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Article(id=1241699535978811956, tenantId=1146029695717560320, journalId=1227999626482147330, issueId=1241699531444769296, articleNumber=null, orderNo=null, doi=10.16579/j.issn.1001.9669.2025.04.013, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1685376000000, receivedDateStr=2023-05-30, revisedDate=1687795200000, revisedDateStr=2023-06-27, acceptedDate=null, acceptedDateStr=null, onlineDate=1773973839039, onlineDateStr=2026-03-20, pubDate=1744646400000, pubDateStr=2025-04-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1773973839039, onlineIssueDateStr=2026-03-20, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1773973839039, creator=13701087609, updateTime=1773973839039, updator=13701087609, issue=Issue{id=1241699531444769296, tenantId=1146029695717560320, journalId=1227999626482147330, year='2025', volume='47', issue='4', pageStart='1', pageEnd='157', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1773973837957, creator=13701087609, updateTime=1773974092709, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241700600002433947, tenantId=1146029695717560320, journalId=1227999626482147330, issueId=1241699531444769296, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241700600006628252, tenantId=1146029695717560320, journalId=1227999626482147330, issueId=1241699531444769296, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=105, endPage=111, ext={EN=ArticleExt(id=1241699536268218935, articleId=1241699535978811956, tenantId=1146029695717560320, journalId=1227999626482147330, language=EN, title=Design of a spur gear based on the relative curvature control and analysis of its stress and mesh stiffness, columnId=1241699532912784251, journalTitle=Journal of Mechanical Strength, columnName=∙Optimization∙Reliability∙, runingTitle=null, highlight=null, articleAbstract=

The relative curvature of the spur gear mesh point is one of the key geometric parameters of the tooth profile,which has a significant impact on the stress distribution and mesh stiffness of the gear. Starting from the relevant research on constant relative curvature (CRC) gears, three types of non-involute spur gear tooth profiles were constructed based on the relative curvature control strategy considering the time-varying mesh characteristics of gears, and the mesh simulation of gear pairs was implemented. The influence of relative curvature control on the maximum contact stress, maximum bending stress,and mesh stiffness of gears was analyzed. The effectiveness of the control strategy was verified. This work provides some reference for the design of spur gears based on the relative curvature control.

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LIU Lei, E-mail：
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直齿轮啮合点的相对曲率是齿廓的关键几何参数之一，对齿轮应力分布和啮合刚度有着显著影响。从等相对曲率（Constant Relative Curvature, CRC）齿轮的相关研究出发，基于考虑齿轮时变啮合特性的相对曲率控制策略，构建了3种非渐开线直齿轮齿廓，并对其进行啮合仿真，分析了相对曲率控制对齿轮最大接触应力、最大弯曲应力及啮合刚度的影响，验证了控制策略的有效性，为基于相对曲率控制的直齿轮设计提供了一定的参考。

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刘雷，男，1978年生，江苏连云港人，博士，副教授；主要研究方向为齿轮传动；E-mail：。
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陈峙玮，男，1999年生，江苏镇江人，硕士研究生；主要研究方向为齿轮传动；E-mail：。

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陈峙玮，男，1999年生，江苏镇江人，硕士研究生；主要研究方向为齿轮传动；E-mail：。

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陈峙玮，男，1999年生，江苏镇江人，硕士研究生；主要研究方向为齿轮传动；E-mail：。

