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Article(id=1228635270350832141, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228635264860484432, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2024.09.013, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1665763200000, receivedDateStr=2022-10-15, revisedDate=1681488000000, revisedDateStr=2023-04-15, acceptedDate=null, acceptedDateStr=null, onlineDate=1770859075390, onlineDateStr=2026-02-12, pubDate=1727452800000, pubDateStr=2024-09-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770859075390, onlineIssueDateStr=2026-02-12, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770859075390, creator=13701087609, updateTime=1770859075390, updator=13701087609, issue=Issue{id=1228635264860484432, tenantId=1146029695717560320, journalId=1225147924628267009, year='2024', volume='37', issue='9', pageStart='1451', pageEnd='1624', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770859074081, creator=13701087609, updateTime=1770859151707, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1228635590502055946, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228635264860484432, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1228635590502055947, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228635264860484432, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=1564, endPage=1574, ext={EN=ArticleExt(id=1228635271277773338, articleId=1228635270350832141, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Dynamic characteristics of different rack vehicles, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

In order to improve the climbing ability of rack vehicles，the gear-rack system is added to the traditional rail vehicles. Aiming at the problem that there are many kinds of gear-rack systems in the world at present，and the diversity of gear-rack systems equipped with them leading to the dynamic characteristics difference of rack vehicles，this paper considers the impact of gear-rack meshing on the basis of analyzing the generation mechanism of gear-rack meshing excitation，the rack vehicle coupled dynamic models with two kinds of Strub system，double row teeth Abt system and Locher system are established，and experimental verification on the model are carried out； Based on the model，the gear-rack meshing behavior of rack vehicle running at different speeds on the engagement section of the ramp is analyzed，and the influence of track irregularities on the gear-rack meshing center distance error is explored； On this basis，the wheel/rail action and car body acceleration of the rack vehicle are studied，and the rack vehicle safety are analyzed as well as stabitity. The results show that there are significant differences in the dynamic characteristics of rack vehicles with different gear-rack systems，and the Locher system has the best dynamic characteristics； The gear-rack meshing behavior of coaxial Strub system and double row teeth Abt system is poor and affected by the track irregularity obviously. The maximum impact value of the gear-rack contact force is 20.3 kN，and the meshing center distance error is 3.73 cm； The safety of coaxial Strub system and double row teeth Abt system is poor，and the maximum wheel/rail vertical force of double row teeth Abt system is 51.7 kN； The car body stability of the differential shaft Strub system is the worst. The maximum car body acceleration is 0.033 m/s2，and the stability index is 1.27. The conclusions offer theoretical support for the design，safe operation，and maintenance of mountain rack railways in China.

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为了提高轨道车辆的爬坡能力，齿轨车辆在传统轨道车辆的基础上加入了齿轮-齿轨啮合系统。针对目前世界上的齿轨铁路系统种类较多，其配备齿轮-齿轨啮合系统多样性所导致的齿轨车辆动态特性差异的问题，本文在解析齿轮-齿轨啮合激扰产生机理的基础上，考虑齿轮-齿轨啮合冲击，建立了两类Strub系统、双排齿Abt系统以及Locher系统四种不同形式的齿轨车辆动力学模型，并对模型进行了实验验证；采用所建立的模型，分析了齿轨车辆以不同速度在坡道啮合路段运行时的齿轮-齿轨啮合行为，并探究了线路不平顺对齿轨车辆齿轮-齿轨啮合中心距误差的影响；在此基础上研究了齿轨车辆的轮轨作用以及车辆加速度，由此分析了齿轨车辆的安全性与平稳性。结果表明：不同形式的齿轨车辆动态特性存在显著差异，Locher系统齿轨车辆动态特性最佳；同轴Strub系统与双排齿Abt系统的齿轮啮合行为较差且受线路不平顺影响明显，齿轮-齿轨啮合力最大冲击值达20.3 kN，啮合中心距误差达3.73 cm；同轴Strub系统与双排齿Abt系统车辆安全性较差，双排齿Abt系统的轮轨垂向力最大达51.7 kN；异轴Strub系统的车辆平稳性最差，最大车体垂向加速度为0.033 m/s2，平稳性指标为1.27。

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袁密奥（1998—），男，硕士研究生。E-mail： 。
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陈兆玮（1988—），男，博士，副教授。E-mail： 。

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陈兆玮（1988—），男，博士，副教授。E-mail： 。

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陈兆玮（1988—），男，博士，副教授。E-mail： 。

