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Article(id=1241784307329597621, tenantId=1146029695717560320, journalId=1241715181215068175, issueId=1241784304989175988, articleNumber=null, orderNo=null, doi=10.6052/1672-6553-2025-059, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1747497600000, receivedDateStr=2025-05-18, revisedDate=1749052800000, revisedDateStr=2025-06-05, acceptedDate=null, acceptedDateStr=null, onlineDate=1773994050104, onlineDateStr=2026-03-20, pubDate=null, pubDateStr=null, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1773994050104, onlineIssueDateStr=2026-03-20, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1773994050104, creator=13701087609, updateTime=1773994050104, updator=13701087609, issue=Issue{id=1241784304989175988, tenantId=1146029695717560320, journalId=1241715181215068175, year='2025', volume='23', issue='10', pageStart='1', pageEnd='96', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1773994049546, creator=13701087609, updateTime=1773994239413, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241785101412012589, tenantId=1146029695717560320, journalId=1241715181215068175, issueId=1241784304989175988, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241785101412012590, tenantId=1146029695717560320, journalId=1241715181215068175, issueId=1241784304989175988, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=10, endPage=17, ext={EN=ArticleExt(id=1241784307660947642, articleId=1241784307329597621, tenantId=1146029695717560320, journalId=1241715181215068175, language=EN, title=Vibration Control of a Timoshenko Fluid-Conveying Pipe Based on Nonlinear Energy Sink, columnId=1241784307421872311, journalTitle=Journal of Dynamics and Control, columnName=Research Articles, runingTitle=null, highlight=null, articleAbstract=

Fluid-conveying pipes hold significant engineering value. In practical applications, pipes are often subjected to vibrations due to various factors. Excessive vibration amplitudes may lend to damage to the pipe itself and its supporting structures, while even small-amplitude vibrations can cause cumulative fatigue over time. Therefore, mitigating pipe vibrations has become a critical issue that needs to be addressed.In this study, a fluid-conveying pipe model is established based on the Timoshenko beam theory. The nonlinear energy sink (NES) cell, as a novel vibration suppression concept, is applied to reduce pipe vibrations. The governing equations of the system are derived using the generalized Hamilton’s principle, and the system’s natural frequencies are obtained through the complex modal method. The system’s response is solved using the harmonic balance method and numerical simulations. Furthermore, the effects of different NES cell quantities and installation configurations on vibration-suppression performance are investigated. The results indicate that when the external excitation is near specific frequencies, a single-point concentrated distribution exhibits superior vibration reduction performance, whereas multi-point concentrated and uniform distributions provide superior suppression performance under broadband excitation.

, correspAuthors=Hu Ding, 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=Yufei Shao, Yingbin Gu, Hu Ding), CN=ArticleExt(id=1241784316418654529, articleId=1241784307329597621, tenantId=1146029695717560320, journalId=1241715181215068175, language=CN, title=基于非线性能量汇的Timoshenko输流管道振动控制, columnId=1241784307690307771, journalTitle=动力学与控制学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

输流管道具有重要的工程价值.工程管道往往会受到各种因素影响而产生振动.振动幅值过大会对管道本身以及支撑带来损害，即使是小幅振动也会带来累积性的损伤.因此如何减弱输流管道的振动成了一个亟须研究的课题.非线性能量汇（nonlinear energy sink，NES）胞元作为一种新型的减振概念被应用于本文中管道的减振.本文采用广义哈密顿原理，基于Timoshenko梁模型建立输流管道与NES胞元减振系统的耦合振动控制方程，使用复模态法求解系统的固有频率.系统的响应由谐波平衡法和数值方法求解，研究了不同NES胞元数量以及不同安装方式对振动抑制效率的影响，研究发现外激励在特定频率附近时，单点集中式分布拥有更出色的减振性能，而多点集中式和均匀平铺式分布对于宽频激励有更好的减振效果.

