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Article(id=1244336750416474930, tenantId=1146029695717560320, journalId=1244311425741537314, issueId=1244336743298740932, articleNumber=null, orderNo=null, doi=10.16450/j.cnki.issn.1004-6801.2025.05.013, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1673280000000, receivedDateStr=2023-01-10, revisedDate=1677427200000, revisedDateStr=2023-02-27, acceptedDate=null, acceptedDateStr=null, onlineDate=1774602599958, onlineDateStr=2026-03-27, pubDate=1759248000000, pubDateStr=2025-10-01, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774602599958, onlineIssueDateStr=2026-03-27, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774602599958, creator=13701087609, updateTime=1774602599958, 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=951, endPage=960, ext={EN=ArticleExt(id=1244336751578297154, articleId=1244336750416474930, tenantId=1146029695717560320, journalId=1244311425741537314, language=EN, title=Influence of Wheel Polygon on Low‑Frequency Vibration of Subway Vehicle Body, columnId=1244336744728998604, journalTitle=Journal of Vibration,Measurement and Diagnosis, columnName=PAPER, runingTitle=null, highlight=null, articleAbstract=

To study the influence of subway wheel polygons on low-frequency vibration of the car body，the polygon wear of wheels of a subway line is investigated，on the basis of grasping the distribution characteristics of wheel polygons of subway lines. A vertical dynamic model of elastic car body considering wheel polygons is established，by the time-domain integral solution method. The relationship between wheel polygon excitation frequency and low frequency vibration of vehicle body is studied. By comparing the low-frequency vibration of wheel polygons of different orders，the effect of changes in the operating speed of metro vehicles under service conditions and changes in the radius of wheel wear on the polygonal action of the wheel body is discussed separately. It is shown that a wheel polygon of order 1—3 at common operating speeds generate a low-frequency excitation frequency of 0—20 Hz，and when the excitation frequency is close to the first-order droop frequency of 10.2 Hz resonance will occur. At the beginning of service，the influence of the second order wheel polygon becomes severe on the low frequency vibration of the car body. With the wear of the wheel radius during service，the influence of the first three order wheel polygon changes on the vibration law of the car body. This paper provides a good reference value for service subway operation and wheel maintenance.

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为了研究地铁车轮多边形对车体低频振动的影响，首先，调研了某地铁线路车轮多边形磨耗情况，在掌握地铁线路车轮多边形分布特征的基础之上，建立了考虑车轮多边形的弹性车体垂向动力学模型；其次，基于时域积分求解方法，研究了车轮多边形激励频率与车体低频振动的关系；最后，通过对比不同阶次的车轮多边形对车体低频振动的结果，讨论了服役条件下地铁车辆运行速度及车轮磨耗半径的变化对车轮多边形作用车体的影响规律。结果表明：1~3阶次的车轮多边形产生0~20 Hz的低频激励频率，当激励频率接近于车体1阶垂弯频率10.2 Hz时会产生共振，致使车体加速度响应增大；服役初期2阶车轮多边形对车体低频振动影响最大；随着服役过程中车轮半径的磨损，前3阶车轮多边形影响车体振动规律均发生改变。

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文永蓬，男，1979年10月生，博士、副教授。主要研究方向为城市轨道车辆系统动力学轨道结构服役性能演化、轮轨系统智能运维等。E-mail：
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徐祖翰，男，1998年7月生，硕士。主要研究方向为城市轨道交通车辆车轮多边形磨损。E-mail：

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徐祖翰，男，1998年7月生，硕士。主要研究方向为城市轨道交通车辆车轮多边形磨损。E-mail：

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徐祖翰，男，1998年7月生，硕士。主要研究方向为城市轨道交通车辆车轮多边形磨损。E-mail：

