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Article(id=1228295922677908169, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228295917908980268, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2025.03.018, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1706025600000, receivedDateStr=2024-01-24, revisedDate=1713283200000, revisedDateStr=2024-04-17, acceptedDate=null, acceptedDateStr=null, onlineDate=1770778168601, onlineDateStr=2026-02-11, pubDate=1741536000000, pubDateStr=2025-03-10, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770778168601, onlineIssueDateStr=2026-02-11, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770778168601, creator=13701087609, updateTime=1770778168601, updator=13701087609, issue=Issue{id=1228295917908980268, tenantId=1146029695717560320, journalId=1225147924628267009, year='2025', volume='38', issue='3', pageStart='449', pageEnd='662', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770778167464, creator=13701087609, updateTime=1770949125482, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1229012967862235389, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228295917908980268, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1229012967862235390, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228295917908980268, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=612, endPage=622, ext={EN=ArticleExt(id=1228295923030229715, articleId=1228295922677908169, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Free vibration of GPLs/Al composite with an array cutouts, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

In this paper，we propose a general displacement field function method，which consists of the highly freely chosen basic function and a series of undetermined weighted coefficients to study the free vibration characteristics of GPLs/Al composite plate with an array cutouts. This general displacement field function method can solve the boundary constraint dependence when choosing the displacement field function to obtain the analytical solution in classical plate theory. There is a linear correlation between the weight coefficients. The system of linear equations is constructed based on the boundary constraints，thus the fundamental system of solutions can then be determined. By changing the system of linear equations as well as the fundamental system of solutions，the type of boundary constraint can be easily transferred. This proposed semi-analytic method not only solves the problem of assumed field function dependence on boundary constraints in the classical solution，but also has the superiority of fast conversion of boundary conditions. Meanwhile，by introducing the scatter integral method，a much more efficient and robust method is obtained other than the continuous integral method on the study of free vibration for the structure with an array cutouts. In this paper，we study the free vibration of GPLs/Al composite plates with a single and array cutouts. From the natural frequency and modal analysis，it is found that the generalized mass matrix and stiffness matrix of the open structure decrease synchronously when the cutouts distributed uniformly，and its free vibration characteristics tend to that of the complete plate.

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针对板经典解析解中不可避免的场函数假设依赖边界约束问题，提出一种具有高自由选择性的一般性场函数方法，研究GPLs/Al复合材料板阵列开口结构的自由振动特性。场函数由高自由度的基函数和待定权重系数求积后求和得到，其中权重系数具有线性相关性，其基础解系恰好由边界约束的位移函数线性方程组确定，因此边界约束的改变只需要修改边界位移函数线性方程组。由此建立的一般性场函数半解析法在处理边界约束上既解决了场函数依赖边界问题，又具备了快速转换边界条件的优点，同时在开口结构和阵列结构上以微分求和替代积分的计算方式具有更高的计算效率和鲁棒性。本文在GPLs/Al复合材料板单开口及阵列开口的自由振动研究中，从固有频率和模态分析发现，随着开口分布的均匀性增加，开口结构的质量矩阵和刚度矩阵同步减小，其自由振动特性趋于完整板的自由振动特性。

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颜建伟（1986—），男，博士，教授。E-mail：
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tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=CN, label=图1, caption=阵列矩形开口方形板模式, figureFileSmall=rnq5/LEzcg6I/DQ5gcDAiA==, figureFileBig=IfSTtPdI5bYmypXKkax7oA==, tableContent=null), ArticleFig(id=1228312958850236478, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=EN, label=Fig.2, caption=Natural frequencies of SSSS GPLs/Al composite plate（0. 5 m×0. 5 m×0. 01 m）with an array rectangular cutouts，of which area ratio is 25%, figureFileSmall=8SIHhCAO+Ou/EiXbjKAQLA==, figureFileBig=Tr9Qfn/u648bEIJE4eSWxA==, tableContent=null), ArticleFig(id=1228312960276299844, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=CN, label=图2, caption=简支边界阵列方形开口石墨烯/Al复合板（0.5 m×0.5 m×0.01 m）的固有频率，开口面积占比为25%, figureFileSmall=8SIHhCAO+Ou/EiXbjKAQLA==, figureFileBig=Tr9Qfn/u648bEIJE4eSWxA==, tableContent=null), ArticleFig(id=1228312960389546060, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=EN, label=Fig.3, caption=The first five natural frequencies of GPLs/Al composite plate with cutouts，of which area ratio is 25%, figureFileSmall=nzc5cBcP74YfwiJJoBZIcQ==, figureFileBig=2C3QWeaWh+pBoXmkE1L3gw==, tableContent=null), ArticleFig(id=1228312960469237842, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=CN, label=图3, caption=GPLs/Al复合板开口面积为25%条件下，前5阶频率随着开口数目的变化, figureFileSmall=nzc5cBcP74YfwiJJoBZIcQ==, figureFileBig=2C3QWeaWh+pBoXmkE1L3gw==, tableContent=null), ArticleFig(id=1228312960582484057, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=EN, label=Fig.4, caption=The first 12 vibration modes of GPLs/Al composite in three different opening modes with the same cutout area ratio of 25% and unopened plate, figureFileSmall=GttyhYVNGwXC+o7gqP2FCQ==, 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ratio is 25%, figureFileSmall=zcLGPuOTRYxHfJe2KiYs4w==, figureFileBig=QLLKma0mpAH4A+F8t4pRWQ==, tableContent=null), ArticleFig(id=1228312961647837327, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=CN, label=图7, caption=CFCF边界阵列方形开口石墨烯/Al复合板的固有频率，开口面积占比25%, figureFileSmall=zcLGPuOTRYxHfJe2KiYs4w==, figureFileBig=QLLKma0mpAH4A+F8t4pRWQ==, tableContent=null), ArticleFig(id=1228312961723334804, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=EN, label=Tab.1, caption=

