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Article(id=1244336745307816780, tenantId=1146029695717560320, journalId=1244311425741537314, issueId=1244336743298740932, articleNumber=null, orderNo=null, doi=10.16450/j.cnki.issn.1004-6801.2025.05.002, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1702224000000, receivedDateStr=2023-12-11, revisedDate=1721145600000, revisedDateStr=2024-07-17, acceptedDate=null, acceptedDateStr=null, onlineDate=1774602598739, onlineDateStr=2026-03-27, pubDate=1759248000000, pubDateStr=2025-10-01, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774602598739, onlineIssueDateStr=2026-03-27, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774602598739, creator=13701087609, updateTime=1774602598739, updator=13701087609, issue=Issue{id=1244336743298740932, tenantId=1146029695717560320, journalId=1244311425741537314, year='2025', volume='45', issue='5', pageStart='855', pageEnd='1056', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1774602598261, creator=13701087609, updateTime=1774603435030, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1244340253042000577, tenantId=1146029695717560320, journalId=1244311425741537314, issueId=1244336743298740932, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1244340253042000578, tenantId=1146029695717560320, journalId=1244311425741537314, issueId=1244336743298740932, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=869, endPage=876, ext={EN=ArticleExt(id=1244336747157504856, articleId=1244336745307816780, tenantId=1146029695717560320, journalId=1244311425741537314, language=EN, title=Dynamic Characteristics for Three‑Dimensional Tip Clearance of Rotor System with Blade Cracks, columnId=1244336744728998604, journalTitle=Journal of Vibration,Measurement and Diagnosis, columnName=PAPER, runingTitle=null, highlight=null, articleAbstract=

To investigate the fault mechanism of blade cracks and to analyze the effects of blade crack on the three-dimensional （3D） tip clearance of the rotor system，while comprehensively considering blade radial deformation，flap-wise bending，and chordwise bending，this paper develops a novel dynamic model of the rotor system based on continuum theory. With the blade breathing crack model under the three-dimensional stress state，a 3D tip clearance dynamic response model of rotor system with blade cracks is further established. The accuracy of the dynamic model is validated by comparing it with the finite element model and experiments. On this basis，the effect of blade crack depth and location on the 3D tip clearance in rotor system is further analyzed. The results show that the amplitudes of the high frequency doubling component of the 3D tip clearance increase with crack depth，while both the fundamental frequency and the high frequency doubling component of the 3D tip clearance show a non-monootonic trend as the relative crack location increases. The research results provide theoretical guidance for research on monitoring and diagnosis method of aero-engine blade crack based on 3D tip clearance.

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为了研究叶片裂纹故障机理，分析叶片裂纹对转子系统三维叶尖间隙的影响，并综合考虑叶片的径向变形、摆振方向和弦长方向的弯曲变形，基于连续梁理论建立了转子系统动力学模型。首先，基于三维应力状态的叶片呼吸裂纹模型，进一步建立了含叶片裂纹的转子系统三维叶尖间隙动态响应模型；其次，通过与有限元模型和实验结果进行对比，验证了动力学模型的准确性；最后，分析了叶片裂纹深度和位置对转子系统三维叶尖间隙的影响。结果表明，转子系统三维叶尖间隙高倍频分量的幅值随着叶片裂纹深度的增加而增加，三维叶尖间隙基频分量和高倍频分量的幅值均随着叶片相对裂纹位置增大而呈现非单调的变化趋势。该研究结果对基于三维叶尖间隙的航空发动机叶片裂纹监测诊断方法研究具有理论指导作用。

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凃成雄，男，2001年2月生，硕士生。主要研究方向为光纤检测及故障诊断。E‑mail：
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张小栋，男，1967年4月生，博士、教授、博士生导师。主要研究方向为智能检测诊断与控制技术、智能机器人技术等。曾发表《Dynamic modeling of rotary blade crack with regard to three‑dimensional tip clearance》（《Journal of Sound and Vibration》2023，Vol.544）等论文。 E‑mail：

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张小栋，男，1967年4月生，博士、教授、博士生导师。主要研究方向为智能检测诊断与控制技术、智能机器人技术等。曾发表《Dynamic modeling of rotary blade crack with regard to three‑dimensional tip clearance》（《Journal of Sound and Vibration》2023，Vol.544）等论文。 E‑mail：

