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Article(id=1244239606342857010, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1244239603624952467, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2023.030, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1687622400000, receivedDateStr=2023-06-25, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1774579439005, onlineDateStr=2026-03-27, pubDate=1708790400000, pubDateStr=2024-02-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774579439005, onlineIssueDateStr=2026-03-27, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774579439005, creator=13701087609, updateTime=1774579439005, updator=13701087609, issue=Issue{id=1244239603624952467, tenantId=1146029695717560320, journalId=1241755870837649424, year='2024', volume='45', issue='1', pageStart='1', pageEnd='144', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1774579438358, creator=13701087609, updateTime=1774590203812, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1244284757283025531, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1244239603624952467, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1244284757283025532, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1244239603624952467, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=74, endPage=87, ext={EN=ArticleExt(id=1244239607848612151, articleId=1244239606342857010, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=Analysis of Circumferential Free Vibration of Functionally Graded Joined Conical-cylindrical Shells, columnId=1244229834482757770, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Paper, runingTitle=null, highlight=null, articleAbstract=

This paper focuses on analyzing the circumferential free vibration of the functionally graded joined conical-cylindrical shell to enhance the vibration performance and stability of the structure, particularly in the aerospace field. First, the properties of the functionally graded materials (FGMs) are described using the Voigt model and the four-parameter power function volume fraction. The energy expressions for the conical shell and cylindrical shell are derived based on the previously obtained displacement-strain relationships formulated utilizing the Donnell thin shell theory. Then, artificial springs are introduced to simulate the continuity conditions and boundary conditions. The displacement function is constructed using Chebyshev polynomials to enable a more accurate analysis of the structural response and performance. The modal frequencies of the functionally graded joined conical-cylindrical shell are calculated employing the Rayleigh-Ritz method with this displacement function. Hence, the influence of gradient exponent, boundary conditions, and geometric parameters on the modal frequencies is analyzed to reveal the vibration characteristics of the structure. The main results indicate that increasing the volume fraction of ceramics effectively enhances the modal frequencies of the structure, while higher gradient exponents lead to a decrease in the modal frequencies. Stronger boundary constraints result in higher modal frequencies for the functionally graded joined conical-cylindrical shell. With an increase in the circumferential wave number, the influence of boundary conditions on the structural modal frequencies diminishes. The effect of boundary constraints is more pronounced on the cylindrical shell compared to the conical shell. Additionally, the axial spring stiffness has a more significant impact on the modal frequencies of the structure compared to the circumferential and radial spring stiffnesses. When the circumferential wave number is greater than 3, the modal frequency of the structure exhibits a linear increase with increasing shell thickness, whereas increasing the conical and cylindrical shell length ratio leads to a decrease in modal frequency. Finally, when the length ratio of the conical and cylindrical shell is fixed, increasing the cone angle initially results in an increase in the modal frequencies of the structure until it reaches a peak value, after which it starts to decrease.

, correspAuthors=Long Cheng, 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=Lei Pang, Long Cheng, Wenguang Liu, Yuhang Zhang, Zhipeng Lv, Runhao Wan), CN=ArticleExt(id=1244239625452106471, articleId=1244239606342857010, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=功能梯度锥-柱连接壳的环向自由振动分析, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

论文旨在分析功能梯度锥-柱连接壳的环向自由振动，以提高其结构的振动性能和稳定性. 采用Voigt模型和四参数幂函数体积分数描述功能梯度材料属性，基于Donnell薄壳理论推导出锥壳和柱壳的位移与应变关系，分别得出锥壳和柱壳的能量表达式. 引入人工弹簧模拟边界和壳体间的连接条件，依据Chebyshev多项式构造位移函数，基于Rayleigh-Ritz法求解FGMs锥-柱连接壳模态频率，分析梯度指数、边界条件和几何参数对模态频率的影响. 结果表明：增加陶瓷体积分数能有效提高结构的模态频率，而增大梯度指数则会降低结构的模态频率；边界约束条件越强，FGMs锥-柱连接壳的模态频率越高；随着环向波数的增大，边界条件对结构模态频率的影响越来越弱，边界约束效果作用于圆柱壳明显强于圆锥壳；当环向波数大于3时，随着壳体厚度增大，结构的模态频率呈线性提高，而增大锥柱壳长度比会降低结构模态频率；在锥柱壳长度比一定时，随着锥角的增大会使结构的模态频率先增加到峰值后减小.

