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Article(id=1241838552397447967, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241838547544641612, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2024.056, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1730736000000, receivedDateStr=2024-11-05, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1774006983136, onlineDateStr=2026-03-20, pubDate=1745337600000, pubDateStr=2025-04-23, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774006983136, onlineIssueDateStr=2026-03-20, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774006983136, creator=13701087609, updateTime=1774006983136, updator=13701087609, issue=Issue{id=1241838547544641612, tenantId=1146029695717560320, journalId=1241755870837649424, year='2025', volume='46', issue='2', pageStart='149', pageEnd='296', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1774006981978, creator=13701087609, updateTime=1774007061443, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241838880882758461, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241838547544641612, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241838880882758462, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241838547544641612, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=286, endPage=296, ext={EN=ArticleExt(id=1241838552632328994, articleId=1241838552397447967, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=A Scaled Model of Geometric and Material Distortion in the Free Vibration of Clamped Shallow Spherical Shells, columnId=1241831201674171363, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Papers, runingTitle=null, highlight=null, articleAbstract=

Shallow spherical shells are widely used in large structures, such as launch vehicles, nuclear containments, and gasholders. Studying scaled models of geometric and material distortion in the free vibration of large, thin-walled clamped shallow spherical shells is essential. First, frequency equations and natural frequency solutions for clamped shallow spherical shells are utilized to apply similarity transformation, leading to the derivation of generalized similarity conditions and scaling laws for the free vibration of these shells. The numerical approach is then validated by comparing the theoretical natural frequencies of clamped shallow spherical shells. Finally, geometric and material distortion similarity predictions for the free vibration of clamped shallow spherical shells with different rise-to-thickness ratios are investigated by numerical simulations. Results indicate that scaled models for clamped shallow spherical shells with different rise-to-thickness ratios, designed according to similarity conditions and incorporating the corresponding generalized similarity condition of mode shape and the natural frequency scaling relationship, can accurately predict the free vibration characteristics of the prototypes. For the free vibration of clamped shallow spherical shells with medium, small, and large rise-to-thickness ratios, in addition to meet the same mode shape, the distorted scaling laws also need to ensure that the scaling factors of rise-to-thickness ratio and Poisson's ratio equal 1, the scaling factor of Poisson's ratio is 1, and the rise-to-thickness ratios of scaled models and the prototype are no less than 25. The method and results of this study offer valuable insights for the design and experiments of scaled models for similar shells with thickness and material distortion.

, correspAuthors=Wei Yu, 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=Wei Yu, Shuai Wang), CN=ArticleExt(id=1241838579547177331, articleId=1241838552397447967, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=固支扁球壳自振的几何和材料畸变缩比模型研究, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

扁球壳广泛应用于运载火箭、核电安全壳和煤气柜等大型结构，研究大型薄壁固支扁球壳自振的几何和材料畸变缩比模型具有重要意义. 基于固支扁球壳的自振频率方程和固有频率解分别进行相似转换，获得结构自振的广义相似条件与固有频率缩比关系. 通过固支扁球壳的自振频率理论计算结果验证了数值模拟的准确性，并利用数值模拟开展不同矢厚比固支扁球壳自振的几何和材料畸变缩比模型研究. 结果表明：不同矢厚比固支扁球壳按照其相似条件设计的缩比模型，结合对应的广义相似条件振型模态关系和固有频率缩比关系能够准确预测其原型的自振特性；一般情形、矢厚比较小和矢厚比较大的固支扁球壳自振畸变相似条件，除均需满足振型相同，还需分别满足矢厚比与泊松比的缩比因子均为1，泊松比的缩比因子为1，及缩比模型和原型的矢厚比均大于等于25. 本文采用的方法和给出的结论可为类似壳体厚度与材料畸变缩比模型的设计及实验提供参考.