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张小栋，男，1967年4月生，博士、教授、博士生导师。主要研究方向为智能检测诊断与控制技术、智能机器人技术等。曾发表《Dynamic modeling of rotary blade crack with regard to three‑dimensional tip clearance》（《Journal of Sound and Vibration》2023，Vol.544）等论文。 E‑mail：

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journalId=1244311425741537314, articleId=1244336745307816780, language=EN, label=Fig.9, caption=Three‑dimensional tip clearance dynamic response of rotor system under different blade crack locations, figureFileSmall=LaNSmWQpG0ow42Z9Rbjc6w==, figureFileBig=nWPSpCukYBLW7mHtn0oZVQ==, tableContent=null), ArticleFig(id=1244351826007540042, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336745307816780, language=CN, label=图9, caption=不同叶片裂纹位置条件下的转子系统三维叶尖间隙动态响应, figureFileSmall=LaNSmWQpG0ow42Z9Rbjc6w==, figureFileBig=nWPSpCukYBLW7mHtn0oZVQ==, tableContent=null), ArticleFig(id=1244351826087231824, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336745307816780, language=EN, label=Tab.1, caption=

The material parameters of the rotor system

, figureFileSmall=null, figureFileBig=null, tableContent=
部件名称ρ/（kg⋅m-3）弹性模量/Pa泊松比
叶片2 7707.1×10100.33
轮盘7 9002.0×10110.30
转轴7 9002.0×10110.30
), ArticleFig(id=1244351826183700821, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336745307816780, language=CN, label=表1, caption=

转子系统材料参数

, figureFileSmall=null, figureFileBig=null, tableContent=
部件名称ρ/（kg⋅m-3）弹性模量/Pa泊松比
叶片2 7707.1×10100.33
轮盘7 9002.0×10110.30
转轴7 9002.0×10110.30
), ArticleFig(id=1244351826250809690, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336745307816780, language=EN, label=Tab.2, caption=

The geometric parameters of the rotor system

, figureFileSmall=null, figureFileBig=null, tableContent=
部件名称参数数值
叶片叶片长度/mm50
叶片宽度/mm18
叶片厚度/mm4
叶片安装角/(°)45
叶片扭转角/(°)0
转子叶片数量6
轮盘轮盘半径/mm20
轮盘位置/mm150
轮盘厚度/mm25
转轴转轴长度/mm460
转轴半径/mm5
), ArticleFig(id=1244351826317918558, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336745307816780, language=CN, label=表2, caption=

转子系统几何参数

, figureFileSmall=null, figureFileBig=null, tableContent=
部件名称参数数值
叶片叶片长度/mm50
叶片宽度/mm18
叶片厚度/mm4
叶片安装角/(°)45
叶片扭转角/(°)0
转子叶片数量6
轮盘轮盘半径/mm20
轮盘位置/mm150
轮盘厚度/mm25
转轴转轴长度/mm460
转轴半径/mm5
), ArticleFig(id=1244351826393416035, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336745307816780, language=EN, label=Tab.3, caption=

The natural frequency comparison of the rotor system

, figureFileSmall=null, figureFileBig=null, tableContent=
模态阶数模态fFEM/Hzfdynamic/Hze/%
1转轴1阶弯曲56.1358.454.13
256.1358.454.13
3转轴2阶弯曲295.77291.061.59
4295.78291.061.59
5转轴扭转‑叶片弯曲324.37341.195.19
6转轴3阶弯曲723.21751.173.87
7723.32751.173.85
8转轴4阶弯曲1 255.8——
91 255.8——
10叶片1阶弯曲1 324.31 308.551.19
111 324.31 308.551.19
121 324.31 308.551.19
131 359.41 308.553.74
141 359.41 308.553.74
), ArticleFig(id=1244351826468913512, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336745307816780, language=CN, label=表3, caption=

转子系统的固有频率对比

, figureFileSmall=null, figureFileBig=null, tableContent=
模态阶数模态fFEM/Hzfdynamic/Hze/%
1转轴1阶弯曲56.1358.454.13
256.1358.454.13
3转轴2阶弯曲295.77291.061.59
4295.78291.061.59
5转轴扭转‑叶片弯曲324.37341.195.19
6转轴3阶弯曲723.21751.173.87
7723.32751.173.85
8转轴4阶弯曲1 255.8——
91 255.8——
10叶片1阶弯曲1 324.31 308.551.19
111 324.31 308.551.19
121 324.31 308.551.19
131 359.41 308.553.74
141 359.41 308.553.74
), ArticleFig(id=1244351826540216685, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336745307816780, language=EN, label=Tab.4, caption=