, correspAuthors=成龙, authorNote=null, correspAuthorsNote=
**E-mail：.
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Si3N4/SUS304 temperature coefficient[15]

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材料属性K0K-1K1K2K3
Si3N4E348.43×1090-3.070×10-42.160×10-7-8.946×10-11
ρ23700000
ν0.240000
SUS304E201.04×10903.079×10-4-6.534×10-70
ρ81660000
ν0.32620-2.002×10-43.797×10-70
), ArticleFig(id=1244274774680126458, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1244239606342857010, language=CN, label=表1, caption=

Si3N4/SUS304温敏系数[15]

, figureFileSmall=null, figureFileBig=null, tableContent=
材料属性K0K-1K1K2K3
Si3N4E348.43×1090-3.070×10-42.160×10-7-8.946×10-11
ρ23700000
ν0.240000
SUS304E201.04×10903.079×10-4-6.534×10-70
ρ81660000
ν0.32620-2.002×10-43.797×10-70
), ArticleFig(id=1244274774768206847, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1244239606342857010, language=EN, label=Table 2, caption=

Material parameters in reference documents

, figureFileSmall=null, figureFileBig=null, tableContent=
MaterialsElastic module E（109 Pa）Passion ratio νDensity ρ（kg/m3）
Ref. [17]Zirconia1680.35700
Aluminum700.32707
Ref. [1]Stainless steel207.7880.31778166
Ni205.0980.38900
), ArticleFig(id=1244274774852091908, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1244239606342857010, language=CN, label=表2, caption=

对比文献中的材料参数

, figureFileSmall=null, figureFileBig=null, tableContent=
MaterialsElastic module E（109 Pa）Passion ratio νDensity ρ（kg/m3）
Ref. [17]Zirconia1680.35700
Aluminum700.32707
Ref. [1]Stainless steel207.7880.31778166
Ni205.0980.38900
), ArticleFig(id=1244274774948560908, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1244239606342857010, language=EN, label=Table 3, caption=

Stiffness coefficient of boundary spring

, figureFileSmall=null, figureFileBig=null, tableContent=
Boundary conditionFSDSSC
Stiffness coefficientk1=0k1=0k1=1×1013 N/mk1=1×1013 N/m
k2=0k2=1×1013 N/mk2=1×1013 N/mk2=1×1013 N/m
k3=0k3=1×1013 N/mk3=1×1013 N/mk3=1×1013 N/m
k4=0k4=0k4=0k4=1×1013 N/rad
), ArticleFig(id=1244274775019864077, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1244239606342857010, language=CN, label=表3, caption=

边界弹簧刚度系数

, figureFileSmall=null, figureFileBig=null, tableContent=
Boundary conditionFSDSSC
Stiffness coefficientk1=0k1=0k1=1×1013 N/mk1=1×1013 N/m
k2=0k2=1×1013 N/mk2=1×1013 N/mk2=1×1013 N/m
k3=0k3=1×1013 N/mk3=1×1013 N/mk3=1×1013 N/m
k4=0k4=0k4=0k4=1×1013 N/rad
), ArticleFig(id=1244274775091167249, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1244239606342857010, language=EN, label=Table 4, caption=

Influence of truncated terms on the dimensionless modal frequency of structures

, figureFileSmall=null, figureFileBig=null, tableContent=
nMFEM
4681012
10.062230.055770.055300.055290.055290.05364
20.056130.044410.043820.043800.043800.04203
30.062480.050750.050400.050380.050380.04544
), ArticleFig(id=1244274775175053334, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1244239606342857010, language=CN, label=表4, caption=

截断项数对结构无量纲模态频率的影响

, figureFileSmall=null, figureFileBig=null, tableContent=
nMFEM
4681012
10.062230.055770.055300.055290.055290.05364
20.056130.044410.043820.043800.043800.04203
30.062480.050750.050400.050380.050380.04544
), ArticleFig(id=1244274775292493849, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1244239606342857010, language=EN, label=Table 5, caption=

Comparison of dimensionless frequencies of FGMs joined conical-cylindrical shell under different boundary conditions (R2=1 m, R1=0.4226 m, Lcy=2.5 m, α0=30°, h=0.1 m, a=1, b=0)

, figureFileSmall=null, figureFileBig=null, tableContent=
nkboundary condition
C-CSS-SSSS-C
PresentRef. [19]PresentRef. [19]PresentRef. [19]
10.50.055780.054020.054770.053220.055090.05354
20.054750.053070.053840.052280.054120.05254
50.054110.052390.053100.051570.053420.05187
100.053730.051980.052640.051120.052990.05148
20.50.044150.043100.042950.042020.043910.04283
20.043520.042550.042550.041610.043360.04226
50.043400.042340.042240.041290.043170.04206
3100.043130.042080.041840.040910.042890.04180
0.50.050690.049590.049720.048770.050680.04958
20.050740.049540.049910.048850.050730.04952
50.051620.050340.050690.049560.051610.05033
100.051470.050230.050430.049360.051460.05022
), ArticleFig(id=1244274775372185630, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1244239606342857010, language=CN, label=表5, caption=