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Mechanical Systems and Signal Processing，2018，111：331-347., articleTitle=Improved analytical models for mesh stiffness and load sharing ratio of spur gears considering structure coupling effect, refAbstract=null)], funds=[Fund(id=1241699554941260002, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241699535978811956, awardId=52075238, language=EN, fundingSource=National Natural Science Foundation of China(52075238), fundOrder=null, country=null), Fund(id=1241699555050311914, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241699535978811956, awardId=52075238, language=CN, fundingSource=国家自然科学基金项目(52075238), fundOrder=null, country=null), Fund(id=1241699555138392308, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241699535978811956, awardId=HTL-A-21G04, language=EN, fundingSource=Foundation of National Key Laboratory of Science and Technology on Helicopter Transmission of Nanjing University of Aeronautics and 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ArticleFig(id=1241699553494225025, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241699535978811956, language=CN, label=图14, caption=CRC齿轮1应力云图, figureFileSmall=khIr0VCOrbks4wGLgNX6OQ==, figureFileBig=3dTxvct84nbhW/tX54d2uw==, tableContent=null), ArticleFig(id=1241699553590694025, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241699535978811956, language=EN, label=Fig. 15, caption=Hertz contact stress distribution, figureFileSmall=pMqWhtaNiicECGMBOxdWEA==, figureFileBig=6/u+7TWzLJqS4rlB/IwKAw==, tableContent=null), ArticleFig(id=1241699553666191501, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241699535978811956, language=CN, label=图15, caption=赫兹接触应力分布, figureFileSmall=pMqWhtaNiicECGMBOxdWEA==, figureFileBig=6/u+7TWzLJqS4rlB/IwKAw==, tableContent=null), ArticleFig(id=1241699553787826327, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241699535978811956, language=EN, label=Tab. 1, caption=

Gear parameters

, figureFileSmall=null, figureFileBig=null, tableContent=
齿数比
Gear ratio		模数
Modulus/ mm		齿顶高系数
Coefficient of addendum		顶隙系数
Coefficient of bottom clearance		节点处压力角
Pressure angle at pitch point /（°）
23/47310.2520
), ArticleFig(id=1241699553888489630, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241699535978811956, language=CN, label=表1, caption=

齿轮参数

, figureFileSmall=null, figureFileBig=null, tableContent=
齿数比
Gear ratio		模数
Modulus/ mm		齿顶高系数
Coefficient of addendum		顶隙系数
Coefficient of bottom clearance		节点处压力角
Pressure angle at pitch point /（°）
23/47310.2520
), ArticleFig(id=1241699554001735845, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241699535978811956, language=EN, label=Tab. 2, caption=

Relative curvature control function

, figureFileSmall=null, figureFileBig=null, tableContent=
PRC齿轮1
PRC gear 1		PRC齿轮2
PRC gear 2		SRC齿轮
SRC gear
控制函数
Control function		Kr=0.001 38r2+0.111 4Kr=0.001 08r2+0.111 4Kr=0.04/(1+e-5(r-2.5))+0.12
), ArticleFig(id=1241699554085621933, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241699535978811956, language=CN, label=表2, caption=

相对曲率控制函数

, figureFileSmall=null, figureFileBig=null, tableContent=
PRC齿轮1
PRC gear 1		PRC齿轮2
PRC gear 2		SRC齿轮
SRC gear
控制函数
Control function		Kr=0.001 38r2+0.111 4Kr=0.001 08r2+0.111 4Kr=0.04/(1+e-5(r-2.5))+0.12
), ArticleFig(id=1241699554198868148, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241699535978811956, language=EN, label=Tab. 3, caption=

Parameters of the finite element model

, figureFileSmall=null, figureFileBig=null, tableContent=
弹性模量
Modulus of elasticity/GPa		泊松比
Poisson ratio		小齿轮负载
Pinion load/（N‧m）		小齿轮孔半径
Radius of pinion hole/ mm		大齿轮孔半径
Radius of gear hole/ mm		齿宽
Tooth width/mm
2060.26950153020
), ArticleFig(id=1241699554349863097, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241699535978811956, language=CN, label=表3, caption=

有限元模型参数

, figureFileSmall=null, figureFileBig=null, tableContent=
弹性模量
Modulus of elasticity/GPa		泊松比
Poisson ratio		小齿轮负载
Pinion load/（N‧m）		小齿轮孔半径
Radius of pinion hole/ mm		大齿轮孔半径
Radius of gear hole/ mm		齿宽
Tooth width/mm
2060.26950153020
), ArticleFig(id=1241699554454720706, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241699535978811956, language=EN, label=Tab. 4, caption=