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Advances in formation mechanism and mitigation measures of out-of-round railway vehicle wheels［J］. Journal of Mechanical Engineering， 2021， 57（6）： 106-120.（in Chinese）, articleTitle=Advances in formation mechanism and mitigation measures of out-of-round railway vehicle wheels, refAbstract=null), Reference(id=1244351849134928006, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336750416474930, doi=null, pmid=null, pmcid=null, year=2021, volume=57, issue=6, pageStart=106, pageEnd=120, url=null, language=null, rfNumber=6, rfOrder=10, authorNames=TAO G Q, WEN Z F, LIANG X, journalName=Vehicle System Dynamics, refType=null, unstructuredReference=TAO G Q， WEN Z F， LIANG X， et al. An investigation into the mechanism of the out-of-round wheels of the metro train and its mitigation measures［J］. 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Journal of the China Railway Society， 2022， 44（9）： 26-32.（in Chinese）, articleTitle=Vibration behavior analysis of high-speed bogie frame under wheel polygonal excitations, refAbstract=null)], funds=[Fund(id=1244351848077963359, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336750416474930, awardId=TPL2103, language=CN, fundingSource=西南交通大学轨道交通运载系统全国重点实验室开放课题资助项目(TPL2103), fundOrder=null, country=null), Fund(id=1244351848145072226, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336750416474930, awardId=R202204, language=CN, fundingSource=上海市轨道交通结构耐久与系统安全重点实验室开放基金资助项目(R202204), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1244351839328645995, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336750416474930, xref=1., ext=[AuthorCompanyExt(id=1244351839341228908, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336750416474930, companyId=1244351839328645995, 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figureFileSmall=d/zroYzGoI2HVBkIklUwaw==, figureFileBig=aL755DzynImZ1oddN3i5nQ==, tableContent=null), ArticleFig(id=1244351847566258252, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336750416474930, language=CN, label=图15, caption=车体加速度均方根随车速、车轮半径变化图, figureFileSmall=d/zroYzGoI2HVBkIklUwaw==, figureFileBig=aL755DzynImZ1oddN3i5nQ==, tableContent=null), ArticleFig(id=1244351847666921552, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336750416474930, language=EN, label=Tab.1, caption=

Parameters of rigid‑flexible coupled model for an urban rail vehicle

, figureFileSmall=null, figureFileBig=null, tableContent=
物理参数数值符号
车体总长度/m21.88L
车辆定距之半/m7.85Lb
转向架轴距之半/m1.25Lw
车体转动惯量/(kg·m2)7.434 6×105Jc
转向架转动惯量/(kg·m2)2.937×103Js
一系悬挂垂向刚度/(N·m-1)0.911×106Kp
二系悬挂垂向刚度/(N·m-1)0.3×106Ks
一系悬挂垂向阻尼/(N·s·m-1)3×104Cp
二系悬挂垂向阻尼/(N·s·m-1)4×104Cs
车体质量/kg2.068 5×104Mc
转向架质量/kg3.525×103Mb
轮对质量/kg1.3×103Mw
车体截面等效抗弯刚度/(kN·m2)1.85EI
车体单位长度等效质量/(t·m-1)0.95ρA
), ArticleFig(id=1244351847767584851, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336750416474930, language=CN, label=表1, caption=

某城市轨道车辆刚柔耦合模型参数

, figureFileSmall=null, figureFileBig=null, tableContent=
物理参数数值符号
车体总长度/m21.88L
车辆定距之半/m7.85Lb
转向架轴距之半/m1.25Lw
车体转动惯量/(kg·m2)7.434 6×105Jc
转向架转动惯量/(kg·m2)2.937×103Js
一系悬挂垂向刚度/(N·m-1)0.911×106Kp
二系悬挂垂向刚度/(N·m-1)0.3×106Ks
一系悬挂垂向阻尼/(N·s·m-1)3×104Cp
二系悬挂垂向阻尼/(N·s·m-1)4×104Cs
车体质量/kg2.068 5×104Mc
转向架质量/kg3.525×103Mb
轮对质量/kg1.3×103Mw
车体截面等效抗弯刚度/(kN·m2)1.85EI
车体单位长度等效质量/(t·m-1)0.95ρA
), ArticleFig(id=1244351847847276630, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336750416474930, language=EN, label=Tab. 2, caption=

The excitation frequency caused by the wheel polygon

, figureFileSmall=null, figureFileBig=null, tableContent=
不圆阶次车速/(km•h-1)
4050607080
14.25.256.37.358.4
28.410.5012.614.7016.8
312.615.7518.922.0525.2
), ArticleFig(id=1244351847935357017, tenantId=1146029695717560320, journalId=1244311425741537314, articleId=1244336750416474930, language=CN, label=表2, caption=

车轮多边形引起的激励频率

, figureFileSmall=null, figureFileBig=null, tableContent=
不圆阶次车速/(km•h-1)
4050607080
14.25.256.37.358.4
28.410.5012.614.7016.8
312.615.7518.922.0525.2
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车轮多边形对地铁车辆车体低频振动的影响
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徐祖翰 1 , 文永蓬 1, 2, 3 , 宗志祥 4 , 程瑞平 1, 3 , 周慧 1, 3
振动、测试与诊断 | 论文 2025,45(5): 951-960
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振动、测试与诊断 | 论文 2025, 45(5): 951-960
车轮多边形对地铁车辆车体低频振动的影响
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徐祖翰1 , 文永蓬1, 2, 3 , 宗志祥4, 程瑞平1, 3, 周慧1, 3
作者信息
  • 1上海工程技术大学城市轨道交通学院　上海，201620
  • 2西南交通大学轨道交通运载系统全国重点实验室　成都，610031
  • 3上海市轨道交通振动与噪声控制技术工程研究中心　上海，201620
  • 4上海地铁维护保障有限公司车辆分公司　上海，200235

