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Article(id=1243220381847371807, tenantId=1146029695717560320, journalId=1242844143416819734, issueId=1243220377556595432, articleNumber=null, orderNo=null, doi=10.15959/j.cnki.0254-0053.2025.03.014, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1744300800000, receivedDateStr=2025-04-11, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1774336436950, onlineDateStr=2026-03-24, pubDate=1758729600000, pubDateStr=2025-09-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774336436950, onlineIssueDateStr=2026-03-24, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774336436950, creator=13701087609, updateTime=1774336436950, updator=13701087609, issue=Issue{id=1243220377556595432, tenantId=1146029695717560320, journalId=1242844143416819734, year='2025', volume='46', issue='3', pageStart='541', pageEnd='810', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1774336435926, creator=13701087609, updateTime=1774336648182, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1243221267898282005, tenantId=1146029695717560320, journalId=1242844143416819734, issueId=1243220377556595432, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1243221267898282006, tenantId=1146029695717560320, journalId=1242844143416819734, issueId=1243220377556595432, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=724, endPage=734, ext={EN=ArticleExt(id=1243220382321328166, articleId=1243220381847371807, tenantId=1146029695717560320, journalId=1242844143416819734, language=EN, title=Nonlinear Dynamics of Graphene-Reinforced Inclined Two-Phase Flow Pipeline, columnId=null, journalTitle=Chinese Quarterly of Mechanics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

A study is conducted on the nonlinear dynamics of inclined flow pipes reinforced with graphene composite materials conveying fluid under two-phase flow conditions. Based on the von Karman nonlinear strain-displacement relationship and Hamilton's principle, the dynamic equations for inclined two-phase flow pipelines are established. The nonlinear dynamic model was solved using Galerkin method and fourth-order Runge-Kutta method to analyze the influence of the distribution pattern, weight fraction, and gas volume fraction of graphene platelets on the natural frequency and nonlinear dynamics of pipes conveying fluid. The experimental results show that the vibration amplitude of graphene reinforced pipes with V-shaped distribution is the smallest, followed by X-shaped distribution and A-shaped distribution. In addition, increasing the gas volume fraction can help alleviate fluid-induced vibration phenomena in pipelines. The above conclusion provides a theoretical basis for the application of graphene-reinforced inclined two-phase flow pipelines in practical engineering.

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对石墨烯增强复合材料倾斜输流管道在两相流情况下的非线性动力学问题进行了研究．基于von Karman非线性应变位移关系与Hamilton原理，建立了倾斜两相流输流管道的动力学方程．通过伽辽金法和四阶龙格-库塔法求解非线性动力学模型，分析了石墨烯纳米片的分布形式、质量分数以及气体体积分数对输流管道固有频率和非线性动力学的影响．实验结果表明，V型分布的石墨烯增强输流管道振动幅度最小，其次是X型分布、A型分布．此外，增加气体体积分数有助于缓解管道的流体诱导振动现象．上述结论为石墨烯增强倾斜两相流输流管道在实际工程中的应用提供了理论依据．

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王爱文，博士，教授．研究方向：非线性动力学与控制．E-mail：
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徐勇胜，硕士生．研究方向：非线性动力学与控制．E-mail：

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徐勇胜，硕士生．研究方向：非线性动力学与控制．E-mail：

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徐勇胜，硕士生．研究方向：非线性动力学与控制．E-mail：

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figureFileBig=Maauyt76UxKtpxRdnNZQmw==, tableContent=null), ArticleFig(id=1243220413262709734, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220381847371807, language=CN, label=图16, caption=不同气体体积分数的频域分析, figureFileSmall=ynovDkFfSigG9fH+PzyHGA==, figureFileBig=Maauyt76UxKtpxRdnNZQmw==, tableContent=null), ArticleFig(id=1243220413359178732, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220381847371807, language=EN, label=Tab.1, caption=

Comparison of critical flow velocity results of flow pipelines under different boundary conditions

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Boundary conditionPaidoussis[19]Ni et al[34]Bahaadini et al[35]Presentεr1εr2εr3
Clamped-clamped2π6.283 26.283 66.281 30.3%0.3%0.37%
Clamped-simply4.494.493 44.493 94.493 00.067%0.009%0.02%
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不同边界条件下输流管道的临界流速结果对比