Experimental results of natural frequency of rotor system with blade cracks

, figureFileSmall=null, figureFileBig=null, tableContent=
实验次数1阶固有频率/Hz2阶固有频率/Hz3阶固有频率/Hz
156185339
256192335
357170322
456186340
556196336
657171330
757203320
857171331
957172319
1057211318
), ArticleFig(id=1244351826636685685, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336745307816780, language=CN, label=表4, caption=

含叶片裂纹的转子系统固有频率实验结果

, figureFileSmall=null, figureFileBig=null, tableContent=
实验次数1阶固有频率/Hz2阶固有频率/Hz3阶固有频率/Hz
156185339
256192335
357170322
456186340
556196336
657171330
757203320
857171331
957172319
1057211318
), ArticleFig(id=1244351826728960378, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336745307816780, language=EN, label=Tab.5, caption=

Comparison of the first three natural frequencies of rotor systems with blade cracks

, figureFileSmall=null, figureFileBig=null, tableContent=
固有频率阶数计算结果/Hz实验结果/Hz相对误差/%
1阶58.6456.63.60
2阶192.59185.73.71
3阶367.36329.011.66
), ArticleFig(id=1244351826829623682, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336745307816780, language=CN, label=表5, caption=

含叶片裂纹的转子系统前3阶固有频率对比

, figureFileSmall=null, figureFileBig=null, tableContent=
固有频率阶数计算结果/Hz实验结果/Hz相对误差/%
1阶58.6456.63.60
2阶192.59185.73.71
3阶367.36329.011.66
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转子系统叶片裂纹三维叶尖间隙动态特性研究
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张小栋 1, 2 , 凃成雄 1 , 范博超 1 , 熊逸伟 1 , 戴飞 1 , 黄鑫 1
振动、测试与诊断 | 论文 2025,45(5): 869-876
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振动、测试与诊断 | 论文 2025, 45(5): 869-876
转子系统叶片裂纹三维叶尖间隙动态特性研究
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张小栋1, 2 , 凃成雄1 , 范博超1, 熊逸伟1, 戴飞1, 黄鑫1
作者信息
  • 1西安交通大学机械工程学院　西安，710049
  • 2西安交通大学现代设计及转子轴承系统教育部重点实验室　西安，710049

    • 张小栋，男，1967年4月生，博士、教授、博士生导师。主要研究方向为智能检测诊断与控制技术、智能机器人技术等。曾发表《Dynamic modeling of rotary blade crack with regard to three‑dimensional tip clearance》（《Journal of Sound and Vibration》2023，Vol.544）等论文。 E‑mail：

通讯作者:

凃成雄，男，2001年2月生，硕士生。主要研究方向为光纤检测及故障诊断。E‑mail：
Dynamic Characteristics for Three‑Dimensional Tip Clearance of Rotor System with Blade Cracks
Xiaodong ZHANG1, 2 , Chengxiong TU1 , Bochao FAN1, Yiwei XIONG1, Fei DAI1, Xin HUANG1
Affiliations
  • 1.School of Mechanical Engineering，Xi'an Jiaotong University　Xi'an，710049，China
  • 2.Key Laboratory of Education Ministry for Modern Design and Rotor-Bearing System，Xi'an Jiaotong University　Xi'an，710049，China
出版时间: 2025-10-01 doi: 10.16450/j.cnki.issn.1004-6801.2025.05.002
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摘要
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为了研究叶片裂纹故障机理，分析叶片裂纹对转子系统三维叶尖间隙的影响，并综合考虑叶片的径向变形、摆振方向和弦长方向的弯曲变形，基于连续梁理论建立了转子系统动力学模型。首先，基于三维应力状态的叶片呼吸裂纹模型，进一步建立了含叶片裂纹的转子系统三维叶尖间隙动态响应模型；其次，通过与有限元模型和实验结果进行对比，验证了动力学模型的准确性；最后，分析了叶片裂纹深度和位置对转子系统三维叶尖间隙的影响。结果表明，转子系统三维叶尖间隙高倍频分量的幅值随着叶片裂纹深度的增加而增加，三维叶尖间隙基频分量和高倍频分量的幅值均随着叶片相对裂纹位置增大而呈现非单调的变化趋势。该研究结果对基于三维叶尖间隙的航空发动机叶片裂纹监测诊断方法研究具有理论指导作用。

