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Article(id=1241029725909480176, tenantId=1146029695717560320, journalId=1227999626482147330, issueId=1241029724655383270, articleNumber=null, orderNo=null, doi=10.16579/j.issn.1001.9669.2025.02.008, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1690473600000, receivedDateStr=2023-07-28, revisedDate=1692201600000, revisedDateStr=2023-08-17, acceptedDate=null, acceptedDateStr=null, onlineDate=1773814143874, onlineDateStr=2026-03-18, pubDate=1739548800000, pubDateStr=2025-02-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1773814143874, onlineIssueDateStr=2026-03-18, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1773814143874, creator=13701087609, updateTime=1773814143874, updator=13701087609, issue=Issue{id=1241029724655383270, tenantId=1146029695717560320, journalId=1227999626482147330, year='2025', volume='47', issue='2', pageStart='1', pageEnd='158', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1773814143575, creator=13701087609, updateTime=1773840259815, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241139264176574915, tenantId=1146029695717560320, journalId=1227999626482147330, issueId=1241029724655383270, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241139264176574916, tenantId=1146029695717560320, journalId=1227999626482147330, issueId=1241029724655383270, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=61, endPage=67, ext={EN=ArticleExt(id=1241029726203081458, articleId=1241029725909480176, tenantId=1146029695717560320, journalId=1227999626482147330, language=EN, title=High-cycling fatigue life assessment of vehicle hypoid gears based on load spectrum, columnId=1241029725523604201, journalTitle=Journal of Mechanical Strength, columnName=Fatigue∙Damage∙Fracture∙Failure Analysis, runingTitle=null, highlight=null, articleAbstract=

For the contact fatigue failure problem of vehicles’ hypoid gears under complex conditions, the rain-flow counting method and Goodman’s average stress equation were used to establish a contact statics model.The load-time history of the contact gear surface was extracted. And the load spectrum of hypoid gears was produced.The research focused on predicting the high-cycling fatigue life of hypoid gears based on the load spectrum.The study also utilized the finite element method to simulate gear teeth’s meshing or contact behaviour under loading conditions. Moreover, the influence mechanism of fatigue damage criterion on the gear fatigue life prediction was revealed.The proposed method is highly significant in assessing and predicting the high-cycling fatigue life of vehicles’ hypoid gears.

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LI Yao, E-mail：
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针对复杂工况下汽车后桥准双曲面齿轮接触疲劳失效问题，采用雨流计数法和Goodman平均应力方程，建立轮齿接触静力学模型，提取接触齿面载荷-时间历程，编制准双曲面齿轮载荷谱；研究基于载荷谱的准双曲面齿轮高周疲劳寿命预测方法，采用有限元方法模拟轮齿在载荷作用下的啮合/接触过程，揭示疲劳损伤准则对齿轮疲劳寿命预测的影响机制。所提方法对指导汽车后桥准双曲面齿轮高周疲劳寿命评估与预测具有重要意义。

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李垚（通信作者），男，1989年生，湖北十堰人，博士，讲师，硕士研究生导师；主要研究方向为传动系统动力学与可靠性；E-mail：。
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熊广林，男，2003年生，湖北孝感人，本科生；主要研究方向为传动系统动力学与可靠性；E-mail：。

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熊广林，男，2003年生，湖北孝感人，本科生；主要研究方向为传动系统动力学与可靠性；E-mail：。

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熊广林，男，2003年生，湖北孝感人，本科生；主要研究方向为传动系统动力学与可靠性；E-mail：。

