mathrm{s}} $ /（kg·m−3）
20060020050150503060.481200
), ArticleFig(id=1281323839355326800, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=EN, label=Tab.2, caption=

Natural frequency of cylindrical shells under classical boundary conditions （Hz）

, figureFileSmall=null, figureFileBig=null, tableContent=
边界条件模态阶数理论结果FEM
结果		误差
（%）
k/K=109k/K=1010k/K=1011k/K=1012k/K=1013k/K=1014
C-C1722.34792.64811.03813.18813.39813.41810.640.31
2792.79934.30964.54967.92968.26968.30967.160.08
31095.711118.661125.661126.511126.601126.601119.200.65
41336.791425.631449.581452.431452.721452.751448.600.26
51375.561468.561484.221486.161486.361486.381476.600.65
61418.831595.661629.761633.371633.731633.771633.200.01
71698.031706.371709.181709.541709.571709.571692.800.99
81771.741924.391932.761933.841933.951933.961913.501.06
S-S1698.60773.90794.20796.58796.82796.85794.760.23
2753.19909.24942.36946.06946.43946.47946.250.02
31084.201108.921116.911117.891117.991118.001111.000.62
41259.001372.801403.601407.251407.621407.661405.400.13
51342.971427.531447.261449.731449.991450.011441.900.54
61369.691580.471616.731620.561620.941620.981621.000.03
71676.541700.121703.531703.971704.011704.021687.600.97
81690.921881.671906.721908.121908.261908.271889.301.00
), ArticleFig(id=1281323839426629969, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=CN, label=表2, caption=

经典边界条件下圆柱壳的固有频率（Hz）

, figureFileSmall=null, figureFileBig=null, tableContent=
边界条件模态阶数理论结果FEM
结果		误差
（%）
k/K=109k/K=1010k/K=1011k/K=1012k/K=1013k/K=1014
C-C1722.34792.64811.03813.18813.39813.41810.640.31
2792.79934.30964.54967.92968.26968.30967.160.08
31095.711118.661125.661126.511126.601126.601119.200.65
41336.791425.631449.581452.431452.721452.751448.600.26
51375.561468.561484.221486.161486.361486.381476.600.65
61418.831595.661629.761633.371633.731633.771633.200.01
71698.031706.371709.181709.541709.571709.571692.800.99
81771.741924.391932.761933.841933.951933.961913.501.06
S-S1698.60773.90794.20796.58796.82796.85794.760.23
2753.19909.24942.36946.06946.43946.47946.250.02
31084.201108.921116.911117.891117.991118.001111.000.62
41259.001372.801403.601407.251407.621407.661405.400.13
51342.971427.531447.261449.731449.991450.011441.900.54
61369.691580.471616.731620.561620.941620.981621.000.03
71676.541700.121703.531703.971704.011704.021687.600.97
81690.921881.671906.721908.121908.261908.271889.301.00
), ArticleFig(id=1281323839493738834, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=EN, label=Tab.3, caption=

First eight natural frequencies of different noise reduction structures

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频率类型不同模态阶数的固有频率 /（Hz）
12345678
刚性背腔1112.21621.81815.02592.42945.53213.93750.03874.9
弹性背腔304.6321.7464.2477.3558.4584.8623.2674.5
), ArticleFig(id=1281323839565042003, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=CN, label=表3, caption=

不同消声结构的前八阶固有频率

, figureFileSmall=null, figureFileBig=null, tableContent=
频率类型不同模态阶数的固有频率 /（Hz）
12345678
刚性背腔1112.21621.81815.02592.42945.53213.93750.03874.9
弹性背腔304.6321.7464.2477.3558.4584.8623.2674.5
), ArticleFig(id=1281323839711842644, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=EN, label=Tab.4, caption=

Effect of Young’s modulus on natural frequency

, figureFileSmall=null, figureFileBig=null, tableContent=
杨氏模量
$ {E}_{\mathrm{s}} $ /GPa		不同模态阶数的固有频率 /（Hz）
12345678
10175.9185.9268.2277.4322.8360.2367.7390.9
30304.6321.7464.2477.3558.4584.8623.2674.5
210802.9843.9965.91192.51221.91460.31578.11634.5
), ArticleFig(id=1281323839791534421, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=CN, label=表4, caption=

杨氏模量$ {E}_{\mathrm{s}} $对固有频率的影响

, figureFileSmall=null, figureFileBig=null, tableContent=
杨氏模量
$ {E}_{\mathrm{s}} $ /GPa		不同模态阶数的固有频率 /（Hz）
12345678
10175.9185.9268.2277.4322.8360.2367.7390.9
30304.6321.7464.2477.3558.4584.8623.2674.5
210802.9843.9965.91192.51221.91460.31578.11634.5
), ArticleFig(id=1281323839862837590, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=EN, label=Tab.5, caption=

Effect of density on natural frequency

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密度
$ {\rho }_{\mathrm{s}} $ /（kg·m−3）		不同模态阶数的固有频率 /（Hz）
12345678
1200304.6321.7464.2477.3558.4584.8623.2674.5
2700280.9301.6421.2454.7513.0563.8580.5630.1
7800228.6253.8332.9395.8414.5443.2523.8525.9
), ArticleFig(id=1281323839938335063, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=CN, label=表5, caption=