不同边界条件下FGMs锥-柱连接壳的无量纲频率对比（R2=1 m，R1=0.4226 m，Lcy=2.5 m，α0=30°，h=0.1 m，a=1，b=0）

, figureFileSmall=null, figureFileBig=null, tableContent=
nkboundary condition
C-CSS-SSSS-C
PresentRef. [19]PresentRef. [19]PresentRef. [19]
10.50.055780.054020.054770.053220.055090.05354
20.054750.053070.053840.052280.054120.05254
50.054110.052390.053100.051570.053420.05187
100.053730.051980.052640.051120.052990.05148
20.50.044150.043100.042950.042020.043910.04283
20.043520.042550.042550.041610.043360.04226
50.043400.042340.042240.041290.043170.04206
3100.043130.042080.041840.040910.042890.04180
0.50.050690.049590.049720.048770.050680.04958
20.050740.049540.049910.048850.050730.04952
50.051620.050340.050690.049560.051610.05033
100.051470.050230.050430.049360.051460.05022
), ArticleFig(id=1244274775443488801, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1244239606342857010, language=EN, label=Table 6, caption=

Comparison of frequency for FGMs cylindrical shell under SD-SD boundary conditions (R2=1, h/R2=0.002, L/R=20)

, figureFileSmall=null, figureFileBig=null, tableContent=
kn
1234567
1Ref. [1]13.2114.4804.15697.038411.24116.45522.635
Present13.2174.4824.15837.041511.24616.46322.646
15Ref. [1]12.9334.38344.06536.885610.99916.10122.148
Present12.9394.38514.06666.888611.00416.10922.159
), ArticleFig(id=1244274775527374885, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1244239606342857010, language=CN, label=表6, caption=

SD-SD边界条件下FGMs圆柱壳的频率对比（R2=1，h/R2=0.002，L/R=20）

, figureFileSmall=null, figureFileBig=null, tableContent=
kn
1234567
1Ref. [1]13.2114.4804.15697.038411.24116.45522.635
Present13.2174.4824.15837.041511.24616.46322.646
15Ref. [1]12.9334.38344.06536.885610.99916.10122.148
Present12.9394.38514.06666.888611.00416.10922.159
), ArticleFig(id=1244274775636426794, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1244239606342857010, language=EN, label=Table 7, caption=

Variation of dimensionless modal frequency with circumferential wave number under different gradient exponent

, figureFileSmall=null, figureFileBig=null, tableContent=
Boundary conditionnk
0.30.5135
C-C10.055210.050150.043660.036360.03435
20.027060.025170.021460.017920.01695
30.014970.013620.011900.009950.00941
40.009840.008970.007840.006590.00623
50.008430.007680.006740.005710.00542
60.009450.008620.007580.006470.00616
70.011910.010850.009540.008190.00781
80.015210.013850.012190.010470.00999
SD-SD10.040310.036670.031990.026740.02530
20.018780.017090.014920.012480.01180
30.010200.009290.008120.006790.00643
40.007070.006440.005650.004760.00451
50.007020.006400.005630.004800.00456
60.008820.008040.007080.006070.00578
70.011610.010580.009310.008000.00763
80.015060.013720.012070.010370.00990
), ArticleFig(id=1244274775724507182, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1244239606342857010, language=CN, label=表7, caption=

不同梯度指数下无量纲模态频率随环向波数的变化

, figureFileSmall=null, figureFileBig=null, tableContent=
Boundary conditionnk
0.30.5135
C-C10.055210.050150.043660.036360.03435
20.027060.025170.021460.017920.01695
30.014970.013620.011900.009950.00941
40.009840.008970.007840.006590.00623
50.008430.007680.006740.005710.00542
60.009450.008620.007580.006470.00616
70.011910.010850.009540.008190.00781
80.015210.013850.012190.010470.00999
SD-SD10.040310.036670.031990.026740.02530
20.018780.017090.014920.012480.01180
30.010200.009290.008120.006790.00643
40.007070.006440.005650.004760.00451
50.007020.006400.005630.004800.00456
60.008820.008040.007080.006070.00578
70.011610.010580.009310.008000.00763
80.015060.013720.012070.010370.00990
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功能梯度锥-柱连接壳的环向自由振动分析
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庞磊 1 , 成龙 1, ** , 刘文光 1 , 张宇航 1 , 吕志鹏 2 , 宛润豪 1
固体力学学报 | 研究论文 2024,45(1): 74-87
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固体力学学报 | 研究论文 2024, 45(1): 74-87
功能梯度锥-柱连接壳的环向自由振动分析
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庞磊1, 成龙1, ** , 刘文光1, 张宇航1, 吕志鹏2, 宛润豪1
作者信息
  • 1南昌航空大学航空制造工程学院，南昌，330063
  • 2东北大学机械工程与自动化学院，沈阳，110819