动力学模型  /  叶片裂纹  /  三维叶尖间隙  /  转子系统  /  故障机理

To investigate the fault mechanism of blade cracks and to analyze the effects of blade crack on the three-dimensional （3D） tip clearance of the rotor system，while comprehensively considering blade radial deformation，flap-wise bending，and chordwise bending，this paper develops a novel dynamic model of the rotor system based on continuum theory. With the blade breathing crack model under the three-dimensional stress state，a 3D tip clearance dynamic response model of rotor system with blade cracks is further established. The accuracy of the dynamic model is validated by comparing it with the finite element model and experiments. On this basis，the effect of blade crack depth and location on the 3D tip clearance in rotor system is further analyzed. The results show that the amplitudes of the high frequency doubling component of the 3D tip clearance increase with crack depth，while both the fundamental frequency and the high frequency doubling component of the 3D tip clearance show a non-monootonic trend as the relative crack location increases. The research results provide theoretical guidance for research on monitoring and diagnosis method of aero-engine blade crack based on 3D tip clearance.

dynamic model  /  blade crack  /  3D tip clearance  /  rotor system  /  fault mechanism
张小栋, 凃成雄, 范博超, 熊逸伟, 戴飞, 黄鑫. 转子系统叶片裂纹三维叶尖间隙动态特性研究. 振动、测试与诊断, 2025 , 45 (5) : 869 -876 . DOI: 10.16450/j.cnki.issn.1004-6801.2025.05.002
Xiaodong ZHANG, Chengxiong TU, Bochao FAN, Yiwei XIONG, Fei DAI, Xin HUANG. Dynamic Characteristics for Three‑Dimensional Tip Clearance of Rotor System with Blade Cracks[J]. Journal of Vibration,Measurement and Diagnosis, 2025 , 45 (5) : 869 -876 . DOI: 10.16450/j.cnki.issn.1004-6801.2025.05.002
正文
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引言
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叶片作为航空发动机的核心零件，其健康状态直接影响发动机的安全性。旋转叶片在运行过程中由于巨大的工作负荷，导致叶片极易出现疲劳裂纹，甚至发生断裂，严重危及发动机运行安全［1］。开展对航空发动机叶片裂纹故障机理及监测诊断方法的研究极为重要。