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journalId=1225147924628267009, articleId=1228635270350832141, language=EN, label=Fig.12, caption=Rack vehicle dynamics performance at different slopes, figureFileSmall=6D4dabXinktqptluoNmXvA==, figureFileBig=l3Cj9v7rCDY/7pVCLWRdmA==, tableContent=null), ArticleFig(id=1228635298725298603, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228635270350832141, language=CN, label=图12, caption=不同坡度下齿轨车辆动力学性能, figureFileSmall=6D4dabXinktqptluoNmXvA==, figureFileBig=l3Cj9v7rCDY/7pVCLWRdmA==, tableContent=null), ArticleFig(id=1228635298838544817, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228635270350832141, language=EN, label=Tab.1, caption=

DoF of rack vehicle

, figureFileSmall=null, figureFileBig=null, tableContent=
车辆系统纵向垂向横向侧滚摇头点头
同轴和异轴Strub、双排齿Abt驱动齿轨(i=1~4)XgZg---θg
Locher驱动齿轮Xg-Yg-θg-
轮对(i=1~4)-ZwYwϕwψwβw
构架(i=1,2)-ZtYtϕtψtβt
车体-ZcYcϕcψcβc
), ArticleFig(id=1228635298930819512, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228635270350832141, language=CN, label=表1, caption=

车辆系统自由度

, figureFileSmall=null, figureFileBig=null, tableContent=
车辆系统纵向垂向横向侧滚摇头点头
同轴和异轴Strub、双排齿Abt驱动齿轨(i=1~4)XgZg---θg
Locher驱动齿轮Xg-Yg-θg-
轮对(i=1~4)-ZwYwϕwψwβw
构架(i=1,2)-ZtYtϕtψtβt
车体-ZcYcϕcψcβc
), ArticleFig(id=1228635299014705597, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228635270350832141, language=EN, label=Tab.2, caption=

Research conditons

, figureFileSmall=null, figureFileBig=null, tableContent=
齿轨系统坡度/‰运行速度/（km·h-1）
Strub，Abt，Locher100，150，20030
), ArticleFig(id=1228635299098591681, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228635270350832141, language=CN, label=表2, caption=

研究工况

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齿轨系统坡度/‰运行速度/（km·h-1）
Strub，Abt，Locher100，150，20030
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Rack vehicle parameters

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项目数值
最大运行速度/（km·h-1）30
车体尺寸/m316×3.2×3.4
车体重量/kg3.2×104
车体转动惯量/（kg·m2）2×106
中心距/m11.6
构架宽/m1.8
构架重量/kg3000
轮对重量/kg1000
二系悬挂x刚度/(kN·m-1)150
二系悬挂y刚度/(kN·m-1)150
二系悬挂z刚度/(kN·m-1)450
二系悬挂x阻尼/(kN·s·m-1)60
二系悬挂y阻尼/(kN·s·m-1)60
二系悬挂z阻尼/(kN·s·m-1)80
一系悬架x刚度/(kN·m-1)104
一系悬架y刚度/(kN·m-1)104
一系悬架z刚度/(kN·m-1)600
一系悬架x阻尼/(kN·s·m-1)6
一系悬架y阻尼/(kN·s·m-1)6
一系悬架z阻尼/(kN·s·m-1)6
轴距/m2.8
齿轮质量/kg452
齿轮模数/mm32
齿轮刚度/(N·m-1)5×108
齿条刚度/(N·m-1)5×108
齿数22
压力角/（°）20
齿轮宽/mm60
齿轨宽/mm80
滚动圆跨距/mm1058
名义滚动圆半径/m0.42
二系悬挂跨距/m1.72
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车辆参数

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项目数值
最大运行速度/（km·h-1）30
车体尺寸/m316×3.2×3.4
车体重量/kg3.2×104
车体转动惯量/（kg·m2）2×106
中心距/m11.6
构架宽/m1.8
构架重量/kg3000
轮对重量/kg1000
二系悬挂x刚度/(kN·m-1)150
二系悬挂y刚度/(kN·m-1)150
二系悬挂z刚度/(kN·m-1)450
二系悬挂x阻尼/(kN·s·m-1)60
二系悬挂y阻尼/(kN·s·m-1)60
二系悬挂z阻尼/(kN·s·m-1)80
一系悬架x刚度/(kN·m-1)104
一系悬架y刚度/(kN·m-1)104
一系悬架z刚度/(kN·m-1)600
一系悬架x阻尼/(kN·s·m-1)6
一系悬架y阻尼/(kN·s·m-1)6
一系悬架z阻尼/(kN·s·m-1)6
轴距/m2.8
齿轮质量/kg452
齿轮模数/mm32
齿轮刚度/(N·m-1)5×108
齿条刚度/(N·m-1)5×108
齿数22
压力角/（°）20
齿轮宽/mm60
齿轨宽/mm80
滚动圆跨距/mm1058
名义滚动圆半径/m0.42
二系悬挂跨距/m1.72
), ArticleFig(id=1228635299354444242, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228635270350832141, language=EN, label=Tab.4, caption=