, correspAuthors=丁虎, authorNote=null, correspAuthorsNote=
† E-mail:
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Weight reduction design of floating raft vibration isolation structure by using nonlinear energy sink cell [J]. Journal of Vibration Engineering，2024，37(10):1739-1746. （in Chinese）, articleTitle=Weight reduction design of floating raft vibration isolation structure by using nonlinear energy sink cell, refAbstract=null), Reference(id=1241784334894564073, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307329597621, doi=null, pmid=null, pmcid=null, year=2024, volume=12, issue=4, pageStart=6809, pageEnd=6819, url=null, language=null, rfNumber=[31], rfOrder=50, authorNames=LIU E G, SHAO Y F, DOU B, journalName=Journal of Vibration Engineering & Technologies, refType=null, unstructuredReference=LIU E G，SHAO Y F，DOU B，et al. High-order modal vibration control of Timoshenko beams based on nonlinear energy sink cells [J]. 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Analysis of damping efficiency of nonlinear energy sink cell [J]. Chinese Journal of Theoretical and Applied Mechanics，2023，55(11):2614-2623. （in Chinese）, articleTitle=Analysis of damping efficiency of nonlinear energy sink cell, refAbstract=null)], funds=[Fund(id=1241784324882760202, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307329597621, awardId=12025204, language=EN, fundingSource=National Science Fund for Distinguished Young Scholars(12025204), fundOrder=null, country=null), Fund(id=1241784324991812110, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307329597621, awardId=12025204, language=CN, fundingSource=国家杰出青年科学基金资助项目(12025204), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1241784316645146954, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307329597621, xref=1., ext=[AuthorCompanyExt(id=1241784316653535563, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307329597621, companyId=1241784316645146954, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1. 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new single-point concentrated distribution position based on the second-order vibration mode, figureFileSmall=hqv/m4c7DchuJx61ZgtblA==, figureFileBig=YfUtrKQc6gsMA3gT4SbiIQ==, tableContent=null), ArticleFig(id=1241784322701722094, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307329597621, language=CN, label=图8, caption=基于二阶振型新的单点集中分布位置, figureFileSmall=hqv/m4c7DchuJx61ZgtblA==, figureFileBig=YfUtrKQc6gsMA3gT4SbiIQ==, tableContent=null), ArticleFig(id=1241784322806579699, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307329597621, language=EN, label=Fig.9, caption=The amplitude-frequency response curves and vibration reduction efficiency of the system under different NES distribution patterns （Near the second-order natural frequency）, figureFileSmall=y4cTS40VGlBSzH5bdjHgUw==, figureFileBig=BYXb5c2xisbOjcY0ikqJcQ==, tableContent=null), ArticleFig(id=1241784322974351862, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307329597621, language=CN, label=图9, caption=不同NES安装方式下系统的幅频响应曲线与减振效率（二阶固有频率附近）, figureFileSmall=y4cTS40VGlBSzH5bdjHgUw==, figureFileBig=BYXb5c2xisbOjcY0ikqJcQ==, tableContent=null), ArticleFig(id=1241784324454941177, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307329597621, language=EN, label=Table 1, caption=

The first four natural frequencies of the fluid-conveying pipe

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OrderResults/Hz
CMMFEM
1213.10 215.52
2544.70 538.73
3960.05 952.12
41440.421414.6
), ArticleFig(id=1241784324576575999, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307329597621, language=CN, label=表1, caption=

输流管道前四阶固有频率

, figureFileSmall=null, figureFileBig=null, tableContent=
OrderResults/Hz
CMMFEM
1213.10 215.52
2544.70 538.73
3960.05 952.12
41440.421414.6
), ArticleFig(id=1241784324664656386, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307329597621, language=EN, label=Table 2, caption=

Example parameters

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SymbolValue and unitSymbol Value and unit
L 1.6 m F0 100 N
D 0.14 m ρf 1000 kg/m3
d 0.132 m V 1 m/s
ρp 7930 kg/m3 l 1 m
E 194 GPa m 0.5 kg
μ0.3 kN 5×109 N/m3
G 74.6 GPa cNNs/m 3
Ip 3.955×10-6 m4 k 500 N/m
c0 10 Ns/m H 0.01 m
s 0.667 m N 10
), ArticleFig(id=1241784324765319684, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307329597621, language=CN, label=表2, caption=