密度$ {\rho }_{\mathrm{s}} $对固有频率的影响

, figureFileSmall=null, figureFileBig=null, tableContent=
密度
$ {\rho }_{\mathrm{s}} $ /（kg·m−3）		不同模态阶数的固有频率 /（Hz）
12345678
1200304.6321.7464.2477.3558.4584.8623.2674.5
2700280.9301.6421.2454.7513.0563.8580.5630.1
7800228.6253.8332.9395.8414.5443.2523.8525.9
), ArticleFig(id=1281323840055775576, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=EN, label=Tab.6, caption=

Effect of thickness on natural frequency

, figureFileSmall=null, figureFileBig=null, tableContent=
厚度
$ {h}_{\mathrm{s}} $ /（mm）		不同模态阶数的固有频率 /（Hz）
12345678
4215.6261.6278.3394.9418.6432.4437.2492.8
6304.6321.7464.2477.3558.4584.8623.2674.5
8374.9405.1544.2654.1674.7687.8778.0855.0
), ArticleFig(id=1281323840122884441, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1281323822972375305, language=CN, label=表6, caption=

厚度$ {h}_{\mathrm{s}} $对固有频率的影响

, figureFileSmall=null, figureFileBig=null, tableContent=
厚度
$ {h}_{\mathrm{s}} $ /（mm）		不同模态阶数的固有频率 /（Hz）
12345678
4215.6261.6278.3394.9418.6432.4437.2492.8
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弹性背腔扩张式消声器声学特性分析
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侯九霄 , 袁苏伟 , 朱海潮
船舶力学 | 流体与结构声学 2026,30(3): 488-497
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船舶力学 |流体与结构声学 2026 , 30 (3) : 488 -497
弹性背腔扩张式消声器声学特性分析
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侯九霄 , 袁苏伟 , 朱海潮
作者信息
  • 海军工程大学 船舶振动噪声重点实验室，武汉 430033
通讯作者:
侯九霄（1990–），男，讲师，通讯作者，E-mail：
作者简介:

袁苏伟（1997–），男，博士研究生，E-mail：

Acoustic characteristics analysis of expansion muffler with flexible back cavity
Jiu-xiao HOU , Su-wei YUAN , Hai-chao ZHU
Affiliations
  • National Key Laboratory on Ship Vibration and Noise, Naval University of Engineering, Wuhan 430033, China
出版时间: 2026-03-15 doi: 10.3969/j.issn.1007-7294.2026.03.013
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摘要
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弹性背腔扩张式消声器相较于刚性背腔扩张式消声器具有更优良的低频消声性能。目前针对弹性背腔扩张式消声器的计算方法多为基于平面波假设的传递矩阵法，此类方法的计算误差随着声固耦合作用的增强而变大，且传统计算方法难以获得弹性边界条件下弹性结构的模态函数。因此，本文提出了一种基于能量原理、不依赖于平面波假设的计算方法。通过子结构分解法将弹性背腔扩张式消声器模型分解为三个子声腔，随后将声场的声压函数及弹性结构的位移函数分别展开为三维与二维切比雪夫级数形式，结合Rayleigh-Ritz法求解切比雪夫级数中的未知系数，从而得到消声器内部声压及传递损失，通过与有限元结果的对比验证了理论模型的正确性。最后，分析了其耦合特性并研究了边界约束和消声器参数对传递损失的影响规律。结果表明，弹性背腔扩张式消声器相较于刚性背腔具有更低的固有频率和更强的低频耦合作用。边界约束对传递损失的影响主要体现在1000 Hz以上，释放边界约束可使传递损失峰值向低频移动并增大峰值，从而提升消声性能。此外，背腔长度或半径的增大会使传递损失曲线向低频移动，其中背腔半径对消声器的声学性能影响更为显著；弹性壁厚度或杨氏模量的减小会使传递损失曲线进一步向低频移动。

弹性背腔  /  耦合特性  /  边界约束  /  传递损失  /  能量原理

Compared with an expansion muffler with a rigid back cavity, the expansion muffler with a flexible back cavity has better low-frequency noise reduction performance. However, most of the current calculation methods for expansion mufflers with the flexible back cavity are based on transfer matrix methods under the plane wave assumption. The calculation error of such methods increases with acoustic-structure coupling, and it is difficult to obtain the modal function of the flexible structure under elastic boundary conditions with traditional calculation methods. Therefore, this paper proposed a calculation method based on the energy principle that does not rely on the plane wave assumption. The model of the muffler is decomposed into three sub-acoustic cavities, and these cavities are coupled to each other through coupling surfaces. Then, the sound pressure function of the sound field and the displacement function of the flexible structure are expanded into three-dimensional and two-dimensional Chebyshev series respectively, and the Rayleigh-Ritz method was used to solve the unknown coefficients in the Chebyshev series. The sound pressure and transmission loss of the muffler were obtained, and the correctness of the theoretical model was verified by comparing it with the FEM results. Finally, the coupling characteristics were analyzed, and the effects of boundary constraints and muffler parameters on transmission loss were studied. The results show that the expansion muffler with a flexible back cavity has a lower natural frequency and stronger low-frequency coupling effect than that with a rigid back cavity. The impact of boundary constraints on transmission loss is mainly reflected above 1000 Hz. When boundary constraints are released, the peak value of transmission loss moves to low frequencies and increases. This shift is conducive to improving sound attenuation performance. As the back cavity’s length or radius increases, the transmission loss curve moves to the low frequency, and the influence of back cavity’s radius on the acoustic performance of the muffler is more obvious. As the thickness of the flexible wall or Young's modulus decreases, the transmission loss curve will move further to the low frequency.