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articleId=1241029725909480176, language=EN, label=Fig.1, caption=Geometric model of the hypoid gear pairs, figureFileSmall=yAuzDdkBVA4Hak+zTuQ+Jw==, figureFileBig=B716zjtzKbOu8rh5Hgxbvw==, tableContent=null), ArticleFig(id=1241029737439621331, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=CN, label=图1, caption=准双曲面齿轮几何模型, figureFileSmall=yAuzDdkBVA4Hak+zTuQ+Jw==, figureFileBig=B716zjtzKbOu8rh5Hgxbvw==, tableContent=null), ArticleFig(id=1241029737640947936, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=EN, label=Fig.2, caption=Mesh model of the hypoid gear, figureFileSmall=CNHPuIcAOHOzBrcc6qZxBQ==, figureFileBig=3ZnRowNSCToU8FGWb/YfBw==, tableContent=null), ArticleFig(id=1241029737758388455, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=CN, label=图2, caption=准双曲面齿轮网格模型, figureFileSmall=CNHPuIcAOHOzBrcc6qZxBQ==, figureFileBig=3ZnRowNSCToU8FGWb/YfBw==, tableContent=null), ArticleFig(id=1241029737825497323, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=EN, label=Fig.3, caption=Output torque of the reducer, figureFileSmall=001X9VNatFKKDU2BD41MzA==, figureFileBig=CdAkoc4itF04TA4DyvCEIA==, tableContent=null), ArticleFig(id=1241029737926160621, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=CN, label=图3, caption=减速器输出转矩, figureFileSmall=001X9VNatFKKDU2BD41MzA==, figureFileBig=CdAkoc4itF04TA4DyvCEIA==, tableContent=null), ArticleFig(id=1241029737984880883, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=EN, label=Fig.4, caption=Frequency histogram of the torque’s mean value and amplitude, figureFileSmall=NYpvxEe0ZI7Cw8nWhCkeKA==, figureFileBig=tMLP9RfHWkJNO8cJ9XrRlw==, tableContent=null), ArticleFig(id=1241029738060378359, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=CN, label=图4, caption=转矩均值、幅值频率直方图, figureFileSmall=NYpvxEe0ZI7Cw8nWhCkeKA==, figureFileBig=tMLP9RfHWkJNO8cJ9XrRlw==, tableContent=null), ArticleFig(id=1241029738156847357, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=EN, label=Fig.5, caption=Contact stress of the wheel in the meshing process under 2 000 N·m, figureFileSmall=BtEacgJhM8dD/T8/GvLRJA==, figureFileBig=REo6mOTXqb4I7j+jgtjxwg==, tableContent=null), ArticleFig(id=1241029738240733443, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=CN, label=图5, caption=2 000 N·m下大齿轮啮入和啮出过程齿面接触应力, figureFileSmall=BtEacgJhM8dD/T8/GvLRJA==, figureFileBig=REo6mOTXqb4I7j+jgtjxwg==, tableContent=null), ArticleFig(id=1241029738366562568, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=EN, label=Fig.6, caption=Curve of the gear rotation radian and contact stress under 2 000 N·m, figureFileSmall=cN1fjb0uT1ANuHwIuWwiCg==, figureFileBig=unCexYRok53fOF63GjkEQw==, tableContent=null), ArticleFig(id=1241029738458837264, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=CN, label=图6, caption=2 000 N∙m下大齿轮旋转弧度与接触应力曲线, figureFileSmall=cN1fjb0uT1ANuHwIuWwiCg==, figureFileBig=unCexYRok53fOF63GjkEQw==, tableContent=null), ArticleFig(id=1241029738538529048, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=EN, label=Fig.7, caption=Curve of the gear torque and maximum contact stress, figureFileSmall=kF1LPENNKzX/Zh9jN/6QQw==, figureFileBig=i53VVi45UzlBb1Y9rjiyQg==, tableContent=null), ArticleFig(id=1241029738618220831, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=CN, label=图7, caption=齿轮转矩与最大接触应力曲线, figureFileSmall=kF1LPENNKzX/Zh9jN/6QQw==, figureFileBig=i53VVi45UzlBb1Y9rjiyQg==, tableContent=null), ArticleFig(id=1241029738714689831, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=EN, label=Tab.1, caption=

Geometric parameter of the hypoid gear

, figureFileSmall=null, figureFileBig=null, tableContent=
参数Parameter小齿轮Pinion大齿轮Wheel
齿数Number of teeth z941
法向模数Normal module mmn/mm1212
齿面宽Width of tooth b/mm7670
大端分度圆直径Pitch diameter d/mm129.8492
法向压力角Normal pressure angle/(°)2020
螺旋角Spiral angle/(°)4646
旋向Direction左旋Left右旋Right
刀盘半径Cutter radius /mm177.8
), ArticleFig(id=1241029738811158832, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=CN, label=表1, caption=