算例参数

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SymbolValue and unitSymbol Value and unit
L 1.6 m F0 100 N
D 0.14 m ρf 1000 kg/m3
d 0.132 m V 1 m/s
ρp 7930 kg/m3 l 1 m
E 194 GPa m 0.5 kg
μ0.3 kN 5×109 N/m3
G 74.6 GPa cNNs/m 3
Ip 3.955×10-6 m4 k 500 N/m
c0 10 Ns/m H 0.01 m
s 0.667 m N 10
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基于非线性能量汇的Timoshenko输流管道振动控制
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邵宇飞 1, 2 , 顾颖宾 1 , 丁虎 2, †
动力学与控制学报 | 研究论文 2025,23(10): 10-17
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动力学与控制学报 | 研究论文 2025, 23(10): 10-17
基于非线性能量汇的Timoshenko输流管道振动控制
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邵宇飞1, 2, 顾颖宾1, 丁虎2, †
作者信息
  • 1.中国核能电力股份有限公司，北京　100000
  • 2.上海大学　力学与工程科学学院，上海　200444

通讯作者:

Vibration Control of a Timoshenko Fluid-Conveying Pipe Based on Nonlinear Energy Sink
Yufei Shao1, 2, Yingbin Gu1, Hu Ding2, †
Affiliations
  • 1. China National Nuclear Power Co., Ltd., Beijing　100000, China
  • 2. School of Mechanics and Engineering Science, Shanghai University, Shanghai　200444, China
doi: 10.6052/1672-6553-2025-059
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摘要
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输流管道具有重要的工程价值.工程管道往往会受到各种因素影响而产生振动.振动幅值过大会对管道本身以及支撑带来损害，即使是小幅振动也会带来累积性的损伤.因此如何减弱输流管道的振动成了一个亟须研究的课题.非线性能量汇（nonlinear energy sink，NES）胞元作为一种新型的减振概念被应用于本文中管道的减振.本文采用广义哈密顿原理，基于Timoshenko梁模型建立输流管道与NES胞元减振系统的耦合振动控制方程，使用复模态法求解系统的固有频率.系统的响应由谐波平衡法和数值方法求解，研究了不同NES胞元数量以及不同安装方式对振动抑制效率的影响，研究发现外激励在特定频率附近时，单点集中式分布拥有更出色的减振性能，而多点集中式和均匀平铺式分布对于宽频激励有更好的减振效果.

输流管道  /  非线性能量汇  /  胞元  /  振动控制

Fluid-conveying pipes hold significant engineering value. In practical applications, pipes are often subjected to vibrations due to various factors. Excessive vibration amplitudes may lend to damage to the pipe itself and its supporting structures, while even small-amplitude vibrations can cause cumulative fatigue over time. Therefore, mitigating pipe vibrations has become a critical issue that needs to be addressed.In this study, a fluid-conveying pipe model is established based on the Timoshenko beam theory. The nonlinear energy sink (NES) cell, as a novel vibration suppression concept, is applied to reduce pipe vibrations. The governing equations of the system are derived using the generalized Hamilton’s principle, and the system’s natural frequencies are obtained through the complex modal method. The system’s response is solved using the harmonic balance method and numerical simulations. Furthermore, the effects of different NES cell quantities and installation configurations on vibration-suppression performance are investigated. The results indicate that when the external excitation is near specific frequencies, a single-point concentrated distribution exhibits superior vibration reduction performance, whereas multi-point concentrated and uniform distributions provide superior suppression performance under broadband excitation.

