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Article(id=1228046470142689824, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228046469559681568, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2024.02.001, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1652371200000, receivedDateStr=2022-05-13, revisedDate=1655568000000, revisedDateStr=2022-06-19, acceptedDate=null, acceptedDateStr=null, onlineDate=1770718694481, onlineDateStr=2026-02-10, pubDate=1709049600000, pubDateStr=2024-02-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770718694481, onlineIssueDateStr=2026-02-10, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770718694481, creator=13701087609, updateTime=1770718694481, updator=13701087609, issue=Issue{id=1228046469559681568, tenantId=1146029695717560320, journalId=1225147924628267009, year='2024', volume='37', issue='2', pageStart='191', pageEnd='364', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770718694343, creator=13701087609, updateTime=1770795432451, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1228368332575928712, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228046469559681568, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1228368332575928713, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228046469559681568, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=191, endPage=199, ext={EN=ArticleExt(id=1228046470339822114, articleId=1228046470142689824, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Dynamic characteristics of planetary gear transmission system with tooth profile modification, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

In order to meet the requirements of high transmission efficiency and low noise in the planetary gear transmission system，the bending-torsion-shaft coupling power of helical planetary gear transmission system was established by employing the lumped parameter method，with considerations for eccentricity error and installation error. Using the fourth-order Runge-Kutta method to solve the dynamic equation，the dynamic characteristics of the planetary gear，such as the dynamic meshing force，dynamic transmission error，and dynamic load coefficient，were obtained. Based on this，the planetary gear modification research is carried out，and the tooth profile modification is established. Upon establishing the tooth surface equation and the dynamic equation that takes tooth surface modification into account，an analysis was conducted on the dynamic characteristics corresponding to varying degrees of modification. The research results show that with the increase of the modification amount，the dynamic meshing force，dynamic transmission error and dynamic load coefficient of the planetary gear all decrease to varying degrees，and then increase after reaching the lowest value. Performance testing of the entire machine revealed a reduction in vibration and noise of the planetary reducer，as well as an improvement in transmission performance providing theoretical support for the design of vibration reduction，noise reduction and transmission efficiency improvement of the planetary gear transmission system.

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基于对行星齿轮传动系统高传动效率和低噪音等要求，以行星齿轮传动系统为研究对象，采用集中参数法建立了考虑偏心误差和安装误差的斜齿行星齿轮传动系统的弯-扭-轴耦合动力学模型，利用四阶Runge-Kutta法求解动力学方程得到行星齿轮的动态啮合力、动态传动误差和动载系数等动态特性，并以此为评价指标展开行星齿轮的修形研究。建立了齿廓修形后的齿面方程和考虑齿面修形的动力学方程，研究不同修形量的行星齿轮传动系统动态特性变化。研究结果表明，随着修形量的增大，行星齿轮的动态啮合力、动态传动误差和动载系数都有不同程度的降低，降到最低值后又开始增大，最终确定齿轮的合理修形量。经过整机性能测试发现合理修形下的行星减速器振动噪声降低、传动性能提升，为行星齿轮传动系统的减振降噪、传动效率提升设计提供了理论支撑。

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莫 帅（1987—），男，博士，教授，博士生导师。E-mail： 。

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莫 帅（1987—），男，博士，教授，博士生导师。E-mail： 。

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莫 帅（1987—），男，博士，教授，博士生导师。E-mail： 。

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Chinese Journal of Mechanical Engineering， 2015， 28（6）： 1194-1203., articleTitle=Statistical modification analysis of helical planetary gears based on response surface method and Monte Carlo simulation, refAbstract=null)], funds=[Fund(id=1228046492443803886, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046470142689824, awardId=52265004, language=CN, fundingSource=国家自然科学基金资助项目(52265004), fundOrder=null, country=null), Fund(id=1228046492515107059, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046470142689824, awardId=2018QNRC001, language=CN, fundingSource=中国科协青年人才托举工程资助项目(2018QNRC001), fundOrder=null, country=null), Fund(id=1228046492582215930, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046470142689824, awardId=HTL-0-21G07, language=CN, fundingSource=直升机传动技术国防科技重点实验室开放基金资助项目(HTL-0-21G07), fundOrder=null, country=null), Fund(id=1228046492649324800, tenantId=1146029695717560320, 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Design parameters of planetary gear transmission system