flexible back cavity  /  coupling characteristics  /  boundary constraint  /  transmission loss  /  energy principle
侯九霄, 袁苏伟, 朱海潮. 弹性背腔扩张式消声器声学特性分析. 船舶力学, 2026 , 30 (3) : 488 -497 . DOI: 10.3969/j.issn.1007-7294.2026.03.013
Jiu-xiao HOU, Su-wei YUAN, Hai-chao ZHU. Acoustic characteristics analysis of expansion muffler with flexible back cavity[J]. Journal of Ship Mechanics, 2026 , 30 (3) : 488 -497 . DOI: 10.3969/j.issn.1007-7294.2026.03.013
正文
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0　引　言
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管路系统广泛应用于船舶领域，目前船舶管路多使用刚性壁扩张式消声器进行管路消声，然而其尺寸过大且低频消声性能有待提升。Huang[1−2]提出了一种管道膜式消声器，将管路中的刚性管壁替换为一段张紧的弹性薄膜，弹性薄膜与声场的耦合作用增强了内部声波的反射作用，在低频取得了较好的宽带消声效果且提高了消声器的流通性。由于膜结构需要非常大的张紧力，在工程应用中较难实现，Huang[3]将弹性薄膜替换为弹性板并建立了板式消声器的理论模型；随后Wang等[4−5]对固支边界条件下板式消声器进行了优化设计及试验验证。Du等[6]分析了不规则背腔板式消声器的声学性能，发现通过改变背腔倾角，可以降低最低吸声频率。Liu等[7]的研究表明，在非均匀边界条件下，弹性板侧边的约束是影响板式消声器耦合效应及优化设计时的重要参数。何涛等[8−9]基于无限长管道和封闭空间的声格林函数法，结合弹性板的简正模态理论，建立了矩形截面低频宽带板式水动力噪声消声理论，随后进行了参数化分析。刘晓昂等[10]针对车用进气管路中的噪声问题，以柔性壁扩张式消声器为研究对象，对比分析了刚性壁消声器与柔性壁消声器的消声性能，并基于正交试验分析了各个参数对传递损失的影响。侯九霄等[11]基于模态叠加法和Kirchhoff-Helmholtz积分公式建立了水介质圆形截面弹性壁扩张式消声器的结构–声耦合模型。
现有文献多基于平面波假设来求解弹性壁消声结构。然而随着声固耦合作用的增强，平面波假设的适用性降低，导致求解结果存在偏差。另外，模态叠加法局限于经典边界条件，弹性边界条件下弹性结构的模态函数难以获取，难以处理实际工程中存在的弹性边界。本文采用能量原理，将声压和位移展开为Chebyshev级数，结合Rayleigh-Ritz法求解消声器内部声压及传递损失，并通过有限元结果对理论结果进行验证。此外，将本文方法的求解结果与传统的模态叠加法的结果进行对比。最后分析了结构边界和参数对传递损失的影响规律。本文提出的计算方法为消声器优化设计提供了一定的理论指导。
1　理论推导
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1.1　模型描述
弹性背腔扩张式消声器结构示意图如图1所示。采用结构分解法将消声器结构分解为3个子声腔（$ {V}_{1} $，$ {V}_{2} $，$ {V}_{3} $），1个弹性圆柱壳$ {S}_{1} $，2个子声腔耦合面（$ {\varOmega }_{1} $，$ {\varOmega }_{2} $）。其中，入口声腔$ {V}_{1} $与主声腔$ {V}_{2} $通过耦合面$ {\varOmega }_{1} $相互耦合，主声腔$ {V}_{2} $与出口声腔$ {V}_{3} $通过耦合面$ {\varOmega }_{2} $相互耦合，弹性圆柱壳$ {S}_{1} $与主腔室$ {V}_{2} $相互作用构成弹性背腔。采用柱坐标系（o-x, $ \theta $, r），子声腔的长度和半径分别为$ {L}_{i} $和$ {R}_{i} $（i=1，2，3），介质声速为$ {c}_{0} $，密度为$ {\rho }_{0} $。弹性圆柱壳密度为$ {\rho }_{\mathrm{s}} $，杨氏模量为$ {E}_{\mathrm{s}} $，泊松比为$ {\mu }_{\mathrm{s}} $，厚度为$ {h}_{\mathrm{s}} $。圆柱壳在轴向、周向和径向的位移分别用$ u $、$ v $和$ w $表示。在子声腔$ {V}_{1} $的入口端面设置源强为$ {Q}_{0} $的面声源来获得平面波激励[12]。子声腔$ {V}_{3} $的出口端面定义为无反射边界，可通过设置出口阻抗为$ {\rho }_{0}{c}_{0} $来实现。
壳体两端分别设置四组均匀分布的弹簧约束，三组线性弹簧（$ {k}_{x} $，$ {k}_{\theta } $，$ {k}_{r} $）和一组旋转弹簧（$ {K}_{r} $），在$ {x}_{2}=0 $和$ {x}_{2}={L}_{2} $处分别记为$ {k}_{x0} $，$ {k}_{\theta 0} $，$ {k}_{r0} $，$ {K}_{r0} $和$ {k}_{x1} $，$ {k}_{\theta 1} $，$ {k}_{r1} $，$ {K}_{r1} $。通过改变弹簧刚度系数的大小来实现任意边界的设置，$ k=K=0 $ N/m表示自由边界，$ k={10}^{12} $ N/m且$ K=0 $ N/m表示简支边界，$ k=K={10}^{12} $ N/m表示固支边界，当$ k $和$ K $取中间值时可模拟弹性边界条件[7]。