fluid-conveying pipe  /  nonlinear energy sink  /  cell  /  vibration reduction
邵宇飞, 顾颖宾, 丁虎. 基于非线性能量汇的Timoshenko输流管道振动控制. 动力学与控制学报, 2025 , 23 (10) : 10 -17 . DOI: 10.6052/1672-6553-2025-059
Yufei Shao, Yingbin Gu, Hu Ding. Vibration Control of a Timoshenko Fluid-Conveying Pipe Based on Nonlinear Energy Sink[J]. Journal of Dynamics and Control, 2025 , 23 (10) : 10 -17 . DOI: 10.6052/1672-6553-2025-059
正文
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引言
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输流管道广泛应用于航空航天[1-4]、海洋工程[5-7]、核电[8]及城市供排水[9]等工程领域，其安全稳定运行至关重要.然而，在流体输送过程中，管道系统常因内部流体的流动激励、压力脉动、机械扰动等因素产生振动，严重时可能导致结构疲劳、连接件松动甚至管道破裂，引发安全事故.因此，研究输流管道的振动特性，尤其是其固有频率及减振方法，具有重要的工程价值[10-13].以往的研究中，学者们较多地使用基于Euler-Bernoulli梁理论的管道模型[13-15].该模型在求解细长管道时具有良好的计算精度.然而随着管道变得短粗或者需要计算高阶模态时，考虑管道剪切变形和横截面扭转运动的基于Timoshenko梁理论的管道模型更加精准[16-21].
在振动抑制领域，被动控制由于其结构较为简单、成本较低以及安装和维护方便的特点备受青睐[22,23].非线性能量汇（NES）是一种常用的被动减振结构，因为其具有宽频减振以及自适应频率的特点广泛应用于工程中[24-28].然而，NES也存在一些问题，比如对于多模态结构，单个NES难以兼顾多个模态的共振抑制，而且在剧烈振动时NES结构的可靠性不够强.随之，Ding等提出了NES胞元的概念，将单个NES通过方便拆卸的外壳相互连接，通过灵活组合NES胞元的数量和连接方式实现对主系统梁的减振[29].通过分布式布置NES胞元的减振策略，能够兼顾多模态共振，并提高NES对强激励振动的抑制效率.随后，NES胞元被应用于更多场景的减振研究[30-32].
结合输流管道的振动控制需求，以及NES胞元对多模态结构共振的有效抑制能力，本文聚焦于Timoshenko模型的输流管道在安装了NES胞元后简谐激励下的非线性强迫振动问题.首先，基于哈密顿原理建立输流管道耦合NES胞元的非线性振动微分方程，并使用伽辽金截断法将偏微分方程离散为非线性的常微分方程.使用复模态法求解系统的固有频率，并且与有限元数值计算的输流管道固有频率进行比较验证.然后，利用谐波平衡法和数值方法求解非线性常微分方程得到系统的频率响应曲线，研究NES胞元对管道减振效率的影响.
1　输流管道与NES胞元力学模型
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根据实际管道，建立安装非线性能量汇（NES）胞元的两端固支输流管道的非线性振动模型.图1(a)为输流管道系统的力学模型示意图.输流管道采用Timoshenko梁理论建模，考虑其黏弹性、横向和横截面旋转耦合.管道的长度为L，外径和内径分别为D和d，密度为ρp，剪切模量为G，弹性模量为E，截面绕中性轴转动惯量为Ip.假定管道中的流体以恒定速度V流动且不可压缩，流体的密度为ρf.管道在距离左端s处受到一个周期的力激励F=F0 sin（ωt），在距离左端l处安装了数个NES胞元.管道横向位移和轴向位移分别用v（x，t）和u（x，t）表示，使用φ（x，t）表示Timoshenko输流管横截面的旋转角度，如图1(b)所示.
单个NES胞元模型如图2所示.第i个NES胞元的振子质量为mi，非线性刚度为kNi，阻尼为cNi，振子竖直方向的位移用Yi（t）表示.胞元内添加了刚度为ki的限幅弹簧，限幅距离为Hi.
第i个NES胞元的控制方程为
其中vli表示v（li，t），限幅弹簧恢复力函数G可以表示为
在输流管道上安装N个NES胞元，管道与NES胞元的相互作用力可以表示为
该作用于输流管道上的力所做虚功的变分为
基于广义哈密顿原理能够导出Timoshenko输液管道横向非线性振动的偏微分—积分控制方程[16]：
两端的固支边界条件为
在后续的计算中，分段函数在分段点处的不连续导数会给计算带来不便.双曲正切函数由于其奇函数具有上下界的特性，在将分段函数处理为连续函数方面有很好的效果.取连续函数为
其中ε是精度参数.
2　系统响应近似解和数值验证
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首先，使用复模态假设法（CMM）求解系统的固有频率，计算结果如下表所示，并使用有限元（finite element method，FEM）软件ANSYS建模检验计算结果.从表1中可以看出，二者计算结果吻合良好.