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参数太阳轮行星轮内齿圈
齿数303090
模数/mm0.850.850.85
压力角/(°)202020
螺旋角/(°)121212
齿宽/mm171618
质量/kg0.0480.0480.752
转动惯量/(kg·m2)5.865.860.002
), ArticleFig(id=1228046491575582905, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046470142689824, language=CN, label=表1, caption=

行星齿轮传动系统设计参数

, figureFileSmall=null, figureFileBig=null, tableContent=
参数太阳轮行星轮内齿圈
齿数303090
模数/mm0.850.850.85
压力角/(°)202020
螺旋角/(°)121212
齿宽/mm171618
质量/kg0.0480.0480.752
转动惯量/(kg·m2)5.865.860.002
), ArticleFig(id=1228046491667857599, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046470142689824, language=EN, label=Tab.2, caption=

Variation of dynamic meshing forces of planetary gear transmission system

, figureFileSmall=null, figureFileBig=null, tableContent=
外啮合(s-p)动态啮合力/N内啮合(r-p)动态啮合力/N
均值最大值极差均值最大值极差
下降率10.33%12.54%19.94%16.28%14.13%11.52%
修形前233230381550248830641163
修形后209126571241208326311029
), ArticleFig(id=1228046491755937985, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046470142689824, language=CN, label=表2, caption=

行星齿轮传动系统动态啮合力变化

, figureFileSmall=null, figureFileBig=null, tableContent=
外啮合(s-p)动态啮合力/N内啮合(r-p)动态啮合力/N
均值最大值极差均值最大值极差
下降率10.33%12.54%19.94%16.28%14.13%11.52%
修形前233230381550248830641163
修形后209126571241208326311029
), ArticleFig(id=1228046491860795590, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046470142689824, language=EN, label=Tab.3, caption=

Variation of dynamic transmission errors of planetary gear transmission system

, figureFileSmall=null, figureFileBig=null, tableContent=
外啮合(s-p)动态传动误差/μm内啮合(r-p)动态传动误差/μm
均值最大值极差均值最大值极差
下降率10.49%16.54%28.70%16.29%19.64%30.63%
修形前15.6420.439.8610.513.755.68
修形后1417.057.038.7911.053.94
), ArticleFig(id=1228046491923710157, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046470142689824, language=CN, label=表3, caption=

行星齿轮传动系统动态传动误差变化

, figureFileSmall=null, figureFileBig=null, tableContent=
外啮合(s-p)动态传动误差/μm内啮合(r-p)动态传动误差/μm
均值最大值极差均值最大值极差
下降率10.49%16.54%28.70%16.29%19.64%30.63%
修形前15.6420.439.8610.513.755.68
修形后1417.057.038.7911.053.94
), ArticleFig(id=1228046492032762071, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046470142689824, language=EN, label=Tab.4, caption=

Variation of load coefficients of planetary gear transmission system

, figureFileSmall=null, figureFileBig=null, tableContent=
外啮合(s-p)动载系数Kv内啮合(r-p)动载系数Kv
均值最大值极差均值最大值极差
下降率10.35%12.53%19.92%16.25%32%11.49%
修形前0.71230.92790.47350.75990.93590.3552
修形后0.63860.81160.37920.63640.80370.3144
), ArticleFig(id=1228046492133425372, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046470142689824, language=CN, label=表4, caption=

行星齿轮传动系统动载系数变化

, figureFileSmall=null, figureFileBig=null, tableContent=
外啮合(s-p)动载系数Kv内啮合(r-p)动载系数Kv
均值最大值极差均值最大值极差
下降率10.35%12.53%19.92%16.25%32%11.49%
修形前0.71230.92790.47350.75990.93590.3552
修形后0.63860.81160.37920.63640.80370.3144
), ArticleFig(id=1228046492213117153, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046470142689824, language=EN, label=Tab.5, caption=

Test data of prototype performance

, figureFileSmall=null, figureFileBig=null, tableContent=
测试台数噪声/dB负载扭矩/(N·m)
修形前修形后修形前修形后
162.554264.6291.1
26354.5258.7289.4
36555263.3292.3
46453259.8287.6
563.555261.3290.8
), ArticleFig(id=1228046492330557674, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228046470142689824, language=CN, label=表5, caption=