    • 徐祖翰，男，1998年7月生，硕士。主要研究方向为城市轨道交通车辆车轮多边形磨损。E-mail：

通讯作者:

文永蓬，男，1979年10月生，博士、副教授。主要研究方向为城市轨道车辆系统动力学轨道结构服役性能演化、轮轨系统智能运维等。E-mail：
Influence of Wheel Polygon on Low‑Frequency Vibration of Subway Vehicle Body
Zuhan XU1 , Yongpeng WEN1, 2, 3 , Zhixiang ZONG4, Ruiping CHENG1, 3, Hui ZHOU1, 3
Affiliations
  • 1.School of Urban Railway Transportation，Shanghai University of Engineering Science　Shanghai，201620，China
  • 2.State Key Laboratory of Rail Transit Vehicle System，Southwest Jiaotong University　Chengdu，610031，China
  • 3.Shanghai Engineering Research Centre for Vibration and Noise Control Technologies in Railway Transportation　Shanghai，201620，China
  • 4.Vehicle Branch of Shanghai Metro Maintenance and Guarantee Co.，Ltd.　Shanghai，200235，China
出版时间: 2025-10-01 doi: 10.16450/j.cnki.issn.1004-6801.2025.05.013
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摘要
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为了研究地铁车轮多边形对车体低频振动的影响，首先，调研了某地铁线路车轮多边形磨耗情况，在掌握地铁线路车轮多边形分布特征的基础之上，建立了考虑车轮多边形的弹性车体垂向动力学模型；其次，基于时域积分求解方法，研究了车轮多边形激励频率与车体低频振动的关系；最后，通过对比不同阶次的车轮多边形对车体低频振动的结果，讨论了服役条件下地铁车辆运行速度及车轮磨耗半径的变化对车轮多边形作用车体的影响规律。结果表明：1~3阶次的车轮多边形产生0~20 Hz的低频激励频率，当激励频率接近于车体1阶垂弯频率10.2 Hz时会产生共振，致使车体加速度响应增大；服役初期2阶车轮多边形对车体低频振动影响最大；随着服役过程中车轮半径的磨损，前3阶车轮多边形影响车体振动规律均发生改变。

车轮多边形  /  径向跳动量  /  低频振动  /  车轮磨耗  /  同频共振

To study the influence of subway wheel polygons on low-frequency vibration of the car body，the polygon wear of wheels of a subway line is investigated，on the basis of grasping the distribution characteristics of wheel polygons of subway lines. A vertical dynamic model of elastic car body considering wheel polygons is established，by the time-domain integral solution method. The relationship between wheel polygon excitation frequency and low frequency vibration of vehicle body is studied. By comparing the low-frequency vibration of wheel polygons of different orders，the effect of changes in the operating speed of metro vehicles under service conditions and changes in the radius of wheel wear on the polygonal action of the wheel body is discussed separately. It is shown that a wheel polygon of order 1—3 at common operating speeds generate a low-frequency excitation frequency of 0—20 Hz，and when the excitation frequency is close to the first-order droop frequency of 10.2 Hz resonance will occur. At the beginning of service，the influence of the second order wheel polygon becomes severe on the low frequency vibration of the car body. With the wear of the wheel radius during service，the influence of the first three order wheel polygon changes on the vibration law of the car body. This paper provides a good reference value for service subway operation and wheel maintenance.