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Boundary conditionPaidoussis[19]Ni et al[34]Bahaadini et al[35]Presentεr1εr2εr3
Clamped-clamped2π6.283 26.283 66.281 30.3%0.3%0.37%
Clamped-simply4.494.493 44.493 94.493 00.067%0.009%0.02%
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石墨烯增强倾斜两相流管道的非线性动力学研究
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徐勇胜 1 , 王爱文 1 , 钟新华 1 , 李贺章 1, 2
力学季刊 | 2025,46(3): 724-734
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力学季刊 | 2025, 46(3): 724-734
石墨烯增强倾斜两相流管道的非线性动力学研究
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徐勇胜1 , 王爱文1 , 钟新华1, 李贺章1, 2
作者信息
  • 1.北京信息科技大学　理学院，北京　100192
  • 2.北京宣房楼宇设备有限公司，北京　100053

    • 徐勇胜，硕士生．研究方向：非线性动力学与控制．E-mail：

通讯作者:

王爱文，博士，教授．研究方向：非线性动力学与控制．E-mail：
Nonlinear Dynamics of Graphene-Reinforced Inclined Two-Phase Flow Pipeline
Yongsheng XU1 , Aiwen WANG1 , Xinhua ZHONG1, Hezhang LI1, 2
Affiliations
  • 1.School of Science, Beijing Information Science and Technology University, Beijing 100192, China
  • 2.Beijing Xuanfang Building Equipment Co. Ltd., Beijing 100053, China
出版时间: 2025-09-25 doi: 10.15959/j.cnki.0254-0053.2025.03.014
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摘要
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对石墨烯增强复合材料倾斜输流管道在两相流情况下的非线性动力学问题进行了研究．基于von Karman非线性应变位移关系与Hamilton原理，建立了倾斜两相流输流管道的动力学方程．通过伽辽金法和四阶龙格-库塔法求解非线性动力学模型，分析了石墨烯纳米片的分布形式、质量分数以及气体体积分数对输流管道固有频率和非线性动力学的影响．实验结果表明，V型分布的石墨烯增强输流管道振动幅度最小，其次是X型分布、A型分布．此外，增加气体体积分数有助于缓解管道的流体诱导振动现象．上述结论为石墨烯增强倾斜两相流输流管道在实际工程中的应用提供了理论依据．

石墨烯增强复合材料  /  倾斜输流管道  /  两相流  /  非线性动力学

A study is conducted on the nonlinear dynamics of inclined flow pipes reinforced with graphene composite materials conveying fluid under two-phase flow conditions. Based on the von Karman nonlinear strain-displacement relationship and Hamilton's principle, the dynamic equations for inclined two-phase flow pipelines are established. The nonlinear dynamic model was solved using Galerkin method and fourth-order Runge-Kutta method to analyze the influence of the distribution pattern, weight fraction, and gas volume fraction of graphene platelets on the natural frequency and nonlinear dynamics of pipes conveying fluid. The experimental results show that the vibration amplitude of graphene reinforced pipes with V-shaped distribution is the smallest, followed by X-shaped distribution and A-shaped distribution. In addition, increasing the gas volume fraction can help alleviate fluid-induced vibration phenomena in pipelines. The above conclusion provides a theoretical basis for the application of graphene-reinforced inclined two-phase flow pipelines in practical engineering.