Analysis of torsional stiffness results

, figureFileSmall=null, figureFileBig=null, tableContent=
渐开线齿轮
Involute gear		CRC齿轮1
CRC gear 1		CRC齿轮2
CRC gear 2		PRC齿轮1
PRC gear 1		PRC齿轮2
PRC gear 2		SRC齿轮
SRC gear
传动重合度
Contact ratio		1.667 71.523 41.805 31.618 61.599 41.687 1
最大啮合冲击比率
Maximum mesh impact ratio		1.3961.4051.4511.4041.3961.394
), ArticleFig(id=1241699554580549834, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241699535978811956, language=CN, label=表4, caption=

扭转刚度结果分析

, figureFileSmall=null, figureFileBig=null, tableContent=
渐开线齿轮
Involute gear		CRC齿轮1
CRC gear 1		CRC齿轮2
CRC gear 2		PRC齿轮1
PRC gear 1		PRC齿轮2
PRC gear 2		SRC齿轮
SRC gear
传动重合度
Contact ratio		1.667 71.523 41.805 31.618 61.599 41.687 1
最大啮合冲击比率
Maximum mesh impact ratio		1.3961.4051.4511.4041.3961.394
), ArticleFig(id=1241699554689601746, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241699535978811956, language=EN, label=Tab. 5, caption=

Maximum bending stress and maximum contact stress

, figureFileSmall=null, figureFileBig=null, tableContent=
渐开线齿轮
Involute gear		CRC齿轮1
CRC gear 1		CRC齿轮2
CRC gear 2		PRC齿轮1
PRC gear 1		PRC齿轮2
PRC gear 2		SRC齿轮
SRC gear
最大弯曲应力
Maximum bending stress/MPa		63.4360.2964.3658.2760.2459.92
最大接触应力
Maximum contact stress / MPa		617.1557.5648.1565.6561.9575.2
), ArticleFig(id=1241699554798653657, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241699535978811956, language=CN, label=表5, caption=

最大弯曲应力和最大接触应力

, figureFileSmall=null, figureFileBig=null, tableContent=
渐开线齿轮
Involute gear		CRC齿轮1
CRC gear 1		CRC齿轮2
CRC gear 2		PRC齿轮1
PRC gear 1		PRC齿轮2
PRC gear 2		SRC齿轮
SRC gear
最大弯曲应力
Maximum bending stress/MPa		63.4360.2964.3658.2760.2459.92
最大接触应力
Maximum contact stress / MPa		617.1557.5648.1565.6561.9575.2
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基于相对曲率控制的直齿轮设计及其应力、啮合刚度分析
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陈峙玮 , 刘雷 , 孔荣
机械强度 | ·优化·可靠性· 2025,47(4): 105-111
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机械强度 | ·优化·可靠性· 2025, 47(4): 105-111
基于相对曲率控制的直齿轮设计及其应力、啮合刚度分析
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陈峙玮 , 刘雷 , 孔荣
作者信息
  • 南京航空航天大学 机电学院，南京　210016

    • 陈峙玮，男，1999年生，江苏镇江人，硕士研究生；主要研究方向为齿轮传动；E-mail：。

通讯作者:

刘雷，男，1978年生，江苏连云港人，博士，副教授；主要研究方向为齿轮传动；E-mail：。
Design of a spur gear based on the relative curvature control and analysis of its stress and mesh stiffness
Zhiwei CHEN , Lei LIU , Rong KONG
Affiliations
  • College of Mechanical & Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
出版时间: 2025-04-15 doi: 10.16579/j.issn.1001.9669.2025.04.013
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摘要
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直齿轮啮合点的相对曲率是齿廓的关键几何参数之一，对齿轮应力分布和啮合刚度有着显著影响。从等相对曲率（Constant Relative Curvature, CRC）齿轮的相关研究出发，基于考虑齿轮时变啮合特性的相对曲率控制策略，构建了3种非渐开线直齿轮齿廓，并对其进行啮合仿真，分析了相对曲率控制对齿轮最大接触应力、最大弯曲应力及啮合刚度的影响，验证了控制策略的有效性，为基于相对曲率控制的直齿轮设计提供了一定的参考。