叶尖间隙是航空发动机的关键参数之一，其变化会影响发动机的效率并能反映叶片状态。Agarwal等［2］通过建立涡轮叶片、轮盘和机匣的数学模型，得到了叶尖间隙简化模型。房友龙等［3］通过建立热载荷和离心载荷耦合作用下的叶尖间隙模型，研究了其变化规律。目前，关于叶尖间隙变化机理的研究大多只考虑径向间隙，其包含的叶片状态信息相对有限。当叶片出现裂纹时，叶片会在空间内发生弯曲和扭转变形，导致叶尖间隙呈现出空间三维特征。文献［4‑5］提出了三维叶尖间隙，并利用有限元方法研究了叶片裂纹对三维叶尖间隙变化规律的影响。Zhang等［6］研究了三维叶尖间隙的光纤动态检测方法并搭建了检测系统。
为了研究三维叶尖间隙的动态特性，Zhang等［7］建立了基于三维应力状态的叶片呼吸裂纹模型，研究了叶片裂纹对三维叶尖间隙动态响应特性的影响。在以往研究中，对叶片裂纹三维叶尖间隙动态特性的研究主要集中在单个叶片上［4，7］，忽略了转子振动对三维叶尖间隙的影响。此外，叶片早期叶片裂纹较微弱，对三维叶尖间隙的影响容易被转子振动所淹没。因此，尽可能减小转子振动对三维叶尖间隙的影响，将有利于提高叶片裂纹诊断的精度。为此，建立旋转叶片‑轮盘‑转轴（转子）耦合系统动力学模型是分析含裂纹叶片转子系统三维叶尖间隙动态特性的必要条件。
在转子耦合系统动力学建模方面，研究人员基于连续梁理论建立了转子系统的解析模型。Ma等［8］将Timoshenko梁理论与集中质量模型相结合，建立了考虑转轴弯曲和扭转变形的柔性转子系统动力学模型。Al‑Bedoor［9］建立了转子系统的耦合模型，研究了转轴扭转‑叶片弯曲耦合情况下转子系统的固有频率。Yang等［10］建立了考虑转轴扭转、轮盘弯曲和叶片弯曲变形的转子系统简化模型，研究了系统的自由振动特性。然而，目前的研究大多集中在转子系统的自由振动、耦合振动和叶片‑机匣摩擦等方面，叶片裂纹对转子系统动态响应特性的影响研究相对较少。为了探讨叶片裂纹对转子系统的影响，Chiu［11］将叶片开裂纹模型与转子系统动力学模型相结合，研究了叶片裂纹对转子系统振动响应的影响。Yang［12］基于转子系统动力学模型揭示了转子系统受迫振动响应的内在机理。目前的研究大多集中在叶片裂纹对转轴振动的影响上，忽略了叶片振动，且叶片周向振动仅关注叶片裂纹的一维故障信息，难以全面分析叶片裂纹引起的三维故障特征。
本研究综合考虑了叶片三维变形和叶片裂纹的三维应力状态，建立了含叶片裂纹的转子系统三维叶尖间隙动态响应模型，研究叶片裂纹对转子系统三维叶尖间隙动态响应特性的影响，为基于三维叶尖间隙的转子系统叶片裂纹故障诊断提供理论基础。
1　含叶片裂纹的转子系统三维叶尖间隙动态响应模型
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1.1　含叶片裂纹的转子系统动力学模型
将转子系统简化为由叶片、轮盘和转轴3部分组成。叶片简化为具有安装角和扭转角的等截面悬臂梁；轮盘简化为刚性轮盘；转轴简化为简支梁，不考虑各部分之间的接触非线性，转子系统结构如图1所示。笔者在文献［13］的基础上，建立了转子系统的动力学模型，并做出改进：①引入叶片扭转角模拟真实叶片的扭转形态；②考虑叶片弦向位移以便计算三维叶尖间隙动态响应。图1中：下标s表示转轴；下标d表示轮盘；下标b表示叶片；下标r表示转子。
全局坐标Oxyz的原点固定在转轴一端。轮盘局部坐标Odxdydzd与Oxyz平行，位于轮盘中心。Orxryrzr表示位于轮盘中心的旋转坐标。叶片局部坐标Obxbybzb固定在叶根，Obxb轴方向与Orxr轴方向相同，代表叶片径向。其中：γ0为叶片安装角；γ1为叶片扭转角。叶片安装角和扭转角如图2所示。
对于距离叶根x的叶片截面，其扭转角可表示为γ（x）=γ0+（γ1- γ0）x/Lb，其中，Lb为叶片长度。
基于拉格朗日方程建立转子系统的动力学模型，需要计算转子系统的总动能Ttotal和总势能Utotal，分别为
转轴动能Ts、轮盘动能Td和转轴势能Us的推导过程见文献［13］。由于笔者考虑了叶片扭转角和叶片弦向位移，叶片动能Tb和叶片势能Ub与文献［13］有所不同。根据图1将叶片简化为悬臂梁模型，则第i个叶片上任意一点Q的位移向量rQ可表示为
其中：B1为轮盘扭转位移ψd引起的坐标变换矩阵；B2为坐标系Odx2y2z2相对于坐标系Odx1y1z1的坐标变换矩阵；B3为坐标系Odx1y1z1相对于轮盘坐标系Odxdydzd的坐标变换矩阵；B4为旋转坐标系Orxryrzr相对于坐标系Odx2y2z2的坐标变换矩阵；B5为叶片局部坐标系Obxbybzb相对于旋转坐标系Orxryrzr的坐标变换矩阵；u、v和w分别为叶片径向、摆振方向和弦长方向的位移。