准双曲面齿轮几何参数

, figureFileSmall=null, figureFileBig=null, tableContent=
参数Parameter小齿轮Pinion大齿轮Wheel
齿数Number of teeth z941
法向模数Normal module mmn/mm1212
齿面宽Width of tooth b/mm7670
大端分度圆直径Pitch diameter d/mm129.8492
法向压力角Normal pressure angle/(°)2020
螺旋角Spiral angle/(°)4646
旋向Direction左旋Left右旋Right
刀盘半径Cutter radius /mm177.8
), ArticleFig(id=1241029738907627827, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=EN, label=Tab.2, caption=

Material parameter of gears

, figureFileSmall=null, figureFileBig=null, tableContent=
弹性模量
Modulus of elasticity/GPa		泊松比
Poisson ratio		抗拉强度
Tensile strength/MPa		屈服强度
Yield strength/MPa
2060.31 600785
), ArticleFig(id=1241029738999902520, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=CN, label=表2, caption=

齿轮材料参数

, figureFileSmall=null, figureFileBig=null, tableContent=
弹性模量
Modulus of elasticity/GPa		泊松比
Poisson ratio		抗拉强度
Tensile strength/MPa		屈服强度
Yield strength/MPa
2060.31 600785
), ArticleFig(id=1241029739075399999, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=EN, label=Tab.3, caption=

Two-dimensional load spectrum

, figureFileSmall=null, figureFileBig=null, tableContent=
均值
Mean value/ (N·m)		幅值Amplitude/ (N·m)
6091 3412 0712 8033 5344 1444 6314 875
3981 54328213470371997
79631 6105 7862 7501 441749386184143
1 194186 33034 10616 2128 4934 4172 2741 086841
1 592320 36058 63927 87314 6027 5943 9021 8671 446
1 989162 28029 70414 1197 3973 8471 980946732
2 38023 9194 3722 0811 090567292139108
2 7859981838746241265
3 183112110000
), ArticleFig(id=1241029739180257606, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=CN, label=表3, caption=

二维载荷谱

, figureFileSmall=null, figureFileBig=null, tableContent=
均值
Mean value/ (N·m)		幅值Amplitude/ (N·m)
6091 3412 0712 8033 5344 1444 6314 875
3981 54328213470371997
79631 6105 7862 7501 441749386184143
1 194186 33034 10616 2128 4934 4172 2741 086841
1 592320 36058 63927 87314 6027 5943 9021 8671 446
1 989162 28029 70414 1197 3973 8471 980946732
2 38023 9194 3722 0811 090567292139108
2 7859981838746241265
3 183112110000
), ArticleFig(id=1241029739268337997, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=EN, label=Tab.4, caption=

One-dimensional load spectrum

, figureFileSmall=null, figureFileBig=null, tableContent=
输出转矩
Output torque/(N·m)		4801 0531 6252 2212 8043 2863 6753 860
循环次数
Number of cycles		727 050133 08063 25633 14017 2348 8724 2733 281
), ArticleFig(id=1241029739343835473, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=CN, label=表4, caption=

一维载荷谱

, figureFileSmall=null, figureFileBig=null, tableContent=
输出转矩
Output torque/(N·m)		4801 0531 6252 2212 8043 2863 6753 860
循环次数
Number of cycles		727 050133 08063 25633 14017 2348 8724 2733 281
), ArticleFig(id=1241029739427721559, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=EN, label=Tab.5, caption=

Material mechanical property parameter of gears

, figureFileSmall=null, figureFileBig=null, tableContent=
参数ParameterSeeger准则Seeger rule20CrNiMo
1.5σu1 950
0.59a0.59
b-0.087-0.087
c-0.58-0.58
n'0.150.15
K′/MPa1.65σu2 145
), ArticleFig(id=1241029739507413339, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=CN, label=表5, caption=