wheel polygon  /  radial runout  /  low-frequency vibration  /  wheel wear  /  co-frequency resonance
徐祖翰, 文永蓬, 宗志祥, 程瑞平, 周慧. 车轮多边形对地铁车辆车体低频振动的影响. 振动、测试与诊断, 2025 , 45 (5) : 951 -960 . DOI: 10.16450/j.cnki.issn.1004-6801.2025.05.013
Zuhan XU, Yongpeng WEN, Zhixiang ZONG, Ruiping CHENG, Hui ZHOU. Influence of Wheel Polygon on Low‑Frequency Vibration of Subway Vehicle Body[J]. Journal of Vibration,Measurement and Diagnosis, 2025 , 45 (5) : 951 -960 . DOI: 10.16450/j.cnki.issn.1004-6801.2025.05.013
正文
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引言
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车轮在制造过程中会因为切削加工工艺及材料分布不均等原因出现车轮初始非圆化现象［1］。随着车轮进入服役状态，经过长期运行后车轮初始非圆化进一步发展，形成车轮多边形磨损并引起车辆性能劣化，不仅造成轮轨强烈的冲击振动和噪声污染，而且导致车辆与轨道动态相互作用日益突出［2］，严重时甚至引起关键零部件疲劳断裂，危及地铁车辆运营安全［3］。
车轮多边形是车轮在经过一定的运行里程发生磨耗后，沿着周向呈现出周期性不圆的现象。多边形以一定的波长延伸至整个圆周表面，相应的波数称为多边形阶数，多边形的峰与谷半径差称为波深［4］。针对车轮多边形产生的机理及影响，学者们开展了大量系统而深入的研究［5‑16］。陶功权等［5］将车轮非圆化磨耗的形成机理进行分类，认为车轮产生的初始缺陷和车辆运行时的固有振动，以及制动时的热弹性失稳问题均可能是形成车轮多边形的原因。Tao等［6］通过现场试验研究了地铁车轮的多边形磨损机理，发现5~8阶谐波车轮失圆的波长固定机制为P2共振。赵新利等［7］根据理论研究和试验分析，发现轮轨表面不平顺会加剧系统固有模态频率的模态共振，诱发车轮多边形磨耗的产生。车轮多边形在车辆运行过程中会逐渐加剧车轮损伤，并发展为不可忽视的轮轨激励。丁军君等［8］对高速列车车轮多边形磨耗的演变过程进行数值模拟，结果表明，车轮初始3阶多边形会演变成以3的整数倍阶为主的多阶混合多边形，其中初始的3阶多边形占主导地位。
国内外学者针对车轮多边形磨损建立了不同的仿真模型。Peng等［9］从车轮磨损的角度出发对代表性磨损模型进行综合比较，讨论了现有磨损模型在铁路车轮多边形化模拟中的适用性。刘孟奇等［10］考虑轮对和轨道的不同建模方法，探讨了中高频车轮多边形磨耗激励下不同模型轮轨动态作用的差异。车轮多边形的阶次直接影响其激励频率范围。当研究10阶以下低阶车轮多边形时，其激励频率在100 Hz以下，采用传统车辆模型是适用的；当研究10阶以上的高阶车轮多边形时，其激励频率在100 Hz以上，则要建立车辆‑轨道耦合模型。车轮多边形主要影响车辆系统的垂向振动。Lan等［11］计算了失圆缺陷主导轮轨界面处的法向接触力，对比了车轮有无多边形磨损的动态响应。由于车辆与轨道是一体的，车轮的损伤必然也会影响到钢轨。Ma等［12］建立了考虑轮轨变摩擦特性的多车辆‑轨道耦合动力学模型，提出了用于评价多边形磨损程度的失圆增长率，并解释了夏季频繁降水与高温环境带来的湿摩擦条件会加剧摩擦功率共振现象。雷晓燕等［13］基于有限元法建立了“车体‑多边形化车轮‑轨道”模型，拟合实测数据获得锯齿波函数，可有效识别实际线路中的车轮多边形分布情况。车轮多边形作用于轮轨产生的不平顺，严重时可影响到车辆运行的平稳性和安全性。尹振坤等［14］计算了不同速度下1~23阶车轮多边形对应幅值的最大轮轨作用力，给出了不同阶次车轮多边形的镟修幅值，为地铁运营维保提出镟轮建议。Mu等［15］发现低阶车轮多边形可激发轮对的1阶弯曲模态和P2共振，从而在相应速度下产生较大的轮轨接触力，影响行车安全。韩光旭等［16］对运营中的高速列车进行车内噪声测试，结果表明，车轮径跳幅值及多边形阶次均可导致高速列车车内振动和噪声偏大。