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Analysis of Circumferential Free Vibration of Functionally Graded Joined Conical-cylindrical Shells
Lei Pang1, Long Cheng1, ** , Wenguang Liu1, Yuhang Zhang1, Zhipeng Lv2, Runhao Wan1
Affiliations
  • 1School of Aeronautical Manufacturing Engineering, Nanchang Hangkong University, Nanchang, 330063
  • 2School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819
出版时间: 2024-02-25 doi: 10.19636/j.cnki.cjsm42-1250/o3.2023.030
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摘要
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论文旨在分析功能梯度锥-柱连接壳的环向自由振动，以提高其结构的振动性能和稳定性. 采用Voigt模型和四参数幂函数体积分数描述功能梯度材料属性，基于Donnell薄壳理论推导出锥壳和柱壳的位移与应变关系，分别得出锥壳和柱壳的能量表达式. 引入人工弹簧模拟边界和壳体间的连接条件，依据Chebyshev多项式构造位移函数，基于Rayleigh-Ritz法求解FGMs锥-柱连接壳模态频率，分析梯度指数、边界条件和几何参数对模态频率的影响. 结果表明：增加陶瓷体积分数能有效提高结构的模态频率，而增大梯度指数则会降低结构的模态频率；边界约束条件越强，FGMs锥-柱连接壳的模态频率越高；随着环向波数的增大，边界条件对结构模态频率的影响越来越弱，边界约束效果作用于圆柱壳明显强于圆锥壳；当环向波数大于3时，随着壳体厚度增大，结构的模态频率呈线性提高，而增大锥柱壳长度比会降低结构模态频率；在锥柱壳长度比一定时，随着锥角的增大会使结构的模态频率先增加到峰值后减小.

锥-柱连接壳  /  功能梯度材料  /  Chebyshev多项式  /  模态频率

This paper focuses on analyzing the circumferential free vibration of the functionally graded joined conical-cylindrical shell to enhance the vibration performance and stability of the structure, particularly in the aerospace field. First, the properties of the functionally graded materials (FGMs) are described using the Voigt model and the four-parameter power function volume fraction. The energy expressions for the conical shell and cylindrical shell are derived based on the previously obtained displacement-strain relationships formulated utilizing the Donnell thin shell theory. Then, artificial springs are introduced to simulate the continuity conditions and boundary conditions. The displacement function is constructed using Chebyshev polynomials to enable a more accurate analysis of the structural response and performance. The modal frequencies of the functionally graded joined conical-cylindrical shell are calculated employing the Rayleigh-Ritz method with this displacement function. Hence, the influence of gradient exponent, boundary conditions, and geometric parameters on the modal frequencies is analyzed to reveal the vibration characteristics of the structure. The main results indicate that increasing the volume fraction of ceramics effectively enhances the modal frequencies of the structure, while higher gradient exponents lead to a decrease in the modal frequencies. Stronger boundary constraints result in higher modal frequencies for the functionally graded joined conical-cylindrical shell. With an increase in the circumferential wave number, the influence of boundary conditions on the structural modal frequencies diminishes. The effect of boundary constraints is more pronounced on the cylindrical shell compared to the conical shell. Additionally, the axial spring stiffness has a more significant impact on the modal frequencies of the structure compared to the circumferential and radial spring stiffnesses. When the circumferential wave number is greater than 3, the modal frequency of the structure exhibits a linear increase with increasing shell thickness, whereas increasing the conical and cylindrical shell length ratio leads to a decrease in modal frequency. Finally, when the length ratio of the conical and cylindrical shell is fixed, increasing the cone angle initially results in an increase in the modal frequencies of the structure until it reaches a peak value, after which it starts to decrease.