样机性能测试数据

, figureFileSmall=null, figureFileBig=null, tableContent=
测试台数噪声/dB负载扭矩/(N·m)
修形前修形后修形前修形后
162.554264.6291.1
26354.5258.7289.4
36555263.3292.3
46453259.8287.6
563.555261.3290.8
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考虑齿廓修形的行星齿轮传动系统动力学特性研究
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莫帅 1, 4 , 刘志鹏 2 , 罗炳睿 2 , 岑国建 3 , 徐家科 3
振动工程学报 | 2024,37(2): 191-199
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振动工程学报 | 2024, 37(2): 191-199
考虑齿廓修形的行星齿轮传动系统动力学特性研究
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莫帅1, 4 , 刘志鹏2, 罗炳睿2, 岑国建3, 徐家科3
作者信息
  • 1广西大学机械工程学院，广西　南宁　530004
  • 2天津工业大学机械工程学院，天津　300387
  • 3宁波中大力德智能传动股份有限公司，浙江　宁波　315301
  • 4华中科技大学数字制造装备与技术国家重点实验室，湖北　武汉　430074

    • 莫 帅（1987—），男，博士，教授，博士生导师。E-mail： 。

Dynamic characteristics of planetary gear transmission system with tooth profile modification
Shuai MO1, 4 , Zhi-peng LIU2, Bing-rui LUO2, Guo-jian CEN3, Jia-ke XU3
Affiliations
  • 1School of Mechanical Engineering，Guangxi University，Nanning 530004，China
  • 2School of Mechanical Engineering， Tiangong University，Tianjin 300387，China
  • 3Ningbo Zhongda Leader Transmission Equipment Co.，Ltd.，Ningbo 315301，China
  • 4State Key Laboratory of Digital Manufacturing Equipment and Technology，Huazhong University of Science and Technology，Wuhan 430074，China
出版时间: 2024-02-28 doi: 10.16385/j.cnki.issn.1004-4523.2024.02.001
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摘要
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基于对行星齿轮传动系统高传动效率和低噪音等要求，以行星齿轮传动系统为研究对象，采用集中参数法建立了考虑偏心误差和安装误差的斜齿行星齿轮传动系统的弯-扭-轴耦合动力学模型，利用四阶Runge-Kutta法求解动力学方程得到行星齿轮的动态啮合力、动态传动误差和动载系数等动态特性，并以此为评价指标展开行星齿轮的修形研究。建立了齿廓修形后的齿面方程和考虑齿面修形的动力学方程，研究不同修形量的行星齿轮传动系统动态特性变化。研究结果表明，随着修形量的增大，行星齿轮的动态啮合力、动态传动误差和动载系数都有不同程度的降低，降到最低值后又开始增大，最终确定齿轮的合理修形量。经过整机性能测试发现合理修形下的行星减速器振动噪声降低、传动性能提升，为行星齿轮传动系统的减振降噪、传动效率提升设计提供了理论支撑。

行星齿轮传动系统  /  齿轮修形  /  动态啮合力  /  动态传动误差

In order to meet the requirements of high transmission efficiency and low noise in the planetary gear transmission system，the bending-torsion-shaft coupling power of helical planetary gear transmission system was established by employing the lumped parameter method，with considerations for eccentricity error and installation error. Using the fourth-order Runge-Kutta method to solve the dynamic equation，the dynamic characteristics of the planetary gear，such as the dynamic meshing force，dynamic transmission error，and dynamic load coefficient，were obtained. Based on this，the planetary gear modification research is carried out，and the tooth profile modification is established. Upon establishing the tooth surface equation and the dynamic equation that takes tooth surface modification into account，an analysis was conducted on the dynamic characteristics corresponding to varying degrees of modification. The research results show that with the increase of the modification amount，the dynamic meshing force，dynamic transmission error and dynamic load coefficient of the planetary gear all decrease to varying degrees，and then increase after reaching the lowest value. Performance testing of the entire machine revealed a reduction in vibration and noise of the planetary reducer，as well as an improvement in transmission performance providing theoretical support for the design of vibration reduction，noise reduction and transmission efficiency improvement of the planetary gear transmission system.