The effect of ni on the vibrational modes when m=n=10

, figureFileSmall=null, figureFileBig=null, tableContent=
ni 简支ni 固定
56
77
88
), ArticleFig(id=1228312961819803801, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=CN, label=表1, caption=

m=n=10时，ni取值对模态的影响

, figureFileSmall=null, figureFileBig=null, tableContent=
ni 简支ni 固定
56
77
88
), ArticleFig(id=1228312961941438620, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=EN, label=Tab.2, caption=

The effect of ni on the natural frequencies when m=n=10

, figureFileSmall=null, figureFileBig=null, tableContent=
边界条件模态固有频率/GHz
10976
简支118.9213.2113.2113.21
242.4433.0333.0313.32
342.6033.0333.0313.32
465.2252.8552.8527.93
580.5266.0666.0633.03
680.6166.0666.0633.03
7102.6785.8785.8752.85
8102.8485.8785.8766.06
9132.85112.28112.2866.06
10132.90112.28112.2885.87
11139.80118.88118.8885.87
12154.78132.08132.08112.28
固定126.4826.4827.2513.35
253.8554.9550.8824.26
353.8554.9550.8849.58
479.5681.7573.2249.58
598.0998.09101.8773.22
698.4798.47102.2288.64
7122.98124.11123.4389.03
8122.98124.11123.43111.45
9155.98160.88143.16111.45
10155.98160.88143.16141.94
11165.51165.51163.70141.94
12179.89186.12164.27148.76
), ArticleFig(id=1228312962075656358, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=CN, label=表2, caption=

m=n=10时，ni取值对固有频率的影响

, figureFileSmall=null, figureFileBig=null, tableContent=
边界条件模态固有频率/GHz
10976
简支118.9213.2113.2113.21
242.4433.0333.0313.32
342.6033.0333.0313.32
465.2252.8552.8527.93
580.5266.0666.0633.03
680.6166.0666.0633.03
7102.6785.8785.8752.85
8102.8485.8785.8766.06
9132.85112.28112.2866.06
10132.90112.28112.2885.87
11139.80118.88118.8885.87
12154.78132.08132.08112.28
固定126.4826.4827.2513.35
253.8554.9550.8824.26
353.8554.9550.8849.58
479.5681.7573.2249.58
598.0998.09101.8773.22
698.4798.47102.2288.64
7122.98124.11123.4389.03
8122.98124.11123.43111.45
9155.98160.88143.16111.45
10155.98160.88143.16141.94
11165.51165.51163.70141.94
12179.89186.12164.27148.76
), ArticleFig(id=1228312962180513959, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=EN, label=Tab.3, caption=

The effect of ni on the natural frequencies when m=n=20

, figureFileSmall=null, figureFileBig=null, tableContent=
边界条件模态固有频率/GHz
2019171514
简支117.4713.2113.2113.2113.21
239.1833.0333.0333.0333.03
339.3233.0333.0333.0333.03
460.2352.8552.8552.8552.85
574.2466.0666.0666.0666.06
674.3166.0666.0666.0666.06
794.7585.8785.8785.8785.87
894.9085.8785.8785.8785.87
9122.25112.28112.28112.28112.20
10122.29112.28112.28112.28112.20
11129.03118.88118.88118.88112.28
12142.60132.08132.08132.08112.28
固定125.1725.1725.3024.5724.25
251.3051.5050.4349.6049.47
351.3051.5050.4349.6049.47
475.6776.0773.9372.9572.95
592.2692.2692.8289.8588.64
692.6892.6893.2390.2889.06
7115.71115.91115.37112.26111.19
8115.71115.91115.37112.26111.19
9147.43148.21144.08141.98112.34
10147.43148.21144.08141.98141.87
11154.60154.60155.62150.36141.87
12169.71170.71165.47163.21148.25
), ArticleFig(id=1228312962352480428, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=CN, label=表3, caption=

m=n=20时，ni取值对固有频率的影响

, figureFileSmall=null, figureFileBig=null, tableContent=
边界条件模态固有频率/GHz
2019171514
简支117.4713.2113.2113.2113.21
239.1833.0333.0333.0333.03
339.3233.0333.0333.0333.03
460.2352.8552.8552.8552.85
574.2466.0666.0666.0666.06
674.3166.0666.0666.0666.06
794.7585.8785.8785.8785.87
894.9085.8785.8785.8785.87
9122.25112.28112.28112.28112.20
10122.29112.28112.28112.28112.20
11129.03118.88118.88118.88112.28
12142.60132.08132.08132.08112.28
固定125.1725.1725.3024.5724.25
251.3051.5050.4349.6049.47
351.3051.5050.4349.6049.47
475.6776.0773.9372.9572.95
592.2692.2692.8289.8588.64
692.6892.6893.2390.2889.06
7115.71115.91115.37112.26111.19
8115.71115.91115.37112.26111.19
9147.43148.21144.08141.98112.34
10147.43148.21144.08141.98141.87
11154.60154.60155.62150.36141.87
12169.71170.71165.47163.21148.25
), ArticleFig(id=1228312962474115250, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=EN, label=Tab.4, caption=