joined conical-cylindrical shells  /  functionally graded material  /  Chebyshev polynomial  /  modal frequency
庞磊, 成龙, 刘文光, 张宇航, 吕志鹏, 宛润豪. 功能梯度锥-柱连接壳的环向自由振动分析. 固体力学学报, 2024 , 45 (1) : 74 -87 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2023.030
Lei Pang, Long Cheng, Wenguang Liu, Yuhang Zhang, Zhipeng Lv, Runhao Wan. Analysis of Circumferential Free Vibration of Functionally Graded Joined Conical-cylindrical Shells[J]. Chinese Journal of Solid Mechanics, 2024 , 45 (1) : 74 -87 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2023.030
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0　引言
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锥壳和柱壳作为壳体基本单元，在航天飞行器中得到广泛应用，航天飞行器在运行过程中，由于与大气的摩擦以及外部激励的影响，会导致温度急剧升高，并且不可避免地引起振动现象. 功能梯度材料（Functionally Graded Materials，FGMs）是一种结构和成分在空间上呈现连续梯度变化的新型复合材料. 相较于传统复合材料，在高温环境中，FGMs能有效缓解材料内部热应力，具备出色的防热、耐腐蚀和高强度性能，同时还具备良好的可设计性. 为了促进功能梯度材料在航天飞机结构设计上的应用，FGMs壳体的动力学设计成了研究者关注的焦点.
围绕FGMs壳体的动力学问题，学者们进行了大量的研究. 基于Love薄壳理论，Loy等[1]采用瑞利里兹方法推导了FGMs圆柱壳的特征频率方程. 李文达[2,3]等利用Fourier级数方法展开位移场，通过能量原理推导了旋转FGMs圆柱壳的自由振动方程. 陈金晓等[4]基于波动法在Flügge理论的基础上推导出了弹性边界条件下FGMs圆柱壳的模态频率方程，探究了不同几何参数对振动特性的影响. 黄小林等[5]基于经典薄壳理论建立了弹性地基上均匀和非均匀分布孔隙FGMs圆锥壳振动方程，进而分析了孔隙和几何参数下壳体的动力学响应. Zheng等[6]基于Donnell壳体理论，利用傅里叶级数来辅助处理弹性边界约束不连续的问题，探究了轴向厚度和斜率等对阶跃圆柱壳固有频率的影响. Qin等[7]通过一阶剪切变形理论和人工弹簧技术得到了石墨烯增强FGMs层合扁壳一般方程，通过采用正交多项式扩展壳体位移场以及Rayleigh-Ritz方法求解了壳体的模态频率，研究了边界条件和层数等对振动特性的影响. 曹洲等[8]在三阶剪切变形理论基础上，利用Hamilton原理和Navier法得到了不同夹层类型的系统固有频率，研究了几何参数和温度场对结构自由振动固有频率的影响.
然而，工程实际中的壳体结构通常以连接壳的方式广泛存在于不同应用场合中. 由于壳体间的相互作用，连接壳的振动特性会更加复杂，因此有必要对其动力学特性展开研究. Bagheri等[9]借助一阶剪切变形理论分析了FGMs圆柱-半球连接壳的自由振动特性，通过利用半解析广义微分求积法离散运动方程，探讨了幂律指数和几何参数对壳体振动的影响. Damercheloo等[10,11]利用一阶剪切变形理论建立了石墨烯（GNP）加固锥-锥连接壳的动力学模型，使用Hamilton原理推导了振动控制微分方程，采用微分求积法确定了壳体的模态频率和相应振型，分析了不同参数对GNP加固圆锥-圆锥连接壳模态频率的影响，继而考虑离心力和科氏力，研究了GNPs加固旋转锥-锥连接壳的行波模态频率. Zarei等[12]利用Donnell薄壳理论推导了加肋圆锥-圆柱复合材料连接壳控制方程，采用幂级数求解了所建模型的模态频率和振型. 在一阶剪切变形理论的框架下，Sobhani等[13]研究了复合三项材料制成的半球-圆柱-圆锥连接壳体的模态频率，基于Donnell壳体理论和Hamilton原理推导了球-柱-锥连接壳结构的相关运动控制方程，利用广义微分求积法（GDQM）求解并研究了不同参数对连接壳振动特性的影响. Fu等[14]建立了弹性地基下的圆锥壳的振动控制方程，使用GDQM离散并求解控制方程，分析了弹性地基环绕FGMs夹层圆锥-圆锥连接壳振动特性，研究了夹层分布和梯度指数对FGMs夹层圆锥-圆锥连接壳振动特性的影响.