Dynamic characteristics of rack vehicle

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项目齿轮接触力/kN啮合误差/m轮轨垂向力/kN车体加速度/（m·s-2）轮重减载率平稳性指标
同轴Strub20.20.037351.52.84×10-40.01020.304
Abt14.70.037351.73.21×10-40.01310.312
异轴Strub13.30.029451.40.03323.68×10-31.27
Locher12.10.008751.21.05×10-73.87×10-50.0283
最优LocherLocherLocherLocherLocherLocher
最差同轴Strub同轴Strub，AbtAbt异轴StrubAbt异轴Strub
), ArticleFig(id=1228635299459301847, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228635270350832141, language=CN, label=表4, caption=

齿轨系统车辆动态特性

, figureFileSmall=null, figureFileBig=null, tableContent=
项目齿轮接触力/kN啮合误差/m轮轨垂向力/kN车体加速度/（m·s-2）轮重减载率平稳性指标
同轴Strub20.20.037351.52.84×10-40.01020.304
Abt14.70.037351.73.21×10-40.01310.312
异轴Strub13.30.029451.40.03323.68×10-31.27
Locher12.10.008751.21.05×10-73.87×10-50.0283
最优LocherLocherLocherLocherLocherLocher
最差同轴Strub同轴Strub，AbtAbt异轴StrubAbt异轴Strub
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不同形式齿轨车辆动态特性研究
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陈兆玮 1 , 袁密奥 1 , 李世辉 1 , 易虹雨 1 , 杨吉忠 2 , 陈志辉 2
振动工程学报 | 2024,37(9): 1564-1574
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振动工程学报 | 2024, 37(9): 1564-1574
不同形式齿轨车辆动态特性研究
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陈兆玮1 , 袁密奥1 , 李世辉1, 易虹雨1, 杨吉忠2, 陈志辉2
作者信息
  • 1重庆交通大学机电与车辆工程学院, 重庆　400074
  • 2中铁二院工程集团有限责任公司科学技术研究院, 四川　成都　610031

    • 陈兆玮（1988—），男，博士，副教授。E-mail： 。

通讯作者:

袁密奥（1998—），男，硕士研究生。E-mail： 。
Dynamic characteristics of different rack vehicles
Zhao-wei CHEN1 , Mi-ao YUAN1 , Shi-hui LI1, Hong-yu YI1, Ji-zhong YANG2, Zhi-hui CHEN2
Affiliations
  • 1School of Mechanotronics and Vehicle Engineering, Chongqing Jiaotong University, Chongqing 400074, China
  • 2Science and Technology Research Institute, China Railway Eryuan Engineering Group Co., Ltd., Chengdu 610036, China
出版时间: 2024-09-28 doi: 10.16385/j.cnki.issn.1004-4523.2024.09.013
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为了提高轨道车辆的爬坡能力，齿轨车辆在传统轨道车辆的基础上加入了齿轮-齿轨啮合系统。针对目前世界上的齿轨铁路系统种类较多，其配备齿轮-齿轨啮合系统多样性所导致的齿轨车辆动态特性差异的问题，本文在解析齿轮-齿轨啮合激扰产生机理的基础上，考虑齿轮-齿轨啮合冲击，建立了两类Strub系统、双排齿Abt系统以及Locher系统四种不同形式的齿轨车辆动力学模型，并对模型进行了实验验证；采用所建立的模型，分析了齿轨车辆以不同速度在坡道啮合路段运行时的齿轮-齿轨啮合行为，并探究了线路不平顺对齿轨车辆齿轮-齿轨啮合中心距误差的影响；在此基础上研究了齿轨车辆的轮轨作用以及车辆加速度，由此分析了齿轨车辆的安全性与平稳性。结果表明：不同形式的齿轨车辆动态特性存在显著差异，Locher系统齿轨车辆动态特性最佳；同轴Strub系统与双排齿Abt系统的齿轮啮合行为较差且受线路不平顺影响明显，齿轮-齿轨啮合力最大冲击值达20.3 kN，啮合中心距误差达3.73 cm；同轴Strub系统与双排齿Abt系统车辆安全性较差，双排齿Abt系统的轮轨垂向力最大达51.7 kN；异轴Strub系统的车辆平稳性最差，最大车体垂向加速度为0.033 m/s2，平稳性指标为1.27。

