准双曲面齿轮传动正向设计方法研究进展与趋势

doi:10.3901/JME.2025.19.018

摘要/Abstract

摘要： 准双曲面齿轮作为一种在航空、特种车辆、精密传动领域广泛应用的复杂齿面空间传动，其啮合质量直接决定整机服役性能。尽管该型齿轮传动在设计理论、创成机理、形貌优化、加工制造等方面取得了显著进展，但高端装备日益苛刻的服役性能需求，对该型传动的正向设计提出了更高挑战。结合文献剖析、市场调研、课题研究，详细阐述了该型传动主动设计方法研究进展与发展趋势，着重介绍了该型传动的几何参数设计、制造参数计算、接触特性分析、齿面形貌优化等内容，详细分析了该型传动在人工智能大背景下面向高端装备服役需求的发展趋势，期望为该领域研究人员与技术人员提供相关理论技术支撑，推动我国准双曲面齿轮正向设计技术的发展。

关键词: 准双曲面齿轮, 正向设计方法, 齿面创成技术, 齿面形貌优化

Abstract: The hypoid gear, as a complex spatial transmission widely utilized in aviation, specialized vehicles, and precision drive systems, has its meshing quality directly impacting the service performance of the entire machine. Although significant progress has been made in the design theory, generation mechanism, surface optimization, and manufacturing of this type of gear transmission, the increasingly performance requirements of high-level equipment present greater challenges for the active design of such transmissions. A detailed exposition of the research progress and development trends in the forward design methodologies of this type of transmission considering literature review, market research, and project studies has been provided. It focused on the configuration design, geometric parameter design, manufacturing parameter design, contact analysis, and tooth surface geometrical optimization of hypoid gears. Moreover, it systematically outlines the development trends of this transmission type to meet the service demands of high-level equipment and artificial intelligence. The aim is to provide theoretical and technical support for researchers and engineers in this field and to promote the advancement of hypoid gear forward design technology in China.

Key words: hypoid gear, active design methodology, tooth surface generating technology, tooth surface geometrical optimization

中图分类号:

TG156

刘思远, 宋朝省, 朱才朝, 梁成成, 牛强. 准双曲面齿轮传动正向设计方法研究进展与趋势[J]. 机械工程学报, 2025, 61(19): 18-42.

LIU Siyuan, SONG Chaosheng, ZHU Caichao, LIANG Chengcheng, NIU Qiang. State of Art and Trend of Hypoid Gear Drive of Active Design Methodology[J]. Journal of Mechanical Engineering, 2025, 61(19): 18-42.