在本节中，采用龙格—库塔法（Runge-Kutta method，RKM）计算了管道在外激励下的幅频曲线，并与谐波平衡法（harmonic balance method，HBM）对比验证.然后，计算不同胞元数量下系统的响应.
在本文中，对所有NES胞元使用相同的参数.令mi=m，kNi=kN，cNi=cN，ki=k，Hi=H.用于计算系统固有频率和响应的物理参数如表2所示.
接下来，使用伽辽金截断法对输流管道连续体系统离散化处理.用NT表示截断阶数，为满足边界条件，方程（5）的解设为
其中ξj =[sinh（βj）-sin（βj）]/[cos（βj）-cosh（βj）]，βj为方程cos（β）·cosh（β）=1的第j个解.设权函数为
将解式（8）代入方程（5）再乘以权函数并在x=0~L上进行积分，从而得到2NT+1个离散的控制方程.后续RKM以及HBM的求解将基于截断后的方程计算.
综合计算精度与计算速度，选取三阶截断阶数和一阶谐波展开计算系统的幅频响应曲线，即NT=3，计算结果如图3所示，两种安装模式下两种计算方法得到的结果吻合较为良好.
从图4中可以看到，随着安装NES胞元个数的增加，系统在一阶固有频率处的响应不断降低，减振效果随着NES胞元个数的增加不断提高.
从图5中可以看出，随着NES胞元数量的增加，系统的响应逐渐降低，并且能够更快地到达稳态阶段.
3　NES胞元不同分布方式
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在安装NES胞元时，考虑三种不同的分布方式，如图6所示.
使用三阶截断阶数和一阶谐波展开计算系统的幅频响应曲线.从图7(a)中可以看出，随着NES胞元数量的增加，系统的响应在不断降低，并且单点集中的安装方式具有更好的减振效果；多点集中式和均布式安装二者相比多点集中式安装的响应略低一些.为了进一步探明减振效果，可以通过安装NES后管道响应幅值均方根的降低率来定义减振效率，如式（10）所示.
图7(b)为三种安装模式下输流管道安装不同数量NES胞元数量时的减振效率.可以看到，随着胞元数量的增加，系统的减振效率不断提高.然而随着胞元数量进一步增加，减振效率的提升速率在变缓慢.其中，中点集中式安装模式拥有最高的减振效率，随之是多点集中式安装方式.
当外激励频率来到二阶固有频率附近时，中间集中分布的方式因为所处位置为振动节点，从而失去减振效果.因此此时单点集中安装NES的位置变更至二阶振型峰值处.根据复模态假设法得到系统二阶振型在x=0.4688 m处振型取得最值.因此在计算二阶频率附近的幅频响应时，将单点集中安装的位置调整到该处，即l=0.4688 m，如图8所示.从图9中可以得到与图7中类似的现象与结论，即减振效率随着胞元数量的增加而提升，并且NES胞元单点集中式分布具有最佳的减振效果，其次是多点集中式安装方法，最后是均匀分布安装方法.
值得注意的是，在图9中，为了应对二阶固有频率附近的外激励所带来的振动，单点集中分布的位置做了调整，安放在了振型的峰值处.这样的安装策略势必带来不便，因为需要针对不同频率外激励做出调整.而多点集中式分布与均匀平铺分布的安装方法更为普适和实用.
4　结论
收起
本文着重NES胞元对于基于Timoshenko梁理论的输流管道的减振研究，通过建立理论模型和控制方程，使用谐波平衡法与数值方法相结合，研究了不同数量NES胞元以及不同分布安装方式对减振性能的影响，得出了以下结论：
（1）NES胞元具有良好的减振性能，并且减振效率随着胞元数量增加而提高，并且更多数量的NES胞元能让系统更早地进入稳态振动；
（2）NES胞元在达到一定数量时，减振效率的提升速度变缓，这意味着可以根据实际情况灵活选择安装不同数量的胞元以平衡减振效率与其他客观因素之间的关系；
（3）NES胞元单点集中式分布针对特定固有频率附近的激励拥有出色的减振效果，而多点并联式分布和均匀平铺式分布安装对于宽频激励的减振有更好的适应性.
本文为工程实际中输流管道的减振提供了指导，并且在未来将着重于研究如何将NES胞元应用于更多的实际场景.
基金
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  • 国家杰出青年科学基金资助项目(12025204)
文献
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2025年第23卷第10期
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doi: 10.6052/1672-6553-2025-059
  • 接收时间：2025-05-18
  • 首发时间：2026-03-20
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  • 收稿日期：2025-05-18
  • 修回日期：2025-06-05
基金
National Science Fund for Distinguished Young Scholars(12025204)
国家杰出青年科学基金资助项目(12025204)
作者信息
    1.中国核能电力股份有限公司，北京　100000
    2.上海大学　力学与工程科学学院，上海　200444

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