齿轮材料力学属性参数

, figureFileSmall=null, figureFileBig=null, tableContent=
参数ParameterSeeger准则Seeger rule20CrNiMo
1.5σu1 950
0.59a0.59
b-0.087-0.087
c-0.58-0.58
n'0.150.15
K′/MPa1.65σu2 145
), ArticleFig(id=1241029739620659553, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=EN, label=Tab.6, caption=

Gear contact fatigue life under different loads

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输出转矩
Output torque /(N·m)		最大接触应力
Maximum contact stress/ MPa		接触疲劳寿命
Contact fatigue life
480536.68.61×109
1 053692.36.27×108
1 625783.81.81×108
2 221879.25.74×107
2 804972.52.10×107
3 2861 049.69.82×106
3 6751 111.85.45×106
3 8601 141.44.22×106
), ArticleFig(id=1241029739691962727, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=CN, label=表6, caption=

齿轮八级载荷下的接触疲劳寿命

, figureFileSmall=null, figureFileBig=null, tableContent=
输出转矩
Output torque /(N·m)		最大接触应力
Maximum contact stress/ MPa		接触疲劳寿命
Contact fatigue life
480536.68.61×109
1 053692.36.27×108
1 625783.81.81×108
2 221879.25.74×107
2 804972.52.10×107
3 2861 049.69.82×106
3 6751 111.85.45×106
3 8601 141.44.22×106
), ArticleFig(id=1241029739788431724, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=EN, label=Tab.7, caption=

Contact fatigue life of the hypoid gear

, figureFileSmall=null, figureFileBig=null, tableContent=
损伤理论Damage theoryMinerMansonCorten-Dolan
疲劳寿命Fatigue life2.2×1081.64×1081.62×108
), ArticleFig(id=1241029739889095032, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029725909480176, language=CN, label=表7, caption=

准双曲面齿轮接触疲劳寿命

, figureFileSmall=null, figureFileBig=null, tableContent=
损伤理论Damage theoryMinerMansonCorten-Dolan
疲劳寿命Fatigue life2.2×1081.64×1081.62×108
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基于载荷谱的汽车后桥准双曲面齿轮高周疲劳寿命评估
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熊广林 1 , 李垚 1, 2, 3 , 邓月 1 , 黄东 2 , 吴行 2 , 朱才朝 3
机械强度 | 疲劳·损伤·断裂·失效分析 2025,47(2): 61-67
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机械强度 | 疲劳·损伤·断裂·失效分析 2025, 47(2): 61-67
基于载荷谱的汽车后桥准双曲面齿轮高周疲劳寿命评估
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熊广林1 , 李垚1, 2, 3 , 邓月1, 黄东2, 吴行2, 朱才朝3
作者信息
  • 1.三峡大学 机械与动力学院，宜昌　443002
  • 2.重庆青山工业有限责任公司 技术中心，重庆　402776
  • 3.重庆大学 高端装备机械传动全国重点实验室，重庆　400044

    • 熊广林，男，2003年生，湖北孝感人，本科生；主要研究方向为传动系统动力学与可靠性；E-mail：。

通讯作者:

李垚（通信作者），男，1989年生，湖北十堰人，博士，讲师，硕士研究生导师；主要研究方向为传动系统动力学与可靠性；E-mail：。
High-cycling fatigue life assessment of vehicle hypoid gears based on load spectrum
Guanglin XIONG1 , Yao LI1, 2, 3 , Yue DENG1, Dong HUANG2, Hang WU2, Caichao ZHU3
Affiliations
  • 1.College of Mechanical & Power, China Three Gorges University, Yichang 443002, China
  • 2.Department of Technology, Chongqing Tsingshan Industrial Co., Ltd., Chongqing 402776, China
  • 3.State Key Laboratory of Mechanical Transmission for Advanced Equipment,Chongqing University, Chongqing 400044, China
出版时间: 2025-02-15 doi: 10.16579/j.issn.1001.9669.2025.02.008
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摘要
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针对复杂工况下汽车后桥准双曲面齿轮接触疲劳失效问题，采用雨流计数法和Goodman平均应力方程，建立轮齿接触静力学模型，提取接触齿面载荷-时间历程，编制准双曲面齿轮载荷谱；研究基于载荷谱的准双曲面齿轮高周疲劳寿命预测方法，采用有限元方法模拟轮齿在载荷作用下的啮合/接触过程，揭示疲劳损伤准则对齿轮疲劳寿命预测的影响机制。所提方法对指导汽车后桥准双曲面齿轮高周疲劳寿命评估与预测具有重要意义。

