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Article(id=1246031926000206471, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246031922707677827, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2024.017, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1713888000000, receivedDateStr=2024-04-24, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1775006761312, onlineDateStr=2026-04-01, pubDate=1724515200000, pubDateStr=2024-08-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1775006761312, onlineIssueDateStr=2026-04-01, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1775006761312, creator=13701087609, updateTime=1775006761312, updator=13701087609, issue=Issue{id=1246031922707677827, tenantId=1146029695717560320, journalId=1241755870837649424, year='2024', volume='45', issue='4', pageStart='427', pageEnd='564', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1775006760527, creator=13701087609, updateTime=1775006830786, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1246032217470779944, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246031922707677827, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1246032217470779945, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246031922707677827, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=520, endPage=532, ext={EN=ArticleExt(id=1246031926251864714, articleId=1246031926000206471, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=Research on Vibration Characteristics of Functionally Graded Stepped Cylindrical Shells with Arbitrary Boundary Conditions, columnId=1244229834482757770, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Paper, runingTitle=null, highlight=null, articleAbstract=

Research on vibration characteristics of functionally graded materials (FGMs) in the aerospace field is a hot topic of current research. In this paper, the vibration characteristics of metal-ceramic functionally graded (FG) stepped cylindrical shells under arbitrary boundary conditions are studied. To conduct this research, a mechanical model of a metal-ceramic FG stepped cylindrical shell based on the axial segmentation concept is developed. First, the properties of metal-ceramic FGMs are obtained using the Voigt model and power function volume fraction. Second, the artificial spring technique is introduced to simulate continuous coupling conditions of shell segments and arbitrary boundary conditions at the ends of the shell. The energy expression for the cylindrical shell is then derived based on the first-order shear deformation theory. Finally, the admissible function is constructed via the Chebyshev polynomial, and the dynamic differential equations of the metal-ceramic FG stepped cylindrical shell under arbitrary boundary conditions are calculated using the Rayleigh-Ritz method. The validity and convergence of the method are verified through comparison with existing literature, and the effects of boundary conditions, volume fraction, geometric parameters, and spring stiffness on modal frequencies are analyzed. It is found that the natural frequency of the FG stepped cylindrical shell initially decreases and then increases with increased number of circumferential waves under classical boundary conditions, and it increases with the number of circumferential waves under both elastic boundary conditions. The natural frequency of the shell increases exponentially with volume fraction. The effects of length-to-radius ratio and thickness-to-radius ratio on the vibration characteristics of the shell differ, with the natural frequency of the shell decreasing with increased length-to-radius ratio and increasing with thickness-to-radius ratio. Additionally, the stiffness of translational springs significantly influences the vibration characteristics of the shell compared to rotational springs.

, 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=Chong Tang, Yu Wang, Xuehui Li, Ziqiang Xu, Zhihong Yang), CN=ArticleExt(id=1246031943935050075, articleId=1246031926000206471, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=任意边界条件下功能梯度阶梯圆柱壳的振动特性研究, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

对金属-陶瓷功能梯度材料阶梯圆柱壳在任意边界条件下的振动特性展开了研究. 首先，采用Voigt模型和幂函数体积分数得到金属-陶瓷功能梯度材料属性. 其次，引入人工弹簧技术，模拟各壳段间的连续耦合及壳体两端的边界条件，并基于一阶剪切变形理论推导出壳体的能量表达式. 最后，选取Chebyshev多项式构造容许函数，基于Rayleigh-Ritz法对任意边界条件下壳体的动力学微分方程进行求解计算，通过与现有文献对比，验证了本文方法的有效性与收敛性. 结果表明：体积分数指数增长，壳体固有频率也随之增长；而长径比和厚径比对壳体的振动特性影响均不同，随着长径比的增加壳体的固有频率减小，随着厚径比的增加壳体的固有频率增大；且相较于旋转弹簧，平动弹簧的刚度值对壳体振动特性的影响显著.