, correspAuthors=于伟, authorNote=null, correspAuthorsNote=
**E-mail：.
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(in Chinese)), articleTitle=Structural similitude for axiaymmetric free vibration of stiffened shallow spherical shells by energy method, refAbstract=null)], funds=[Fund(id=1241838585100435984, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, awardId=51868013; 52068015, language=CN, fundingSource=国家自然科学基金项目(51868013; 52068015), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1241838579832390015, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, xref=1, ext=[AuthorCompanyExt(id=1241838579840778624, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, companyId=1241838579832390015, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1Guangxi Key Laboratory of Green Building Materials and Construction Industrialization, Guilin University of Technology, Guilin, 541004), AuthorCompanyExt(id=1241838579849167233, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, companyId=1241838579832390015, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1桂林理工大学广西绿色建材与建筑工业化重点实验室，桂林，541004)]), AuthorCompany(id=1241838579920470405, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, xref=2, ext=[AuthorCompanyExt(id=1241838579928859014, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, companyId=1241838579920470405, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2Guangxi Key Laboratory of Geomechanics and Geotechnical Engineering, Guilin University of Technology, Guilin, 541004), AuthorCompanyExt(id=1241838579937247623, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, companyId=1241838579920470405, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2桂林理工大学广西岩土力学与工程重点实验室，桂林，541004)])], figs=[ArticleFig(id=1241838581619163591, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=EN, label=Fig.1, caption=Cross-sectional view of a shallow spherical shell, figureFileSmall=hq0crO7fa8bplAHKIUiWiA==, figureFileBig=8whM09ITDql0tOsFQivlAg==, tableContent=null), ArticleFig(id=1241838581686272458, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=CN, label=图1, caption=光滑扁球壳剖面图, figureFileSmall=hq0crO7fa8bplAHKIUiWiA==, figureFileBig=8whM09ITDql0tOsFQivlAg==, tableContent=null), ArticleFig(id=1241838581895987666, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=EN, label=Table 1, caption=

Geometrical and material parameters of the shallow spherical shell

, figureFileSmall=null, figureFileBig=null, tableContent=
几何参数λ底面半径b（m）矢高H（m）厚度h（m）弹性模量E（Pa）密度ρ（kg·m-3）泊松比ν
7.510.10.01182.1×101178500.3
), ArticleFig(id=1241838582021816791, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=CN, label=表1, caption=

一般光滑扁球壳的几何和材料参数

, figureFileSmall=null, figureFileBig=null, tableContent=
几何参数λ底面半径b（m）矢高H（m）厚度h（m）弹性模量E（Pa）密度ρ（kg·m-3）泊松比ν
7.510.10.01182.1×101178500.3
), ArticleFig(id=1241838583498211803, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=EN, label=Table 2, caption=

Comparison of simulated and theoretical results for the first six order modals of the clamped shallow spherical shell with λ=7.5

, figureFileSmall=null, figureFileBig=null, tableContent=
自振模态波数（m，n） αb模拟值（Hz）理论值（Hz）误差（%）模拟振型图
（0，1）5.555184.956187.969-1.60
（0，2）6.346200.041202.776-1.35
（0，3）7.364226.28229.159-1.26
（1，1）7.956244.74248.443-1.49
（0，4）8.468264.163267.456-1.23
（1，2）9.279297.702301.923-1.40
), ArticleFig(id=1241838583582097888, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=CN, label=表2, caption=

λ=7.5时固支扁球壳前六阶模态分析的模拟与理论结果对比

, figureFileSmall=null, figureFileBig=null, tableContent=
自振模态波数（m，n） αb模拟值（Hz）理论值（Hz）误差（%）模拟振型图
（0，1）5.555184.956187.969-1.60
（0，2）6.346200.041202.776-1.35
（0，3）7.364226.28229.159-1.26
（1，1）7.956244.74248.443-1.49
（0，4）8.468264.163267.456-1.23
（1，2）9.279297.702301.923-1.40
), ArticleFig(id=1241838583674372579, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=EN, label=Table 3, caption=

Geometric and material parameters of the shallow spherical shell prototype and scaled models

, figureFileSmall=null, figureFileBig=null, tableContent=
底面半径b（m）矢高H（m）厚度h（m）弹性模量E（Pa）密度ρ（kg·m-3）泊松比ν
原型10.10.01182.1×101178500.3
缩比模型110.020.002362.1×101178500.3
缩比模型20.10.01250.0014752.1×101178500.3
), ArticleFig(id=1241838583749870051, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=CN, label=表3, caption=

扁球壳原型与缩比模型的几何和材料参数

, figureFileSmall=null, figureFileBig=null, tableContent=
底面半径b（m）矢高H（m）厚度h（m）弹性模量E（Pa）密度ρ（kg·m-3）泊松比ν
原型10.10.01182.1×101178500.3
缩比模型110.020.002362.1×101178500.3
缩比模型20.10.01250.0014752.1×101178500.3
), ArticleFig(id=1241838583825367526, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=EN, label=Table 4, caption=

Similarity prediction for free vibration of the clamped shallow spherical shell prototype by the scaled model 1