相对曲率  /  直齿轮设计  /  啮合过程  /  应力分析

The relative curvature of the spur gear mesh point is one of the key geometric parameters of the tooth profile,which has a significant impact on the stress distribution and mesh stiffness of the gear. Starting from the relevant research on constant relative curvature (CRC) gears, three types of non-involute spur gear tooth profiles were constructed based on the relative curvature control strategy considering the time-varying mesh characteristics of gears, and the mesh simulation of gear pairs was implemented. The influence of relative curvature control on the maximum contact stress, maximum bending stress,and mesh stiffness of gears was analyzed. The effectiveness of the control strategy was verified. This work provides some reference for the design of spur gears based on the relative curvature control.

Relative curvature  /  Spur gear design  /  Mesh process  /  Stress analysis
陈峙玮, 刘雷, 孔荣. 基于相对曲率控制的直齿轮设计及其应力、啮合刚度分析. 机械强度, 2025 , 47 (4) : 105 -111 . DOI: 10.16579/j.issn.1001.9669.2025.04.013
Zhiwei CHEN, Lei LIU, Rong KONG. Design of a spur gear based on the relative curvature control and analysis of its stress and mesh stiffness[J]. Journal of Mechanical Strength, 2025 , 47 (4) : 105 -111 . DOI: 10.16579/j.issn.1001.9669.2025.04.013
正文
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0　引言
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在平行轴齿轮传动中，渐开线齿轮因其突出的优点被广泛应用，但也存在缺点，如凸‑凸齿廓啮合，啮合点相对曲率较大，接触承载能力受到限制；齿面间滑动摩擦较大，易导致磨损、发热等。采用新型齿廓是克服渐开线齿轮缺点的有效方法之一，相关学者对新型齿轮传动进行了大量研究，如余弦曲线齿轮[1]、分阶式三圆弧齿轮[2]、圆弧齿线齿轮[3]、S型齿轮[4]、抛物线齿轮[5]、非Hertz接触齿轮[6]、基于包络特征的非圆齿轮[7]等。LIU等[8]144-158提出了一种基于啮合点相对曲率控制的共轭齿廓设计方法，并构建了等相对曲率（Constant Relative Curvature, CRC）齿轮，与相同参数的渐开线齿轮相比，该齿轮具有接触、弯曲应力较低，齿间滑动系数较小，啮合刚度较大等优点，但其传动重合度有所下降，目前采用慢走丝线切割方法完成了齿轮试制并开展了传动性能初步测试。