综上所述，虽然研究者已经开展了大量关于FGMs壳体振动问题的研究，但是对于连接壳的动力学问题求解仍较为复杂. 因此，本工作以FGMs锥-柱连接壳为研究对象，基于Donnell薄壳理论并引入人工弹簧推导出圆锥壳、圆柱壳以及边界条件和壳体连接条件的能量方程，基于能量原理分析FGMs锥-柱连接壳的自由振动特性. 研究结果可为航空航天等实际工程的动力学设计提供一定的理论支撑.
1　FGMs锥-柱连接壳模型
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如图1所示的锥-柱连接壳几何模型，设定锥角方向为正方向，其中圆柱壳的长度为Lcy，圆锥壳的长度为Lco、圆锥角为α0、圆锥壳小端的内径为R1，大端的内径和圆柱壳的内径为R2，连接壳体的厚度均为h. 在连接壳中面建立图示坐标系（x，θ，z），圆锥壳和圆柱壳中面上任意一点的轴向、周向和法向位移分别表示为uco、vco、wco和ucy、vcy、wcy，通过引入人工弹簧模拟壳体两端边界条件以及壳体之间的连接条件.
假设连接壳是由金属陶瓷FGMs制成，壳体外表面为纯金属，内表面为纯陶瓷，金属和陶瓷沿厚度方向的分布服从Voigt模型，FGMs连接壳的有效材料参数表达式如式1所示.
式中：P1和P2分别为陶瓷和金属的材料属性，主要指材料的弹性模量E、质量密度ρ和泊松比ν. 陶瓷体积分数V1和金属体积分数V2服从幂函数分布形式如式2所示. 图2展示了体积分数在厚度方向上的变化. 因而，可通过调整梯度指数k来改变材料体积分数和材料物性参数，进而改变结构的模态频率，尽可能避免实际工程中共振现象的发生.
式中：a，b，c为分布参数.
假设材料属性具有与温度相关的性能[15]：
式中：P表示弹性模量E（Pa）、密度ρ（kg/m3）和泊松比ν. K0、K-1、K1、K2、K3表示温敏系数. 表1给出了Si3N4/SUS304温敏系数.
2　FGMs锥-柱连接壳能量方程
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对于圆锥壳中面上任意一点的半径R是关于坐标x的函数关系式：
其中，当x=Lco时，表示圆柱端半径.
FGMs圆柱壳和圆锥壳动能T分别表示为：
式中：（·）表示位移对时间t的求导，ρi为单位长度的质量密度.
基于Donnell薄壳理论[16]，可得到薄壳上任意点的应变表达式：
根据胡克定律，得到壳体应力与应变的关系为：
式中：Qpj（p，j=1、2、6）为刚度系数，具体表示如下：
式中：α和β分别表示FGMs壳体的轴向和周向，、和表示壳体中曲面应变分量，、和表示壳体中曲面曲率. 上标i指co和cy，分别表示锥壳和柱壳. ui、vi和wi分别表示壳体上轴向、环向和径向上任意一点的位移.
圆锥薄壳的中面应变与位移的关系可表示为：
圆柱薄壳中面应变与位移的关系表示为：
对于FGMs圆柱和圆锥薄壳，其力与力矩可表示为：
FGMs圆柱壳与圆锥壳应变能可表示为：
式中：S为简化刚度矩阵，ε为壳体中曲面应变组成的向量，可表示为：
式中：、、（p，j=1、2、6）分别为拉伸，耦合和弯曲刚度：
引入约束弹簧模拟连接壳的边界条件和连接条件，假设k1、k2、k3和k4四组弹簧和k5、k6、k7和k8四组弹簧分别是施加于圆锥壳的左端和圆柱壳右端x、θ和z方向的线弹簧以及z方向的扭转弹簧. 因此，储存在边界弹簧的弹性势能可表示为：
假设ku、kv、kw和kφ分别为壳段连接处x、θ和z方向的线弹簧以及z方向的扭转弹簧，通过分析几何模型可得到圆锥壳和圆柱壳连接条件为：
因此，储存在连接处的弹簧弹性势能为：
3　FGMs锥-柱连接壳频率方程
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将FGMs连接壳轴向、环向和径向三个方向上的振动位移ui、vi和wi表示成环向模态和环向波数的位移函数的乘积形式：
式中：An（ζ）、Bn（ζ）、Cn（ζ）是环向波数为n时的振型函数. 引入Chebyshev多项式：
式中：Aa、Ba、Ca为待定系数，M为截断项数，Ta（·）是Chebyshev多项式，表达式为[17,18]：
由于切比雪夫多项式所定义的区间是[-1，1]，定义无量纲：
因此，FGMs连接壳结构的拉格朗日能量泛函L为：
式中：Ui、Ti和Vi分别为FGMs圆柱壳和圆锥壳的最大应变能、最大动能以及约束弹簧的最大弹性势能.