graphene reinforced composite materials  /  pipes conveying fluid  /  two-phase flow  /  nonlinear dynamics
徐勇胜, 王爱文, 钟新华, 李贺章. 石墨烯增强倾斜两相流管道的非线性动力学研究. 力学季刊, 2025 , 46 (3) : 724 -734 . DOI: 10.15959/j.cnki.0254-0053.2025.03.014
Yongsheng XU, Aiwen WANG, Xinhua ZHONG, Hezhang LI. Nonlinear Dynamics of Graphene-Reinforced Inclined Two-Phase Flow Pipeline[J]. Chinese Quarterly of Mechanics, 2025 , 46 (3) : 724 -734 . DOI: 10.15959/j.cnki.0254-0053.2025.03.014
正文
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输流管道[1]作为能源输送的生命线，其安全运行直接关系到国家战略安全与公共利益．然而，输流管道在复杂情况下频发的振动失稳[2]问题，尤其是气液两相流[3]导致的非线性失稳现象，已成为制约工业发展的挑战．
近年来，随着工程领域对高性能材料的迫切需求，石墨烯增强复合材料因其独特的力学性能（高强度和高模量[4]）、出色的阻尼特性[5]和良好的热传导性能，已成为国内外学者广泛关注的研究热点．其中在壳体结构方面，Yang等[6]开创性地研究了横向载荷与面内力联合作用下石墨烯增强圆锥壳的非线性动力学行为；Li等[7]系统分析了含初始几何缺陷的石墨烯片增强圆柱壳在移动载荷作用下的非线性动力响应；在梁结构的研究方面，Yang等[8]深入探讨了表面应力类型及其尺寸效应对功能梯度层压复合材料曲面纳米梁非线性面内稳定性的影响．Gao等[9]对弹性支撑约束的三维石墨烯泡沫多跨功能梯度梁进行了屈曲特性研究；Liang等[10]基于三阶剪切变形梁理论和von Karman非线性几何关系，建立了轴向功能梯度石墨烯片增强复合材料弯曲梁的线性和非线性等几何有限元模型．Zhao等[11]创新性地研究了流体环境中变厚度功能梯度石墨烯折纸增强超材料梁的振动特性；在板结构研究方面，Mao等[12]通过对比实验，分析了横向激励与参量激励联合作用下纯聚合物板与三种石墨烯片增强聚合物复合板的非线性振动差异；Yang等[13]重点研究了功能梯度石墨烯片增强金属泡沫板的低速冲击响应特性；Teng等[14]研究了简支边界条件下石墨烯增强功能梯度多孔纳米复合薄板的非线性强迫振动行为；Zhang等[15]采用广义微分正交有限元法，深入研究了功能梯度石墨烯片增强多孔复合材料板的屈曲特性；在组合结构研究方面，Zhang等[16]研究了耦合双圆柱壳-环形-矩形板的自由振动和稳态响应；Gao等[17]在空气动力学和热环境下对功能梯度矩形和圆柱组合壳结构进行自由和稳态随机振动分析；此外，Liu等[18]创新性地分析了热力耦合环境下石墨烯增强输流管道在爆炸载荷作用下的非线性瞬态响应特性．这些研究成果系统揭示了石墨烯增强复合材料在壳、梁、板与组合结构中的非线性动力学行为特征，为该类材料在航空航天、海洋工程、土木建筑等领域的工程应用提供了重要的理论支撑．
在输流管道动力学研究领域，国内外学者已取得一系列重要成果．Paidoussis[19]系统建立了细长管、柔性管及悬臂管在不同激励条件下的线性与非线性振动理论体系；Wang等[20]研究了带旋转弹簧的铰接式柔性输流管道的流固耦合稳定性问题；高云等[21]探讨了固-液两相内流激励下悬臂输流管道的稳定性特征；Wang等[22]研究了受单侧点约束作用的输流管道的屈曲行为；Chang等[23]将非线性能量汇技术应用于悬臂式输流管道的振动控制研究；Hao等[24]探讨了具有初始几何缺陷的输流管道的参数共振；Monette等[25]研究了两相流中柔性软管的弹性失稳机理；王美玲等[26]对输流管道系统两相流体诱导振动现象进行了研究．An等[27]基于广义积分变换技术，建立了气液两相流输流管道动力学行为的数值分析方法；Adeogoke等[28]探讨了热力耦合作用下两相流悬臂管的非线性振动特性；Guo等[29]系统比较了单层和双层输流管道在两相流工况下的自由振动响应特性．值得注意的是，上述研究主要集中于水平或垂直布置的输流管道，而工程实际中大量存在的倾斜管道研究相对不足．针对这一现状，An等[30]率先建立了倾斜海底输流管道的振动数学模型并求得解析解；Jin等[31]通过实验研究揭示了倾斜管中段塞流的力学特性；Fang等[32]进一步阐明了水流速度对倾斜管道动力学行为的影响机制．然而，关于石墨烯增强复合材料倾斜输流管道在两相流条件下的非线性动力学研究仍属空白．
本文基于von Karman非线性应变位移关系和Hamilton原理，建立石墨烯增强倾斜两相流输流管道的非线性动力学模型，采用伽辽金法和四阶龙格-库塔法进行数值求解与可视化分析，研究成果将为相关工程应用提供重要的理论支撑和技术储备．
1　材料物性参数