The first twelve natural frequencies of a SSSS 10 nm×10 nm graphene

, figureFileSmall=null, figureFileBig=null, tableContent=
模态(m, n)固有频率/GHz误差/%
原子-连续法[20]经典板解析解本文方法
(1, 1)13.195013.214313.21370.14
(1, 2)或(2, 1)33.001133.035733.03200.09
(2, 2)53.440952.857152.8476‒1.11
(1, 3)或(3, 1)65.669866.071466.05650.59
(2, 3)或(3, 2)87.105585.892885.8677‒1.42
(1, 4)或(4, 1)110.6689112.3213112.27851.45
(3, 3)122.3142118.9285118.8804‒2.81
(2, 4)或(4, 2)133.3478132.1427132.0834‒0.95
), ArticleFig(id=1228312962616721588, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=CN, label=表4, caption=

四边简支的单层石墨烯（10 nm×10 nm）前12阶固有频率

, figureFileSmall=null, figureFileBig=null, tableContent=
模态(m, n)固有频率/GHz误差/%
原子-连续法[20]经典板解析解本文方法
(1, 1)13.195013.214313.21370.14
(1, 2)或(2, 1)33.001133.035733.03200.09
(2, 2)53.440952.857152.8476‒1.11
(1, 3)或(3, 1)65.669866.071466.05650.59
(2, 3)或(3, 2)87.105585.892885.8677‒1.42
(1, 4)或(4, 1)110.6689112.3213112.27851.45
(3, 3)122.3142118.9285118.8804‒2.81
(2, 4)或(4, 2)133.3478132.1427132.0834‒0.95
), ArticleFig(id=1228312962742550712, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=EN, label=Tab.5, caption=

The first twelve natural frequencies of a CCCC 10 nm×10 nm graphene

, figureFileSmall=null, figureFileBig=null, tableContent=
模态(m, n)固有频率/GHz误差/%
原子-连续法[20]经典板解析解本方法
(1, 1)23.140024.917124.56586.16
(1, 2)或(2, 1)46.926067.667250.12116.81
(2, 2)70.480599.668373.92554.89
(1, 3)或(3, 1)82.8536143.137789.85508.45
(2, 3)或(3, 2)107.2780167.8438112.72195.07
(1, 4)或(4, 1)129.9146249.6375143.797310.69
(3, 3)145.6631224.2537150.36183.23
(2, 4)或(4,)155.2559270.6687165.47436.58
), ArticleFig(id=1228312962864185531, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=CN, label=表5, caption=

四边固定的单层石墨烯（10 nm×10 nm）前12阶固有频率

, figureFileSmall=null, figureFileBig=null, tableContent=
模态(m, n)固有频率/GHz误差/%
原子-连续法[20]经典板解析解本方法
(1, 1)23.140024.917124.56586.16
(1, 2)或(2, 1)46.926067.667250.12116.81
(2, 2)70.480599.668373.92554.89
(1, 3)或(3, 1)82.8536143.137789.85508.45
(2, 3)或(3, 2)107.2780167.8438112.72195.07
(1, 4)或(4, 1)129.9146249.6375143.797310.69
(3, 3)145.6631224.2537150.36183.23
(2, 4)或(4,)155.2559270.6687165.47436.58
), ArticleFig(id=1228312962985820350, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=EN, label=Tab.6, caption=

The first twelve natural frequencies of a SSSS 10 nm square graphene with a square cutout（10/13 nm）

, figureFileSmall=null, figureFileBig=null, tableContent=
模态(m, n)固有频率/GHz误差/%
原子-连续法本方法
(1, 1)13.165313.1325‒0.2491
(1, 2)或(2, 1)33.186033.0300‒0.4701
(2, 2)54.203152.6282‒2.9056
(1, 3)或(3, 1)65.279965.71650.6688
(2, 3)或(3, 2)88.378285.8530‒2.8573
(1, 4)或(4, 1)109.9323112.24962.1079
(3, 3)124.5802118.8215‒4.6225
(2, 4)或(4, 2)134.1927131.4207‒2.0657
), ArticleFig(id=1228312963082289345, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=CN, label=表6, caption=

四边简支单层石墨烯（10 nm）中心开口（10/13 nm）前12阶固有频率

, figureFileSmall=null, figureFileBig=null, tableContent=
模态(m, n)固有频率/GHz误差/%
原子-连续法本方法
(1, 1)13.165313.1325‒0.2491
(1, 2)或(2, 1)33.186033.0300‒0.4701
(2, 2)54.203152.6282‒2.9056
(1, 3)或(3, 1)65.279965.71650.6688
(2, 3)或(3, 2)88.378285.8530‒2.8573
(1, 4)或(4, 1)109.9323112.24962.1079
(3, 3)124.5802118.8215‒4.6225
(2, 4)或(4, 2)134.1927131.4207‒2.0657
), ArticleFig(id=1228312963208118469, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=EN, label=Tab.7, caption=

Mechanical properties of GPLs/Al composite with an addition of 1%（mass friction）GPLs

, figureFileSmall=null, figureFileBig=null, tableContent=
密度ρ/(g·cm‒3)泊松比μ杨氏模量E/GPa
2.65770.302976.7285
), ArticleFig(id=1228312963333947592, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=CN, label=表7, caption=