准双曲面齿轮  /  接触疲劳  /  载荷谱  /  疲劳寿命

For the contact fatigue failure problem of vehicles’ hypoid gears under complex conditions, the rain-flow counting method and Goodman’s average stress equation were used to establish a contact statics model.The load-time history of the contact gear surface was extracted. And the load spectrum of hypoid gears was produced.The research focused on predicting the high-cycling fatigue life of hypoid gears based on the load spectrum.The study also utilized the finite element method to simulate gear teeth’s meshing or contact behaviour under loading conditions. Moreover, the influence mechanism of fatigue damage criterion on the gear fatigue life prediction was revealed.The proposed method is highly significant in assessing and predicting the high-cycling fatigue life of vehicles’ hypoid gears.

Hypoid gear  /  Contact fatigue  /  Load spectrum  /  Fatigue life
熊广林, 李垚, 邓月, 黄东, 吴行, 朱才朝. 基于载荷谱的汽车后桥准双曲面齿轮高周疲劳寿命评估. 机械强度, 2025 , 47 (2) : 61 -67 . DOI: 10.16579/j.issn.1001.9669.2025.02.008
Guanglin XIONG, Yao LI, Yue DENG, Dong HUANG, Hang WU, Caichao ZHU. High-cycling fatigue life assessment of vehicle hypoid gears based on load spectrum[J]. Journal of Mechanical Strength, 2025 , 47 (2) : 61 -67 . DOI: 10.16579/j.issn.1001.9669.2025.02.008
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0　引言
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准双曲面齿轮具有机械效率高、承载能力强、噪声低、传动平稳等优点，被广泛应用于汽车驱动后桥等重要场合。针对高速、重载等复杂工况下汽车后桥准双曲面齿轮出现的接触疲劳失效问题，研究准双曲面齿轮在全寿命周期疲劳寿命评估方法，对齿轮疲劳寿命进行评估，对提高齿轮服役可靠性具有重要意义。
在齿轮疲劳寿命方面，李学飞等[1]研究了齿面接触应力计算时摩擦因数对接触疲劳寿命的影响。郭都等[2]基于修正的P-S-N曲线和累积疲劳损伤理论，预测了斜齿轮接触疲劳寿命。陈兴彬等[3]基于齿轮疲劳失效理论，建立了齿轮CAE模型，分析了齿轮接触动力学特性和疲劳寿命。魏冰阳等[4]计算了锥齿轮弯曲疲劳寿命，验证了疲劳寿命仿真与加速试验手段评价锥齿轮弯曲疲劳强度的可行性。郭玉梁等[5]采用有限元方法，使用nCode软件对齿轮进行疲劳寿命仿真，验证了加速疲劳试验评估齿轮弯曲疲劳强度的可行性。赵龙等[6]建立了控制尺寸链，研究了齿隙对齿轮弯曲疲劳寿命的影响。唐鑫等[7]修正了国家标准夹具计算方法，开展了齿轮弯曲疲劳试验，得到了齿轮弯曲疲劳极限结果。LEE等[8]建立了某桥壳传动单元有限元模型，提出了基于全系统的准双曲面齿轮齿根疲劳寿命预测有限元模型。LIU等[9]基于Dang Van多轴疲劳准则，研究了残余应力对齿轮接触疲劳寿命的影响。HE等[10]研究了外部载荷对渐开线齿轮接触疲劳寿命的影响。