齿轨车辆  /  车辆动力学  /  齿轨铁路系统  /  齿轮-齿轨啮合  /  动力学性能

In order to improve the climbing ability of rack vehicles，the gear-rack system is added to the traditional rail vehicles. Aiming at the problem that there are many kinds of gear-rack systems in the world at present，and the diversity of gear-rack systems equipped with them leading to the dynamic characteristics difference of rack vehicles，this paper considers the impact of gear-rack meshing on the basis of analyzing the generation mechanism of gear-rack meshing excitation，the rack vehicle coupled dynamic models with two kinds of Strub system，double row teeth Abt system and Locher system are established，and experimental verification on the model are carried out； Based on the model，the gear-rack meshing behavior of rack vehicle running at different speeds on the engagement section of the ramp is analyzed，and the influence of track irregularities on the gear-rack meshing center distance error is explored； On this basis，the wheel/rail action and car body acceleration of the rack vehicle are studied，and the rack vehicle safety are analyzed as well as stabitity. The results show that there are significant differences in the dynamic characteristics of rack vehicles with different gear-rack systems，and the Locher system has the best dynamic characteristics； The gear-rack meshing behavior of coaxial Strub system and double row teeth Abt system is poor and affected by the track irregularity obviously. The maximum impact value of the gear-rack contact force is 20.3 kN，and the meshing center distance error is 3.73 cm； The safety of coaxial Strub system and double row teeth Abt system is poor，and the maximum wheel/rail vertical force of double row teeth Abt system is 51.7 kN； The car body stability of the differential shaft Strub system is the worst. The maximum car body acceleration is 0.033 m/s2，and the stability index is 1.27. The conclusions offer theoretical support for the design，safe operation，and maintenance of mountain rack railways in China.