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倾斜输流管道由相同厚度的完全粘合的石墨烯纳米片增强复合材料层组成，层数为NL，每层由各向同性聚合物基质和石墨烯纳米片的混合物制成．石墨烯的含量沿厚度方向梯度分布．如图1所示，根据石墨烯含量的分布模式分别定义为X型分布、V型分布、A型分布．
X型分布表示石墨烯质量分数从内外表面层到中间层逐层逐渐增加．A型分布表示石墨烯质量分数从管道内表面到外表面逐层单调增加．V型分布模式与A型分布相反．根据修正的Halpin-Tsai力学模型[32]，第k层复合材料的杨氏模量为
其中，参数ηL、ηW表达式如下
式中，EM和EG分别为填充基质和石墨烯的杨氏模量，lG、wG和hG分别为石墨烯片的长度、宽度和厚度．第k层的石墨烯体积分数表示为
其中，表示石墨烯在第k层的质量分数，不同分布模式的表达式如下[26]
X型分布模式
V型分布模式
A型分布模式
其中，k=1，2，…，NL，WG为石墨烯总体质量分数．石墨烯增强倾斜输流管道的等效材料性能用以下混合规则表示[31-33]
其中，材料属性分别表示质量密度，泊松比，热膨胀系数．此外，参数下标“G”，“M”，和“c”分别代表石墨烯，基材料和石墨烯增强复合材料．
2　石墨烯增强倾斜输流管道的动力学方程
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基于von Karman应变位移关系和欧拉-伯努利梁模型，应变位移关系如下
其中，εx为横向应变．u1和u3分别为位移场沿输流管道横向和轴向的分量．根据欧拉-伯努利梁理论，输流管道位移场的任意一点位移都可以表示为
其中，u和w分别为沿轴向和横向的位移．因此，输流管道的应变位移关系可以表示为
输流管道的温度变化用ΔT表示．根据广义胡克定律，应力σx可以表示为以下形式
其中，热应力表示为εT=α（r）ΔT
将式（8）~式（10）代入式（11）得到倾斜输流管道的势能和内部流体的重力势能，表示为
则应变能为
将式（12）代入式（13），U可表示为
其中
式中，A11和D11分别为拉伸刚度系数和弯曲刚度系数，NT和是热应力．
石墨烯增强倾斜输流管道的动能为
内部流体的动能为
石墨烯增强倾斜输流管道内部的流体横向运动所产生的粘性阻尼所引起的虚功表达式为
当石墨烯增强倾斜输流管道受到外力载荷影响时，外力虚功计算如下
在上述方程中，m，Ml，Mg，Cw，p，V和G分别为管道质量、流体质量、气体质量、粘度系数、外部载荷、流体速度和重力参数．利用Hamilton原理
石墨烯增强倾斜输流管道的控制方程表示为
对式（24）无量纲化，引入无量纲参数
将无量纲参数式（25）代入方程（24），得到石墨烯增强倾斜输流管道平面耦合强迫振动的无量纲控制方程
3　数值算法
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使用伽辽金截断法将式（26）离散化为一组具有有限自由度的非线性耦合常微分方程．为了得到运动控制方程的离散形式，选择一组正交函数族
通过伽辽金方法，可得一组非线性耦合常微分方程组
其中，质量矩阵M
阻尼矩阵C
刚度矩阵K
第n个横向运动的广义坐标qn（τ）
式（28）可以通过下面的过程将其简化为一阶微分形式
其中
结构的固有频率可以通过求解方程（33）中矩阵[E]（t）的特征值来获得，具体步骤如下：
假设Z（τ）表示为如下形式
则降阶方程（33）产生以下特征值问题
其中，I表示单位矩阵
将流速、体积分数、阻尼系数、重力参数等代入无量纲控制方程中，计算系数矩阵M、C、K和G，最后通过求解系统固有频率的值可以使用式（36）获得．
4　数值实验
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本文选用环氧树脂和石墨烯作为聚合物基体和增强材料，流体选择水，其密度为ρf=1 000 kg/m3，几何参数分别为，ri=0.2 m，r0=0.225 m，L=5 m，IG=2.5×10-6 m，wG=1.5×10-6 m，t14=1.4×10-6 m，hG=6×10-6 m；石墨烯的密度、弹性模量、泊松比和热膨胀系数分别为ρG=1.06 g/cm3，EG=1.01×1012 Pa，vG=0.186，αG=6×10-6；环氧树脂的密度、弹性模量、泊松比和热膨胀系数分别为ρM=1.2 g/cm3，EM=3×1010 Pa，VM=0.34，αM=6×10-6；石墨烯的总重量比为fG=0.01，总层数NL=10，环境温度为T0=300 K，黏性阻尼系数为cW=0.02，除特别说明外，数值实验的几何和材料性能参数保持不变．
为了验证所用模型和求解方法的准确性，与文献[19]、文献[34]和文献[35]在相同参数下的临界流速进行了对比．表1相对误差结果显示，误差较小，吻合良好，从而验证了本文模型和实验方法的可靠性．
石墨烯增强倾斜输流管道不同模态下的固有频率随无量纲流速的变化如图2所示，这与Ni等[34]结果的数据具有很好的一致性．
不同气体体积分数下两相流管道自由振动复数频率对比如图3所示，忽略阻尼效应，同时保持其他参数不变．如果流速为零，则位于实部轴上的单相和两相流的所有频率都表示非流动管道的相应固有频率．在较低的流速下，由于单相流和两相流之间的动量差可以忽略不计，两个系统的行为几乎相同．当流速越高，相应的分支彼此分离得越多．此外，气体体积分数的增加会导致与单相流显著偏差，随后分叉点的数量增加．