由Rayleigh-Ritz法，对待定系数求偏导，可推出模态频率方程：
式中：K和M分别为刚度阵和质量阵，Ai、Bi和Ci为待定系数组成的向量，为模态频率. 求解频率方程可得到FGMs锥-柱连接壳的特征频率，Ai、Bi和Ci的具体表达式为：
定义无量纲模态频率为：
4　分析与讨论
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4.1　模型验证
为了更好的达到计算条件，探究梯度指数k=2，环向波数n=2和3，T=300 K下由陶瓷氧化锆金属铝构造的FGMs锥-柱连接壳，当设置边界弹簧刚度为0，连接处弹簧刚度如图3（a）所示变化，在大于等于1×1012时，结构无量纲频率分别达到0.01360和0.03723且不再变化，通过与文献[19]比较，最大误差仅为2.14%. 因此连接弹簧刚度大于等于1×1012时可较好的模拟连接条件，同时也说明边界弹簧刚度等于0时，其可模拟自由边界；当设置连接弹簧刚度为1×1013，边界弹簧刚度如图3（b）所示，在大于等于1×1012时，无量纲频率分别为0.04358和0.05075且不再变化，通过与文献[19]比较，最大误差为2.4%. 因此，边界弹簧刚度大于等于1×1012，可更好的模拟固支边界. 综上所述，自由（F）、软简支（SD）、硬简支（SS）和固支（C）四种边界弹簧刚度系数设置如表3所示，以圆锥左端边界为例.
壳体位移结构是无穷项切比雪夫多项式的总和，在实际计算过程中，为了便于提高计算效率，只需要计算有限项. 因此表4分析了C-C边界条件下，梯度指数为1的FGMs锥-柱连接壳的收敛性并与有限元进行对比结果较吻合，产生的误差主要原因是由于有限元网格大小的选取. 图4展示了根据线性离散分层思想并基于ABAQUS软件建立FGMs锥-柱连接壳有限元模型，环向波数为1、2和3时的振型如图5所示. 结果表明：在截断数为第10项时，频率已经达到收敛，说明切比雪夫对于求解结果具有较好的收敛性. 因此本文后续计算均采用截断10.
表5研究了C-C、SS-SS和SS-C三种边界条件下FGMs锥-柱连接壳的模态频率，表6研究了金属钢和陶瓷镍制成的SD-SD边界条件FGMs圆柱壳模态频率. 结果表明：在各种边界条件下，本文基于能量原理求解与文献[19]和文献[1]的计算结果基本吻合，总体误差小于3%，产生误差的主要原因是由于不同壳体理论和求解方法. 这也说明Chebyshev多项式可用于分析FGMs锥-柱连接壳的振动特性.
4.2　FGMs锥-柱连接壳的模态频率分析
如未特殊说明，则选取T=300 K下a=1，b=0陶瓷Si3N4和金属SUS304作为FGMs锥-柱连接壳组成材料. 壳体的几何参数为：R2=1 m，R1=0.5 m，α0=30°，h=0.003 m，Lcy=5 m.
4.2.1　梯度指数的影响
表7分析了C-C和SD-SD两种边界条件不同梯度指数下模态频率随环向波数的变化，对比可得：边界约束效果越强，结构刚度越大，结构模态频率越高. 随着梯度指数的不断增大，结构的整体刚度不断减小，导致结构频率逐渐减小. 随着环向波数的不断增大，结构的模态频率呈现出先减小后增大的趋势.
图6（a）和（b）分别对C-C，SD-SD两种边界条件下频率在不同环向波数下随梯度指数变化的影响进行了分析. 结果表明：不同边界条件下，当梯度指数趋近于1×10-2时，可将结构看作由内表面材料构成的连接壳结构，随着梯度指数的不断增大，最终模态频率会逐渐减小并收敛趋于稳定. 当梯度指数趋于1×102时，可将结构看作是由外表面材料制成的连接壳结构. 同时也说明FGMs连接壳采用四参数幂函数体积分数，可以通过改变指数k，满足工程实际要求. 后续如未说明，则研究主要采用梯度指数为1进行讨论分析.
4.2.2　边界条件的影响
图7（a）研究了C-C、C-F、F-C、SD-SD、SD-C、C-SD六种经典边界条件对FGMs连接壳模态频率的影响，结果表明：当环向波数小于6时，边界条件对于FGMs连接壳的频率影响较大，且约束条件越大，结构的整体刚度越大，频率越高；当环向波数大于6时，边界条件对结构的频率影响较小；通过对比SD-C、C-SD和F-C、C-F边界条件，说明边界条件作用于圆柱壳对于结构的约束效果大于圆锥壳；不同边界条件下，FGMs连接壳结构频率随着环向波数的增大呈先减小后增大的趋势，图7（b）所示当环境温度升高时，降低了结构的弹性模量，因而减小了结构刚度. 从而会使结构模态频率会降低但变化范围较小. 在外部激励条件下，低频比较容易发生共振，因此主要对T=300 K下环向波数大于1的结构振动特性进行分析.