rack vehicle  /  vehicle dynamics  /  gear-rack system  /  gear-rack meshing  /  dynamic characteristics
陈兆玮, 袁密奥, 李世辉, 易虹雨, 杨吉忠, 陈志辉. 不同形式齿轨车辆动态特性研究. 振动工程学报, 2024 , 37 (9) : 1564 -1574 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.09.013
Zhao-wei CHEN, Mi-ao YUAN, Shi-hui LI, Hong-yu YI, Ji-zhong YANG, Zhi-hui CHEN. Dynamic characteristics of different rack vehicles[J]. Journal of Vibration Engineering, 2024 , 37 (9) : 1564 -1574 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.09.013
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1　概述
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齿轨铁路的应用至今已经有了100多年的历史，近年来，国外有许多新建与改建的齿轨铁路与传统铁路结合在一起，用于交通运输以及旅游观光［1-2］。在重庆大足建立的齿轨铁路首次载人测试已经完成，主要依托都江堰-四姑娘山山地轨道交通项目进行编制的四川省地方标准《山地（齿轨）轨道交通技术规范》已经颁布［3］。现今齿轨铁路系统形式较多，所配备的齿轨车辆的驱动结构具有多样性［4］，齿轨车辆动态特性也有可能会由此产生一定差异。因此，有必要对不同形式的齿轨车辆进行动态特性分析。目前已知的齿轨铁路系统形式主要为Marsh，Riggenbach，Abt，Strub，Locher，Margon，Vonroll七种。Marsh，Riggenbach，Margon，Strub，Vonroll齿轨铁路系统主要差别体现在齿轨铁路上，而齿轨车辆驱动齿轮均为垂向啮合单一渐开线齿轮［5］，与车辆上固定横轴铰接，车辆结构相似，其中Strub应用最为广泛，又分为驱动齿轮与车轴或构架相连的同轴Strub齿轨车辆和异轴Strub齿轨车辆。此外，Abt齿轨系统为多个齿轮交错并排连接，实现与下方多个齿条交错啮合，Locher齿轨系统为两个齿轮水平对称设置，与中间齿轨横向啮合。因此本研究选择对同轴Strub齿轨系统、Abt齿轨系统、异轴Strub齿轨系统以及Locher齿轨系统展开动力学分析，几种齿轨结构如图1所示。
齿轨车辆的牵引与制动主要靠齿轮啮合实现，而对于齿轮啮合行为相关的研究已经十分成熟，其中蒋建政等［6］基于齿轮啮合刚度解析计算方法、SIMPACK225号力元以及有限元法建立了三种不同时变啮合刚度计算模型，对比分析了不同速度工况下的直齿圆柱齿轮的啮合力以及动态传递误差。Yu等［7］以动态分形齿隙为内激励建立了包含表面形貌和齿轮中心运动的齿轮动力学模型，分析了齿侧间隙对齿轮副的影响。Neusser等［8］基于多体动力学建立了齿轮传动系统模型，计算了不同齿面的齿轮系统传动误差并对结构噪声进行了预测。陈春俊等［9］以时变啮合刚度与轨道不平顺为激励，探究了在变载荷作用下机车传动齿轮的振动特性。夏茂浩等［10］分析了载荷因素与安装误差对小模数螺旋锥齿轮啮合特性的影响。康博强等［11］建立了5自由度弯-扭-轴耦合传动系统动力学模型，在考虑齿轮齿面摩擦的基础上对斜齿轮系统展开了动力学特性分析。可以得知齿轮啮合系统是一个较为复杂的耦合动力学系统，其啮合行为受传动结构影响较为显著。
目前国内外学者主要对齿轨车辆相关领域做出了以下相关研究，陈双喜等［12］通过SIMPACK软件建立了单节齿轨车辆动力学模型，对比分析了不同齿轮参数以及驱动方式对山地齿轨车辆动力学特性的影响。张乾等［13］基于有限元理论，建立了齿轨（钢轨）-轨枕-桥梁-墩台空间耦合计算模型，分析了在典型荷载作用下的齿轨铁路轨道-简支梁桥相互作用，并对轨缝位置提出了建议。Chen等［14］通过建立齿轨车辆耦合动力学模型对比分析了不同电机布置模式下的齿轨车辆动态特性。赵冠闯等［15］分析了转动惯量以及重心高度对齿轨车辆动力学性能的影响。井国庆等［16］介绍了不同齿轨铁路系统及其轨下基础结构，从工艺性以及经济性上提出了齿轨铁路系统的选取考虑因素。Seo等［17］设计并通过实验开发了带有弹性齿轮的齿轨车辆转向架，并对其动力学性能进行了评估。索蓓蓓等［18］分析了采煤机齿轨轮和销轨在不同齿轨轮节距以及中心距工况下的齿轨轮-销轨啮合特性。上述研究从多个角度分析了齿轨车辆的齿轮-齿轨啮合行为以及车辆动态特性，可以证明齿轮啮合激扰对车辆动态特性有一定影响。
为探究不同形式齿轨车辆的动态特性，首先，本文在解析齿轮-齿轨啮合激扰产生机理的基础上，根据齿轮-齿轨啮合方式建立了齿轮-齿轨啮合模型；其次，考虑不同形式齿轨车辆的驱动结构以及功能特点，建立了齿轨车辆动力学模型，并对模型进行了实验对比验证；然后，采用所建立的模型，计算了不同齿轨系统的齿轨车辆齿轮动态接触力，探究齿轮-齿轨啮合系统动态特性，并展开了线路不平顺下的齿轮-齿轨啮合中心距误差分析；最后，研究了不同齿轨系统齿轨车辆的动态响应，在此基础上对比分析齿轨车辆的运行安全性与舒适性，并给出了最佳齿轨铁路系统。