planetary gear transmission system  /  gear modification  /  dynamic meshing force  /  dynamic transmission error
莫帅, 刘志鹏, 罗炳睿, 岑国建, 徐家科. 考虑齿廓修形的行星齿轮传动系统动力学特性研究. 振动工程学报, 2024 , 37 (2) : 191 -199 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.02.001
Shuai MO, Zhi-peng LIU, Bing-rui LUO, Guo-jian CEN, Jia-ke XU. Dynamic characteristics of planetary gear transmission system with tooth profile modification[J]. Journal of Vibration Engineering, 2024 , 37 (2) : 191 -199 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.02.001
正文
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引言
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行星齿轮传动系统因承载能力高、运转平稳而广泛应用于航空航天、汽车船舶、精密机器人等对传动稳定性和传动效率要求较高的智能装备领域。行星齿轮在传动过程中往往会产生振动噪声，引起传动精度降低等问题，使行星齿轮传动系统的性能下降，可通过对齿轮齿面微观修形来改善啮合性能［1］。影响齿轮传动系统动态特性的主要因素为动态传动误差，降低动态传动误差的大小和波动可以减小齿轮系统的振动和噪声。此外降低齿面动态啮合力和动载系数可改善齿轮受载情况、避免齿轮过早失效，提升齿轮使用寿命。
齿轮系统动力学是研究齿轮传动过程中产生动态激励的基础。相关学者对一般的圆柱直齿轮副和行星齿轮传动系统的动力学进行了研究。MO等［2-5］建立了人字齿轮、面齿轮和行星齿轮的静力学和动力学模型，研究了传动系统的载荷分配均匀性机理。杨通强等［6］建立了考虑误差和时变刚度的斜齿行星齿轮系统的平移-扭转耦合动力学模型。在齿轮修形方面，Bahk等［7］通过摄动分析研究了齿廓修形对行星齿轮动态响应的影响。魏静等［8］建立了考虑齿轮修形后刚度与误差的非线性动力学模型，分析了不同修形参数对啮合刚度和动态特性的影响。陈洪月等［9］建立了齿廓修形前后的传动误差数学模型，采用多种数值分析方法分析了修形参数对传动误差的影响。CHEN等［10-11］提出了考虑齿轮修形、承载变形和安装偏差的时变啮合刚度和传动误差的计算方法。
目前的文献多是针对单一齿轮副或是直齿轮展开修形研究，针对斜齿行星齿轮修形的动力学特性研究还较少。因此，本文构建了斜齿行星齿轮的弯-扭-轴耦合动力学模型，采用四阶Runge-Kutta法求解分析行星齿轮传动系统的动态特性。基于此，将齿廓修形耦合到动力学方程中，分析齿廓修形对动态啮合力、动态传动误差和动载系数的影响。
1　行星齿轮传动系统动力学建模
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1.1　行星齿轮传动系统结构设计
以智能装备用某型号精密行星减速器为研究对象，建立三维数字化样机，研究该系统的输出级行星轮系，其结构如图1所示。该传动系统以太阳轮输入、行星架输出，输入扭矩为18 N·m，输入转速为3000 r/min，太阳轮-行星轮、行星轮-内齿圈的平均啮合刚度Km分别为5×109 N/m，8×109 N/m，行星架的质量为0.621 kg，转动惯量为0.047 kg·m2。其他设计参数如表1所示。
采用集中参数法构建如图2所示的行星齿轮传动系统弯-扭-轴耦合动力学模型，xj，yj，zj为平移位移，uzj为扭转位移（j=s，r，c，pi），其中，s，r，c和pi（i=1，2，3）分别代表太阳轮、内齿圈、行星架和行星轮。
斜齿行星齿轮的动力学方程可以表示为：
式中　 M为质量矩阵；C为阻尼矩阵；K为刚度矩阵；x=［xs ys zs uzs xpi ypi zpi uzpi xr yr zr uzr xc yc zc uzc］T为广义坐标矩阵，其中，uzs，uzpi，uzr和uzc分别为太阳轮绕z轴、第j个行里轮、内齿圈和行星架的扭转线位移；F为外载荷矩阵。