现有研究往往针对中高频的车辆振动，并将车体考虑为刚体，对车体的低频弹性振动重视不足。为此，笔者根据地铁车辆低频振动的实际情况，基于某地铁的车轮多边形及车轮磨耗特征，建立含车轮多边形的车辆低频垂向振动模型，获取该地铁车轮多边形的主要阶次与幅值分布，分析了车轮多边形激励频率与车体低频垂向振动频率的关系，研究了车轮多边形作用下车轮半径的磨耗对车体振动加速度以及对车辆低频垂向振动的影响规律。
1　地铁车轮现场跟踪调研
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1.1　车轮不圆度测试及表征
笔者对某地铁线路的车轮不圆度进行了跟踪测量。车轮多边形测量现场如图1所示。
地铁车辆段一般采用车轮径跳量作为车轮镟修的指标，车轮径跳量为车轮最大半径与最小半径之差［17］。对车轮不圆度测试仪获取的车轮数据进行径跳量统计，给出编号为14、08、29、27的测试车辆的48个车轮在0~33×104 km内不同运行里程车辆车轮径跳量，如图2所示。由图可知，车轮的径跳量随着车辆运行里程的增大成正相关变化。在地铁运行初期车轮径跳量发展缓慢，当车辆运行里程达到20×104 km以后车轮的径跳量出现大幅增长，且主要分布在0.2~0.6 mm之间，个别车轮径跳量甚至超过了0.6 mm，说明该线路车辆普遍存在较为严重的车轮失圆现象。
目前，国内对车轮径跳量标准没有明确规定。根据上海地铁维保部门维修经验，车轮径向跳动量标准不应超过0.6 mm。因此，针对27号测试车辆48个车轮运行33×104 km时的实测结果，选取径跳量超过0.6 mm的3个严重不圆车轮以及正常磨损车轮进行车轮不圆度磨耗测试和对比分析，获得严重不圆车轮极坐标表现形式，如图3所示。
图3中，相较于正常磨损车轮，1号严重不圆车轮总体表现为心形，即明显的3阶多边形现象，而2号和3号不圆车轮则表现为偏心和椭圆现象。为了解该线路车轮多边形的阶次构成及对应的不平顺水平分布，需对车轮进行不圆度幅值与不平顺水平换算［18］。将典型严重失圆车轮的周向磨损数据进行傅里叶级数拟合，得到前阶车轮不圆度幅值为
其中：为傅里叶级数的阶数，即车轮多边形的阶数；为最大阶数（地铁车辆车轮多边形波长一般大于66 mm，通常令即可满足计算要求）；、分别为第阶展开式对应的正弦波和余弦波的幅值；为车轮半径的倒数；为车轮周向位置。
根据同名三角函数公式，式（1）可表示为仅含正弦波的表达式，即
其中：为第阶谐波的幅值，；为第阶谐波的波长，；为第阶谐波的相位，。
实测的车轮周向不圆顺可由幅值为、波长为的阶正弦函数叠加组成，进一步进行车轮不平顺水平转换，即
其中：为车轮不平顺水平；为参考值，取1 μm。
利用式（3）得到车轮不平顺水平随多边形阶次分布图，如图4所示。由图可知，实测地铁车辆车轮的非圆化磨耗以低阶为主，且不平顺水平较大的阶次主要分布在1~3阶，不同阶次车轮多边形表现的不平顺水平整体上随着阶次的增大而减小。3个严重失圆车轮的前3阶车轮多边形不平顺水平均达到40 dB/μm（即0.1 mm的车轮不圆度幅值有效值）。当车轮多边形阶次为10阶时，不平顺水平级多分布于20 dB/μm附近，此时的车轮不圆度有效幅值为0.01 mm，磨损程度已远小于前3阶车轮多边形。4号正常磨损车轮的各阶次不平顺水平普遍小于前3个严重失圆车轮，说明这3个典型车轮存在较为严重的低阶车轮多边形问题。
车轮多边形作为一种外源激励，低阶车轮多边形表现为偏心、椭圆和心形特征，1~3阶车轮多边形所对应的激励波长范围为880~2 639 mm，属于轨道长波不平顺。根据波长频率计算公式，当地铁车辆以80 km/h的速度运行时，将给车辆带来0~20 Hz的低频激扰。因此，需要进一步对1~3阶车轮多边形引起的车辆0~20 Hz低频垂向振动进行研究。
1.2　车轮磨损
随着地铁运行里程的增加，车轮磨损不断累积加剧，从而导致车轮的半径逐渐减小。根据地铁实际运营数据，得到了如图5所示的车轮半径磨损随行驶距离变化图。
由图5可知，随着运行里程的增加，车轮半径的磨损并非完全符合线性减小的趋势，所有车辆的运营前期车轮磨损相对剧烈，在新车轮运行5×104~10×104 km后会出现一个拐点。这是因为新车轮与钢轨初始接触匹配时，二者相互磨损严重；随着车辆服役里程增加，车轮钢轨型面进行一定程度的磨合后，车轮的磨损速度渐趋于平缓。不同车辆的车轮磨损略有差异，同车辆不同轴的车轮半径磨损趋势接近，同轴的左右车轮半径磨损基本一致。
综上所述，车辆长期服役过程中不仅在车轮周向产生不圆顺现象，在径向也存在不同程度的磨损。车轮周向和径向的磨损具有动态随机性和时变性，将会引起车辆性能的劣化。因此，车辆服役条件下需同时考虑车轮不圆和车轮磨耗对车辆低频垂向振动的影响。
2　考虑车轮多边形的车辆动力学模型
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2.1　含车轮多边形的车辆低频垂向模型