2　齿轨车辆系统模型
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本节将简要阐述齿轨车辆的结构，首先建立了齿轮-齿轨啮合模型，然后根据驱动齿轮的布置方式建立齿轨车辆模型，并在此基础上对齿轨车辆进行数学与力学表达，形成齿轨车辆动力学模型。
2.1　齿轮-齿轨啮合模型
本节通过齿轮动力学，在不考虑齿侧间隙等时变因素影响的情况下，根据不同齿轨铁路系统的齿轮-齿轨啮合驱动装置结构和功能特点，将齿轮-齿轨系统进行局部细节简化，建立两种Strub系统与双排齿Abt系统对应的五自由度和Locher系统对应的八自由度齿轮-齿轨啮合模型，如图2（a），（b）所示，其中y轴为啮合线方向。
驱动齿轮以弹性旋转铰接的方式安装在齿轨车辆转向架零部件上。Ig，Tg，θg分别为驱动齿轮的转动惯量、转矩和扭转角；Kxg与Cxg分别为齿轮与转向架弹性联结的纵向刚度和阻尼；Kzg和Czg分别为齿轮与转向架弹性联结的垂向（Locher齿轨系统模式下为横向）刚度和阻尼；为驱动齿轮基圆半径；Kxh，Cxh，Kzh，Czh分别为齿轨与轨道板的纵向和垂向扣件刚度和阻尼。Kt，Ct分别为传动系统的啮合刚度与啮合阻尼：
式中　 δg，δh分别为驱动齿轮与齿轨弹性变形量；Kg，Kh分别为驱动齿轮与齿轨单齿刚度；εg为轮齿啮合的阻尼比。
根据牛顿运动定律建立动力学微分方程如下：
驱动齿轮运动方程为：
五自由度齿条运动方程为：
八自由度齿条运动方程为：
式中　 Zg，Xg分别为驱动齿轮在垂向与纵向上的位移；Zh，Xh分别为驱动齿轨垂向与纵向的位移；rbg为驱动齿轮的基圆半径；Fdpi，Ffpi分别为齿轮-齿轨啮合的动态接触力与动态摩擦力；为驱动齿轮质量；为齿轨质量。
此外，齿轮-齿轨啮合受线路激励以及转向架自身振动等误差激励的影响，使得驱动齿轮在进入和退出啮合时，啮入啮出点的位置偏离理论啮合点，产生啮合误差［19］，由此产生啮入冲击。
式中　 Tz为理论啮入时间，Tz=2π/（z1ω1），z1为驱动齿轮齿数，ω1为驱动齿轮转动角速度；t0为实际的啮入时间。
一般情况下，Δt约为Tz的5%~10%，可得出任意t时刻的啮入冲击力：
式中　 为后一轮齿进入啮合前，前一轮齿的动态接触力；K1（t*），K2（t*）分别为驱动齿轮与齿轨在t*时刻啮合点的齿节刚度；Δφ为驱动齿轮因啮入点的变化引起的啮合角的变化量；Δγ为在相应情况下驱动齿轮与齿轨啮合角度变化量之和。
2.2　齿轨车辆耦合动力学模型
详细考虑不同齿轨铁路系统的齿轨车辆结构，并可将齿轨车辆各零部件模拟为多刚体系统的一部分［20］。然后在齿轮-齿轨啮合模型的基础上，将其所受的载荷加入车辆耦合动力学模型。所得齿轨车辆系统动力学模型如图3（a）所示，图中，，分别为车体、构架质量；，分别为车体、构架的点头转动惯量；，分别为转向架一侧二系悬挂垂向刚度、垂向阻尼；，分别为每轴箱一系悬挂垂向刚度、垂向阻尼；为构架的点头角度；，，和分别为车体、构架、轮对和钢轨的垂向位移；，，和分别为第一轮对至第四轮对与钢轨间的轮轨力。根据各齿轨系统的齿轨车辆走行系统的装置结构和功能特点的区别，将其分为与车轴相连的同轴Strub系统和双排齿Abt系统，与构架相连的异轴Strub系统和Locher系统，基于此分别建立各走行系统的耦合动力学模型如图3（b）所示。图3（c）为同轴Strub与双排齿Abt系统走行部的齿轮-齿轨啮合部分模型端视图，其中同轴Strub为单一渐开线齿轮齿条，而双排齿Abt系统齿轮结构由两排齿盘交错排列，齿轨结构由两根实心齿条并排组成，错动量与齿轮保持一致。模型自由度为47，如表1所示。
根据图2和表1可以看出，齿轨车辆动力学模型与传统客车的主要区别体现在走行系统部分，其中同轴Strub系统、双排齿Abt系统差异表现在轮对运动方程上，异轴Strub系统、Locher系统差异表现在构架运动方程上，其余各部件运动方程一致。因此依据达朗贝尔原理对与传统客车相差异的各部件建立运动微分方程。
2.2.1　同轴Strub系统、双排齿Abt系统轮对运动方程(i=1~4)
垂向运动：
横向运动：
侧滚运动：
摇头运动：
点头运动：
式中　 Mw为轮对质量；Ywi为轮对横向位移；FxfLi和FxfRi分别为一系悬挂左、右纵向力；FyfLi和FyfRi分别为一系悬挂左、右横向力；FzfLi和FzfRi分别为一系悬挂左、右垂向力；Iwx，Iwy和Iwz分别为轮对侧滚、点头、摇头转动惯量；Igx，Igz分别为驱动齿轮侧滚、摇头转动惯量；ϕwi，βwi和ψwi分别为轮对侧滚、点头、摇头角度；MLzi和MRzi分别为第i轮对左轮、右轮所受蠕滑力矩在z轴上的分量；MLyi和MRyi分别为第i轮对左轮、右轮所受蠕滑力矩在y轴上的分量；FLxi，FRxi，FLyi，FRyi，FLzi，FRzi分别为第i轮对左、右轮所受蠕滑力在x轴、y轴和z轴上的分量；NLxi，NRxi，NLyi，NRyi，NLzi，NRzi分别为第i轮对左、右轮所受法向力在x轴、y轴和z轴上的分量；ϕsewi，Rwi分别为轨道线路的超高角、曲率半径；r0为车轮的名义滚动圆半径；a0为左右轮接触点定距的一半；Fgz和Fgy为驱动齿轮传递到车轴上沿z轴和y轴方向的力； v为车辆运行速度；g为重力加速度；Ω为轮对的名义滚动圆角度；rLi，rRi分别为第i对轮对左、右轮的滚动半径；dw为一系悬挂横向距离的一半。