1.2　斜齿轮副时变啮合刚度
根据Maatar等［12］提出的齿轮副接触线长度理论计算斜齿轮的时变啮合刚度为：
式中　 Km为平均啮合刚度；L（τ）为总接触线长度；τ=t/Tm，其中，t为时间，Tm为啮合周期。
L（τ）计算如下：
式中　 γsp=zsφn/（2π）和γrp=zrφn/（2π）分别为齿轮外啮合副和内啮合副的初相位，其中，φn=2π（n-1）/N为第n个行星轮的位置角，N为行星轮个数；k为傅里叶级数。
Ak，Bk和Lm分别计算如下：
式中　 εβ为齿轮副轴向重合度；εα为齿轮副端面重合度；βb为齿轮基圆螺旋角；b为齿宽。
1.3　传动系统啮合线等效位移
将各构件的平移和扭转位移向啮合线方向投影，得到各构件的等效位移如图3所示。
外啮合、内啮合的等效位移分别为：
式中　 ψspi=φsp-αpi和ψrpi=φrp-αpi分别为太阳轮-行星轮和行星轮-内齿圈的啮合角，其中，φsp和φrp分别为两者的位置角，αpi为行星轮端面压力角；，和分别为太阳轮、内齿圈和第j个行星轮的扭转角度；espi，erpi分别为外啮合、内啮合偏心误差和安装误差引起的综合偏差；rs为太阳轮分度圆半径；rr为内齿圈分度圆半径。
1.4　行星齿轮传动系统动力学方程
太阳轮平衡方程为：
式中　 ms和Isz分别为太阳轮的质量和转动惯量；ωc为行星架加速度；Ks，Ksz和Ksu分别为径向支撑刚度、轴向支撑刚度和扭转刚度，Cs，Csz和Csu为对应的阻尼；Fspi=Ksp·δspi为外啮合啮合力，Dspi=Csp·spi为其阻尼力，其中，Ksp为啮合刚度，Csp为啮合阻尼；为太阳轮的扭转位移；为太阳轮上的输入扭矩；β为螺旋角。
内齿圈平衡方程为：
式中　 mr和Irz分别为内齿圈的质量和转动惯量；Kr，Krz和Kru分别为径向支撑刚度、轴向支撑刚度和扭转刚度，Cr，Crz和Cru为对应的阻尼；Frpi=Krp·δrpi为行星轮与内齿圈的啮合力，Drpi=Crp·pi为其阻尼力，其中，Krp为啮合刚度，Crp为啮合阻尼；为内齿圈的扭转角度。
行星架平衡方程为：
式中　 mc和Ic分别为行星架的质量和转动惯量；和分别为第j个行星轮啮合线上位移沿x和y方向的投影；Kc和Kcz分别为径向支撑刚度和轴向支撑刚度，Cc和Ccz为对应阻尼；为行星架的扭转角度；为行星架上的输出扭矩；rc为行星架上滚柱与太阳轮的中心距离。
行星轮平衡方程为：
式中　 mp和Ipz分别为行星轮的质量和转动惯量；和分别为行星轮与太阳轮和内齿圈公切线垂线与两齿轮中心连线之间的夹角；Kcpix，Kcpiy和Kcpiz分别为径向和轴向支撑刚度，Ccpix，Ccpiy和Ccpiz为对应阻尼；表示第j个行星轮的扭转角加速度；rpi为行星轮分度圆半径。
利用四阶Runge-Kutta法求解上述方程得到修形前的行星齿轮传动系统动态特性如图4所示。其中，动载系数Kv是齿轮强度计算中考虑齿轮振动引起的啮合力增大的影响系数，ISO定义的计算公式为：Kv=F/F0，F为齿面动态啮合力，F0为静态啮合力，由静力学计算得到。由图4可见，外啮合最大啮合力达到了3038 N，均值为2332 N，内啮合最大啮合力达到了3064 N，均值为2488 N；外啮合动载系数极差为0.4735，内啮合动载系数极差为0.3552，可见修形前齿轮啮合力较高，齿面承受较大载荷，动载系数波动范围较大；而外啮合动态传动误差最大值为20.43 μm，极差为9.86 μm，内啮合动态传动误差最大值为13.75 μm，极差为5.68 μm，可见修形前动态传动误差波动较大，会引起较大的振动噪声，因此进行齿面微观修形显得尤为必要。
2　行星齿轮传动系统修形研究
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行星齿轮在制造、安装和受载时会产生误差和变形，导致行星齿轮传动误差和齿面载荷发生变化，造成齿轮系统的振动和噪声。齿轮微观修形可以改善齿面受载情况，降低齿轮传动误差的波动，从而有效抑制行星齿轮传动系统的振动和噪声。
2.1　修形齿面方程构建
对行星轮齿顶进行齿廓修形研究，采用齿轮啮合原理推导修形函数在啮合线上的相对位移，建立修形前后的齿面模型，如图5所示。图5齿廓曲线中实线为修形前的渐开线齿廓，虚线为修形后的渐开线齿廓，Camax为最大修形量，Lc为修形长度，M和M'分别为修形前、后渐开线齿廓的起点。建立修形函数［13-14］如下：
修形前的齿面方程为：
式中　 为渐开线展角和压力角之和。