, correspAuthors=王宇, authorNote=null, correspAuthorsNote=
** E-mail：.
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Material properties of FG stepped cylindrical shell

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参数结构钢氮化硅
弹性模量（GPa）207.788322.2715
密度（kg/m3）81662370
泊松比0.317560.24
), ArticleFig(id=1246042394022495169, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246031926000206471, language=CN, label=表1, caption=

FG阶梯圆柱壳的材料属性

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参数结构钢氮化硅
弹性模量（GPa）207.788322.2715
密度（kg/m3）81662370
泊松比0.317560.24
), ArticleFig(id=1246042394097992642, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246031926000206471, language=EN, label=Table 2, caption=

Values of spring stiffness with different boundary conditions

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边界条件（N/m）（N/m）（N/m）（N·m/rad）（N·m/rad）
F00000
C10121012101210121012
S01012101200
E1108108108108108
E2001081080
E3010810800
), ArticleFig(id=1246042394202850243, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246031926000206471, language=CN, label=表2, caption=

不同边界条件下的弹簧刚度值

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边界条件（N/m）（N/m）（N/m）（N·m/rad）（N·m/rad）
F00000
C10121012101210121012
S01012101200
E1108108108108108
E2001081080
E3010810800
), ArticleFig(id=1246042394286736324, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246031926000206471, language=EN, label=Table 3, caption=

Influence of number of shell segment on dimensionless frequency of cylindrical shell with S-S boundary conditions

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n壳段数文献[15]df
2345
10.016100.016100.016100.016100.016100
20.009390.009390.009390.009390.009380
30.022100.022100.022100.022100.022100
40.042090.042090.042090.042090.042090
50.067980.067980.067980.067980.068010.044
60.099670.099670.099670.099670.099730.060
), ArticleFig(id=1246042394353845189, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246031926000206471, language=CN, label=表3, caption=

S-S边界条件下壳段数对圆柱壳无量纲频率的影响

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n壳段数文献[15]df
2345
10.016100.016100.016100.016100.016100
20.009390.009390.009390.009390.009380
30.022100.022100.022100.022100.022100
40.042090.042090.042090.042090.042090
50.067980.067980.067980.067980.068010.044
60.099670.099670.099670.099670.099730.060
), ArticleFig(id=1246042394437731270, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246031926000206471, language=EN, label=Table 4, caption=

Convergence of natural frequency of FG stepped cylindrical shells with different boundary conditions

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边界条件n固有频率（HZ）
截断数M
M=5M=7M=9M=11M=13
C-C12866.5762834.1652827.0512827.0482827.045
21564.0231539.0041531.5861531.5761531.571
31250.2661241.0021237.0531237.0431237.040
41565.7671556.8721552.7161552.7081552.702
51969.5771959.5281955.7931955.7841955.770
S-S12094.0332093.7892093.7622093.7582093.756
2933.541933.475933.475933.475933.475
3983.712983.201982.857982.851982.850
41390.5311388.6511388.0811388.0791388.075
51846.6971843.9301843.1371843.1341843.131
C-F11169.0101163.3921161.7141161.4161161.132
2498.131496.450495.836495.827495.822
3611.957611.123610.398610.386610.373
41032.1251030.4951028.9881028.9711028.969
51620.0611617.5271615.3021615.3021615.302
), ArticleFig(id=1246042394534200263, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246031926000206471, language=CN, label=表4, caption=

不同边界条件下FG阶梯圆柱壳固有频率的收敛性

, figureFileSmall=null, figureFileBig=null, tableContent=
边界条件n固有频率（HZ）
截断数M
M=5M=7M=9M=11M=13
C-C12866.5762834.1652827.0512827.0482827.045
21564.0231539.0041531.5861531.5761531.571
31250.2661241.0021237.0531237.0431237.040
41565.7671556.8721552.7161552.7081552.702
51969.5771959.5281955.7931955.7841955.770
S-S12094.0332093.7892093.7622093.7582093.756
2933.541933.475933.475933.475933.475
3983.712983.201982.857982.851982.850
41390.5311388.6511388.0811388.0791388.075
51846.6971843.9301843.1371843.1341843.131
C-F11169.0101163.3921161.7141161.4161161.132
2498.131496.450495.836495.827495.822
3611.957611.123610.398610.386610.373
41032.1251030.4951028.9881028.9711028.969
51620.0611617.5271615.3021615.3021615.302
), ArticleFig(id=1246042394622280648, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246031926000206471, language=EN, label=Table 5, caption=

Comparisons of dimensionless frequency of cylindrical shells with arbitrary boundary conditions