添加均布1%质量分数石墨烯纳米片的石墨烯/铝复合材料的力学性能参数

, figureFileSmall=null, figureFileBig=null, tableContent=
密度ρ/(g·cm‒3)泊松比μ杨氏模量E/GPa
2.65770.302976.7285
), ArticleFig(id=1228312964747428046, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=EN, label=Tab.8, caption=

The first twelve natural frequencies of SSSS GPLs/Al composite plate（0. 5 m×0. 5 m×0. 01 m）with a central opening

, figureFileSmall=null, figureFileBig=null, tableContent=
模态(m, n)固有频率/GHz误差/%
有限元本方法
(1, 1)199.49202.571.5439
(1, 2)或(2, 1)504.20510.721.2931
(2, 2)797.63810.561.6211
(1, 3)或(3, 1)1002.01011.170.9152
(2, 3)或(3, 2)1302.91325.771.7553
(1, 4)或(4, 1)1694.61731.572.1816
(3, 3)1799.81838.392.1441
(2, 4)或(4, 2)1984.82019.631.7548
), ArticleFig(id=1228312966701973712, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228295922677908169, language=CN, label=表8, caption=

简支GPLs/Al复合材料板（0. 5 m×0. 5 m×0. 01 m）中心开口（0. 05 m）的前12阶固有频率

, figureFileSmall=null, figureFileBig=null, tableContent=
模态(m, n)固有频率/GHz误差/%
有限元本方法
(1, 1)199.49202.571.5439
(1, 2)或(2, 1)504.20510.721.2931
(2, 2)797.63810.561.6211
(1, 3)或(3, 1)1002.01011.170.9152
(2, 3)或(3, 2)1302.91325.771.7553
(1, 4)或(4, 1)1694.61731.572.1816
(3, 3)1799.81838.392.1441
(2, 4)或(4, 2)1984.82019.631.7548
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GPLs/Al复合材料阵列开口结构的自由振动
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颜建伟 1, 2 , 江思岑 1, 2 , 何陵辉 3
振动工程学报 | 2025,38(3): 612-622
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振动工程学报 | 2025, 38(3): 612-622
GPLs/Al复合材料阵列开口结构的自由振动
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颜建伟1, 2 , 江思岑1, 2, 何陵辉3
作者信息
  • 1.华东交通大学轨道交通基础设施性能监测与保障国家重点实验室，江西　南昌　330013
  • 2.华东交通大学土木建筑学院，江西　南昌　330013
  • 3.中国科学技术大学中国科学院材料力学行为和设计重点实验室，安徽　合肥　230027

通讯作者:

颜建伟（1986—），男，博士，教授。E-mail：
Free vibration of GPLs/Al composite with an array cutouts
Jianwei YAN1, 2 , Sicen JIANG1, 2, Linghui HE3
Affiliations
  • 1.State Key Laboratory of Performance Monitoring and Protecting of Rail Transit Infrastructure，East China Jiaotong University，Nanchang 330013，China
  • 2.School of Civil Engineering and Architecture，East China Jiaotong University，Nanchang 330013，China
  • 3.CAS Key Laboratory of Mechanical Behavior and Design of Materials，University of Science and Technology of China，Hefei 230027，China
出版时间: 2025-03-10 doi: 10.16385/j.cnki.issn.1004-4523.2025.03.018
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摘要
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针对板经典解析解中不可避免的场函数假设依赖边界约束问题，提出一种具有高自由选择性的一般性场函数方法，研究GPLs/Al复合材料板阵列开口结构的自由振动特性。场函数由高自由度的基函数和待定权重系数求积后求和得到，其中权重系数具有线性相关性，其基础解系恰好由边界约束的位移函数线性方程组确定，因此边界约束的改变只需要修改边界位移函数线性方程组。由此建立的一般性场函数半解析法在处理边界约束上既解决了场函数依赖边界问题，又具备了快速转换边界条件的优点，同时在开口结构和阵列结构上以微分求和替代积分的计算方式具有更高的计算效率和鲁棒性。本文在GPLs/Al复合材料板单开口及阵列开口的自由振动研究中，从固有频率和模态分析发现，随着开口分布的均匀性增加，开口结构的质量矩阵和刚度矩阵同步减小，其自由振动特性趋于完整板的自由振动特性。

自由振动  /  GPLs/Al复合材料  /  阵列开口  /  一般性场函数法

In this paper，we propose a general displacement field function method，which consists of the highly freely chosen basic function and a series of undetermined weighted coefficients to study the free vibration characteristics of GPLs/Al composite plate with an array cutouts. This general displacement field function method can solve the boundary constraint dependence when choosing the displacement field function to obtain the analytical solution in classical plate theory. There is a linear correlation between the weight coefficients. The system of linear equations is constructed based on the boundary constraints，thus the fundamental system of solutions can then be determined. By changing the system of linear equations as well as the fundamental system of solutions，the type of boundary constraint can be easily transferred. This proposed semi-analytic method not only solves the problem of assumed field function dependence on boundary constraints in the classical solution，but also has the superiority of fast conversion of boundary conditions. Meanwhile，by introducing the scatter integral method，a much more efficient and robust method is obtained other than the continuous integral method on the study of free vibration for the structure with an array cutouts. In this paper，we study the free vibration of GPLs/Al composite plates with a single and array cutouts. From the natural frequency and modal analysis，it is found that the generalized mass matrix and stiffness matrix of the open structure decrease synchronously when the cutouts distributed uniformly，and its free vibration characteristics tend to that of the complete plate.