1.2　能量原理求解
（1）声压与结构位移的级数形式
按照瑞利-里兹法，需选择合适的试函数对耦合模型的拉格朗日函数进行求解。Chebyshev级数具有良好的收敛性[12]，因此本文选用Chebyshev级数作为半径r及轴线x方向上的容许函数。然而，Chebyshev级数不具有对称性，而圆柱形耦合声腔在圆周$ \theta $方向具有对称模态和反对称模态，单纯使用Chebyshev级数无法表征这种对称模态和反对称模态，所以在圆周$ \theta $方向选用具有奇偶对称性的傅里叶级数，从而，圆柱形声腔内的声压及圆柱壳结构位移函数可表示为
$ {p}_{i}\left(x，r，\theta \right)=\sum\limits_{m=0}^{\mathrm{\infty }}\sum\limits_{n=0}^{\mathrm{\infty }}\sum\limits_{l=0}^{\mathrm{\infty }}A_{mnl}^{i}{T}_{m}\left(x\right){T}_{n}\left(r\right)\cos \left(l\theta \right)+\sum\limits_{m=0}^{\mathrm{\infty }}\sum\limits_{n=0}^{\mathrm{\infty }}\sum\limits_{l=1}^{\mathrm{\infty }}\overline{A}_{mnl}^{i}{T}_{m}\left(x\right){T}_{n}\left(r\right)\sin \left(l\theta \right) $
$ \begin{aligned}u(x，\theta )&=\sum\limits_{m=0}^{\mathrm{\infty }}\sum\limits_{l=0}^{\mathrm{\infty }}B_{ml}^{1}{T}_{m}\left(x\right)\cos \left(l\theta \right)+\sum\limits_{m=0}^{\mathrm{\infty }}\sum\limits_{l=1}^{\mathrm{\infty }}\overline{B}_{ml}^{1}{T}_{m}\left(x\right)\sin \left(l\theta \right)\\v(x，\theta )&=\sum\limits_{m=0}^{\mathrm{\infty }}\sum\limits_{l=0}^{\mathrm{\infty }}B_{ml}^{2}{T}_{m}\left(x\right)\cos \left(l\theta \right)+\sum\limits_{m=0}^{\mathrm{\infty }}\sum\limits_{l=1}^{\mathrm{\infty }}\overline{B}_{ml}^{2}{T}_{m}\left(x\right)\sin \left(l\theta \right)\\w(x，\theta )&=\sum\limits_{m=0}^{\mathrm{\infty }}\sum\limits_{l=0}^{\mathrm{\infty }}B_{ml}^{3}{T}_{m}\left(x\right)\cos \left(l\theta \right)+\sum\limits_{m=0}^{\mathrm{\infty }}\sum\limits_{l=1}^{\mathrm{\infty }}\overline{B}_{ml}^{3}{T}_{m}\left(x\right)\sin \left(l\theta \right)\end{aligned} $
其中，$ A_{mnl}^{i} $和$ \overline{A}_{mnl}^{i} $为声腔$ {V}_{i} $声压容许函数中的未知系数，$ B_{ml}^{1} $、$ \overline{B}_{ml}^{1} $、$ B_{ml}^{2} $、$ \overline{B}_{ml}^{2} $、$ B_{ml}^{3} $、$ \overline{B}_{ml}^{3} $分别为圆柱壳体在轴向x，周向$ \theta $及径向r的位移容许函数的未知系数。$ {T}_{m}(x) $和$ {T}_{n}(r) $为Chebyshev多项式，以$ {T}_{m}(x) $为例，其表达式为$ {T}_{m}(x)=\cos (m\arccos x) $。
此外，由于Chebyshev多项式定义区间为[−1，1]，需要进行坐标变换
$ \begin{array}{c}{\alpha }_{i}=\dfrac{2{x}_{i}}{{L}_{i}}-1\\{\gamma }_{i}=\dfrac{2{r}_{i}}{{R}_{i}}-1\end{array} $
（2）耦合结构的拉格朗日方程
分别构建圆柱声腔及圆柱壳的拉格朗日函数
$ \begin{array}{c}{L}_{c1}={U}_{c1}-{T}_{c1}-{W}_{12}-{W}_{Q}\\{L}_{c2}={U}_{c2}-{T}_{c2}-{W}_{s2c}+{W}_{21}-{W}_{23}\\{L}_{c3}={U}_{c3}-{T}_{c3}+{W}_{32}-{W}_{\mathrm{Zout}}\\{L}_{\mathrm{s}}={U}_{\mathrm{s}}+{U}_{\mathrm{sp}}-{T}_{\mathrm{s}}+{W}_{\mathrm{c}2\mathrm{s}}\end{array} $