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边界条件n=1n=2n=3n=4
S-S文献[10]0.01610.039270.10980.21028
本文0.01610.039270.10950.209015
df000.0030.006
C-C文献[10]0.032850.040640.109970.21032
本文0.032970.040670.109640.20907
df0.00400.0030.006
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任意边界条件下圆柱壳无量纲频率比较

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边界条件n=1n=2n=3n=4
S-S文献[10]0.01610.039270.10980.21028
本文0.01610.039270.10950.209015
df000.0030.006
C-C文献[10]0.032850.040640.109970.21032
本文0.032970.040670.109640.20907
df0.00400.0030.006
), ArticleFig(id=1246042394815218634, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246031926000206471, language=EN, label=Table 6, caption=

Comparisons of natural frequency of FG cylindrical shells with S-S boundary conditions

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n固有频率（HZ）
N=0.5N=2
文献[9]本文文献[9]本文
113.32113.32313.10313.103
24.52994.51694.44354.4433
34.25864.1914.12354.1234
47.21667.09196.9826.9818
511.51211.33711.15111.151
), ArticleFig(id=1246042394894910411, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246031926000206471, language=CN, label=表6, caption=

S-S边界条件下FG圆柱壳的固有频率比较

, figureFileSmall=null, figureFileBig=null, tableContent=
n固有频率（HZ）
N=0.5N=2
文献[9]本文文献[9]本文
113.32113.32313.10313.103
24.52994.51694.44354.4433
34.25864.1914.12354.1234
47.21667.09196.9826.9818
511.51211.33711.15111.151
), ArticleFig(id=1246042394970407884, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246031926000206471, language=EN, label=Table 7, caption=

Comparisons of dimensionless frequency of stepped cylindrical shells

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nS-SC-C
文献[16]文献[14]本文文献[16]文献[14]本文
10.056540.056540.056540.097650.097660.09864
20.020740.020740.020750.041360.041360.04181
30.020540.020530.020550.027740.027730.02787
40.029140.029130.029160.033740.033720.03384
50.038190.038180.038200.041380.041360.04145
), ArticleFig(id=1246042395108819917, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246031926000206471, language=CN, label=表7, caption=

阶梯圆柱壳无量纲频率比较

, figureFileSmall=null, figureFileBig=null, tableContent=
nS-SC-C
文献[16]文献[14]本文文献[16]文献[14]本文
10.056540.056540.056540.097650.097660.09864
20.020740.020740.020750.041360.041360.04181
30.020540.020530.020550.027740.027730.02787
40.029140.029130.029160.033740.033720.03384
50.038190.038180.038200.041380.041360.04145
), ArticleFig(id=1246042395180123086, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246031926000206471, language=EN, label=Table 8, caption=

Comparisons of natural frequency of FG stepped cylindrical shells withdifferent boundary conditions

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n固有频率（Hz）
S-SC-CE1-E1E1-E2
120932826736622
29331530753709
39821236976947
41387155213901347
51842195518551823
), ArticleFig(id=1246042395259814863, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246031926000206471, language=CN, label=表8, caption=

不同边界条件下FG阶梯圆柱壳的固有频率比较

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n固有频率（Hz）
S-SC-CE1-E1E1-E2
120932826736622
29331530753709
39821236976947
41387155213901347
51842195518551823
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任意边界条件下功能梯度阶梯圆柱壳的振动特性研究
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唐冲 1 , 王宇 1, ** , 李学辉 1 , 徐自强 1 , 杨志宏 2
固体力学学报 | 研究论文 2024,45(4): 520-532
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固体力学学报 | 研究论文 2024, 45(4): 520-532
任意边界条件下功能梯度阶梯圆柱壳的振动特性研究
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唐冲1, 王宇1, ** , 李学辉1, 徐自强1, 杨志宏2
作者信息
  • 1辽宁科技大学机械工程与自动化学院，鞍山，114051
  • 2山西北方晋东化工有限公司，阳泉，045000

通讯作者:

** E-mail：.
Research on Vibration Characteristics of Functionally Graded Stepped Cylindrical Shells with Arbitrary Boundary Conditions
Chong Tang1, Yu Wang1, ** , Xuehui Li1, Ziqiang Xu1, Zhihong Yang2
Affiliations
  • 1School of Mechanical Engineering and Automation, University of Science and Technology Liaoning, Anshan, 114051
  • 2Shanxi North Jindong Chemical Company Limited, Yangquan, 045000
出版时间: 2024-08-25 doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.017
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摘要
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对金属-陶瓷功能梯度材料阶梯圆柱壳在任意边界条件下的振动特性展开了研究. 首先，采用Voigt模型和幂函数体积分数得到金属-陶瓷功能梯度材料属性. 其次，引入人工弹簧技术，模拟各壳段间的连续耦合及壳体两端的边界条件，并基于一阶剪切变形理论推导出壳体的能量表达式. 最后，选取Chebyshev多项式构造容许函数，基于Rayleigh-Ritz法对任意边界条件下壳体的动力学微分方程进行求解计算，通过与现有文献对比，验证了本文方法的有效性与收敛性. 结果表明：体积分数指数增长，壳体固有频率也随之增长；而长径比和厚径比对壳体的振动特性影响均不同，随着长径比的增加壳体的固有频率减小，随着厚径比的增加壳体的固有频率增大；且相较于旋转弹簧，平动弹簧的刚度值对壳体振动特性的影响显著.

功能梯度材料  /  阶梯圆柱壳  /  任意边界条件  /  振动特性  /  人工弹簧技术

Research on vibration characteristics of functionally graded materials (FGMs) in the aerospace field is a hot topic of current research. In this paper, the vibration characteristics of metal-ceramic functionally graded (FG) stepped cylindrical shells under arbitrary boundary conditions are studied. To conduct this research, a mechanical model of a metal-ceramic FG stepped cylindrical shell based on the axial segmentation concept is developed. First, the properties of metal-ceramic FGMs are obtained using the Voigt model and power function volume fraction. Second, the artificial spring technique is introduced to simulate continuous coupling conditions of shell segments and arbitrary boundary conditions at the ends of the shell. The energy expression for the cylindrical shell is then derived based on the first-order shear deformation theory. Finally, the admissible function is constructed via the Chebyshev polynomial, and the dynamic differential equations of the metal-ceramic FG stepped cylindrical shell under arbitrary boundary conditions are calculated using the Rayleigh-Ritz method. The validity and convergence of the method are verified through comparison with existing literature, and the effects of boundary conditions, volume fraction, geometric parameters, and spring stiffness on modal frequencies are analyzed. It is found that the natural frequency of the FG stepped cylindrical shell initially decreases and then increases with increased number of circumferential waves under classical boundary conditions, and it increases with the number of circumferential waves under both elastic boundary conditions. The natural frequency of the shell increases exponentially with volume fraction. The effects of length-to-radius ratio and thickness-to-radius ratio on the vibration characteristics of the shell differ, with the natural frequency of the shell decreasing with increased length-to-radius ratio and increasing with thickness-to-radius ratio. Additionally, the stiffness of translational springs significantly influences the vibration characteristics of the shell compared to rotational springs.