B1~B5的具体表达式为
忽略叶片位移向量中的高阶项，第i个叶片位移向量rQ中的各元素分别简化为
其中：θi为第i个叶片的角位移；Nb为转子叶片数量。
转子系统所有叶片的总动能可表示为
其中：ρb为叶片材料密度；Ab为叶片横截面积。
考虑叶片扭转角和叶片弦向位移，叶片势能表示为
其中：Eb为叶片的弹性模量；fc（x）为叶片的离心力。
叶片任意截面沿yb轴和zb轴周围的截面惯性矩［14］表示为
其中：Iyy和Izz均为叶片在质心主轴周围的截面惯性矩；kb和hb分别为叶片的宽度和厚度。
在得到转子系统的总动能和总势能后，采用伽辽金方法对其进行离散，将式（1）和式（2）代入拉格朗日方程，即
其中：L=Ttotal-Utotal；F为广义外力向量；q为广义位移向量。
转子耦合系统的动态响应模型为
其中：M，D，G，K和KΩ分别为转子系统的质量矩阵、阻尼矩阵、科氏力矩阵、整体刚度矩阵和旋转软化刚度矩阵；Fdynamic为转轴和叶片的动态耦合项引起的耦合激振力［13］。
实际情况中复杂的工作载荷会导致叶片振动，裂纹在叶片振动的作用下交替张开和闭合，表现出裂纹的呼吸效应［7］。引入基于三维应力状态的叶片呼吸裂纹模型，得到含叶片裂纹的转子系统动力学模型，即
其中：Kc和分别为叶片裂纹引起的刚度矩阵和附加外力向量；fbreathing（t）为呼吸函数。
当呼吸函数取值为0和1，分别表示裂纹处于完全闭合状态和完全张开状态。当呼吸函数0<fbreathing（t）<1，表示叶片裂纹处于部分闭合状态。叶片裂纹呼吸函数的推导过程可参考文献［7］。叶片裂纹的变化将导致Kc和的变化，从而导致转子系统整体刚度矩阵和阻尼矩阵的变化，最终影响转轴的振动响应和转子系统的三维叶尖间隙动态响应。
1.2　含叶片裂纹的转子系统三维叶尖间隙动态响应
为了求解含叶片裂纹的转子系统动力学模型，可采用Houbolt法、Wilson‑θ法和Newmark‑β法等数值计算方法。根据文献［15］给出的算例，对于两自由度的质量‑弹簧‑阻尼系统，在简谐激励作用下Newmark‑β算法具有较高的求解精度。笔者设定Newmark‑β算法的参数δ=0.5、β=0.4，确保计算收敛且求解精度在可接受范围内。采用Newmark‑β算法求解式（18），得到叶片在径向、摆振方向和弦长方向的位移。要计算转子系统的三维叶尖间隙，还需要计算叶片摆振方向和弦长方向的偏转角
根据叶尖表面变形前后的几何关系，转子系统的三维叶尖间隙动态响应可表示为
其中：r（t）、α（t）和β（t）分别表示转子系统的径向间隙（radial tip clearance，简称RTC）、轴向偏转角（axial deflection angle，简称ADA）和周向滑移角（circumferential deflection angle，简称CDA）。
含叶片裂纹的转子系统三维叶尖间隙动态响应计算流程如图3所示，具体计算步骤如下：
1） 初始化含叶片裂纹的转子系统三维叶尖间隙动态响应模型的几何参数和材料参数，设定求解时长，将求解过程分解为若干时间步（t1，t2，…，tend），时间间隔为Δt；
2） 计算转子系统的质量矩阵、刚度矩阵和阻尼矩阵，假设叶片呼吸裂纹初始状态为闭合状态，计算叶片裂纹刚度矩阵和附加载荷向量；
3） 根据式（18）构建含叶片裂纹的转子系统动力学模型，并采用Newmark‑β算法进行求解，得到叶片径向、摆振方向和弦长方向的位移；
4） 根据式（20）计算含叶片裂纹的转子系统三维叶尖间隙动态响应；
5） 计算叶片裂纹呼吸函数，并更新含叶片裂纹的转子系统叶片裂纹刚度矩阵和附加载荷向量；
6） 重复步骤3~5，直至循环结束。
2　含叶片裂纹的转子系统动力学模型验证
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2.1　数值验证
为了验证所建立的转子系统动力学模型，将模型计算得到的固有频率与有限元模型进行对比。转子系统有限元模型如图4所示，由6个叶片、轮盘和转轴组成。
将轮盘设置为刚体，与转子系统动力学模型一致。对转轴施加简支梁边界条件，并将叶片、轮盘和转轴之间设置为固定连接。采用六面体单元进行网格划分，有限元模型的单元数量超过23 000。转子系统的材料参数和几何参数分别如表1和表2所示。在动力学模型中，将叶片和轮盘的模态截断数设置为3。为了比较有限元模型和动力学模型得到的结果，将固有频率的相对误差定义为
其中：fFEM和fdynamic分别为由有限元模型和动力学模型得到的固有频率。