其中，$ {U}_{\mathrm{c}i}，{T}_{\mathrm{c}i}，{W}_{Q}，{W}_{ij}，{W}_{\mathrm{Zout}} $分别表示声场$ {V}_{i} $的声势能、声动能、声源做功、耦合面做功和出口阻抗做功，具体表达式为
$ {U}_{\mathrm{c}i}=\frac{{R}_{i}{}^{2}{L}_{i}}{16{\rho }_{0}{c}_{0}{}^{2}}\int\limits_{0}^{2{\text{π}} }\int\limits_{-1}^{1}\int\limits_{-1}^{1}\left({\gamma }_{i}+1\right){p}_{i}{}^{2}{\text{d}} {\alpha }_{i}{\text{d}} {\gamma }_{i}{\text{d}} {\theta }_{i} $
$ {T}_{\mathrm{c}i}=\frac{{R}_{i}{}^{2}{L}_{i}}{4{\rho }_{0}{\omega }^{2}}\int\limits_{0}^{2{\text{π}} }\int\limits_{-1}^{1}\int\limits_{-1}^{1}\left[\frac{\left({\gamma }_{i}+1\right)}{{L}_{i}{}^{2}}{\left(\frac{\partial {p}_{i}}{\partial {\alpha }_{i}}\right)}^{2}+\frac{\left({\gamma }_{i}+1\right)}{{R}_{i}{}^{2}}{\left(\frac{\partial {p}_{i}}{\partial {\gamma }_{i}}\right)}^{2}+\frac{1}{{R}_{i}{}^{2}\left({\gamma }_{i}+1\right)}{\left(\frac{\partial {p}_{i}}{\partial {\theta }_{i}}\right)}^{2}\right]{\mathrm{d}}{\alpha }_{i}{\text{d}} {\gamma }_{i}{\text{d}} {\theta }_{i} $
$ {W}_{Q}=\int\frac{{Q}_{0}{\left.{p}_{1}\right| }_{{{\alpha }_{1}}=-1}}{\text{j}\omega }{\mathrm{d}}s $
$ {W}_{ij}=-\frac{R_{i}^{2}}{4\rho {\omega }^{2}{L}_{i}}\int\limits_{0}^{2{\text{π}} }\int\limits_{-1}^{1}\left({\gamma }_{i}+1\right)\left(\frac{{p}_{i}\partial {p}_{i}}{{L}_{i}\partial {\alpha }_{i}}-\frac{{p}_{j}\partial {p}_{i}}{{L}_{i}\partial {\alpha }_{i}}+\frac{{p}_{i}\partial {p}_{j}}{{L}_{j}\partial {\alpha }_{j}}\right){\mathrm{d}}{\gamma }_{i}{\text{d}} {\theta }_{i} $
$ {W}_{Z\mathrm{out}}=-\frac{{R}_{3}{}^{2}}{8\text{j}\omega {Z}_{\mathrm{out}}}\int\limits_{0}^{2{\text{π}} }\int\limits_{-1}^{1}({\gamma }_{3}+1){p}_{3}{}^{2}(1,{\gamma }_{3},{\theta }_{3}){\mathrm{d}}{\gamma }_{3}{\text{d}} {\theta }_{3} $
Sanders壳体理论能较好地表征薄壁圆柱壳的固有特性[13]，且具有较好的计算精度，因此本文基于Sanders薄壳理论构建圆柱壳耦合结构动力学模型。$ {U}_{\mathrm{s}}、{T}_{\mathrm{s}}、{U}_{\mathrm{sp}} $分别表示圆柱壳应变势能、动能和边界势能，表达式为
$\begin{split} {U}_{\mathrm{s}}=&\frac{{L}_{2}{R}_{2}{E}_{\mathrm{s}}{h}_{\mathrm{s}}}{4\left(1-{\mu }^{2}\right)}\int\limits_{0}^{2{\text{π}} }\int\limits_{-1}^{1}\left\{{\left(\frac{2}{{L}_{2}}\frac{\partial u}{\partial \alpha }+\frac{\partial v}{{R}_{2}\partial \theta }+\frac{w}{{R}_{2}}\right)}^{2}-\frac{4(1-\mu )}{{L}_{2}}\frac{\partial u}{\partial \alpha }\left(\frac{\partial v}{{R}_{2}\partial \theta }+\frac{w}{{R}_{2}}\right)+\frac{(1-\mu )}{2}{\left(\frac{2}{{L}_{2}}\frac{\partial v}{\partial \alpha }+\frac{\partial u}{{R}_{2}\partial \theta }\right)}^{2}\right\}{\mathrm{d}}\alpha{\mathrm{ d}}\theta +\\&\frac{{L}_{2}{R}_{2}{E}_{\mathrm{s}}h_{\mathrm{s}}^{3}}{48\left(1-{\mu }^{2}\right)}\int\limits_{0}^{2{\text{π}} }\int\limits_{-1}^{1}\left\{{\left(\frac{4}{L_{2}^{2}}\frac{{\partial }^{2}w}{\partial {\alpha }^{2}}+\frac{{\partial }^{2}w}{R_{2}^{2}\partial {\theta }^{2}}\right)}^{2}\right.