functionally graded materials  /  stepped cylindrical shells  /  arbitrary boundary conditions  /  vibration characteristics  /  artificial spring technology
唐冲, 王宇, 李学辉, 徐自强, 杨志宏. 任意边界条件下功能梯度阶梯圆柱壳的振动特性研究. 固体力学学报, 2024 , 45 (4) : 520 -532 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.017
Chong Tang, Yu Wang, Xuehui Li, Ziqiang Xu, Zhihong Yang. Research on Vibration Characteristics of Functionally Graded Stepped Cylindrical Shells with Arbitrary Boundary Conditions[J]. Chinese Journal of Solid Mechanics, 2024 , 45 (4) : 520 -532 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.017
正文
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0　引言
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功能梯度材料（Functionally Graded Materials，FGMs）是一种结构和成分呈现连续梯度变化的复合材料，具备高强度、耐高温和耐腐蚀等优点[1,2]. 阶梯圆柱壳应用于实际工程，例如激光平台的减震基座和火箭的燃料箱等[3,4]. 将FGMs应用到阶梯圆柱壳中，在航空航天、军事和化工等领域有着广泛的应用前景，故开展任意边界条件下功能梯度（Functionally Graded，FG）阶梯圆柱壳的振动特性研究具有重要价值.
目前，学者对FG圆柱壳的动力学特性展开了大量的研究，例如：Loy等[5]基于Love壳体理论，研究了由镍和不锈钢组成的FG圆柱壳的振动特性，并分析了几何参数、体积分数指数以及材料组成类型对壳体固有频率的影响. 王宇等[6]基于Sanders壳体理论和传递矩阵方法，对旋转态双FG碳纳米管增强复合材料圆柱壳的行波振动特性开展了研究. 庞福振等[7]引入耦合弹簧技术，采用一阶剪切变形理论和Ritz法，对中厚FG圆柱壳的自由振动进行了研究. 刘超等[8]基于Sanders壳体理论和Rayleigh-Ritz法，推导了含孔隙金属-陶瓷FG圆柱壳的模态频率方程，研究了FG圆柱壳在热荷载、弹性基础以及轴向荷载作用下的模态频率.
在实际工程应用中，难以完全实现经典边界条件，因此，众多研究者对任意边界条件下FG圆柱壳的振动问题展开了研究. 例如：Qin等[9]通过一阶剪切变形理论和人工弹簧技术，分析了任意边界条件下旋转FG碳纳米管增强复合材料圆柱壳的自由振动. Miao等[10]对厚度不均匀的三层FG圆柱壳自由振动展开了研究，利用人工弹簧技术来模拟任意边界条件，讨论了几何参数、材料参数以及弹簧刚度对壳体结构模态频率的影响. 刘超等[11]利用Chebyshev多项式构造容许函数，对旋转FG层合圆柱壳在任意边界条件下的行波模态特性进行了讨论. 王宇等[12]基于一阶剪切变形理论，利用Halpin-Tsai微观力学模型，研究了弹性边界下FG石墨烯增强自旋圆柱壳的振动特性. 张宇航等[13]基于Love薄壳理论和Rayleigh-Ritz法，分析了旋转FG锥-柱连接壳在任意边界条件下的行波模态特性.
关于阶梯圆柱壳相关结构的动力学研究，Zhang等[14]基于Flügge壳体理论，并采用区域分解法，研究了厚度沿轴向阶梯变化的圆柱壳振动的精确解，分析了边界条件和几何参数对固有频率和振型的影响. Tang等[15]采用回转矩阵法，求解了任意边界条件下多级圆柱壳的固有频率和稳态响应问题. Li等[16]利用Flügge壳体理论，通过Jacobi-Ritz法，讨论了任意边界条件下均匀阶梯圆柱壳的自由振动特性. Xu等[18]基于一阶剪切变形理论和Gegenbauer-Ritz法，对石墨烯增强多孔复合材料阶梯圆柱壳的自由振动进行了分析.