齿廓修形也是克服渐开线齿轮缺点的方式之一，根据修形的优化目标有着不同的方法，张金龙等[9]基于齿轮热变形开展了修形研究，夏伯乾等[10]基于弹流摩擦副刚度进行了修形，田晓青等[11]基于齿轮时变啮合过程进行了修形。齿廓修形类似于对部分齿廓根据修形需要进行重新设计，即将啮合线分段并分别设计对应的齿廓。
本文将齿轮修形的理念引入到对相对曲率的控制中，提出了基于时变啮合过程的相对曲率控制方法，并对采用该方法设计出的齿廓进行应力及啮合刚度分析，旨在基于该设计方法在不削弱轮齿强度情况下克服CRC齿轮传动重合度降低的缺点。
1　基于相对曲率控制的共轭齿廓设计
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1.1　相对曲率控制方程
基于齿轮啮合原理[12]建立了如图1所示的坐标系，该方法可通过已知啮合线生成对应的产形齿条齿廓，从而生成对应的共轭齿廓，适用于渐开线齿轮、摆线齿轮、CRC齿轮等齿形。其中，Σ1为小齿轮齿廓，Σ2为大齿轮齿廓，Σ3为产形齿条齿廓，Σm为啮合线。当产形齿条横向移动距离L时，小齿轮与大齿轮也绕各自齿轮轴分别旋转角度φ1与φ2，使这3条线与啮合线的3个交点重合。
在图1中，啮合线Σm的方程可表示为
式中，r、α分别为啮合线极坐标方程中的极径与极角。对于渐开线齿轮，其啮合线为直线，α与r无关，为常数，而CRC齿轮的啮合线为曲线，α、r为变量，受式（1）约束。
由齿轮啮合原理，可推导出产形齿条的方程，表示为
式中，r'为啮合线的极坐标方程中r对α的1阶导数；C为积分常数。
小齿轮齿廓方程为
同理，大齿轮的齿廓方程为
式中，r1、r2分别为齿轮1与齿轮2的节圆半径。根据微分几何原理，齿廓1与齿廓2的相对曲率可分别表示为
在任意啮合点处的相对曲率可表示为
由式（7）可知，各啮合点的相对曲率可视为r与α的函数，反之，可以基于给定的相对曲率逆推出啮合线的极坐标方程，从而生成相应的共轭齿廓。
1.2　相对曲率控制策略
1.2.1　恒定相对曲率
在本文使用的共轭齿廓生成方法中，啮合线Σm由节点分为两段，分别位于第一象限与第三象限，其极坐标方程需分别求解：将相对曲率的约束条件分别代入式（7）中，得到相应的微分方程，采用数值方法求解，并使用多项式函数拟合数值解，以获取两段啮合线的极坐标方程，最后将其分别代入式（1）~式（4）中，即可生成对应的共轭齿廓，其详细过程可参照文献［8］144-158。
以表1参数的标准渐开线齿轮的相对曲率分布为参照，通过前述的相对曲率控制方程创建了相对曲率为恒定值的CRC齿轮，取该渐开线齿轮的相对曲率最小值0.111 4 mm-1设计了CRC齿轮副1，并取相对曲率为0.16 mm-1设计了CRC齿轮副2作为对照，3种齿轮的小齿轮齿廓对比与相对曲率分布如图2、图3所示。
基于LIU等[8]144-158[13-15][16]1349-1364对CRC齿轮的相关研究，在恒定相对曲率取同参数渐开线齿轮最小值的情况下，相比渐开线齿轮，CRC齿轮的齿廓顶部变尖，底部变厚，弯曲应力更小，抗弯强度更大，从而具有更大的啮合刚度，且具有较低的在单对齿啮合区恒定的接触应力，同时滑动系数也低于渐开线齿轮，并具有更大的最小油膜厚度，使其有更高的胶合承载能力。然而，该参数下渐开线齿轮的重合度为1.667 7，CRC齿轮重合度为1. 523 4，随着传动比的降低，单对齿啮合区间也在增加，使得传动过程中的振动与冲击增大。较高的相对曲率可以提升传动重合度，相对曲率取0.16 mm-1时的CRC齿轮重合度为1. 805 3，但此时其各项承载能力较差，不具应用价值。以上分析说明，这种恒定相对曲率的控制方法有待改进。
1.2.2　考虑齿轮啮合特性的相对曲率控制策略
在齿轮的啮合过程中，单对齿啮合时由一对轮齿承担所有的负载，在该啮合区间一般出现弯曲应力与接触应力的最大值，需要将该区间的相对曲率控制在较低的范围以提高承载能力。在双对齿啮合区间，由于有两对轮齿分担负载，单对轮齿的负载大幅降低，对承载能力要求降低。此外，双对齿与单对齿啮合刚度的差异也是齿轮传动过程中振动冲击的来源，提高传动重合度，降低单、双对齿之间啮合刚度的差值都是减少振动冲击的可行方法，因此，可以在双对齿啮合区间适当增大相对曲率以提高传动重合度。