参考文献

[1] STEWART A. WILDHABER E. Design，production and application of the hypoid rear-axle gear[R]. SAE Technical，1926.
[2] WILDHABER E. Gear tooth curvature treated simply[J]. American Mechinst，1945，8(18)：28-35.
[3] WILDHABER E. Basic relationship of hypoid gears[J]. American Machinist，1945，90(20)：24-30.
[4] WILDHABER E. Basic relationship of hypoid gears-II[J]. American Machinist，1946，90(3)：108-111.
[5] WILDHABER E. Basic relationship of hypoid gears-III[J]. American Machinist，1946，90(5)：131-134.
[6] WILDHABER E. Conjugate pitch surface[J]. American Machinist，1946，90(10)：122-135.
[7] WILDHABER E. Hypoid gear：U.S. 1，826，852[P]. 1931-10-13.
[8] WILDHABER E. Machine for generating gears：U.S. 2，145，000[P]. 1939-1-24.
[9] EDWARD S. Cutter and method for cutting nongenerated spiral bevel and hypoid gears：U.S. 3，283，659[P]. 1966-11-8.
[10] COLEMAN W. Paper 13：Bevel and hypoid gear surface durability：Pitting and scuffing[C]// Proceedings of the Institution of Mechanical Engineers，Conference Proceedings. Sage UK：London，England：SAGE Publications，1967：191-204.
[11] PITTS L. Bevel and hypoid gear noise reduction[R]. SAE Technical Paper，1972.
[12] WILCOX L，COLEMAN W. Application of finite elements to the analysis of gear tooth stresses[J]. 1973(1)：1139-1148..
[13] COLEMAN W. Effect of mounting displacements on bevel and hypoid gear tooth strength[R]. SAE Technical Paper，1975.
[14] COLEMAN W. Analysis of mounting deflections on bevel and hypoid gears[R]. SAE Technical Paper，1975.
[15] GABRIELE L A. Gear rolling equipment：U.S. Patent 4，058，999[P]. 1977.
[16] WITHAM W. Hypoid gears in passenger cars and trucks[R]. SAE Technical Paper，1939.
[17] LITVIN F L，ZHANG Y. Determination of settings of a tilted head-cutter for generation of hypoid and spiral bevel gears[R]. NASA report，1988.
[18] DOONER D B，SEIREG A A. The kinematic geometry of gearing：A concurrent engineering approach[M]. London：John Wiley，1995.
[19] DOONER D B. On evolutoids and spatial involutoids to define hypoid flank geometry[J]. Mechanism and Machine Theory，2020，156：104150.
[20] 张波. 高减比准双曲面齿轮几何设计及啮合性能调控方法研究[D]. 洛阳：河南科技大学，2022. ZHANG Bo. Research on profile design and meshing performance regulation method of hypoid gear with high reduction ratio[D]. Luoyang：Henan University of Science and Technology，2022.
[21] 刘思远. 小交错角准双曲面齿轮传动几何设计与啮合特性研究[D]. 重庆：重庆大学，2021. LIU Siyuan. Research on geometry design methodology and mesh characteristic analysis of hypoid gear in low crossed shaft angle configuration[D]. Chongqing：Chongqing University，2021.
[22] ANSI/AGMA. ISO-23509 Bevel and hypoid gear geometry[S]. Geneva：International Organization for Standardization，2016.
[23] DIN 3965，Toleranzen für Kegelradverzahnungen[S]. 1986.
[24] 中华人民共和国国家市场监督管理总局，中国国家标准化管理委员会. GB/T 43146-2023锥齿轮和准双曲面齿轮几何学[S]. 北京：中国标准出版社，2023. General Administration of Quality Supervision，Inspection and Quarantine of the People’s Republic of China，Standardization Administration of the People’s Republic of China. GB/T 43146-2023 Bevel and hypoid gear geometry[S]. Beijing：Standards Press of China，2023.
[25] 中华人民共和国国家市场监督管理总局，中国国家标准化管理委员会. GB/Z 43147-2023锥齿轮设计建议[S]. 北京：中国标准出版社，2023. General Administration of Quality Supervision，Inspection and Quarantine of the People’s Republic of China，Standardization Administration of the People’s Republic of China. GB/Z 43147-2023 Design recommendations for bevel gears[S]. Beijing：Standards Press of China，2023.