综上所述，目前对阶梯圆柱壳的振动展开了大量研究，但大多研究是由单一材料组成，FGMs组成的阶梯圆柱壳的研究较少，此外，关于讨论任意边界条件影响的文献也比较少. 故开展对FG阶梯圆柱壳在任意边界条件下的振动特性研究具有重要的研究意义和应用前景. 本文以金属-陶瓷FG阶梯圆柱壳模型为研究对象. 首先，通过轴向分段思想建立了金属-陶瓷FG阶梯圆柱壳力学模型，引入人工弹簧技术模拟各壳段间的连续耦合及壳体两端的边界条件；其次，采用一阶剪切变形理论、Chebyshev多项式、Rayleigh-Ritz法求解了壳体的固有频率；最后分析了边界条件、体积分数指数、几何参数和弹簧刚度对壳体振动特性的影响.
1　理论方法
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1.1　FG阶梯圆柱壳的力学模型
如图1（a）所示，建立FG阶梯圆柱壳的力学模型，固定柱坐标系（x，θ，z）在壳体表面作为参考，O为坐标原点，x、θ和z分别表示轴向、周向和径向方向，中面任意一点在x、θ和z方向上的位移分别由u、v和w表示. 壳体中面半径为R，壳体总长度为L，如图1（b）所示，第i壳段的长度为Li，第i壳段的厚度为hi. 为了使数学表述更加简单方便，定义轴向无量纲坐标为η=x/L，因此得到∂/∂x=（1/L）∂/∂η.
由图1（c）可知，在壳体两端（即η=0和η=1处）定义五组弹簧，即设置三组线性移动弹簧和两组扭转弹簧，分别用、、、、和、、、、表示在壳体两端单位弧长处的弹簧刚度系数，通过调节弹簧刚度系数值来模拟任意边界条件.
1.2　FG阶梯圆柱壳的材料属性
本文研究的FG阶梯圆柱壳是由金属-陶瓷FGMs制成，金属和陶瓷在厚度方向上分布符合Voigt模型，第i段壳体的有效材料参数可表示为：
式中，、、分别为功能梯度基体的杨氏模量、泊松比和密度，E1、μ1、ρ1和E2、μ2、ρ2分别为金属和陶瓷材料的杨氏模量、泊松比和密度. 表示第i段壳体中金属材料的体积分数，具体表达式为：
式中，N为厚度方向上的幂律指数，当z=hi/2时，，，，即壳体外表面为金属材料；当z=-hi/2时，，，，即壳体内表面为陶瓷材料.
2　方程描述
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2.1　FG阶梯圆柱壳的动力学方程
基于一阶剪切变形理论，假设在整个壳体厚度上横向剪切应变是恒定的，同时考虑横向剪切变形，采用剪切修正系数保证壳体表面的零剪切条件，故圆柱壳第i个壳段上任意一点在x、θ和z方向的位移分量表示为：
式中，、和为壳体中面任意一点在x、θ和z方向上的位移，和为壳体中面任意一点相对于x轴和θ轴的扭转.
第i壳段的应变分量表示为：
式中，、、、、为壳体第i个壳段上任意一点的中面应变，、、、、为中面曲率. 具体表示为：
根据广义胡克定律可以得到如下关系：
式中，、为壳体内任意一点沿x、θ方向上的正应力，、、为xθ、xz、θz平面内的切应力. 为减缩刚度系数，具体表达式为：
在x和θ方向中面上的内力、力矩和横向剪切力表示为：
式中，、、为面内的合力，、、为弯矩和扭矩的合力矩，、是横向剪力合力，κ=5/6为剪切修正系数. 、和分别为第i个壳段的拉伸刚度、耦合刚度和弯曲刚度，表示为：
FG阶梯圆柱壳第i个壳段的应变能可以表示为：
式中，ηi、ηi+1表示第i个壳段左端和右端的无量纲坐标.
第i个壳段的动能可以表示为：
式中，、、为惯性项，其具体表达式如下：
相邻两个壳段之间的耦合势能表示为：
两端的边界弹簧所储存的弹性势能表示为
2.2　FG阶梯圆柱壳的振动方程
考虑轴对称几何体变形的周期性，将FG阶梯圆柱壳的位移场统一表示为：
式中，n为周向波数，ω为固有角频率，Ui（η）、Vi（η）、Wi（η）、ψi（η）、Θi（η）为壳体的轴向振型函数.
采用Chebyshev多项式构造容许位移函数，具体表达式如下：
式中，、、、、是按照Rayleigh-Ritz法思想所假定振型函数的待定系数，m为轴向半波数，M为Chebyshev多项式截断数. ，Tm（·）为第一类Chebyshev多项式，其递推关系式表示为：
在Chebyshev多项式中，η在区间[-1，1]上，而定义的无量纲坐标η∈[0，1]，因此需要对Chebyshev多项式中的η进行坐标变换，如下式：
展开系数向量qr，i（r=u，v，w，ψ，Θ）和函数向量pr，i（r=u，v，w，ψ，Θ）通过Rayleigh-Ritz法定义后，容许位移函数表示为：
式中，qr，i（r=u，v，w，ψ，Θ）和pr，i（r=u，v，w，ψ，Θ）分别为1×（M+1）和（M+1）×1的向量，具体表达式如下：
取各能量表达式的最大值，可将壳体的拉格朗日能量函数表示为：