齿廓修形后的渐开线齿廓如图5的红色虚线所示，修形齿廓的基圆半径变小，各点压力角也随即发生变化。
修形后的齿面方程为：
式中　 γs为齿轮绕z轴的转角；θ为M和M'点向径的夹角；θ׳为M点向径与y轴的夹角。θ和θ'分别计算如下：
式中　 rb为基圆半径；ra为齿顶圆半径；sa为齿顶圆齿厚；r为修形后基圆半径；rj为修形起点向径。r和rj分别计算如下：
式中　 pb为基节；α0和分别为修形前和修形后的分度圆压力角。计算如下：
式中　 Ca为修形曲线上任一点的修形量；x为修形曲线上的点到修形起点的距离；Lc=（εα-1）pb为修形长度，修形起点为单双齿啮合交点，即长齿廓修形。
通过上述齿面方程推导出齿轮修形函数为：
式中　 αk为压力角；xi，yi（i=0，1）为齿面坐标。
2.2　考虑修形的动力学方程
为了考虑齿廓修形对行星齿轮传动系统动态特性的影响，除了式（1）中的外载荷激励F外，将式（15）修形函数引起的额外激励E（x，t）引入动力学方程中。
修形后啮合线上的等效位移为：
式中　 hsp，hrp分别为内、外啮合副上的修形函数。
行星轮修形后的动力学方程为：
式中　 ，分别为修形后内、外啮合齿轮副啮后阻尼力。
修形后的行星轮动力学方程除了由δspi和δrpi变为和，其余部分没有变化。所以修形后行星轮方程右侧的额外激励可以表示为矩阵形式：
类似地，太阳轮和内齿圈的额外激励为：
式中　 和分别为内、外啮合啮合角。
因此，行星齿轮传动系统的全局额外激励可表示为如下的矩阵形式：
引入额外激励后公式（1）变为：
3　计入齿轮修形的传动系统动态特性
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3.1　行星齿轮传动系统动态特性
设定不同的行星轮修形量研究其对行星齿轮传动系统动态特性的影响。图6~8给出了不同修形量下的行星齿轮传动系统动态啮合力、动态传动误差和动载系数曲线。
图6（a）和（c）给出了不同修形量下外啮合和内啮合动态啮合力的变化曲线，图6（b）和（d）给出了动态啮合力的均值、最大值和极差变化。可以看到随着修形量的增大，外啮合、内啮合的动态啮合力逐渐减小，当修形量为15 μm时动态啮合力的均值、最大值和极差达到最小，此后当修形量继续增加，动态啮合力的各项参数又开始增大。从表2中可以看到，外啮合动态啮合力最大值由修形前的3038 N减小到2657 N，下降了12.54%，内啮合动态啮合力最大值从3064 N减小到2631 N，下降了14.13%，此后开始增大。
图7（a）和（c）给出了不同修形量下外啮合和内啮合动态传动误差的变化曲线，图7（b）和（d）为其均值、最大值和极差变化。表3为其修形前后的数值变化，可以看到随着修形量的增大，外啮合和内啮合的动态传动误差波动范围趋于平稳，当修形量为15 μm时均值、最大值和极差达到最小，外啮合动态传动误差极差由修形前的9.86 μm减小到7.03 μm，下降了28.70%，内啮合动态传动误差极差由修形前的5.68 μm减小到3.94 μm，下降了30.63%，此后开始增大，内、外啮合的动态传动误差极差下降率较大，表明动态传动误差的波动量下降明显。
图8（a）和（c）给出了不同修形量下内、外啮合动载系数Kv的变化曲线，图8（b）和（d）给出了Kv的均值、最大值和极差变化。表4为修形前后Kv数值的变化。由图8可见，随着修形量的增大，内、外啮合的Kv值逐渐减小，当修形量为15 μm时动载系数的三项变动量达到最小，外啮合Kv均值由修形前的0.7123减小到0.6386，下降了10.35%，内啮合Kv均值从0.7599减小到0.6364，下降了16.25%，此后开始增大。综上所述，当修形量为15 μm时，行星齿轮传动系统的动态啮合力、动态传动误差、Kv的最大值、均值和极差都达到最小，经过修形后，齿面载荷减小、动态传动误差波动量降低，有利于减小行星齿轮传动系统的振动噪声，对于增加齿面承载能力，延长齿轮的使用寿命具有重要意义。
3.2　行星齿轮传动系统样机性能测试
动态传动误差、动态啮合力、动载系数是衡量行星齿轮传动系统振动噪声的重要参数，通过上述数值理论研究得出了最佳修形量。负载能力和噪声等级是行星齿轮传动系统最严格的出厂检验指标。因此，为了验证修形前后样机的负载能力和噪声水平，对样机行星轮进行齿廓修形，修形量Camax=15 μm。加工装配后在同等测试环境下进行负载和噪声测试，样机如图9（a）所示。