在工程实际中，齿轮由于外部转矩波动，呈现多轴高周疲劳失效现象。陈卓等[11]结合S-N曲线和Miner线性累积损伤准则，评估了三峡升船机齿条疲劳寿命。苏灵[12]通过雨流计数法得到风机叶片叶根疲劳载荷谱和应力谱，计算了基于Miner线性损伤理论的叶片叶根疲劳寿命。JIA等[13]研究了在不同载荷水平下车用锥齿轮接触疲劳寿命与疲劳损伤准则对齿轮疲劳寿命预测的影响。邹喜红等[14]基于汽车实际行驶工况和载荷数据编制载荷谱，研究了减速器齿轮的疲劳损伤。SHINDE等[15]提出了一种改进的雨流计数法，分析了随机载荷下结构疲劳寿命。MAYER等[16]研究了极限循环载荷对合金材料疲劳损伤的影响。LI等[17]基于雨流计数法和非参数外推法编制了二维载荷谱，对汽车减速器进行了疲劳寿命预测。LIU等[18]基于实测风载荷谱、Dang Van多轴疲劳准则和Basquin方程，研究了随机载荷下风电机组疲劳寿命。以上研究多基于线性疲劳损伤准则和商业软件，未考虑齿轮准确几何形状、啮合过程与材料力学特性，疲劳寿命预测结果无法指导工程实际。
因此，本文采用齿轮几何学、啮合理论和有限元理论，建立精确的准双曲面齿轮三维模型和静力性模型，研究齿轮啮合特性和载荷谱编制方法。建立基于载荷谱的准双曲面齿轮接触疲劳寿命预测方法，为汽车后桥准双曲面齿轮接触疲劳寿命分析和预测提供方法支撑。
1　准双曲面齿轮有限元建模
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1.1　准双曲面齿轮三维模型
采用齿轮啮合理论和CAD技术，建立准双曲面齿轮三维模型，如图1所示。表1所示为准双曲面齿轮副主要几何参数。
汽车上准双曲面齿轮常用材料为20CrNiMo，淬火低温回火后的20CrNiMo材料具有良好的耐磨性，且具备疲劳强度高、抗冲击韧性好、硬度高等特点，非常适合高速、重载场合。齿轮力学参数如表2所示。
1.2　齿轮有限元模型
考虑到齿轮为回转体和计算成本，对大齿轮模型进行简化。图2所示为准双曲面齿轮有限元模型，该模型包含大齿轮7个齿。齿轮接触网格足够细密，保证收敛，其他区域网格相对较粗。选用网格类型为线性减缩积分单元CED1OR，大齿轮有237 912个网格，小齿轮有320 411个网格，如图2所示。
2　准双曲面齿轮载荷谱编制
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2.1　采集方案
汽车减速器主要承受发动机输出的波动转矩载荷和实际路面行驶的负载转矩载荷。因此，对汽车实际路面行驶时减速器所承受的载荷进行采集，采用随机载荷谱模拟技术，最大程度模拟汽车减速器承受的载荷谱，提高准双曲面齿轮可靠性和疲劳寿命评估的准确性[19]。图3所示为汽车减速器输出转矩的载荷时间历程。
采用雨流计数法对输出转矩进行统计处理，得到载荷的幅值和均值信息。根据工程经验，载荷幅值服从威布尔分布，载荷均值服从高斯分布[20]，如图4所示。
2.2　载荷谱编制
由于测试采样数据有限，不足以代表全生命周期的载荷历史。因此，将累积加载周期扩展到106周期，该周期可以真实地反映整个生命周期所经历的载荷历史。
1）载荷幅值最大值为
式中，xa为载荷幅值；α为幅值形状参数；β为幅值尺度参数；xamax为最大幅值。
由p(xa)=10-6计算该工况下幅值的最大值xamax=4 875 N·m。
2）载荷均值最大值为
式中，xm为载荷均值；xmmax为载荷均值最大值；μ为载荷均值的均值；σ为载荷均值标准差。
由p(xm)=10-6计算该工况下均值最大值xmmax=3 183 N·m。
载荷幅值和均值可分为8个等级，均值比例系数分别为0.125、0.25、0.375、0.5、0.625、0.75、0.875和1，幅值比例系数分别为1、0.95、0.85、0.725、0.575、0.425、0.275和0.125，且为不等间距分割。采用联合概率密度函数计算各载荷区间的频率：
式中，sa1、sa2分别为幅值积分下限与上限；sm1、sm2分别为均值积分下限与上限；N为累积次数。
由式（1）~式（3）计算得到的二维载荷谱如表3所示。德国弗劳恩霍夫研究所在载荷谱技术应用领域提出8级程序载荷谱，广泛应用于汽车齿轮的疲劳寿命估算[21]。把应力等效转化为转矩等效，表4所示为二维载荷谱对应的等效一维载荷谱。