式中，H为壳段数.
根据最小势能原理，对振型系数q求导将拉格朗日函数进行最小化，如下：
通过式（23）得到FG阶梯圆柱壳的振动特征方程为：
式中，Kε、Kb、Ks和M分别为势能矩阵、连接刚度矩阵、边界刚度矩阵和质量矩阵.
3　收敛性和有效性验证
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3.1　模型参数
为了验证本文方法在求解FG阶梯圆柱壳振动特性的收敛性和正确性，采用轴向分段的思想，建立了金属-陶瓷FG阶梯圆柱壳的模型. 其中，金属材料为结构钢，陶瓷材料为氮化硅，材料性能参考文献[2]，具体材料参数如表1所示. FG阶梯圆柱壳的尺寸为R=0.1 m，L=5R，h2=0.002 m，h1=2h2，L1=L2.
3.2　收敛性验证
人工弹簧技术模拟了壳体两端的边界条件和各壳段间的连续耦合，图2为连接弹簧刚度对圆柱壳无量纲频率的影响. 由图2可知，当弹簧刚度值为1010 N/m时，壳体各阶模态下的固有频率已经收敛，能够模拟各壳段间的连续耦合条件. 因此，为了保证准确性和有效性，后续研究连接弹簧刚度值取1012 N/m.
图3为FG阶梯圆柱壳随边界弹簧刚度的收敛性曲线，验证了不同边界条件下弹簧刚度的收敛性. 由图3可知，当弹簧刚度值小于104 N/m时，壳体的固有频率几乎没有变化，可以视为自由边界条件；当弹簧刚度值在104～1010 N/m时，壳体的固有频率迅速增加，可以视为弹性边界条件；当弹簧刚度值大于1012 N/m时，壳体的固有频率收敛，可以视为固支边界条件. 为了方便表达，将不同的边界条件用不同的字母表示，F表示自由边界条件，S表示简支边界条件，C表示固支边界条件，E1、E2和E3分别表示是三种不同的弹性边界条件. 不同边界条件下的弹簧刚度值如表2所示，其中，j=0，1.
S-S边界条件下壳段数对圆柱壳无量纲频率的影响如表3所示，其中，R=1 m，L/R=20，h/R=0.01，m=1，无量纲频率为Ω=ωR[（1-μ2）ρ/E]1/2. 由表3可知，在保证各个壳段连续的情况下，壳段数对无量纲频率的影响较小. 其中，定义误差公式为：
式中，df为误差值，f文献和f本文分别为文献给出的参考频率和本文计算的频率参数.
不同边界条件下FG阶梯圆柱壳固有频率的收敛性如表4所示，其中，m=1. 由于采用Chebyshev多项式构造容许位移函数，考虑到其M的问题，需要对振型函数展开项M进行分析. 由表4可知，当M大于5时，随着M的增大，壳体的固有频率逐渐趋于稳定，对计算结果产生的影响较小. 因此，在保证计算精度和计算效率的前提下，所讨论Chebyshev多项式的截断数取M=10.
3.3　有效性验证
为了验证本文方法的适用性，对任意边界条件下均匀厚度圆柱壳的固有频率进行了验证. 当壳体各子段的厚度相等且组成材料的体积分数指数N=0时，模型由FG阶梯圆柱壳变为了单一材料的均匀厚度圆柱壳，得到的结果如表5所示，其中，R=1 m，L=20 m，h=0.05 m，m=1，无量纲频率为. 表6为S-S边界条件下FG圆柱壳的固有频率对比，考虑了体积分数指数N=0.5和N=2两种情况，其中，L/R=1，h/R=0.002. 表7为S-S和C-C边界条件下阶梯圆柱壳的无量纲频率对比，其中，h1/R=0.01，h2/h1=0.5，L1/L=1/2，L/R=10，无量纲频率为. 结果表明，分析结果与文献和仿真结果保持良好的一致性.
4　结果分析
收起
4.1　边界条件影响
不同边界条件下，即S-S、C-C、E1-E1和E1-E2四种边界条件下，FG阶梯圆柱壳的固有频率随周向波数的变化关系如表8所示. 随着周向波数的增大，两种经典边界条件下的固有频率都先减小后增大. 而在E1-E1和E1-E2两种弹性边界条件下，固有频率都随着周向波数的增大而增大. 由表8可知，在S-S边界条件下，FG阶梯圆柱壳的最低阶固有频率出现在周向波数n=2处，C-C边界条件下最低阶频率在模态（1，3）处获取，并且由于边界连接刚度的影响，C-C边界条件下FG阶梯圆柱壳的固有频率最大.
4.2　体积分数指数影响
给定的金属-陶瓷FG阶梯圆柱壳模型中，其材料属性在厚度方向上呈现梯度变化，当N=0时，FGMs仅由单一金属材料组成，随着N的增大陶瓷材料的比例逐渐增大. 本文对E1-E1、E1-E3和E2-E3三种弹性边界条件下金属-陶瓷体积分数指数对固有频率的影响展开了分析，其中，选取周向波数n=2为例. 由图4可知，三种弹性边界条件下固有频率都随着体积分数指数的增大而呈现增大的趋势，这是由于随着体积分数指数的增加，陶瓷在FGMs中的比例增加，而固有频率主要受FGMs的杨氏模量和密度的影响. 由式（24）可知，壳体结构的固有频率主要取决于质量矩阵和刚度矩阵，由于边界连接刚度的影响，E1-E1边界条件下对应的固有频率最大. 同时，随着体积分数指数的增大，边界条件对阶梯圆柱壳的固有频率的影响越来越大.