2.2.2　异轴Strub系统构架运动方程(i=1,2)
垂向运动：
横向运动：
侧滚运动：
摇头运动：
点头运动：
2.2.3　Locher系统构架运动方程(i=1,2)
垂向运动：
横向运动：
侧滚运动：
摇动运动：
点头运动：
式中　 Yti为构架横向位移；FxtLi和FxtRi分别为二系悬挂左、右纵向力；FytLi和FytRi分别为二系悬挂左、右横向力；FztLi和FztRi分别为二系悬挂左、右垂向力；Igy为驱动齿轮点头转动惯量；Itx，Itz分别为构架侧滚、摇头转动惯量；ϕti，ψti分别为构架测滚、摇头角度；Hg为构架质心与驱动齿轮中心的垂向距离；dsc为抗蛇行减振器横向距离的一半；FxsLi，FxsRi分别为抗蛇形减振器左、右侧纵向力；ϕseti，Rti分别为轨道上第i转向架中心所对应的外轨超高角、曲率半径；Htw为构架质心与轮对中心线的垂向距离；HBt为构架质心至二系悬挂下平面的距离；Mri为抗侧滚力矩；lt为转向架轮对定距的一半；ds为二系悬挂横向距离的一半；FgxL，FgxR，FgyL，FgyR，FgzL与FgzR分别为左、右侧驱动齿轮沿纵向、横向以及垂向传递到构架上的力； Hg为构架质心与驱动齿轮中心的垂向距离；lg为前后驱动齿轮纵向距离的一半；dg为左右驱动齿轮横向距离的一半。
2.2.4　车体等各部件运动方程
齿轨车辆其余部件与传统客车类似，可以作为多刚体系统来考虑。根据车辆耦合动力学和齿轮动力学可推得齿轨车辆运动方程如下：
式中　 M，C，K分别为整个车辆系统的广义质量矩阵、广义阻尼矩阵与广义刚度矩阵；，，X分别为车辆的广义加速度、速度、位移矩阵；P为车辆的广义载荷向量。详细推导过程及运动方程可参考文献［21-22］。
2.3　数值积分求解方法
本文建立的四种齿轨系统的齿轨车辆耦合动力学模型是基于多体动力学建立的，这是一个大型复杂的非线性动力学微分方程组。本文为提高运算求解速度，采用翟婉明所提出的新型快速显式积分法（翟方法）对车辆模型进行求解，详细计算推导与迭代过程参考文献［20，23］。
2.4　系统模型验证
为进一步验证齿轨车辆模型的准确性，本文于重庆大足某齿轨线路设计并实施了10%坡度Strub齿轨系统齿轨车辆运行实验测试。针对齿轮接触力与车体加速度进行了测试，部分实测现场如图4（a）所示。
将实验测试与模型计算所得结果进行对比，如图4（b），（c）所示。可以看出模型计算与实验测试结果相比，齿轮接触力与车体加速度的计算结果均要略小于实验测度结果，但实验测试结果与模型的计算结果基本接近，因此可以认为模型是准确可行的。
3　工况参数
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本节将对不同齿轨系统的齿轨车辆的运行工况与相关参数进行简要说明。
本文所选取的齿轨铁路系统为以下三种：Strub，Abt以及Locher齿轨铁路系统，其中Strub又分为同轴和异轴Strub齿轨系统。
为了更清晰地研究不同驱动系统的齿轨车辆动力学特性，本文只在齿轮啮合误差处设置了美国五级谱线路不平顺。选择了几种工况进行对比分析，具体工况如表2所示。另外，齿轨车辆在传统列车基础上增加了齿轮啮合系统，且各结构尺寸较小［24］，其主要车辆参数如表3所示。
4　不同齿轨车辆齿轮-齿轨啮合行为
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本节对四种齿轨系统以20 km/h在100‰的坡度上运行时的齿轮接触力展开分析，并探究了冲击较大的Strub系统与Abt系统的振动特性。另外，为确定线路激励对齿轮-齿条非线性啮合的影响，分析了美国五级谱线路不平顺对齿轮-齿条中心距的影响。
图5为不同齿轨系统的齿轨车辆以20 km/h在100‰的坡度上运行时驱动齿轮的动态接触力曲线，图6和7为在啮合过程中的齿轮垂向（Locher为横向）位移和加速度及各自频域分析曲线。
从图中可以看出三种垂向啮合的齿轨系统在啮合过程中冲击现象较明显，其中与车轴相连的同轴Strub系统和双排齿Abt系统冲击较大，最大达到20.2 kN。Locher系统齿轮动态接触力较为平稳，齿轮接触力保持在11 kN左右。结合图6与7可以得知Locher系统的驱动齿轮振动现象不明显，而同轴Strub系统、双排齿Abt系统与异轴Strub系统驱动齿轮振动规律相似，处于周期运动当中。
图8为齿轨车辆在美国五级谱线路不平顺的工况下运行时齿轮-齿轨系统的啮合中心距误差曲线。可以看出，垂向布置的驱动齿轮受线路不平顺影响更加明显，其中安装在车轴上的同轴Strub系统与双排齿Abt系统啮合中心距误差更大，最大达到0.0373 m。Locher系统驱动齿轮为横向布置，其啮合中心距误差主要受左右不平顺影响，因此啮合中心距误差较小。
5　不同齿轨车辆的动态特性
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本节在考虑齿轮-齿轨啮合行为的基础上，研究四种齿轨系统的齿轨车辆在100‰~200‰坡度坡道上以不同速度运行时的轮轨垂向力和车体加速度，并以轮重减载率与平稳性指标为参考对比分析四种齿轨系统的齿轨车辆动力学性能。