图8（a）分析了边界两端轴向和旋转弹簧刚度k1=k5=k4=k8=0时，结构由F-F到SD-SD边界条件的模态频率的变化，结果表明：环向弹簧k2、k6和径向弹簧k3、k7的影响基本一致，当弹簧刚度小于1×104时，模态频率基本没有变化，在弹簧刚度大于等于1×1012后模态频率趋于收敛. 图8（b）研究了两端环向弹簧和径向弹簧k2=k6=k3=k7=1×1013时，由SD-SD到C-C边界条件的模态频率的变化，结果表明：当弹簧刚度小于1×104时，模态频率基本没有变化，大于1×104时，模态频率随轴向弹簧k1、k5的变化比旋转弹簧k4、k8更为明显，说明了轴向弹簧刚度系数对结构的频率变化比旋转弹簧刚度系数影响更大.
4.2.3　厚径比的影响
图9（a）和（b）分别研究了C-C、SD-SD两种边界条件下FGMs连接壳频率随厚径比的影响. 结果表明：随着厚径比的不断增大，结构的刚度不断增大，结构的非线性现象产生愈加困难，因此在图中展现出模态频率随厚度的增加近似呈线性变化. 在环向波数小于3时，厚度的增加对模态频率影响较弱，当环向波数大于3时，模态频率随着结构厚度的增大几乎呈线性升高. 图9（c）和（d）分别分析了在n=2的C-C和SD-SD两种边界条件下，随着梯度指数增加，模态频率在下降，说明梯度指数比厚度对结构的模态频率影响更大.
4.2.4　圆锥角的影响
图10展示了n=2时C-C边界条件模态频率随半锥角的影响，如图10（a）得出：无论如何改变柱壳和锥壳的长度比，结构的模态频率都是先增大后减小. 长度比越大，结构的刚度降低，导致结构频率越低. 作为一个整体，当锥角趋近于±90°时即结构近似于板-壳连接，锥角对结构模态频率影响更大. 而当结构为壳-壳连接时，模态频率对结构的长度参数更为敏感. 当Lcy/Lco分别为1、2和3时，锥角在负方向上，结构的模态频率分别在25°～35°、35°～45°和15°～25°存在峰值；在正方向上，分别在15°～25°、25°～35°和55°～65°存在峰值. 如图10（b）所示，在锥壳半径一定情况下，改变锥角会导致锥壳长度的改变；随着锥角的增加，结构的模态频率先增加达到峰值后减小，并且柱壳越长，模态频率达到峰值所需锥角越大.
5　结论
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本文基于Donnell壳体理论和引入弹簧模拟任意边界条件和连接条件，采用Chebyshev多项式和Ritz法，分析了边界条件、梯度指数和几何参数对FGMs锥-柱连接壳模态频率的影响，主要结论如下：
（1）增大梯度指数能够降低结构的模态频率. 当梯度指数趋近于1×10-2或1×102时，可退化为由单一材料构成的壳体结构.
（2）边界约束条件越强，FGMs锥-柱连接壳的模态频率越高. 随着环向波数的增大，边界条件对结构模态频率的影响越来越弱，边界约束效果作用于圆柱壳明显强于圆锥壳.
（3）改变壳体厚度能够有效的提高结构模态频率. 特别在环向波数小于3时，改变壳体厚度对结构模态频率影响较弱. 随着环向波数的增加，结构模态频率会呈线性提高；即使在增大壳体厚度前提下，梯度指数的增加，结构的模态频率仍会降低.
（4）在锥柱壳长度比一定时，研究发现随着锥角的增大，结构的模态频率呈先增加后减小的趋势；此外，锥角为正方向且锥壳半径一定时，所连接的柱壳越长，需要更大的锥角才能使频率达到峰值.
基金
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  • 国家自然科学基金项目(51965042)
  • 研究生创新专项资金项目(YC2022-030)
文献
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2024年第45卷第1期
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2023.030
  • 接收时间：2023-06-25
  • 首发时间：2026-03-27
  • 出版时间：2024-02-25
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  • 收稿日期：2023-06-25
基金
国家自然科学基金项目(51965042)
研究生创新专项资金项目(YC2022-030)
作者信息
    1南昌航空大学航空制造工程学院，南昌，330063
    2东北大学机械工程与自动化学院，沈阳，110819

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