4.3　几何参数影响
几何参数是影响FG阶梯圆柱壳固有频率的重要因素之一. 本文讨论了弹性边界条件下长径比和厚径比对FG阶梯圆柱壳的固有频率的影响，图5（a）为不同周向波数下固有频率随长径比的变化关系，图5（b）不同周向波数下固有频率随厚径比的变化关系.
由图5（a）可知，保持半径和厚度比值不变的情况下，当长径比从1～10变化时，固有频率随着长径比的增大而呈现减小的趋势，当长径比为1时，各阶固有频率得到最大值. 并且不同周向波数下固有频率的变化趋势有所不同，低阶周向波数下长径比对固有频率的影响相对较小，随着周向波数的增大，固有频率的变化趋势逐渐增大；由图5（b）可知，保持长度和半径不变的情况下，在低阶周向波数下，FG阶梯圆柱壳的固有频率对厚径比的变化不敏感，随着周向波数的增大，固有频率随厚径比的增大而增大，且周向波数越大，固有频率的增大趋势越明显.
4.4　弹簧刚度影响
为了研究弹簧刚度的影响，在壳体两端设置五组弹簧，即三组线性平动弹簧和两组扭转弹簧，通过调节不同弹簧刚度值来实现任意复杂的边界条件. 如图6（a）所示，设置扭转弹簧刚度值为1012 N/m，平动弹簧刚度值从100 N/m增加到1015 N/m. 如图6（b）所示，设置平动弹簧刚度值为1012 N/m，扭转弹簧刚度值从100 N/m增加到1015 N/m. 由图6（a）可知，随着弹簧刚度值的增大，FG阶梯圆柱壳的固有频率逐渐增大，不同体积分数下的固有频率随刚度值的增大呈现近似线性关系. 对于线性平动弹簧而言，弹簧刚度的敏感范围出现在106-1010 N/m. 由图6（b）可知，扭转弹簧对FG阶梯圆柱壳的固有频率的影响较小. 上述现象表明，不同方向上的弹簧对弹簧刚度敏感范围和固有频率的影响程度不同.
5　结论
收起
基于一阶剪切变形理论，以金属-陶瓷FG阶梯圆柱壳为研究对象，提出了一种以Chebyshev多项式为容许函数的半解析法，此方法具有良好的收敛性和有效性，采用人工弹簧技术确定弹簧刚度值，从而得到任意边界条件下的振动特性，主要结论如下：
（1）在经典边界条件下，FG阶梯圆柱壳的固有频率随着周向波数的增大呈现先减小后增大的趋势；在E1-E1和E1-E2两种弹性边界条件下，固有频率都随着周向波数的增大而增大；由于边界连接刚度的影响，C-C边界条件下FG阶梯圆柱壳的固有频率最大.
（2）在不同弹性边界条件下，FG阶梯圆柱壳的固有频率随着体积分数指数的增大而增大；主要由于随着体积分数指数的增大，陶瓷的比例逐渐增大，其固有频率主要受FG的杨氏模量和密度的影响.
（3）FG阶梯圆柱壳的固有频率随着长径比的增大而减小；低阶周向波数下固有频率对厚径比的变化不敏感，而高阶周向波数下固有频率随着厚径比的增加而增大，并且增大的趋势越来越明显.
（4）边界弹簧刚度对壳体频率具有明显影响，平动弹簧对频率的影响较大，旋转弹簧对频率的影响较小.
基金
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  • 国家自然科学基金项目(51775257; 51775258)
  • 辽宁省教育厅基金项目(LJKMZ20220637)
文献
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2024年第45卷第4期
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.017
  • 接收时间：2024-04-24
  • 首发时间：2026-04-01
  • 出版时间：2024-08-25
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  • 收稿日期：2024-04-24
基金
国家自然科学基金项目(51775257; 51775258)
辽宁省教育厅基金项目(LJKMZ20220637)
作者信息
    1辽宁科技大学机械工程与自动化学院，鞍山，114051
    2山西北方晋东化工有限公司，阳泉，045000

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2种不同金属材料的力学参数
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