图9为齿轨车辆以20 km/h运行速度在100‰坡道上行驶时的轮轨垂向力时域曲线以及不同运行速度下的轮轨垂向力最大值。
可以看出Locher系统轮轨垂向力几乎没有变化。双排齿Abt系统的齿轨车辆轮轨垂向力波动较大且存在更多冲击，最大值为51.7 kN，其次分别为同轴Strub系统、异轴Strub系统，且轮轨垂向力最大值随速度的增加呈增长趋势。
图10为齿轨车辆以20 km/h运行速度在100‰坡度上行驶时的车体垂向加速度及其频域分析曲线。图中可以看出异轴Strub系统的车体加速度远大于另外几种齿轨系统，最大加速度为16.1 mm/s2，其主要影响频率为1.92 Hz。
图11为不同速度下的齿轨车辆在100‰坡道运行时的轮重减载率以及车体垂向平稳性指标。可以看出随着速度的增加，轮重减载率与平稳性指标逐渐增大。其中同轴Strub系统与双排齿Abt系统轮重减载率较大，最大值达到0.0131；异轴Strub系统平稳性指标较大，最大为1.27。
图12为齿轨车辆以20 km/h运行速度在不同坡度坡道上运行时的轮重减载率以及车体垂向平稳性指标。可以看出随着坡度的增加，轮重减载率与平稳性指标逐渐增大。
可以认为同轴Strub系统与双排齿Abt系统齿轨车辆安全性较差，而异轴Strub系统齿轨车辆平稳性较差，Locher齿轨系统动力学性能较好。
表4为不同齿轨系统的齿轨车辆以不同速度在100‰坡道上运行时的各动态特性指标，并在表格后方列出了各项指标的最优值与最差值所对应的齿轨铁路系统。通过对比分析可以看出同轴Strub系统与双排齿Abt系统的齿轨车辆齿轮啮合特性和运行安全性较差，而异轴Strub系统齿轨车辆运行平稳性较差，Locher系统齿轨车辆动态特性各指标最佳。
6　结论
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本研究针对齿轨系统对车辆动态特性的影响，在解析齿轮-齿轨非线性啮合激扰的产生机理以及考虑齿轮-齿轨啮合冲击的基础上，建立了齿轨车辆系统耦合动力学模型。研究了爬坡路段下，同轴Strub齿轨系统、Abt双排齿轨系统、异轴Strub齿轨系统以及Locher齿轨系统驱动的齿轨车辆齿轮-齿轨非线性啮合行为以及车辆动态响应，并分析了齿轨车辆系统动力学性能。从以上研究中得出以下结论：
（1）该方法对研究不同齿轨铁路系统齿轨车辆动态特性是有效的，不同齿轨系统的齿轨车辆动态特性有显著差异。Locher系统的齿轨车辆动态特性最佳。
（2）针对本文所建立模型及采用的方法，同轴Strub系统与双排齿Abt系统齿轨车辆齿轮-齿轨啮合特性较差。同轴Strub齿轮接触力最大值为20.2 kN，线路不平顺作用下啮合误差达到0.037 m。
（3）针对本文所建立模型及采用的方法，同轴Strub系统与双排齿Abt系统齿轨车辆运行安全性较差，双排齿Abt系统轮重减载率最大为0.013；异轴Strub系统的运行平稳性较差，最大车体加速度为0.033 m/s2，平稳性指标为1.27。
基金
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  • 国家自然科学基金资助项目(52008067)
  • 重庆市自然科学基金资助项目(CSTB2022NSCQ-MSX1193)
  • 重庆市教育委员会科学技术研究项目(KJZD-M202300701)
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2024年第37卷第9期
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doi: 10.16385/j.cnki.issn.1004-4523.2024.09.013
  • 接收时间：2022-10-15
  • 首发时间：2026-02-12
  • 出版时间：2024-09-28
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  • 收稿日期：2022-10-15
  • 修回日期：2023-04-15
基金
国家自然科学基金资助项目(52008067)
重庆市自然科学基金资助项目(CSTB2022NSCQ-MSX1193)
重庆市教育委员会科学技术研究项目(KJZD-M202300701)
作者信息
    1重庆交通大学机电与车辆工程学院, 重庆　400074
    2中铁二院工程集团有限责任公司科学技术研究院, 四川　成都　610031

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袁密奥（1998—），男，硕士研究生。E-mail： 。
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2种不同金属材料的力学参数
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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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