为了对相对曲率进行所要求的控制，需要确认单对齿啮合区间的范围，本文使用CRC齿轮1计算了单对齿啮合区间对应的极距范围，并以其范围作为参照，分别使用抛物线函数与sigmoid函数对双对齿啮合区间的相对曲率进行控制，生成了对应的啮合线与共轭齿廓，这里将其分别称为PRC（Parabolic Relative Curvature）齿轮与SRC（Sigmoid Relative Curvature）齿轮，相对曲率控制函数如表2所示。不同于CRC齿轮与PRC齿轮，在使用sigmoid函数对相对曲率进行控制时，求解微分方程所获得的数值解无法用多项式获得较好的拟合效果，本文在此使用高斯曲线拟合以获取啮合线的近似解析解，其中PRC齿轮1与SRC齿轮啮合线微分方程在第一象限与第三象限的数值解如图4所示，其横、纵坐标分别为极距值和极角的正切值。共轭齿廓的相对曲率变化趋势以及各齿轮的啮合线形状如图5、图6所示，小齿轮齿廓曲线如图7所示。
2　应力与啮合刚度评价
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2.1　有限元模型设置
为了对所设计齿廓进行评估，使用前述设计的齿廓导入三维建模软件建立有限元模型进行啮合仿真。需要注意的是，在使用有限元模型获取齿轮啮合刚度的过程中，LIANG等[17]指出：为精确计算啮合刚度，齿轮模型需要保留完整的圆环状基体部分，且中心齿轮孔的大小对得到的啮合刚度有直接影响，进行啮合刚度对比需要将模型的相关参数保持一致。本文有限元模型所用的其他参数如表3所示。基于以上参数建立了如图8所示的模型，进行准静态啮合仿真，获取各个啮合位置的弯曲应力、接触应力、啮合刚度等数据。
齿轮轮齿在啮合过程中的变形可以近似视作变截面梁的弯曲变形，为避免出现剪切自锁造成变形量偏小的现象，如图9所示，本文使用线性缩减积分单元C3D8R，沿齿廓方向的网格尺寸取该齿轮各啮合位置最小赫兹接触半宽的1/3，约0.04 mm，以克服沙漏问题，并保证应力分析的准确性。齿轮基体的网格尺寸为1.5 mm，单个齿轮的网格数量约为80 000个，整个模型的网格数量约为160 000个，其中渐开线齿轮模型的网格如图10所示。
2.2　啮合刚度分析
渐开线齿轮的啮合线为直线，在啮合过程中各啮合点的接触力的方向始终一致，如图11所示，而本文的非渐开线齿轮的啮合线为曲线，使其在多对齿啮合时啮合力方向并不一致，导致此时的啮合刚度计算不能沿用渐开线齿轮的方法。为进行有效的数据对比，这里的计量单位不再使用常见的啮合刚度，而是使用扭转刚度进行对比。
在不考虑齿轮变形的情况下，两个齿轮在各个啮合位置的角位移之比应与传动比一致，而由于齿轮变形的影响，在以齿轮1的角位移为参照的情况下，齿轮2的角位移会比其理论值略大，偏差为θ，同时提取齿轮2所受负载转矩，将负载转矩除以角位移偏差值即可得到扭转刚度，原理如图12所示，计算式为
各种齿廓的扭转刚度及传动重合度如图13、表4所示，其中最大啮合冲击比率定义为各种齿廓在双对齿啮合区的首尾处的扭转刚度值除以其邻近单对齿啮合区的扭转刚度值的最大值，以此反映啮入与啮出过程中因刚度变化引起的冲击的程度。
较高相对曲率的CRC齿轮虽然具有更大的重合度，但其扭转刚度相较于其他齿轮也明显降低，且最大啮合冲击也有所增大。在相对曲率为0.111 4 mm-1的CRC齿轮基础上设计的两种PRC齿轮的传动重合度略有提高，但最大啮合冲击比率略大于渐开线齿轮。而SRC齿轮与渐开线齿轮相比，在重合度与扭转刚度上都有提升，且最大啮合冲击比率有所降低，说明该设计方法符合预期。
2.3　接触应力与弯曲应力分析
接触应力与弯曲应力均可通过有限元模型获得，根据LIU等[16]1349-1364的研究，在本文所用的网格尺寸下，通过有限元模型获得的结果与解析计算获得的结果基本一致，其中有限元模型中的弯曲应力通过第四强度理论，即畸变能理论获得等效的Mises应力，而接触应力可直接从有限元模型中的对应场输出获得。