3　准双曲面齿轮有限元接触模型
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3.1　载荷施加
定义接触对：在正常工况下，小齿轮齿面与大齿轮齿面啮合，将参与啮合的5对齿面定义为面-面接触对。
载荷步定义如下：分别约束两个齿轮中心参考点的6个自由度。载荷步①在小齿轮参考点施加小角度转角，令大、小齿轮进入预接触状态。载荷步②释放大齿轮参考点旋转自由度，并在该点施加一个小转矩，使齿轮达到理想的初始啮合状态。载荷步③在小齿轮参考点施加随时间均匀变化的轴向位移，尽可能代表真实齿轮啮合区域，在大齿轮参考点上施加轴向转矩，模拟准双曲面齿轮加载啮合过程。
3.2　基于有限元的齿轮副齿面接触分析
求解该有限元模型，得到准双曲面齿轮齿面接触应力时间历程。图5为2 000 N·m下大齿轮啮入和啮出过程齿面接触应力图。齿轮齿面接触印痕呈形状规则、细长的椭圆状，从大端底部向小端顶部倾斜，在接触过程中应力值平稳。其中，齿面最大接触应力出现在中部，轮齿旋转弧度与齿面接触应力拟合曲线如图6所示。
本文主要选取1 000、2 000、3 000、4 000、5 000 N·m的5个恒定载荷进行接触分析，得到5种恒定载荷下的最大接触应力，分别为725、839、935、1 165、1 350 MPa。如图7所示，计算得到材料在弹性范围内的转矩、应力拟合式为
式中，x为转矩，N·m；y为接触应力，MPa。
4　准双曲面齿轮高周疲劳寿命评估
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4.1　疲劳寿命计算方法
工程实际中常用损伤累积准则有Miner线性准则、Manson双线性准则和Corten-Dolan非线性准则[22-24]。Miner线性准则的疲劳变量D定义为在实际应力下的循环次数n与材料在该应力下的疲劳寿命N之比，表示为
式中，D为损伤总累积量；Di为某级载荷下的损伤累积量；m为经历载荷的级数；ni为在第i级载荷下的循环次数；Ni为第i级载荷下的疲劳寿命。当D接近1时，达到疲劳寿命极限。
Manson双线性准则把每种载荷的疲劳损伤分为两个阶段，对于每个阶段，使用线性累积损伤准则计算疲劳寿命。对于包含两个以上载荷等级的载荷块，载荷的双线性理论损伤曲线为
式中，Ni，Ⅰ、Ni，Ⅱ分别为第i级载荷下第1部分、第2部分疲劳寿命；Ni为第i级载荷下的疲劳寿命；Nmin为载荷谱中最小疲劳寿命；Nmax为载荷谱中最大疲劳寿命。
根据多阶段应力作用下Corten-Dolan非线性准则，高水平应力对结构本身造成损伤，同时影响低水平应力引起的损伤，机械结构在多级应力下的疲劳寿命计算式为
式中，N1为载荷谱中最大载荷下齿轮的疲劳寿命；αi为第i级载荷下循环次数与总循环数之比；σ1为载荷最大应力值；σi为第i级载荷下的应力值；d为物质常数，Corten假设d/m=0.85，m为S-N曲线的斜率。
4.2　齿轮材料的疲劳S-N曲线
采用Seeger法生成材料的S-N曲线，材料相关参数如表5所示。
工程实际中所得材料S-N曲线与齿轮S-N曲线存在差异[25]。为准确模拟齿轮S-N曲线，采用Goodman准则修正平均应力，得到齿轮的S-N曲线[26]。
式中，σ-1为对称循环载荷下的材料疲劳极限；σm为平均应力；σb为材料强度极限。
4.3　齿轮高周疲劳寿命计算
根据得到的一维载荷谱、载荷接触应力拟合曲线及修正后的齿轮S-N曲线，计算不同载荷下的齿轮疲劳寿命，如表6所示。
由式（5）得，Miner线性准则下的疲劳寿命D=0.004 5。当D=1时，发生疲劳破坏。因此，有NMiner=2.2×108。由式（6）~式（8）可计算Manson双线性准则下的疲劳寿命。已知Nmax=8.61×109，Nmin=4.22×106，得到φ=-0.44，Z=-2 430.6。
从而得到
当DI=DⅡ=1，NⅠ=2.6×107，NⅡ=1.38×108，NManson=NⅠ+NⅡ=1.64×108。
由式（9）得Corten-Dolan非线性准则下的疲劳寿命为