, figureFileSmall=null, figureFileBig=null, tableContent=
自振模态波数（m，n）原型频率（Hz）模型频率（Hz）预测原型频率（Hz）误差（%）原型振型图缩比模型1振型图
（0，1）184.95637.5686187.8431.56
（0，2）200.04140.539202.6951.33
（0，3）226.2845.8354229.1771.28
（1，1）244.7449.694248.471.52
（0，4）264.16353.5054267.5271.27
（1，2）297.70260.4158302.0791.47
), ArticleFig(id=1241838583947002347, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=CN, label=表4, caption=

缩比模型1对固支扁球壳原型的自振相似预测

, figureFileSmall=null, figureFileBig=null, tableContent=
自振模态波数（m，n）原型频率（Hz）模型频率（Hz）预测原型频率（Hz）误差（%）原型振型图缩比模型1振型图
（0，1）184.95637.5686187.8431.56
（0，2）200.04140.539202.6951.33
（0，3）226.2845.8354229.1771.28
（1，1）244.7449.694248.471.52
（0，4）264.16353.5054267.5271.27
（1，2）297.70260.4158302.0791.47
), ArticleFig(id=1241838584039277037, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=EN, label=Table 5, caption=

Similarity prediction for free vibration of the clamped shallow spherical shell prototype by the scaled model 2

, figureFileSmall=null, figureFileBig=null, tableContent=
自振模态波数（m，n）原型频率（Hz）模型频率（Hz）预测原型频率（Hz）误差（%）原型振型图缩比模型2振型图
（0，1）184.9562291.1183.2880.9
（0，2）200.0412481.44198.5150.76
（0，3）226.282807.69224.6150.74
（1，1）244.743032.44242.5950.88
（0，4）264.1633277.9262.2320.73
（1，2）297.7023689.91295.19280.84
), ArticleFig(id=1241838584169300466, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=CN, label=表5, caption=

缩比模型2对固支扁球壳原型的自振相似预测

, figureFileSmall=null, figureFileBig=null, tableContent=
自振模态波数（m，n）原型频率（Hz）模型频率（Hz）预测原型频率（Hz）误差（%）原型振型图缩比模型2振型图
（0，1）184.9562291.1183.2880.9
（0，2）200.0412481.44198.5150.76
（0，3）226.282807.69224.6150.74
（1，1）244.743032.44242.5950.88
（0，4）264.1633277.9262.2320.73
（1，2）297.7023689.91295.19280.84
), ArticleFig(id=1241838584278352373, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=EN, label=Table 6, caption=

Geometric and material parameters of the shallow spherical shell prototype and the scaled model with a small rise-to-thickness ratio

, figureFileSmall=null, figureFileBig=null, tableContent=
底面半径b（m）矢高H（m）厚度h（m）弹性模量E（Pa）密度ρ（kg·m-3）泊松比ν
原型10.010.010582.1×101178500.3
缩比模型0.10.0010.0021162.1×101178500.3
), ArticleFig(id=1241838584353849848, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=CN, label=表6, caption=

矢厚比较小时扁球壳原型及缩比模型的材料和几何参数

, figureFileSmall=null, figureFileBig=null, tableContent=
底面半径b（m）矢高H（m）厚度h（m）弹性模量E（Pa）密度ρ（kg·m-3）泊松比ν
原型10.010.010582.1×101178500.3
缩比模型0.10.0010.0021162.1×101178500.3
), ArticleFig(id=1241838584450318843, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=EN, label=Table 7, caption=

Similarity prediction for free vibration of the clamped shallow spherical shell prototype by the scaled model with a small rise-to-thickness ratio

, figureFileSmall=null, figureFileBig=null, tableContent=
自振模态波数（m，n）原型频率（Hz）模型频率（Hz）预测原型频率（Hz）误差（%）原型振型图缩比模型振型图
（0，1）58.7841132.2356.6123.69
（0，2）93.5141840.192.0051.61
（1，1）161.1793194.99159.7500.89
（1，2）223.3834429.77221.4890.85
（2，1）316.8476275.7313.7850.97
（2，2）405.3378013.89400.6951.15
), ArticleFig(id=1241838584555176447, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=CN, label=表7, caption=

矢厚比较小缩比模型对固支扁球壳原型的自振相似预测

, figureFileSmall=null, figureFileBig=null, tableContent=
自振模态波数（m，n）原型频率（Hz）模型频率（Hz）预测原型频率（Hz）误差（%）原型振型图缩比模型振型图
（0，1）58.7841132.2356.6123.69
（0，2）93.5141840.192.0051.61
（1，1）161.1793194.99159.7500.89
（1，2）223.3834429.77221.4890.85
（2，1）316.8476275.7313.7850.97
（2，2）405.3378013.89400.6951.15
), ArticleFig(id=1241838584630673922, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=EN, label=Table 8, caption=