车辆的低频垂向振动分为刚性振动和弹性振动。刚性振动包括车体浮沉和点头运动，其特征频率在0~4 Hz之间；弹性振动以1阶垂弯振动为主，其特征频率在8~20 Hz之间［19］。根据车辆系统动力学理论，建立如图6所示的含车轮多边形的车辆低频垂向振动模型。其中：为车体在x位置处t时刻的垂向位移；、分别为转向架、轮对的垂向位移；、分别为前后转向架位置；为车体中部位置。模型中所受外界激励为作用在4个轮对上的车轮多边形激励。
模型考虑的自由度不仅包括10自由度的刚性振动，即车体和前后2个转向架的浮沉、点头运动及4个轮对的浮沉运动，而且还将车体设定为弹性均质等截面的欧拉梁，并考虑车体的前2阶弹性振动，车体的垂向振动为刚性振动与前2阶弹性振动之和。根据牛顿‑欧拉法列出刚柔耦合模型各自由度的运动方程，并利用龙格库塔法对模型进行时域积分求解。模型中采用了某城市轨道车辆刚柔耦合模型参数，如表1所示。
考虑车体弹性振动，车体的振动偏微分方程为
其中：为前后转向架作用在车体上的垂向力；为前后转向架位置；为Dirac函数。
根据牛顿第二定律，列出车体在处转向架上的受力为
采用模态叠加法求解式（4）的车体振动偏微分方程，设车体的第阶振型和模态坐标分别为和，其中的数学表达式为
当=1时，车体的1阶振型为浮沉运动，此时的振型函数=1；当=2时，车体的2阶振型为点头运动，振型函数为。前2阶振型组成车体振动中的刚性振动，车体的弹性振动从第3阶振型开始算起。因此，车体的垂向振动位移方程在各振型叠加后可表示为
将式（7）代入式（4）并通过积分简化，利用振型的正交性和Dirac函数的性质，得到车体浮沉、点头和弹性振动方程组为
其中：、分别为第阶车体的阻尼比和自振频率。
车轮多边形为轮轨间的一种不平顺且直接作用于轮对，则轮对浮沉运动方程为
其中：为第位转向架的垂向位移；为第轴轮对的垂向位移（当时，；当时，）；为转向架点头运动；为第轴轮对受到轮轨间不平顺产生的轮轨作用力。
根据Hertz接触线性理论，可表示为
其中：为轮轨间接触刚度（根据表1，=1.206 3×109 N/m）；为钢轨垂向位移；为第轴轮对处的车轮多边形不圆度幅值；为第轴轮对处的美国5级谱垂直不平顺幅值。
由于钢轨的垂向振动频率多为高频［20］，而本研究针对的是车体低频振动，因此可以认为Zr=0。构架运动方程参见文献［21］。联立上述车辆垂向模型的振动微分方程，可将车辆系统振动方程统一表示为
其中：、和分别为系统的质量矩阵、阻尼矩阵和刚度矩阵；为含多边形激励的广义载荷矩阵。
2.2　车轮多边形表征
为了确定车轮多边形对车辆系统的影响，将车轮多边形作为模型的输入激励。国际上多采用含有1至阶谐波的傅里叶级数形式的时域位移函数来描述这种激励［21］，因此式（1）可表示为
其中：为车速；为车轮半径。
基于此函数建立的1~3阶车轮多边形激励，施加于模型的第1轴轮对上。将1~3阶的多边形车轮与理想车轮进行对比，车轮多边形示意图如图7所示。可知，理想的1~3阶车轮多边形整体分别表现为偏心、椭圆和心形，这与图1中的前3阶车轮多边形具有较好的对应性。
当出现车轮多边形现象时，会给车辆系统带来不同频率的振动激励［22］。根据车辆运行速度及车轮多边形阶次，激励频率的计算公式为
表2给出了1~3阶、半径为0.42 m的车轮在不同运行速度下，由车轮多边形引起的激励频率。对典型速度下的城市轨道车辆而言，1~3阶车轮多边形的激励频率属于低频振动范畴，其数值更接近于车体弹性垂弯振动频率10.2 Hz。
2.3　验证
由于笔者主要研究考虑车轮多边形激励频率的车体低频振动，因此利用多体动力学软件Simpack对施加的车轮多边形特征频率进行验证。为简化计算，验证部分Simpack模型与仿真模拟结果均采用刚性车体进行计算，使用垂向加速度指标来分析车辆动力学性能。Simpack中添加车轮多边形的方法为在模型上依次选取转向架‑轮对‑轮轨副，在对应的Untruess模块里设置添加车轮多边形的阶次及幅值等参数。
选取车轮多边形激励幅值为0.1 mm，在第1轴轮对和第2轴轮对处分别施加特征阶数为1阶、3阶的车轮多边形激励，考虑垂向美国5级谱的输入，并设置车辆运行速度为80 km/h进行模拟计算。基于Simpack的地铁车辆动力学模型如图8所示。
Simpack求解加速度的方法是在指定位置添加加速度传感器，计算结束后在结果后处理中导出车体垂向加速度时域数据，添加106号频谱过滤器得到车体频域上的加速度。在研究模型中只考虑刚性车体运动部分，施加垂向美国5级谱和相同的车轮多边形激励，对模型进行数值积分求解后得到车体的浮沉运动加速度。车体振动加速度频域对比图如图9所示。
由图9可知，在施加车轮多边形和垂向美国5级谱激励后，2种模型在频域上计算得到的车体加速度值具有较好的对应性，且均出现1阶车轮多边形8.4 Hz和3阶车轮多边形25.2 Hz的激励频率峰值，说明本研究仿真与软件建模较吻合。由于Simpack模型是空间三维模型，理论模型为垂向振动模型，导致二者计算的车体浮沉振动频率有一定的误差。