free vibration  /  GPLs/Al composite  /  cutouts  /  general displacement field function method
颜建伟, 江思岑, 何陵辉. GPLs/Al复合材料阵列开口结构的自由振动. 振动工程学报, 2025 , 38 (3) : 612 -622 . DOI: 10.16385/j.cnki.issn.1004-4523.2025.03.018
Jianwei YAN, Sicen JIANG, Linghui HE. Free vibration of GPLs/Al composite with an array cutouts[J]. Journal of Vibration Engineering, 2025 , 38 (3) : 612 -622 . DOI: 10.16385/j.cnki.issn.1004-4523.2025.03.018
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工程结构的力学性能受到材料属性的影响，工程材料逐步从单一材料向可设计的复合材料发展。与均质材料相比，复合材料在轻质和力学性能方面具有相当的优势[1-2]。复合材料灵活的可设计性为传统功能材料设计开辟了全新的空间，也能够满足不同的功能需求从而具有越来越广泛的应用前景。复合材料具有独特的优点，例如比刚度大、比强度大等。早期金属基复合材料的增强体主要以微尺度纤维或陶瓷颗粒等为主，在力学性能上有显著的提高，但整体的延展性和断裂韧性等则明显不足。现代技术的发展为纳米增强复合材料及其拓扑结构提供了新的视角和源动力，促进了纳米增强复合材料的快速发展。大量研究[3]表明，纳米尺度的增强体能极大地提升金属基复合材料的强度且保持良好的塑性和延展性。近年来，石墨烯和碳纳米材料因其独特的力学特性和高自润滑性而被广泛关注。石墨烯纳米片（GPLs）以极高的杨氏模量、断裂强度及比表面积和延展性[4-5]为金属基复合材料的研究开辟了新方向。板单元构件是工程结构应用中最常见的基本单元呈现形式，广泛应用于土木交通、航空航天、汽车工业等领域。由于安装连接、减重和拓扑结构设计的需要，通常会在板内开口。相对于完整的板结构而言，开口的形式和尺寸会造成板质量和刚度分布的变化，这必然会影响结构的振动特性。对于开口板的振动特性，国内外学者从解析法、半解析法和数值模拟上进行了较多的研究，并取得了很多成果。
AVALO等[6]采用双傅里叶级数近似场函数和Rayleigh-Ritz法数值模拟研究了双方形开口薄板的自由振动特性。LAURA等[7]等采用近似割圆法研究了简支各向同性板圆形开口的横向自由振动。上述数值模拟计算中，不论是方形开口还是圆形开口，求解的中心思想是将因开口造成动能和势能的缺失考虑成对完整板体系的刚度矩阵和质量矩阵的影响，预设场函数由边界约束形式决定。相类似地，CHO等[8]从能量法导出拉格朗日运动方程，随后将系统的固有频率和模态转化成多自由度系统的特征值求解问题从而避开预设场函数，研究了任意边界约束下方形、椭圆形及组合开口形式板的自由振动。KWAK等[9]提出独立坐标耦合法，将开口局部的坐标和整体坐标通过运动学关系耦合在一起，从而简化了复杂积分区域的势能和动能积分。LIEW等[10]采用离散Ritz法和子域分解方式研究了中心开口矩形板的自由振动特性。根据中心开口板结构的中心对称性，可采用四分之一结构即L型板进行研究，L型板可进一步细分成三个矩形子域，各子域之间的边界为对称或反对称边界，从而将有缺失的开口板问题重新转化为完整的矩形板进行求解问题。需要注意的是，上述简化方法的前提是要求结构关于中心对称。为了解决非中心对称开口板的振动问题，CHEN等[11]将分区的边界采用虚拟弹簧模拟，研究了各种非中心对称开口形式的面内和面外振动频率和振型。上述研究方法中，整体坐标系和开口局部坐标系间的转化关系处理是关键。LEE等[12]应用间接BIEMs法将不同开口间的联系用坐标夹角表示，开口系统表示为多个线性代数系统的叠加，研究了多圆开口圆形板的自由振动。HUANG等[13]将开口板处理成变厚度板，采用格林函数对任意开口形状的位移函数进行离散，而后通过积分得到系统的刚度和质量矩阵，从而研究任意形状开口板的自由振动。王平等[14]针对带孔口型特殊结构和边界条件的耦合板振动不易求解问题，将板壳振动功率流理论结合Abaqus有限元软件计算得到了结构共振频率处的传递路径图。陈卫等[15]利用映射技术将三维壳曲面转化为二维无网格离散模型，基于最小移动二乘法提出了求解任意壳结构线性弯曲自由振动的最小二乘无网格方法。彭林欣等[16]基于一阶剪切变形理论和移动最小二乘无网格方法研究了Winkler弹性地基圆形加肋板的自由振动。
尽管包括有限元和无网格法等的数值方法能够避开场函数的预设问题，对于任意开口结构、复杂边界和荷载施加等外部环境都具有广泛适用性，但从理论参数分析、精度控制和快速预测等角度，解析解仍然是相当重要的理论分析方法，在工程应用中具有重要的价值。通常解析解或半解析方法中，针对不同的边界约束条件假设合适的场函数是精确求解自由振动固有频率及相应振型的重要前提条件。位移场函数的选取目前多以改进傅里叶级数[17]、正交多项式[18]及切比雪夫多项式级数[19]为主，它们在分析单一规则开口的板结构时具有很好的收敛性，而在分析较复杂的内部开口如组合开口、不规则开口、曲边开口问题和高阶振型时则需要保留相当多项的级数项来保证收敛性和求解精度。此外，在经典的完整板理论解析求解中，典型简支和固定约束边界条件下的自由振动有成熟的假设场函数，但在悬臂边界条件的场函数假设上则遭遇了相当大的困难。对应地，在数值模拟计算中，简支和固定边界约束施加后刚度矩阵和质量矩阵正定，从而能从特征值和特征向量求解中解出板自由振动的固有频率和振型，然而对于悬臂约束条件则会造成刚度和质量矩阵奇异而无法求解。