\left.-2(1-\mu )[\frac{4}{L_{2}^{2}R_{2}^{2}}\frac{{\partial }^{2}w}{\partial {\alpha }^{2}}\frac{{\partial }^{2}w}{\partial {\theta }^{2}}-{\left(\frac{2}{{L}_{2}{R}_{2}}\frac{{\partial }^{2}w}{\partial \alpha \partial \theta }\right)}^{2}]\right\}{\mathrm{d}}\alpha {\mathrm{d}}\theta+ \\& \frac{{L}_{2}{E}_{\mathrm{s}}h_{\mathrm{s}}^{3}}{48{R}_{2}\left(1-{\mu }^{2}\right)}\int\limits_{0}^{2{\text{π}} }\int\limits_{-1}^{1}\left\{-\frac{8\mu }{L_{2}^{2}}\frac{\partial v}{\partial \theta }\frac{{\partial }^{2}w}{\partial {\alpha }^{2}}-\frac{2}{R_{2}^{2}}\frac{\partial v}{\partial \theta }\frac{{\partial }^{2}w}{\partial {\theta }^{2}}+{\left(\frac{\partial v}{{R}_{2}\partial \theta }\right)}^{2}-\frac{16(1-\mu )}{L_{2}^{2}}\frac{\partial v}{\partial \alpha }\frac{{\partial }^{2}w}{\partial \alpha \partial \theta }+\frac{8(1-\mu )}{L_{2}^{2}}{\left(\frac{\partial v}{\partial \alpha }\right)}^{2}\right\}{\mathrm{d}}\alpha {\mathrm{d}}\theta \end{split} $
其中，$ \mu $和$ {E}_{\mathrm{s}} $分别为泊松比和杨氏模量。
$ {T}_{\mathrm{s}}\text=\,=\frac{{\omega }^{2}{L}_{2}{R}_{2}{\rho }_{\mathrm{s}}{h}_{\mathrm{s}}}{4}\int\limits_{0}^{2{\text{π}} }\int\limits_{-1}^{1}\left[{u}^{2}+{v}^{2}+{w}^{2}\right]{\mathrm{d}}\alpha {\mathrm{d}}\theta $
$ {U}_{\mathrm{sp}}=\frac{1}{2}{\int\limits_{0}^{2{\text{π}} }\left[{k}_{x0}{u}^{2}+{k}_{\theta 0}{v}^{2}+{k}_{r0}{w}^{2}+\frac{4}{L_{2}^{2}}{K}_{r0}{\left(\frac{\partial w}{\partial \alpha }\right)}^{2}\right]}_{\alpha =-1}{R}_{2}{\mathrm{d}}\theta+\frac{1}{2}{\int\limits_{0}^{2{\text{π}} }\left[{k}_{x1}{u}^{2}+{k}_{\theta 1}{v}^{2}+{k}_{r1}{w}^{2}+\frac{4}{L_{2}^{2}}{K}_{r1}{\left(\frac{\partial w}{\partial \alpha }\right)}^{2}\right]}_{\alpha =1}{R}_{2}{\mathrm{d}}\theta $
其中，$ {W}_{\mathrm{s2c}} $为圆柱壳振动对声腔$ {V}_{2} $的做功项，$ {W}_{\mathrm{c2s}} $为声腔$ {V}_{2} $声压变化对圆柱壳的做功项，由牛顿第二定律可知$ {W}_{\mathrm{s2c}}={W}_{\mathrm{c2s}} $，其表达式为
$ {W}_{\mathrm{s}2 \mathrm{c}}={W}_{\mathrm{c}2 \mathrm{s}}=\frac{{L}_{2}{R}_{2}}{2}\int\limits_{0}^{2{\text{π}} }\int\limits_{-1}^{1}w(\alpha ,\theta ){p}_{2}(\alpha ,1,\theta ){\mathrm{d}}\alpha {\mathrm{d}}\theta $
（3）求解过程
采用里兹法对公式中的所有未知系数取极值