3　结果分析与讨论
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3.1　多边形阶次与车体振动的关系
车轮多边形作为一种车轮损伤引起的激励，在列车运行过程中可能会与车辆系统某部件固有频率相近产生共振，从而恶化车辆运行平稳性［23］。因此，需要明确多边形阶次与车体的振动关系。
将表1的车辆参数代入本研究所建立的垂向振动模型，模型输入采用垂向美国5级谱，设定车辆速度为80 km/h，得到车体低频垂向振动加速度功率谱，如图10所示。
由图10可知，车体低频振动包括刚性振动和弹性振动两个特征频段。在刚性振动段内，车体弹性模型与刚体模型计算的加速度功率谱基本一致；在弹性振动段内，弹性车体的振动能量要远大于刚性车体，这说明车体的弹性振动是组成车体低频振动的重要部分，且车体的1阶垂弯频率为10.2 Hz，与低阶车轮多边形激励频率比较接近，将会引起车体低频振动进一步加剧。
将车辆运行速度设为80 km/h，在模型的第1轴轮对处施加幅值为0.1 mm、阶次为1~3阶的低阶车轮多边形激励，观察1~3阶理想车轮多边形与美国5级谱耦合的车体加速度功率谱，其随阶次变化如图11所示。
由图11可知，在考虑车轮多边形作用后，车体加速度功率谱中会出现车轮多边形激励频率的峰值，且该激励频率越接近车体弹性振动频率，其对应数值处的车体加速度功率谱峰值越大。80 km/h运行速度下1阶车轮多边形产生的激励频率为8.4 Hz，相较于2、3阶车轮多边形的激励频率，该频率更接近车体弹性振动频率10.2 Hz，因此1阶车轮多边形作用的激励频率处峰值要大于2、3阶。由此可见，低阶车轮多边形会加剧车体的弹性振动，其激励频率越接近车体弹性振动频率，则该阶车轮多边形引起的车体振动越剧烈。
地铁车辆在运行中同时受到线路固有轨道不平顺激励和车轮服役产生的激励作用，为了研究车轮服役前后车体的振动变化，以美国5级谱作为固有轨道不平顺激励，考虑地铁车辆常用典型运行速度，分别对比车轮多边形与美国5级谱作用的车体功率谱，如图12所示。
图12（a）~（c）为前3阶车轮多边形与美国5级谱耦合的车体加速度功率谱，（d）~（f）为车轮多边形单独作用的结果。车轮多边形作为车辆服役过程中产生的特有激励，计算常用速度下的车体功率谱上均出现了表2中激励频率的峰值。分别对比图（a）和（d）、图（b）和（e）、图（c）和（f）后可知，车轮多边形与美国5级谱耦合后，车轮多边形单独作用的峰值频率仍存在。这说明车轮多边形在车轮服役过程中一旦形成，会逐渐恶化并发展成轮轨间不可忽视的不平顺作用，此时仅考虑轨道谱是不全面的，需结合实际运营地铁的车轮多边形情况进行分析。
若只考虑车轮多边形作用，则能清晰地反映车轮多边形与车体振动的关系。由图12（d）~（f）可知：1阶车轮多边形的激励频率范围为4.2~8.4 Hz，随着速度增加车体功率谱也增大，其频率尚未达到车体1阶垂弯频率，不会引起车体的弹性共振；3阶车轮多边形的激励频率范围为12.6~25.2 Hz，已经大于车体低频振动频率，随着速度增加逐渐远离车体1阶弹性共振频段；当车轮出现2阶多边形时，产生的激励频率处于8.4~16.8 Hz范围内，会覆盖车体弹性共振频率，在这个速度区间内2阶车轮多边形引起的车体振动加速度功率谱最大，更容易激发车体的弹性共振，不利于旅客乘车舒适性。由于2阶车轮多边形在50 km/h速度下的激励频率最接近车体弹性频率，因此车体低频振动对2阶车轮多边形的敏感速度为50 km/h。
为了突出车轮多边形在轮轨不平顺中的贡献，后续讨论均只考虑车轮多边形单独作用的结果。进一步考虑前3阶多边形车轮的车体加速度均方根随车速、车轮阶次的变化，如图13所示。由图可知，当只考虑车轮多边形作用时，2阶车轮多边形作用下车体的加速度均方根出现明显的峰值，因此对于服役初期的车辆需重视2阶车轮多边形问题。
3.2　车轮磨损对车体振动的影响
随着车辆运行里程的增加，车轮磨损日益严重，车轮半径会出现不同程度的减小。为了研究含车轮多边形的磨损车轮对车辆运行平稳性的影响，设置磨损车轮半径为0.41 m，选取车轮多边形幅值为0.1 mm，以车辆运行速度为80 km/h的1阶车轮多边形和速度为50 km/h的2阶车轮多边形为研究对象，对比车轮多边形新旧车轮的车体功率谱，如图14所示。
由图14可知，车轮磨损会增大车轮多边形的激励频率，使得1、2阶车轮多边形激励接近或远离车体1阶垂弯频率10.2 Hz，进而改变车轮多边形对车体低频振动的影响。进一步研究车轮磨损与车轮多边形对车体低频振动的影响，获得了前3阶车轮多边形作用下车体加速度均方根随车速、车轮半径的变化，如图15所示。