此外，Kirchhoff薄板理论在求解自由振动和挠曲变形时通常忽略板的横向剪切变形和转动惯量的影响，一些特定情况下会与实际工程情况产生偏差。针对上述方法问题，本文建立一般性的场函数，场函数由基函数和待定系列权重函数求和确定，基函数具有高度自由性，解决场函数选择时受限边界条件的问题。需要注意的是待定权重函数之间并不是完全独立的，一般具有线性相关性，其相关性恰好由边界约束条件确定。因此，待定权重系数可以由边界约束相关的线性方程组的基础解系线性表达。由此结合本构关系和Hamilton原理建立的石墨烯纳米片铝（GPLs/Al）复合材料板自由振动控制方程的广义刚度矩阵和广义质量矩阵具有正定性，从而根据特征方程的特征值和特征向量得到系统的固有频率和模态。该方法在处理方形开口结构问题时将局部坐标和整体坐标采用统一坐标，以离散积分方式替代连续积分方式能够大幅度提高计算效率和便捷性。本文研究阵列开口分布方式对开口板结构的自由振动特性影响，该研究对板结构的轻量化设计具有一定的指导意义。
1　位移场函数
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不同于经典板理论中场函数的选择需要以边界约束条件为前提，本文中场函数采用一般形式：
式中，f为具有高度自由选择性的二维基函数，例如多项式、三角函数、切比雪夫级数等；为了保留面内惯量对系统自由振动的影响，沿x和y方向的面内位移分量u0、v0和z方向的面外位移w0都采用一般形式的场函数表示；ς、ζ、ξ分别为x、y和z方向位移的权重系数。
二维基函数f为一维基函数p和q的克罗内克积：
式中，
其中，m、n分别为基函数p和q的项数。
上述基函数具有高度自由性，本文中为了位移函数的求导便利，不妨选择三角函数：
式中，mi、ni分别为三角基函数p和q的正弦项的试探项数。
2　Kirchhoff板
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根据Kirchhoff板理论，板上任意点（x，y，z）的位移可以表示为：
式中，u0、v0和w0分别是中平面点在x、y和z方向的位移。根据式（5）得到应变和位移的关系：
应力-应变关系为：
式中，E为杨氏模量，μ为泊松比。
将A和D记作：
式中，。
系统的运动能为：
应变能为：
根据Hamilton原理，Lagrangian函数从t1到t2时刻的时间积分的变分为0，即：
其中，
以Δ记作Δ =[ ς ζ ξ ]，设(u0，v0，w0)(x，y，t ) =fΔeiωt，有：
面内和面外质量矩阵及刚度矩阵如下：
其中，
至此板的自由振动问题转化为方程（14）的特征值和特征向量问题。需要注意的是，上述振动方程中的质量矩阵和刚度矩阵未施加边界约束条件，且保留了面内转动惯量对自由振动特性的影响。
（1）四边固定约束边界
将边界条件式（16）代入到场函数式（1）中得到：
将式（17）写成矩阵形式：
式中，矩阵G为：
解出式（18）的基础解系：
由此，Δ可以由Z的线性组合得到，即
线性组合系数γ待求。将式（21）代入式（14），同时式子左、右都左乘ZT得：
据此得到正定的广义质量矩阵Meq和广义刚度矩阵Keq：
求解式（22）振动控制方程的特征值就得到了系统的固有频率，将式（22）的特征向量代入式（21）中则得到了真实的振型。上述求解过程中可以看出边界约束的施加完全体现在式（18）中，因此变换边界约束条件仅需要简单地对与边界约束相关的G矩阵做相应变换即可实现，如四边简支约束边界和两边简支两边悬臂（SFSF）边界约束。
（2）四边简支约束边界：
将边界条件式（16）代入到场函数式（1）中得到：
矩阵G为：
（3）SFSF约束边界
将边界条件式（16）代入到场函数式（1）中得到：
矩阵G为：
3　结果与讨论
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3.1　方法验证
为了验证本文方法的正确性，以两个算例研究进行对比分析。以完整的单层石墨烯片和中心开口石墨烯片的自由振动为例。需要指出的是，单层石墨烯的材料厚度在众多的文献中存在争议，目前广泛采用的是多层石墨烯的层间距或者平均层厚作为单原子层石墨烯的厚度。本文作者最近的研究工作发现，当采用平均间距作为单原子层石墨烯的材料厚度时会呈量级地放大其抗弯刚度。通过自由振动和非线性弯曲变形的研究，提出了单层石墨烯厚度包括内禀厚度和结构厚度两个部分的概念。当研究单层石墨烯的横向自由振动或者变形时，应当采用内禀厚度，其值为0.0739 nm，据此从原子-连续多尺度理论模型中解析地得到单原子层石墨烯的杨氏模量E=3.1851 TPa，泊松比为0.4123，相对应的密度为9871 kg/m3[5]。对于两层或多层石墨烯的厚度则回归到内禀厚度与结构厚度之和，即层间距0.334 nm，杨氏模量0.7049 TPa，密度为2184 kg/m3。以S表示简支边界约束，C表示固定边界约束，F表示自由边界。