通过准静态啮合仿真分别获取单对齿啮合区间小齿轮弯曲应力最大值与接触应力最大值，此时CRC齿轮1的弯曲应力以及接触应力云图如图14所示。基于赫兹接触原理与XIE等[18]提出的用于计算双对齿啮合情况下载荷分配系数的最小势能原理，计算了6种齿廓沿啮合线的赫兹接触应力分布，如图15所示，并结合有限元模型分析了最大接触应力的情况，各个齿廓的啮合过程最大弯曲应力与最大接触应力如表5所示。
以渐开线齿轮为参照，除去作为对照的Kr=0.16 mm-1的CRC齿轮2，其他几种齿廓均获得了小于同参数渐开线齿轮的弯曲应力。其中，基于Kr=0.111 4 mm-1的CRC齿轮1设计的PRC齿轮的弯曲应力均有所降低，而SRC齿轮的弯曲应力略大于其他齿轮，但仍然低于渐开线齿轮。
由于接触应力主要取决于接触点的接触力与相对曲率，在主要参数相同的情况下，单对齿啮合区间下的相对曲率直接决定了最大接触应力，SRC齿轮的最大接触应力约等于相同参数下渐开线齿轮在单对齿啮合时的最小接触应力，通过有限元模型获取的各齿轮最大接触应力也印证了图14的计算结果，结果表明，在单对齿啮合区间进行相对曲率控制对降低最大接触应力效果最大，而在双对齿接触时接触应力较低，对其进行控制的效果不明显。
3　结论
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通过采用考虑齿轮时变啮合特性的相对曲率控制策略，设计了PRC齿轮与SRC齿轮，并基于有限元模型对其应力及啮合刚度进行了评估。
结果表明，PRC齿轮与SRC齿轮的设计方法能够克服CRC齿轮传动重合度降低的缺点，且所设计的PRC齿轮与SRC齿轮的啮合刚度、最大弯曲应力、最大接触应力也优于同参数的渐开线齿轮。因此，适当提高双对齿啮合区的相对曲率可以在不削弱轮齿强度的情况下实现传动重合度的提升，是一种有效的齿廓设计方法。
基金
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  • 国家自然科学基金项目(52075238)
  • 南京航空航天大学“直升机传动技术重点实验室”基金课题(HTL-A-21G04)
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2025年第47卷第4期
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doi: 10.16579/j.issn.1001.9669.2025.04.013
  • 接收时间：2023-05-30
  • 首发时间：2026-03-20
  • 出版时间：2025-04-15
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  • 收稿日期：2023-05-30
  • 修回日期：2023-06-27
基金
National Natural Science Foundation of China(52075238)
国家自然科学基金项目(52075238)
Foundation of National Key Laboratory of Science and Technology on Helicopter Transmission of Nanjing University of Aeronautics and Astronautics(HTL-A-21G04)
南京航空航天大学“直升机传动技术重点实验室”基金课题(HTL-A-21G04)
作者信息
    南京航空航天大学 机电学院，南京　210016

通讯作者:

刘雷，男，1978年生，江苏连云港人，博士，副教授；主要研究方向为齿轮传动；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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