$ \begin{array}{l}\dfrac{\partial {L}_{\mathrm{c}1}}{\partial A_{mnl}^{1}}=0，\dfrac{\partial {L}_{\mathrm{c}2}}{\partial A_{mnl}^{2}}=0，\dfrac{\partial {L}_{\mathrm{c}3}}{\partial A_{mnl}^{3}}=0，\\\dfrac{\partial {L}_{\mathrm{c}1}}{\partial \overline{A}_{mnl}^{1}}=0，\dfrac{\partial {L}_{\mathrm{c}2}}{\partial \overline{A}_{mnl}^{2}}=0，\dfrac{\partial {L}_{\mathrm{c}3}}{\partial \overline{A}_{mnl}^{3}}=0，\\\dfrac{\partial {L}_{\mathrm{s}}}{\partial B_{ml}^{1}}=0，\,\dfrac{\partial {L}_{\mathrm{s}}}{\partial B_{ml}^{2}}=0\,，\dfrac{\partial {L}_{\mathrm{s}}}{\partial B_{ml}^{3}}=0，\\\dfrac{\partial {L}_{\mathrm{s}}}{\partial \overline{B}_{ml}^{1}}=0，\dfrac{\partial {L}_{\mathrm{s}}}{\partial \overline{B}_{ml}^{2}}=0，\dfrac{\partial {L}_{\mathrm{s}}}{\partial \overline{B}_{ml}^{3}}=0\end{array} $
得到12组线性方程，以矩阵形式描述为
$ (\boldsymbol{K}+\omega \boldsymbol{Z}-{\omega }^{2}\boldsymbol{M})\boldsymbol{E}=\boldsymbol{Q} $
通过求解，系数矩阵为
$ \boldsymbol{E}={(\boldsymbol{K}+\omega \boldsymbol{Z}-{{\omega }^{2}}\boldsymbol{M})}^{-1}\boldsymbol{Q} $
将系数代入公式（1）和（2），即可得到耦合结构内任一点处的声压响应以及圆柱壳上任一点的振动响应。
1.3　传递损失
为了反映消声器的消声性能，采用传递损失作为评价指标，传递损失计算公式为
$ TL=10\mathrm{\lg }\left(\frac{{W}_{{\mathrm{in}}}}{{W}_{{\mathrm{out}}}}\right)=20\mathrm{\lg }\left| \frac{{p}_{{\mathrm{in}}}}{{p}_{{\mathrm{out}}}}\right| $
其中，$ {W}_{{\mathrm{in}}} $和$ {W}_{{\mathrm{out}}} $分别为消声器入口的入射声功率和消声器出口的透射声功率，对于进出口截面相同的消声器，可用入口入射声压$ {p}_{{\mathrm{in}}} $及出口透射声压$ {p}_{{\mathrm{out}}} $来计算。
2　理论验证
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取文献[11]中参数对本文计算方法进行理论模型的验证，具体参数如表1所示。
2.1　边界弹簧刚度值的确定
模拟结构边界的固支或者简支时，对于不同的结构形式，弹簧刚度值有不同的取值，取值不宜过大，否则将导致刚度矩阵演变为病态矩阵。经典边界条件下圆柱壳的固有频率见表2。其中，C-C表示圆柱壳两端固支，即两端的四个边界弹簧均设置为无穷大，S-S表示圆柱壳两端简支，即两端的线性弹簧k为无穷大而旋转弹簧K为零。表中给出了当刚度值取1012时，前八阶固有频率的理论结果与FEM结果的误差值，两端固支时最大误差为1.06%，两端简支时最大误差为1.00%，表明理论计算的准确性。当刚度值取1012时，圆柱壳前八阶固有频率已经收敛，在实际计算时证明该值已经足够大可以模拟无穷大值，无特别说明，后文中均选取1012来模拟刚度值为无穷大的情形。
2.2　有限元验证
为进一步验证理论的准确性，采用COMSOL Multiphysics中声−壳耦合模块，求得边界条件为两端简支时，弹性背腔扩张式消声结构的传递损失曲线如图2所示。消声结构内部介质为水，$ {\rho }_{0} $=1000 kg/m3，$ {c}_{0} $=1500 m/s。通过对比发现，本文方法计算结果与有限元计算结果吻合较好，而传统的传递矩阵法与有限元结果存在偏差，并且这种偏差会随着弹性管壁厚度的减小而增大。因为传递矩阵法基于平面波假设，弹性管壁越薄，与内部声腔的耦合作用越强，非平面波化越严重，从而导致传递损失的计算结果出现偏差。而本文计算方法基于能量法，并没有平面波假设这一前提，因此计算结果更为准确。图3给出了两端固支时，耦合结构传递损失的理论计算结果与有限元计算结果，两者吻合较好，表明本文方法能较好地模拟不同结构边界条件下的耦合结构的耦合作用。在运用传递矩阵法求解弹性边界条件下的耦合振动时，其振动模态函数难以直接表示，而本方法只需要重新计算边界势能对应的刚度矩阵即可，计算更为简单高效。另外，值得一提的是，本文求解方法同样适用于空气介质消声器的声学分析，若将弹性壁的杨氏模量设置为无穷大即可模拟刚性背腔消声器。
3　耦合特性分析
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本节将讨论结构参数对耦合结构固有特性的影响，将公式（15）中声源激励向量$ \boldsymbol{Q} $置零，可计算得到耦合结构的固有频率。结构参数与上一节保持一致，内部介质为水，圆柱壳两端结构边界为固支。分别计算得到弹性背腔扩张式消声结构以及刚性背腔扩张式消声结构的固有频率，结果见表3。可以看出，弹性背腔具有更低的固有频率，低频耦合作用更强，这也印证了文献[11]中弹性背腔扩张式消声结构在低频范围内比刚性背腔扩张式消声结构具有更好的消声性能的结论。