对2个模型分别进行模态分析，得到转子系统的各阶固有频率。转子系统的固有频率对比如表3所示。由于转轴的模态截断数设置为3，因此无法通过动力学模型得到转轴的第4阶弯曲模态。转子系统有6个叶片，但只有5阶叶片弯曲模态具有相似的固有频率，这是由于转子系统的静态耦合项［13］导致叶片弯曲模态向耦合模态转变所致。模态分析结果表明，“转轴扭转‑叶片弯曲”模态的固有频率相对误差为5.19%，其他模态的相对误差均小于5%，表明笔者建立的转子系统动力学模型具有较高的精度。
2.2　实验验证
图5为搭建的含叶片裂纹的转子系统固有频率测量系统。测量系统由含叶片裂纹转子系统、电涡流位移传感器、电源、信号调理器和上位机等组成。在转子系统的叶轮盘上安装1个裂纹叶片和5个正常叶片。叶片裂纹位置靠近叶根，裂纹面到叶根的距离为5 mm，叶片裂纹深度为2 mm，叶片安装角为45°，叶片扭转角为25°，叶片材料为铝合金。
用橡皮锤敲击转轴，通过电涡流传感器采集转轴振动位移信号。转子系统的转轴振动信号及频谱如图6所示。传感器的采样频率设为10 kHz。进行10组实验，保存各组实验的数据，截取时间长度为1 s的转轴振动信号，并对时域信号进行去均值处理，得到转轴振动位移时域波形，如图6（a）所示。如图6（b）所示，频谱中存在56 Hz的转轴1阶固有频率分量，由局部放大图可以看到转轴的2、3阶固有频率分量。
表4为通过10组锤击实验得到含叶片裂纹的转子系统固有频率实验结果。对10组实验数据取平均值，得到含叶片裂纹转子系统前3阶固有频率分别为56.6、185.7和329 Hz。
根据笔者建立的含叶片裂纹的转子系统动力学模型，计算得到含有1个裂纹叶片和5个正常叶片转子系统的前3阶固有频率分别为58.64、192.59和367.36 Hz。将计算结果与实验结果进行对比，含叶片裂纹的转子系统前3阶固有频率对比如表5所示。含叶片裂纹转子系统前3阶固有频率的相对误差分别为3.60%、3.71%和11.66%。
导致相对误差的主要原因包括：①在实验中，叶片、轮盘和转轴是通过螺栓连接的，动力学模型则假定各部件之间为固定连接；②由于模态截断数的存在，动力学模型不可避免地存在截断误差。尽管通过仿真计算和锤击实验得到的含叶片裂纹转子系统固有频率存在误差，但前2阶固有频率相对误差均小于5%，第3阶固有频率的相对误差为11.66%，仍在可接受的范围内，这表明笔者建立的含叶片裂纹转子系统动力学模型具有较好的精度。
3　含叶片裂纹的转子系统三维叶尖间隙动态响应特性分析
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3.1　叶片裂纹深度对转子系统三维叶尖间隙的影响
假设转子系统有6个叶片，其中1号叶片有裂纹。叶片和轮盘的模态截断数均设置为3。设置叶片相对裂纹深度λ为0.1~0.5，间隔为0.1；叶片相对裂纹位置η为0.1，即裂纹靠近叶片根部。转子系统的转速设置为1 440 r/min（即转频fr为24 Hz）。气动载荷频率设为N倍转频，N为静子叶片数量。取N=6，则气动载荷频率为144 Hz。通过求解含叶片裂纹转子系统的三维叶尖间隙动态响应模型，得到不同裂纹深度条件下的转子系统三维叶尖间隙动态响应如图7所示。转子系统的径向间隙动态响应如图7（a）、（b）所示，随着叶片裂纹深度增加，径向间隙的幅值也会增大。由径向间隙的频谱可知，其基频分量为转频的N倍频（6fr），即气动载荷的频率。由于叶片呼吸裂纹的非线性，故径向间隙的频谱中还存在2N、3N和4N等高倍频分量。由径向间隙的2N倍频（12fr）可以看出，当叶片相对裂纹深度大于0.3时，径向间隙的2N倍频幅值明显增大，表明随着叶片裂纹深度增加，径向间隙表现出更强的非线性。转子系统的轴向偏转角和周向滑移角动态响应如图7（c）~（f）所示，轴向偏转角和周向滑移角的基频分量同样为转频的N倍频，随着叶片裂纹深度增加，基频分量的幅值变化不大。由于叶片呼吸裂纹的非线性，轴向偏转角和周向滑移角频谱中也存在高倍频分量，但幅值相对较小，其2N倍频分量的幅值随着叶片裂纹深度增大而增大，相较于径向间隙，轴向偏转角和周向滑移角频谱中高倍频分量幅值变化的线性特征较为明显，能更好地反映叶片裂纹深度信息。
为了进一步分析叶片裂纹对转子系统三维叶尖间隙的影响，将含叶片裂纹转子系统不同叶片的三维叶尖间隙动态响应进行对比。1号叶片为含裂纹叶片，其余叶片均为正常叶片。设置叶片相对裂纹深度λ为0.5，叶片相对裂纹位置η为0.1，含叶片裂纹的转子系统各叶片的三维叶尖间隙动态响应如图8所示。由图8（a）、（b）可知，当叶片有裂纹时，其径向间隙明显大于正常叶片，且裂纹叶片径向间隙的频谱中存在高倍频分量，而正常叶片径向间隙的频谱中只有基频分量。含叶片裂纹转子系统各叶片的轴向偏转角和周向滑移角动态响应分别如图8（c）~（f）所示。与正常叶片相比，当叶片存在裂纹时，轴向偏转角和周向滑移角基频分量的幅值更大，且其频谱中存在高倍频分量。