由图15（a）可知，当速度在40~80 km/h范围内，1阶车轮多边形作用的车体振动加速度均方根与车辆速度和磨损半径均成正相关，说明在高速时地铁车轮的磨损能够加剧1阶车轮多边形对车体振动的作用。由图15（b）可知，车轮服役条件下，2阶车轮多边形作用的车体加速度均方根先降后增，运行初期车体对2阶车轮多边形的敏感速度为50 km/h，随着车轮进一步磨损，车体对2阶车轮多边形的敏感速度变为40 km/h。由图15（c）可知，随着车轮半径减小，3阶车轮多边形对车体振动的影响逐渐增大，3阶车轮多边形在车轮严重磨损后出现了30 km/h的敏感速度，这使得3阶车轮多边形带来的激励频率接近车体低频振动频率，从而导致车体振动加剧。
综上所述，地铁车辆长期服役导致车轮磨损半径减小，从而改变低阶车轮多边形的激励频率。高速运行时能加剧1阶车轮多边形对车体振动的影响，低速运行时会促进3阶车轮多边形对车体的作用，车轮磨损也改变了车体对2阶车轮多边形的敏感速度。
4　结论
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1） 研究了某地铁车辆1~3阶车轮多边形磨损现象，在服役过程中车轮多边形影响车体系统振动特性，使地铁车辆动力学性能逐渐劣化。服役初期2阶车轮多边形对车体低频振动影响最大，随着服役过程中车轮半径的磨损，车体对2阶车轮多边形的敏感速度由50 km/h降至40 km/h。在高速时车轮的磨损能加剧1阶车轮多边形对车体振动的作用。3阶车轮多边形在车轮严重磨损后出现较低的30 km/h敏感速度。
2） 地铁车辆服役过程中车轮周向和径向都会产生不同程度的磨损，周向磨损表现为车轮多边形现象，径向磨损表现为车轮半径减小，而车轮半径减小会增大车轮多边形的激励频率。这两种磨损具有动态随机性和时变性，因此在研究车轮多边形问题时，新车轮主要考虑2阶车轮多边形可能引起的10.2 Hz车体弹性共振问题，车轮半径磨损过程增加了共振的可能性，应重视磨损车轮的前3阶车轮多边形。
3） 随着地铁车辆运行速度的改变，低阶车轮多边形会产生0~20 Hz的低频激扰。当该激扰频率接近于车体1阶垂弯频率10.2 Hz时会产生共振现象，致使车体垂向振动加剧，影响乘客乘坐舒适性。因此，应通过及时镟轮、变速行驶等手段避免低阶多边形的激励频率与车体频率产生同频共振。
基金
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  • 西南交通大学轨道交通运载系统全国重点实验室开放课题资助项目(TPL2103)
  • 上海市轨道交通结构耐久与系统安全重点实验室开放基金资助项目(R202204)
文献
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2025年第45卷第5期
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doi: 10.16450/j.cnki.issn.1004-6801.2025.05.013
  • 接收时间：2023-01-10
  • 首发时间：2026-03-27
  • 出版时间：2025-10-01
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  • 收稿日期：2023-01-10
  • 修回日期：2023-02-27
基金
西南交通大学轨道交通运载系统全国重点实验室开放课题资助项目(TPL2103)
上海市轨道交通结构耐久与系统安全重点实验室开放基金资助项目(R202204)
作者信息
    1上海工程技术大学城市轨道交通学院　上海，201620
    2西南交通大学轨道交通运载系统全国重点实验室　成都，610031
    3上海市轨道交通振动与噪声控制技术工程研究中心　上海，201620
    4上海地铁维护保障有限公司车辆分公司　上海，200235

通讯作者:

文永蓬，男，1979年10月生，博士、副教授。主要研究方向为城市轨道车辆系统动力学轨道结构服役性能演化、轮轨系统智能运维等。E-mail：
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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
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
Percentage of total
species (%)
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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