Geometric and material parameters of the shallow spherical shell prototype and scaled models with a large rise-to-thickness ratio

, figureFileSmall=null, figureFileBig=null, tableContent=
底面半径b（m）矢高H（m）厚度h（m）弹性模量E（Pa）密度ρ（kg·m-3）泊松比ν
原型32.36.160.00752.1×101178500.3
Fe制缩比模型0.80750.1540.002142.1×101178500.3
Al制缩比模型0.80750.1540.002146.875×101027620.3
Cu制缩比模型0.80750.1540.002141.24×101189180.33
PVC制缩比模型0.80750.1540.002143.79×10915000.4
), ArticleFig(id=1241838584769085959, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=CN, label=表8, caption=

矢厚比较大时扁球壳原型和缩比模型的材料与几何参数

, figureFileSmall=null, figureFileBig=null, tableContent=
底面半径b（m）矢高H（m）厚度h（m）弹性模量E（Pa）密度ρ（kg·m-3）泊松比ν
原型32.36.160.00752.1×101178500.3
Fe制缩比模型0.80750.1540.002142.1×101178500.3
Al制缩比模型0.80750.1540.002146.875×101027620.3
Cu制缩比模型0.80750.1540.002141.24×101189180.33
PVC制缩比模型0.80750.1540.002143.79×10915000.4
), ArticleFig(id=1241838584869749259, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=EN, label=Table 9, caption=

Similarity prediction for free vibration of the clamped shallow spherical shell prototype by scaled models with a large rise-to-thickness ratio

, figureFileSmall=null, figureFileBig=null, tableContent=
自振模态波数（m，n）原型频率（Hz）缩比模型频率（Hz）频率缩比系数频率（Hz）预测原型误差（%）原型振型图缩比模型振型图
Fe制缩比模型（1，0）9.255372.0600.0259.3020.50
（1，1）9.311374.8170.0259.3700.64
（1，2）9.337377.3020.0259.4331.02
Al制缩比模型（1，0）9.255358.8910.02599.3010.50
（1，1）9.311361.5510.02599.3640.57
（1，2）9.337363.9480.02599.4260.96
Cu制缩比模型（1，0）9.255268.1430.03479.2980.46
（1，1）9.311270.190.03479.3760.69
（1，2）9.337272.030.03479.4391.10
PVC制缩比模型（1，0）9.255114.2150.08139.2860.33
（1，1）9.311115.1500.08139.3620.54
（1，2）9.337115.9890.08139.4301.00
), ArticleFig(id=1241838584966218254, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838552397447967, language=CN, label=表9, caption=

矢厚比较大时缩比模型对固支扁球壳原型的自振相似预测

, figureFileSmall=null, figureFileBig=null, tableContent=
自振模态波数（m，n）原型频率（Hz）缩比模型频率（Hz）频率缩比系数频率（Hz）预测原型误差（%）原型振型图缩比模型振型图
Fe制缩比模型（1，0）9.255372.0600.0259.3020.50
（1，1）9.311374.8170.0259.3700.64
（1，2）9.337377.3020.0259.4331.02
Al制缩比模型（1，0）9.255358.8910.02599.3010.50
（1，1）9.311361.5510.02599.3640.57
（1，2）9.337363.9480.02599.4260.96
Cu制缩比模型（1，0）9.255268.1430.03479.2980.46
（1，1）9.311270.190.03479.3760.69
（1，2）9.337272.030.03479.4391.10
PVC制缩比模型（1，0）9.255114.2150.08139.2860.33
（1，1）9.311115.1500.08139.3620.54
（1，2）9.337115.9890.08139.4301.00
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固支扁球壳自振的几何和材料畸变缩比模型研究
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于伟 1, 2, ** , 王帅 1
固体力学学报 | 研究论文 2025,46(2): 286-296
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固体力学学报 | 研究论文 2025, 46(2): 286-296
固支扁球壳自振的几何和材料畸变缩比模型研究
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于伟1, 2, ** , 王帅1
作者信息
  • 1桂林理工大学广西绿色建材与建筑工业化重点实验室，桂林，541004
  • 2桂林理工大学广西岩土力学与工程重点实验室，桂林，541004