本节将首先从模态对比定性分析和固有频率定量分析两个角度验证本文方法的收敛性。根据式（3）和（4），取m=n=10，讨论mi=ni取值对简支和固定约束的石墨烯片自由振动模态的影响，如表1所示。从表1列举的振型和顺序，不难发现随着ni增大，假模态逐渐消失，简支和固定边界约束下的模态都趋于经典板理论。换而言之，增大ni可以促进本文方法收敛。
表2和3列出了给定m=n，当取值分别为10和20时石墨烯片的固有频率的变化。当ni取值大于6时，简支约束下的固有频率前12阶完全相等。然而，当ni=6时，由于产生的一些假模态（表1所示）而导致出现一些多余的固有频率。当ni=10时，即意味着基函数采用纯正弦函数时，固有频率值有较大的变化。固定边界约束计算结果的对比情况则显示ni取值越大，计算结果越稳定。
对比表2和3的计算结果发现，n取值的增大对计算精度有明显效果。并且，同样验证了上述结论，即ni取值增大能有效提高本文方法的计算鲁棒性。因此，本文接下来的计算将设定n=20，ni取值不小于17，且不取20。
简支边界约束下经典板的自由振动固有频率解为：
表4列举了四边简支（SSSS）边长为10 nm的方形单层石墨烯分别采用原子-连续多尺度数值模拟、经典板解析解和本文方法的前12阶固有频率。从数值上看，经典板解析解与本文方法吻合得相当好，二者的最大偏不超过0.05%。与原子-连续多尺度全数值模拟的结果相比，由于数值计算边界效应的影响，偏差略大，但即使是高阶频率的误差如振型（1，4）与本文方法或经典解的偏差也小于3%。因此，本文方法具有高精度和高鲁棒性。
固定边界约束下板的自由振动经典解为：
表5列出了四边固定约束（CCCC）边长10 nm的方形单层石墨烯三种方法的前12阶固有频率。由表5可见，经典解计算的固有频率在预测基频时与原子-连续多尺度数值模拟和本文方法的结果吻合得很好，在高阶频的预测上存在严重的失真问题。对比原子-连续多尺度法和本文方法，原子-连续多尺度无网格方法数值模拟避开了场函数的预设问题，预测的固有频率和振型顺序与本文方法偏差限制在6.5%±3.5%范围内，而经典解平均误差为33.9%。因此，本文提出的一般场函数半解析法保证了计算效率的同时具有全数值模拟方法的计算精度。
3.2　中心开口单层石墨烯的自由振动
中心开口的单层石墨烯几何参数具体如下：边长10 nm的方形石墨烯，以正中心开方形口，方形开口的尺寸为10/13 nm。鉴于算例1中在对固定边界约束条件下三种预测方法之间都存在较大偏差的现象，本算例仅考虑简支边界约束下的开口对固有频率的影响。表6列出了原子-连续多尺度无网格方法和本文方法预测的中心开口石墨烯的前12阶频率。与算例1中完整石墨烯两种方法的预测结果情况一致，二者无偏差增大的趋势，再次证明了本文方法的精度和鲁棒性。值得关注的是，与原子-连续多尺度无网格全数值模拟相比，一方面本文方法在两种边界约束的转化中，仅需要对与边界约束相关的G矩阵作修改从而更新基础解系，省去了全数值模拟方法中不同边界约束的重新建模工作，因而本文方法在边界处理上具有更灵活的实用性；另一方面，对于开口板的模拟，全数值模拟除重新建模外，还需要对开口区域的网格（有限元法）或者节点（无网格法）做摘除处理，这一处理工作会改变系统的总刚度矩阵和质量矩阵的维数。本文方法中开口区域仅仅是改变了积分区域，从微分求和的角度而言，仅是求和部分比完整板减少了，可以认为计算量比之前减少了。同时，本文方法建立的广义刚度矩阵和质量矩阵的维数与积分或者微分求和离散的节点数无关，仅依赖于基函数的维数，因而在开口板的求解上矩阵维数与完整板一致。简而言之，从计算角度而言，在研究板的自由振动时，板开口或者多开口对本文方法不会增加任何复杂度。
根据上述算例的验证结果，本文方法应用到开口石墨烯/Al复合材料的自由振动研究中。根据混合法则，添加均布1%质量分数石墨烯纳米片的石墨烯/铝复合材料的力学性能参数如表7所示。表8列举了简支边界约束下GPLs/Al复合材料板（0.5 m×0.5 m×0.01 m）中心开口（0.05 m）的前12阶固有频率有限元和本文方法的对比结果，二者最大偏差仅约为2%，再次验证了本文方法的可行性和计算精度。本节以方形开口为例，按照阵列方式开口，开口示意图如图1所示，包括奇数开口和偶数开口两种模式，二者的主要差别是板中心是否开口，即奇数开口模式在板中心开口，偶数开口模式在板中心不开口。为便于统计，采用i×i表示在方形板两边上都开i个口的阵列开口模式。图2是以总开口面积占比为25%的前提，不同阵列方形开口形式下的石墨烯/Al复合板的前12阶固有频率。图2（a）列出了1×1~4×4四种开口模式的前12阶固有频率。从4条折线的变化来看，高阶频率随着开口数的增加可能呈现跳跃式的变化，这是因为开口造成结构不像完整板一样遵循表1中的模态顺序。例如中心单一开口的频率在多处如5和6阶、9和10阶振型上顺序与完整板出现的顺序相反。但是随着开口数的增加，如3×3和4×4的模态趋于与完整板一致。为证实上述猜想，图2（b）列出了更多的阵列开口数GPLs/Al复合材料板的固有频率的变化。随着开口数的增加，开口板的模态顺序严格趋近于完整板。此外，随着开口数的增加，开口板的固有频率越趋于一致，如13×13和18×18的固有频率折线基本重合。