图9（b）为样机负载测试示意图，将样机安装于负载测试仪，设置电机转速为3000 r/min，启动电源，使电机正反转各90 s，记录每台样机最大负载扭矩如表5所示。修形前负载扭矩平均值为261.54 N·m，修形后为290.24 N·m，提升了约10.97%。随后在空载环境下进行噪声测试，将样机放置于噪声测试台上与电机相连，工业噪声分贝仪与样机距离L=100 cm，如图9（c）所示。使电机正反转各8 s，期间无异音、传动平稳，分别记录5台样机的噪声值如表5所示。修形前最大噪声分贝值为65 dB，平均值为63.6 dB，修形后样机最大噪声分贝值为55 dB，平均值为54.3 dB，平均值降低了约14.62%，噪声降低明显。经过齿轮修形优化后，行星齿轮传动系统样机性能得到改善。
4　结论
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（1） 采用集中参数法建立行星齿轮传动系统考虑偏心误差和安装误差的弯-扭-轴耦合动力学模型，以四阶Runge-Kutta法求解方程分析动态啮合力、动态传动误差和动载系数的变化规律。
（2） 建立了行星轮齿廓修形前、后的齿面方程，推导了齿廓修形函数和修形后的传动误差，将修形函数引起的额外激励引入动力学方程，求解方程得到不同修形量下的动态啮合力、动态传动误差和动载系数，当修形量增加到15 μm时行星齿轮传动系统的动态特性达到最小，此后开始增大，表明修形量太大反而使系统动态特性变差。
（3） 确定最佳修形量为15 μm，此时外啮合啮合力最大值下降了12.54%，内啮合啮合力最大值下降了14.13%；外啮合动态传动误差极差下降了28.70%，内啮合动态传动误差极差下降了30.63%；外啮合Kv均值下降了10.35%，内啮合Kv均值下降了16.25%。各项指标均有不同程度的降低，经过整机性能测试发现行星齿轮传动系统承载能力得到改善，噪声降低，为高端机械装备行星齿轮传动系统的减振降噪、传动效率提升了提供理论依据和技术支撑。
基金
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  • 国家自然科学基金资助项目(52265004)
  • 中国科协青年人才托举工程资助项目(2018QNRC001)
  • 直升机传动技术国防科技重点实验室开放基金资助项目(HTL-0-21G07)
  • 数字制造装备与技术国家重点实验室开放基金资助项目(DMETKF2021017)
  • 江苏省泰州市双创计划资助项目
文献
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2024年第37卷第2期
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文章信息
doi: 10.16385/j.cnki.issn.1004-4523.2024.02.001
  • 接收时间：2022-05-13
  • 首发时间：2026-02-10
  • 出版时间：2024-02-28
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出版历史
  • 收稿日期：2022-05-13
  • 修回日期：2022-06-19
基金
国家自然科学基金资助项目(52265004)
中国科协青年人才托举工程资助项目(2018QNRC001)
直升机传动技术国防科技重点实验室开放基金资助项目(HTL-0-21G07)
数字制造装备与技术国家重点实验室开放基金资助项目(DMETKF2021017)
江苏省泰州市双创计划资助项目
作者信息
    1广西大学机械工程学院，广西　南宁　530004
    2天津工业大学机械工程学院，天津　300387
    3宁波中大力德智能传动股份有限公司，浙江　宁波　315301
    4华中科技大学数字制造装备与技术国家重点实验室，湖北　武汉　430074
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https://castjournals.cast.org.cn/joweb/zdgcxb/CN/10.16385/j.cnki.issn.1004-4523.2024.02.001
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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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