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A Scaled Model of Geometric and Material Distortion in the Free Vibration of Clamped Shallow Spherical Shells
Wei Yu1, 2, ** , Shuai Wang1
Affiliations
  • 1Guangxi Key Laboratory of Green Building Materials and Construction Industrialization, Guilin University of Technology, Guilin, 541004
  • 2Guangxi Key Laboratory of Geomechanics and Geotechnical Engineering, Guilin University of Technology, Guilin, 541004
出版时间: 2025-04-23 doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.056
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摘要
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扁球壳广泛应用于运载火箭、核电安全壳和煤气柜等大型结构，研究大型薄壁固支扁球壳自振的几何和材料畸变缩比模型具有重要意义. 基于固支扁球壳的自振频率方程和固有频率解分别进行相似转换，获得结构自振的广义相似条件与固有频率缩比关系. 通过固支扁球壳的自振频率理论计算结果验证了数值模拟的准确性，并利用数值模拟开展不同矢厚比固支扁球壳自振的几何和材料畸变缩比模型研究. 结果表明：不同矢厚比固支扁球壳按照其相似条件设计的缩比模型，结合对应的广义相似条件振型模态关系和固有频率缩比关系能够准确预测其原型的自振特性；一般情形、矢厚比较小和矢厚比较大的固支扁球壳自振畸变相似条件，除均需满足振型相同，还需分别满足矢厚比与泊松比的缩比因子均为1，泊松比的缩比因子为1，及缩比模型和原型的矢厚比均大于等于25. 本文采用的方法和给出的结论可为类似壳体厚度与材料畸变缩比模型的设计及实验提供参考.

扁球壳  /  自振  /  畸变  /  矢厚比  /  相似律

Shallow spherical shells are widely used in large structures, such as launch vehicles, nuclear containments, and gasholders. Studying scaled models of geometric and material distortion in the free vibration of large, thin-walled clamped shallow spherical shells is essential. First, frequency equations and natural frequency solutions for clamped shallow spherical shells are utilized to apply similarity transformation, leading to the derivation of generalized similarity conditions and scaling laws for the free vibration of these shells. The numerical approach is then validated by comparing the theoretical natural frequencies of clamped shallow spherical shells. Finally, geometric and material distortion similarity predictions for the free vibration of clamped shallow spherical shells with different rise-to-thickness ratios are investigated by numerical simulations. Results indicate that scaled models for clamped shallow spherical shells with different rise-to-thickness ratios, designed according to similarity conditions and incorporating the corresponding generalized similarity condition of mode shape and the natural frequency scaling relationship, can accurately predict the free vibration characteristics of the prototypes. For the free vibration of clamped shallow spherical shells with medium, small, and large rise-to-thickness ratios, in addition to meet the same mode shape, the distorted scaling laws also need to ensure that the scaling factors of rise-to-thickness ratio and Poisson's ratio equal 1, the scaling factor of Poisson's ratio is 1, and the rise-to-thickness ratios of scaled models and the prototype are no less than 25. The method and results of this study offer valuable insights for the design and experiments of scaled models for similar shells with thickness and material distortion.