表4~6给出了不同弹性管壁参数（$ {E}_{\mathrm{s}} $，$ {\rho }_{\mathrm{s}} $，$ {h}_{\mathrm{s}} $）下固有频率。可以看出，单独改变某一参数，耦合结构固有频率的变化规律相似。杨氏模量越低，密度越大，厚度越小，耦合结构的固有频率更低，这与文献[11]中传递损失峰值向低频移动的结论相印证。
4　边界约束分析
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参数与前文保持一致，本节探究圆柱壳两端边界约束对传递损失的影响。令两侧旋转约束K （$ {K}_{r0} $，$ {K}_{r1} $）为零，而线性约束无穷大，即简支边界。从图4中可以看出，传递损失峰值随着旋转约束的释放而向低频移动，但仅在中高频变化较为明显，在1000 Hz以下，旋转约束的影响并不大。
令两侧旋转约束K （$ {K}_{r0} $，$ {K}_{r1} $）为零，改变线性约束k （$ {k}_{x0} $，$ {k}_{\theta 0} $，$ {k}_{r0} $，$ {k}_{x1} $，$ {k}_{\theta 1} $，$ {k}_{r1} $，）。不同k值对应的传递损失曲线如图5所示。结果表明减小线性约束，传递损失曲线将进一步向低频移动，当k值取值范围为106~1010 N/m时，变化最为明显，当k <106 N/m时，边界趋于自由边界，当k >1010 N/m时，边界趋于简支边界，继续改变线性约束刚度值，传递损失曲线基本没有变化。
以上分析表明，圆柱壳结构两端边界条件对低频范围内的传递损失的影响较小，改变边界条件对1000 Hz以上产生一定影响，完全释放边界约束，传递损失峰值将向低频移动，且峰值增大，有利于提升结构的消声性能。
5　背腔参数分析
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5.1　背腔尺寸
设置弹性背腔的长度分别为L2=400 mm、L2=600 mm和L2=800 mm，其余参数与表1保持一致，得到不同背腔长度对弹性背腔扩张式消声器传递损失的影响如图6所示。设置弹性背腔的半径分别为R2=100 mm、R2=150 mm和R2=200 mm，得到不同背腔半径对传递损失的影响如图7所示。随着背腔长度和背腔半径的增大，传递损失曲线向低频移动，背腔半径的变化对声学性能影响较背腔长度更为显著，在安装空间允许的情况下可以适当增大背腔半径。
5.2　弹性壁参数
设置弹性壁的厚度分别为hs=4 mm、hs=6 mm和hs=8 mm，其余参数与表1保持一致，得到不同弹性壁厚度对弹性背腔扩张式消声器传递损失的影响如图8所示。设置弹性壁的杨氏模量分别为Es=20 GPa、Es=30 GPa和Es=40 GPa，得到不同杨氏模量对传递损失的影响如图9所示。综合上述分析可知，相比于刚性背腔消声器，弹性背腔消声器具有更优异的低频消声性能。随着弹性壁的厚度和杨氏模量的减小，传递损失曲线将进一步向低频移动。
6　结　论
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本文建立了一种基于能量原理求解弹性背腔扩张式消声器传递损失的计算方法，首先通过将声场及结构位移场展开为Chebyshev多项式级数，然后结合Rayleigh-Ritz法对消声器动力学方程进行了求解，进而得到传递损失。最后分析了消声结构的耦合特性以及边界约束对传递损失的影响，结果表明：
（1）本文计算方法与有限元结果吻合良好，且相较于传递矩阵法，本方法更加适用于弹性结构与内部声场的强耦合作用。
（2）弹性背腔扩张式消声器比刚性背腔扩张式消声器具有更低的固有频率，低频耦合作用更强；弹性壁的杨氏模量和厚度越小、密度越大，耦合作用越强。
（3）弹性壁的边界约束对低频范围内的传递损失影响较小，释放边界约束，对1000 Hz以上影响明显，传递损失峰值将向低频移动，且峰值增大，有利于提升结构的消声性能。
（4）随着背腔长度和背腔半径的增大，传递损失曲线向低频移动，背腔半径对消声器的声学性能影响更为明显；相比于刚性背腔消声器，弹性背腔消声器具有更优异的低频消声性能，随着弹性壁的厚度和杨氏模量的减小，传递损失曲线将进一步向低频移动。

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2026年第30卷第3期
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doi: 10.3969/j.issn.1007-7294.2026.03.013
  • 接收时间：2024-07-09
  • 首发时间：2026-07-07
  • 出版时间：2026-03-15
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  • 收稿日期：2024-07-09
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    海军工程大学 船舶振动噪声重点实验室，武汉 430033

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侯九霄（1990–），男，讲师，通讯作者，E-mail：
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2种不同金属材料的力学参数
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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
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