3.2　叶片裂纹位置对转子系统三维叶尖间隙的影响
为了分析叶片裂纹位置对转子系统三维叶尖间隙的影响，设置叶片相对裂纹深度λ为0.5，叶片相对裂纹位置η从0.1增大到0.9，间隔为0.2。转子系统的转频fr设置为24 Hz，气动载荷频率设置为144 Hz。通过求解含叶片裂纹的转子系统三维叶尖间隙动态响应模型，得到不同叶片裂纹位置条件下，转子系统1号叶片的三维叶尖间隙动态响应，如图9所示。
由图9（a）、（b）可知，当叶片无裂纹时，其径向间隙变化可忽略不计。当叶片裂纹位置靠近叶根（η=0.1）时，径向间隙幅值最大。随着叶片裂纹位置逐渐远离叶根，径向间隙幅值呈现先减小后增大的变化趋势。根据径向间隙的频谱可知，其基频分量为转频的N倍频（6fr），基频分量幅值随叶片裂纹位置远离叶根而先减小后增大。由于叶片呼吸裂纹的非线性，径向间隙频谱中存在2N倍频（12fr）分量。含叶片裂纹转子系统的轴向偏转角和周向滑移角动态响应如图9（c）~（f）所示。当叶片相对裂纹位置η为0.1、0.5和0.7时，轴向偏转角和周向滑移角小于正常叶片；当η为0.3和0.9时，轴向偏转角和周向滑移角大于正常叶片。随着叶片裂纹位置远离叶根，轴向偏转角和周向滑移角的基频分量和2N倍频分量的幅值均呈现非单调的变化趋势。叶片裂纹位置对转子系统刚度矩阵的影响是通过叶片模态振型函数引入的。随着叶片裂纹位置远离叶根，叶片1阶模态振型函数逐渐增大，但2阶和3阶模态振型函数呈现震荡变化的趋势。在叶片前3阶模态振型函数的共同影响下，转子系统三维叶尖间隙随叶片相对裂纹位置增大呈现非单调的变化趋势。
4　结论
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1） 考虑叶片在径向、摆振方向和弦长方向的变形，基于拉格朗日方程建立的含叶片裂纹转子系统三维叶尖间隙动态响应模型具有较好的精度。
2） 叶片呼吸裂纹的非线性会导致转子系统三维叶尖间隙频谱中出现基频Nfr的倍频分量，并且当转子系统存在叶片裂纹时，其三维叶尖间隙幅值大于正常叶片的三维叶尖间隙。
3） 叶片裂纹深度对三维叶尖间隙的影响主要体现在其频谱中的高倍频分量，三维叶尖间隙高倍频分量的幅值随着叶片裂纹深度的增加而增加。叶片裂纹位置对三维叶尖间隙的影响表现为其基频分量和高倍频分量的幅值均随着叶片相对裂纹位置的增大呈现非单调的变化趋势。三维叶尖间隙动态响应比传统的径向间隙能更好地反映转子系统的叶片裂纹故障信息。
基金
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  • 国家自然科学基金资助项目(52175117)
文献
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2025年第45卷第5期
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doi: 10.16450/j.cnki.issn.1004-6801.2025.05.002
  • 接收时间：2023-12-11
  • 首发时间：2026-03-27
  • 出版时间：2025-10-01
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  • 收稿日期：2023-12-11
  • 修回日期：2024-07-17
基金
国家自然科学基金资助项目(52175117)
作者信息
    1西安交通大学机械工程学院　西安，710049
    2西安交通大学现代设计及转子轴承系统教育部重点实验室　西安，710049

通讯作者:

凃成雄，男，2001年2月生，硕士生。主要研究方向为光纤检测及故障诊断。E‑mail：
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2种不同金属材料的力学参数
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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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