从图4、图5中可以看出，随着石墨烯含量的增加，石墨烯增强倾斜输流管道的临界流速逐渐增大．临界流速的提高意味着系统在更高速度下才会发生失稳，因此石墨烯含量的增大对系统稳定性具有正面影响．
从图6可以证明，V型分布下倾斜输流管道的临界流速最高，表明选取V型分布可以有效扩大稳定性范围．同时，固有频率越大，结构不易振动；在石墨烯含量相同的情况下V型分布下输流管道的固有频率最大，即抑制振动效果最佳．
图7展示了V型分布模式下，流速为5 m/s、外部激励频率为一阶固有频率2倍条件下的频谱分析结果．通过快速傅里叶变换对微分方程解的频域特性进行分析，可以观察到明显的低频峰值，这表明系统对低频振动具有较强的响应能力．相比之下，高频范围内曲线平滑，表明高频振动对系统的影响相对较小．
如图8~图10所示，当气体的初始体积分数为0时，观察到输流管道已经发生分叉．随着气体体积分数ε的增加，输流管道逐渐趋于稳定．显然随气体体积分数增加，临界流速增加，意味着系统在更高的流速下才会经历失稳，从而使得系统稳定性增加．此外，增加体积气体分数ε还会导致混沌分岔发生的临界流速更高，这表明系统进入混沌状态的阈值随着气体含量的增加而提高．对比图8~图10，随着石墨烯含量的增加，输流管道分叉的波动变小，管道就更稳定．综上所述，石墨烯含量的增加与气体体积分数的提升均对石墨烯增强倾斜输流管道的动力学稳定性具有显著影响．并且数据显示可知，当石墨烯含量为1.2%时，观点分叉的幅值降低更多，并且可以看出在一阶模态发散失稳条件下，气体体积分数对输送管的发散失稳位移及振幅存在显著相关性：随着气体体积分数的增大，发散位移的增量呈递减趋势，同时振幅也随之减小．
如图11~图15所示，随着气体体积分数的增加，系统的混沌程度显著降低，表明气体体积分数的提升有助于增强系统的稳定性．
通过图16可以得到，在稳态条件下，对于石墨烯含量为1%的倾斜输流管道，主共振峰和次共振峰都随着气体体积分数的增加而减小．此外，共振频率随着气体体积分数的增加而增加．
5　结论
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本文采用伽辽金法和四阶龙格-库塔法求解非线性动力学方程，研究石墨烯增强倾斜两相流输流管道的非线性动力学行为．分析了石墨烯分布模式、质量分数和两相气体体积分数对石墨烯增强复合材料输流管道非线性动力学的影响，得到以下主要结论
（1）不同的石墨烯分布模式影响输流管道的稳定性．V型分布具有最佳的力学性能，其次是X型分布，而A型分布对于结构固有频率和振动抑制的提升最小．
（2）石墨烯的质量分数提升有助于抑制管道中的振动，并提高系统稳定性．当石墨烯含量较低为0.6%时，系统呈现明显的不规则振动行为；随着石墨烯含量的增加到1.2%时，系统逐渐趋向稳定的振动．
（3）随着气体体积分数的增加，系统进入混沌状态的临界点提高，表明管道动力学行为趋于稳定．
（4）在低于临界流速时，管道呈现周期振荡的多个极限环行为；当流速接近临界流速时，系统出现从周期向混沌的转变．
上述研究成果不仅系统揭示了石墨烯增强复合材料在倾斜两相流输流管道中的动态调控机制，更创新性地建立了“材料-流体-结构”多参数耦合的稳定性判据，为航空航天、海洋工程等领域的倾斜管道系统优化设计提供了理论支撑和技术路线．特别是在深海输运、核反应堆冷却等严苛工况下的管道减振设计方面具有重要的工程指导价值．
基金
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  • 国家自然科学基金(12272058; 11772063)
文献
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2025年第46卷第3期
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doi: 10.15959/j.cnki.0254-0053.2025.03.014
  • 接收时间：2025-04-11
  • 首发时间：2026-03-24
  • 出版时间：2025-09-25
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  • 收稿日期：2025-04-11
基金
国家自然科学基金(12272058; 11772063)
作者信息
    1.北京信息科技大学　理学院，北京　100192
    2.北京宣房楼宇设备有限公司，北京　100053

通讯作者:

王爱文，博士，教授．研究方向：非线性动力学与控制．E-mail：
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2种不同金属材料的力学参数
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Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
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鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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