shallow spherical shells  /  free vibration  /  distortion  /  rise-to-thickness ratio  /  scaling law
于伟, 王帅. 固支扁球壳自振的几何和材料畸变缩比模型研究. 固体力学学报, 2025 , 46 (2) : 286 -296 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.056
Wei Yu, Shuai Wang. A Scaled Model of Geometric and Material Distortion in the Free Vibration of Clamped Shallow Spherical Shells[J]. Chinese Journal of Solid Mechanics, 2025 , 46 (2) : 286 -296 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.056
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0　引言
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扁球壳为矢高远小于底面半径的壳体，该类薄壁扁壳被广泛应用在运载火箭、潜艇、LNG运输船、储氢罐、核电安全壳和煤气柜等结构. 在实际工程中，这些壳体结构的振动问题比较显著. 由于此类结构中扁球壳体积较大，采用结构原型进行试验存在成本高、周期长和难度大等诸多问题，因此常采用缩比模型来预测原型的振动特性. 大型扁球壳的壳体厚度远小于结构的其他尺寸，进行结构缩比模型设计时需考虑厚度的畸变；有时缩比模型材料不同于原型结构，会给试验带来方便和效益；因此开展几何和材料畸变缩比模型设计的研究就十分必要.
关于板壳结构振动理论分析和缩比模型研究，学者们做了许多工作. 曹志远[1]系统介绍了板壳振动分析的一般方法. Reissner[2,3]采用解析法，分析了扁球壳在不同边界条件下的轴对称自由振动；并通过Rayleigh-Ritz方法计算了固支扁球壳最低频率的近似值. 王璠和刘仁怀[4]利用解析法研究了复合材料层合开顶扁球壳在简谐荷载作用下的非线性动态屈曲问题. 杨增涛和徐加初[5]采用数值方法对复合材料层合扁球壳在矩形脉冲作用下的轴对称冲击屈曲开展了研究. 罗忠等[6-8]采用控制方程法和灵敏度分析研究了悬臂板、单层弹性薄板、层合板、涂层板、夹层板及圆环薄板的自振畸变相似，提出了模型畸变参数区间的确定方法，通过模拟或实验对相似关系进行了验证. 罗忠团队[9-11]利用控制方程法对薄壁短圆柱壳、旋转运动的薄壁圆柱壳和考虑强度要求的薄壁圆柱壳进行了振动相似分析，并通过理论分析或实验验证了得到的相似律. 罗忠等[12]采用方程分析与敏感性分析相结合的方法，研究了旋转层合薄壁短圆柱壳振动畸变相似，提出了一种畸变模型的几何尺寸区间确定方法. Rezaeepazhand和Simitses[13,14]使用方程分析法研究了一般层合圆柱壳和双曲层合圆柱壳自振的相似律，并研究了层数、铺层顺序、材料参数等畸变对自振相似的影响. Ungbhakorn等[15,16]利用方程分析法对对称正交层合圆柱壳和反对称正交层合圆柱壳的振动进行相似分析，并采用解析法研究了完全相似模型、铺层顺序畸变模型和材料畸变模型对原型自振相似预测的准确性；结果表明，材料畸变模型预测精度不佳. Torkamani等[17]基于方程分析法，采用数值模拟和结构实验的方法，对正交加筋圆柱壳进行了自振相似预测，并通过改变加筋肋数量、材料、截面形状以解决加筋肋厚度过小的问题. 罗忠等[18]基于Donnell理论和最小热力学能量原理，推导出一端固支另一端自由边界条件下带锥形齿薄壁短圆筒的频率方程，根据量纲分析法建立了该结构的动力学相似律，并设计了锥形齿数量和形状改变的结构畸变模型，采用有限元法对原型和畸变模型的固有特性进行相似分析，验证了所提出相似律的有效性. Rosa和Franco[19]基于密加筋理论，利用经典模态法和能量分配法对圆柱壳与纵环加筋圆柱壳进行了自振及强迫振动相似数值分析. Liu等[20,21]基于变幂法提出了耦合畸变相似方法，对圆柱-圆锥壳组合壳和螺栓连接的圆柱-圆锥壳组合壳进行了自振畸变相似分析，讨论了长度、半径或连接螺栓对建立畸变模型的影响；并利用数值模拟和模态实验验证了新相似方法的有效性.
关于旋转壳和扁壳的振动相似预测也取得了一些进展. 罗忠团队[22,23]基于最小二乘法建立了薄壁圆锥壳和凹形薄壁锥壳的自振畸变相似律，通过数值模拟和模态实验，验证了提出的畸变相似律的正确性. 孟强强等[24]采用方程分析法和敏感性分析法建立了中大型薄壁截锥壳的自振畸变相似律，通过有限元法验证了畸变相似关系的准确性. Ungbhakorn和Singhatanadgid[25]通过能量法推导了层合双曲扁壳结构的自振相似律；结果表明铺层顺序畸变的模型能准确预测原型的振动特性，但材料畸变模型不能准确预测原型的振动特性. Rezaee和Yazdi[26]利用方程分析法对含碳纳米管的层合双曲壳体进行了非线性振动相似分析，并探讨了碳纳米管重量百分比、大挠度、长径比和铺叠顺序对模型设计的影响. 于伟和李正良[27]利用能量法建立了加筋扁球壳自由振动的相似律，并研究了加筋肋截面畸变和材料畸变对结构固有频率的影响.