表7所示为准双曲面齿轮在三种损伤累积理论下的接触疲劳寿命预测结果。由表7可知，Corten-Dolan准则估算的疲劳寿命最小，Manson双线性准则次之，Miner线性准则估算的疲劳寿命最大。这是由于Miner线性准则没有考虑载荷顺序等因素。Manson双线性准则将疲劳损伤分为两个阶段，根据载荷的加载顺序确定。在多级载荷情况下，考虑两个以上载荷水平时，Manson双线性准则计算的疲劳损伤由最高水平载荷和最低水平载荷决定，故与Miner准则相比寿命较短。Corten-Dolan准则考虑了加载顺序与非线性因素影响，导致疲劳寿命预测值较低。三种疲劳累积损伤准则所得到的接触疲劳寿命在同一数量级上，基本满足工程需要。哪种结果更接近真实结果，有待试验的验证。
5　结论
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针对汽车后桥准双曲面齿轮失效机制不明，建立了精确的准双曲面齿轮三维模型和静力学模型，研究了齿轮啮合特性与损伤理论对齿轮疲劳寿命计算的影响机制，主要结论如下：
1）提出一种齿轮接触疲劳寿命评估方法，可根据扩展后的载荷时间历程，对准双曲面齿轮进行接触疲劳寿命评估，该方法工程实用性较强。
2）在波动载荷谱下，Miner线性准则预测的疲劳寿命为2.2×108，Manson双线性准则预测的疲劳寿命为1.64×108，Corten-Dolan非线性准则预测的疲劳寿命为1.62×108。其中Miner线性准则计算结果最乐观，Corten-Dolan准则计算结果相对保守。在上述累积损伤准则中，Miner准则较为简单，在工程应用中发现，对随机载荷的疲劳寿命预测吻合度较高，在二级或多级载荷下，预测寿命偏差较大。Manson双线性规则模型相对简单，并且考虑加载顺序的影响，适用于应力变程不大的多级载荷谱。Corten-Dolan准则充分考虑到加载顺序及非线性因素的影响，但必须由精度较高的试验确定参数，才能保证评估结果的精确性。
基金
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  • 湖北省自然科学基金计划项目(2023AFB066)
  • 水电机械设备设计与维护湖北省重点实验室开放基金项目(2021KJX10)
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2025年第47卷第2期
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doi: 10.16579/j.issn.1001.9669.2025.02.008
  • 接收时间：2023-07-28
  • 首发时间：2026-03-18
  • 出版时间：2025-02-15
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  • 收稿日期：2023-07-28
  • 修回日期：2023-08-17
基金
Hubei Provincial Natural Science Foundation of China(2023AFB066)
湖北省自然科学基金计划项目(2023AFB066)
Opening Foundation of Hubei Key Laboratory of Hydroelectric Machinery Design and Maintenance(2021KJX10)
水电机械设备设计与维护湖北省重点实验室开放基金项目(2021KJX10)
作者信息
    1.三峡大学 机械与动力学院，宜昌　443002
    2.重庆青山工业有限责任公司 技术中心，重庆　402776
    3.重庆大学 高端装备机械传动全国重点实验室，重庆　400044

通讯作者:

李垚（通信作者），男，1989年生，湖北十堰人，博士，讲师，硕士研究生导师；主要研究方向为传动系统动力学与可靠性；E-mail：。
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https://castjournals.cast.org.cn/joweb/jxqd/CN/10.16579/j.issn.1001.9669.2025.02.008
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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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