图3给出了奇数开口、偶数开口及完整复合板的前5阶频率。随着开口数的增加，开口分布越均匀，任意阶次的固有频率和振型趋于完整板的自由振动，并基本遵循单调性增加的规律。奇数开口模式与偶数开口模式相比，偶数开口模式的曲线相对平滑，趋于完整板的自由振动相对更稳定，这主要是因为奇数开口模式中心开口对整体结构的影响更大。图4给出的是三种不同开口模式下同样开口面积占比（25%）的前12阶振型。图4（a）中显示中心单开口板的模态与多开口板和完整板的模态有显著的差异。中心单开口板的前4阶模态与完整板相似；第6~8阶和9~10阶具有比较清晰的完整板自由振动模态，但出现顺序有较大的偏移；其他未标注的模态甚至在完整板自由振动中未找到匹配的模态。图4（b）和（c）显示多开口模式的自由振动模态除了第9和10阶模态与完整板有差异外，其他阶模态都与完整板一致。因此，有理由给出结论如下：随着开口数的增加，GPLs/Al复合板的开口在全板上的分布越来越均匀，广义刚度矩阵和广义质量矩阵同步减小，因此其自由振动行为无论是频率还是模态上都趋向于完整板的自由振动特性。
本文方法除在板的开口处理上具有明显的计算优势，在边界约束上仅仅根据与边界约束相关的线性方程组形成的G矩阵做转化外，非常便利地研究各种约束下的自由振动。图5~7给出了CCCC、SFSF和CFCF三种典型边界约束下的前12阶固有频率与开口模型间的关系。随着开口数量的增加，三种边界约束都呈现出前述结论，即开口数的增加，使得板的刚度和质量分布重新均匀分布，因而系统的固有频率和模态趋近于完整板的固有频率和模态。
4　结论
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本文提出一种独立于边界条件的一般场函数半解析法，场函数由高自由选择的基函数和待定权重系数积求积后求和得到，具有变形形式无差异性表达，如本文将同样的场函数表达式同时用于描述面内拉压变形和面外弯曲变形。由于边界约束的存在，待定权重系数间具有线性相关性，其基础解析能恰好由边界约束的位移函数线性方程组确定。据此建立的振动控制方程中的广义刚度矩阵和质量矩阵具有正定性，因此系统的自由振动转换为特征值和特征方程的求解问题。本方法应用在石墨烯和GPLs/Al复合材料板的阵列开口结构自由振动特性研究上发现：
（1）简支边界约束的完整单层石墨烯片及中心单开口石墨烯片自由振动的前12阶固有频率预测结果证实了本文一般场函数半解析解与原子-连续多尺度数值模拟及经典板解析解高度吻合。本文一般场函数半解析法构造的刚度矩阵和质量矩阵只依赖于基函数的维数，与离散积分无关，因此域内积分可以采用离散积分方式高精度、高计算效率地求出。
（2）对比固定边界约束单层石墨烯的自由振动前12阶固有频率发现，本文提出的一般场函数半解析法与原子-连续多尺度全数值模拟计算结果较经典解吻合得很好，误差为6.5%±3.5%，经典解平均误差为33.9%，再次证明本文方法的鲁棒性。
（3）开口板的自由振动需将积分区域从完整板减去开口部分，本文采用离散积分方式，求和计算量比完整板更少，因此在求解开口类复杂结构时更具计算优势。本文在研究阵列开口GPLs/Al增强复合材料中发现，随着开口的增多，开口结构的固有频率和振型越趋于完整板的固有频率和振型，这是因为开口的增多使得板质量和刚度的分布越来越均匀，系统刚度和质量同步减少。换而言之，开口面积不会影响结构的自由振动特性，开口分布对结构的自由振动特性具有主导作用。
基金
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  • 国家重点研发计划资助项目(2023YFC3009400)
  • 国家自然科学基金资助项目(12072112)
  • 江西省自然科学基金资助项目(20232ACB204028)
文献
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2025年第38卷第3期
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doi: 10.16385/j.cnki.issn.1004-4523.2025.03.018
  • 接收时间：2024-01-24
  • 首发时间：2026-02-11
  • 出版时间：2025-03-10
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  • 收稿日期：2024-01-24
  • 修回日期：2024-04-17
基金
国家重点研发计划资助项目(2023YFC3009400)
国家自然科学基金资助项目(12072112)
江西省自然科学基金资助项目(20232ACB204028)
作者信息
    1.华东交通大学轨道交通基础设施性能监测与保障国家重点实验室，江西　南昌　330013
    2.华东交通大学土木建筑学院，江西　南昌　330013
    3.中国科学技术大学中国科学院材料力学行为和设计重点实验室，安徽　合肥　230027

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颜建伟（1986—），男，博士，教授。E-mail：
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