关于固支扁球壳厚度和材料畸变的自振相似预测研究鲜有报道. 因此，对固支扁球壳的自振频率方程和固有频率解分别进行相似转换，结合振型表达式，并分别获得固支扁球壳自振的广义相似条件和固有频率缩比关系；并利用通过理论验证的数值模拟方法，对固支扁球壳厚度畸变、厚度和材料均畸变的自振缩比模型进行研究.
1　扁球壳的固有频率
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当球壳矢跨比H/b<1/4时可视为扁球壳，壳体中面可用极坐标表示. 固支光滑扁球壳剖面如图1所示.
扁球壳振型方程为[1]：
式中：ρ为壳体密度，h为壳体厚度，E为弹性模量，ν为泊松比，R为曲率半径，ω为壳体振动圆频率，W为z向位移振型函数，F为应力振型函数，弯曲刚度D=Eh3/[12（1-ν2）]，球坐标拉普拉斯算子表达式.
引入新的函数Ф将振型方程式（1）化简成为一个微分方程，并采用分离变量法得到位移振型函数Ф的一般解为[1]：
式中：C1、C3、C5、C7为常参数，Jm（x）为第一类贝塞尔函数，Im（x）为第一类修正贝塞尔函数，m为壳体振型的节圆，n为节径，.
将一般解代入固支边界条件r=b，u=v=w=∂w/∂r=0，其中：b为扁球壳底面半径，u、v、w分别为扁球壳径向、切向和z向的位移，可得固支扁球壳的自振频率方程[1]：
式中[1]：
在扁球壳中存在如下几何关系[2]：
其中：H为扁球壳矢高.
将式（5）代入式（4）中的第一式可得[2]：
求解频率方程式（3）可得（αb）的值，将（αb）值和式（4）中的第三式代入式（4）中的第二式，可得固支扁球壳自振的圆频率表达式：
2　固支扁球壳的固有频率相似关系
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2.1　一般情形下固支扁球壳的固有频率相似关系
将式（6）代入式（3），并令（αb）=x，固支扁球壳的自振频率方程可简化为：
缩比模型和原型的频率方程分别为：
其中下标mo表示缩比模型的参数，下标p表示原型的参数.
假设相似因子：
将相似因子式（11）代入原型的频率方程式（10），可得：
首先研究贝塞尔函数的广义相似条件，m为正整数或零时第一类贝塞尔函数Jm（x）表达式为：
缩比模型和原型的第一类贝塞尔函数表达式分别为：
将相似因子Cx=xp/xmo代入原型的第一类贝塞尔函数式（15）可得：
根据相似理论，原型和缩比模型的第一类贝塞尔方程中的系数对应成比例，则存在相似条件：
第一类修正贝塞尔函数Im（x）表达式为：
同理可得与式（17）相同的相似条件.
由缩比模型频率方程式（9）和原型的频率方程式（12）中的系数对应成比例，以及式（2）和式（17），可得一般情形下固支扁球壳自振的广义相似条件为：
对式（7）进行相似转换，并结合式（19）和工程频率f=ω/（2π），可得一般情形下固支扁球壳固有频率的缩比关系：
2.2　矢厚比较小时固支扁球壳的固有频率相似关系
当球壳形状越来越扁时，b/R越来越小，则λ趋于零，1/λ趋于无穷，此时式（3）中仅保留含1/λ的项，固支扁球壳的自振频率方程化简为：
由式（21）可知，（αb）值由m值决定，并且可以求（αb）的值.
由式（4）中的第二、第三式解出固支扁球壳的圆频率表达式为：
如果满足条件：λ-2（n+m）≤0，其中：m为径向模态波数，n为环向模态波数，λ由式（6）确定；则式（21）就具有足够的精度.
对式（21）进行相似转换并结合位移振型式（2），得到矢厚比较小时固支扁球壳自振的广义相似条件为：
对式（22）进行相似转换，并结合f=ω/（2π），得到矢厚比较小时固支扁球壳的固有频率缩比关系为：
2.3　矢厚比较大时固支扁球壳固有频率相似关系
将式（5）代入式（7）可得固支扁球壳的圆频率表达式：
当H/h大于25时[2]，可忽略式（25）右侧根号中首项，该式简化为：
因忽略了式（25）中的（αb）项，也就忽略了频率方程式（3）对相似条件的影响，根据位移振型式（2），可得矢厚比较大时固支扁球壳自振的广义相似条件为：
对式（26）进行相似转换，并结合f=ω/（2π），可得矢厚比较大时固支扁球壳的固有频率缩比关系为：
3　固支扁球壳自振的有限元算例验证与相似预测
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3.1　一般情形下固支扁球壳自振数值模拟验证与相似预测
对固支边界扁球壳自振进行数值模拟验证，结构具体参数见表1. 通过查表[1]获得（αb）值，并结合式（4）中第二、第三式及f=ω/（2π）求出扁球壳的工程频率. 采用ANSYS软件对固支扁球壳进行模态分析，使用shell181单元，网格采用四边形映射网格. 选取固支扁球壳前六阶模态进行模拟值与理论值的对比，结果列于表2. 其中，相对误差为：（数值模拟结果-理论结果）/理论结果×100%. 由表2可知，固支扁球壳的模拟结果和理论结果吻合较好，说明固支扁球壳的自振数值模拟是准确和可靠的.
以λ=7.5的扁球壳为原型进行自振相似关系的验证，原型和缩比模型的几何与材料参数见表3，令矢高和厚度的缩比因子相同，单独选取底面半径缩比因子，缩比模型和原型的材料相同. 选取的缩比因子分别为：缩比模型1与原型的底面半径相同，矢高与厚度的缩比因子均为1/5；缩比模型2的底面半径缩比因子为1/10，矢高与厚度的缩比因子均为1/8. 采用ANSYS得到原型和缩比模型的工程频率，利用式（20）求得固有频率的缩比系数，利用缩比模型的自振频率得到其原型的自振频率预测结果，并与原型的固有频率结果进行对比，详见表4和表5. 其中，百分比误差为：（原型预测固有频率-原型固有频率）/原型固有频率×100%. 由表4和表5可知，在一般情形下固支扁球壳厚度畸变的情况下，根据广义相似条件式（19）设计的缩比模型，结合固有频率缩比关系式（20），能够准确预测原型的固有频率，且缩比模型和其原型的振型相同.
3.2　矢厚比较小时固支扁球壳自振相似预测
扁球壳原型和缩比模型的几何与材料参数见表6. 扁球壳原型的矢厚比取值为0.9456，原型和缩比模型的材料相同，底面半径和矢高的缩比因子均为1/10，厚度缩比因子为1/5. 采用ANSYS模拟得到原型和模型的工程频率，用式（24）求得固有频率的缩比系数，由缩比模型的自振频率预测得到其原型的自振频率预测结果，并与原型的固有频率结果进行比较，详见表7. 由表7可知，当矢厚比较小且固支扁球壳厚度发生畸变时，依据广义相似条件式（23）设计的缩比模型，结合固有频率缩比关系式（24），能够准确预测原型的固有频率，且缩比模型与其原型的振型一致.
3.3　矢厚比较大时固支扁球壳自振相似预测
大型煤气柜顶为径环加筋扁球壳，现不考虑加筋肋，取光滑扁球壳为例进行验证. 扁球壳原型和缩比模型的几何及材料参数见表8，扁球壳缩比模型的几何参数相同，底边半径和矢高的缩比因子均为1/40，厚度缩比因子为1/3.5，四种缩比模型的材料分别为铁、铝、铜和PVC. 采用ANSYS得到原型和缩比模型的工程频率，用式（28）求得固有频率的缩比系数，根据缩比模型的自振频率得到其原型的自振频率预测结果，并与原型的固有频率结果进行对比，详见表9. 由表9可知，固支扁球壳矢厚比较大时，厚度和材料均畸变的缩比模型能够准确预测原型的固有频率，且缩比模型与其原型的振型相同，表明固有频率的缩比关系式（28）是正确的.
4　结论
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对固支扁球壳的自振频率方程和固有频率解分别进行相似转换，结合振型表达式，建立了不同矢厚比固支扁球壳自振的广义相似条件和缩比关系式. 采用理论分析验证了数值模拟的准确性和可靠性. 根据建立的相似关系设计了固支扁球壳的缩比模型，进而采用数值模拟对固支扁球壳进行厚度畸变、厚度和材料均畸变的自振相似预测，得到如下结论：
（1）一般情形的固支扁球壳自振相似条件要求较多，缩比模型与原型需保持矢高和厚度的缩比因子相同，泊松比缩比因子为1，以及振型相同（Cm=Cn=1），在这三个条件下缩比模型的固有频率能准确预测其原型的固有频率.
（2）矢厚比较小时固支扁球壳自振相似条件要求较少，缩比模型与原型需满足泊松比缩比因子为1，振型相同（Cm=Cn=1），在这两个条件下缩比模型的固有频率能准确预测其原型的固有频率.
（3）矢厚比较大时固支扁球壳自振相似条件要求也较少，但要求缩比模型和原型的矢厚比均大于等于25，振型相同（Cm=Cn=1），在这两个条件下缩比模型的固有频率能准确预测其原型的固有频率.
（4）基于固支扁球壳的自振频率方程并结合振型表达式进行相似转换，提出了结构自振的缩比模型设计方法；根据固支扁球壳自振圆频率解的相似转换建立了结构的自振相似关系；这些为设计厚度和材料均畸变的固支扁球壳自振缩比模型奠定了基础.
基金
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  • 国家自然科学基金项目(51868013; 52068015)
文献
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2025年第46卷第2期
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.056
  • 接收时间：2024-11-05
  • 首发时间：2026-03-20
  • 出版时间：2025-04-23
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  • 收稿日期：2024-11-05
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国家自然科学基金项目(51868013; 52068015)
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    1桂林理工大学广西绿色建材与建筑工业化重点实验室，桂林，541004
    2桂林理工大学广西岩土力学与工程重点实验室，桂林，541004

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