[26] 中华人民共和国国家市场监督管理总局，中国国家标准化管理委员会. GB/T 11365-2019，锥齿轮精度制[S]. 北京：中国标准出版社，2019. General Administration of Quality Supervision，Inspection and Quarantine of the People’s Republic of China，Standardization Administration of the People’s Republic of China. GB/T 11365-2019 Bevel gears—ISO system of accuracy[S]. Beijing：Standards Press of China，2023.
[27] 朱孝录. 齿轮传动设计手册[M]. 北京：化学工业出版社，2005. ZHU Xiaolu. Gear transmission design manual[M]. Beijing：Chemical Industry Press，2005.
[28] LITVIN F L，FUENTES A. Gear geometry and applied theory[M]. Cambridge：Cambridge University Press，2004.
[29] 郑昌启. 弧齿锥齿轮和准双曲面齿轮：啮合原理，齿坯设计，加工调整和齿面分析计算原理[M]. 北京：机械工业出版社，1988. ZHENG Changqi. Spiral bevel gears and hypoid gears：Principle of meshing，gear blank design，principles of calculation for machining adjustment and flank analysis[M]. Beijing：China Machine Press，1988.
[30] 张金良，邓效忠，郭强，等. 弧齿锥齿轮与准双曲面齿轮设计与制造的现状[J]. 拖拉机与农用运输车，2002(5)：9-11. ZHANG Jinliang，DENG Xiaozhong，GUO Qiang，et al. Current status of design and manufacturing of spiral bevel and hypoid gears[J]. Tractor & Farm Transporter，2002(5)：9-11.
[31] 吴序堂. 齿轮啮合原理[M]. 西安：西安交通大学出版社，2009. WU Xutang. Theory of gearing[M]. Xi’an：Xi’an Jiao Tong University Press，2009.
[32] 曾韬. 螺旋锥齿轮设计与加工[M]. 哈尔滨：哈尔滨工业大学出版社，1989. ZENG Tao. Design and manufacturing of spiral bevel gear[M]. Harbi：Harbin Institute of Technology Press，1989.
[33] 董学朱. 齿轮啮合理论基础[M]. 北京：机械工业出版社，1989. DONG Xuezhu. Theoretical basis of gear meshing[M]. Beijing：China Machine Press，1989.
[34] 张金良，方宗德，曹雪梅，等. 准双曲面齿轮的修正节锥设计方法及切齿试验[J]. 机械工程学报，2007， 43(9)：185-189. ZHANG Jinliang，FANG Zongde，CAO Xuemei，et al. Modified pitch cone method in the hypoid gear design and tooth-cutting experiment[J]. Journal of Mechanical Engineering，2007，43(9)：185-189.
[35] STADTFELD H J. Hypoloid gears with small shaft angles and zero-to-large offsets[M]. Gleason Company Publication，2009.
[36] LIU S，ZHU C，FUENTES-AZNAR A，et al. Computerized approach for design and generation of face-milled non-generated hypoid gears with low shaft angle[J]. Mechanism and Machine Theory，2020，155：1040.
[37] LIU S，SONG C，ZHU C，et al. Effects of tooth modifications on mesh characteristics of crossed beveloid gear pair with small shaft angle[J]. Mechanism and Machine Theory，2018，119：142-160.
[38] LIU S，SONG C，ZHU C，et al. Concave and convex modifications analysis for skewed beveloid gears considering misalignments[J]. Mechanism and Machine Theory，2019，133：127-149.
[39] LIU S，SONG C，ZHU C，et al. Concave modifications of tooth surfaces of beveloid gears with crossed axes[J]. Proceedings of the Institution of Mechanical Engineers，Part C：Journal of Mechanical Engineering Science，2019，233(4)：1411-1425.
[40] LITVIN F L，SEOL I H. Computerized determination of gear tooth surface as envelope to two parameter family of surfaces[J]. Computer Methods in Applied Mechanics & Engineering，1996，138(1-4)：213-225.
[41] ARGYRIS J，LITVIN F L，LIAN Q，et al. Determination of envelope to family of planar parametric curves and envelope singularities[J]. Computer Methods in Applied Mechanics & Engineering，1999，175(s1-2)：175-187.
[42] 王延忠，王建甫，周元子，等. 航空弧齿锥齿轮的齿面偏差最小化[J]. 机床与液压，2008，36(6)：6-10. WANG Yanzhong，WANG Jianfu，ZHOU Yuanzi，et al. Minimization of tooth surface deviation in aerospace spiral bevel gears[J]. Machine Tool & Hydraulics，2008，36(6)：6-10.
[43] 王延忠，周云飞，周济，等. 考虑轮齿制造误差的螺旋锥齿轮加载接触分析[J]. 机械科学与技术，2002，21(2)：224-227. WANG Yanzhong，ZHOU Yunfei，ZHOU Ji，et al. Load contact analysis of spiral bevel gears considering tooth manufacturing errors[J]. Journal of Mechanical Science and Technology，2002，21(2)：224-227.
[44] 王延忠，刘溢溥，张祖智，等. 重载弧齿锥齿轮承载特性分析[J]. 新技术新工艺，2014(1)：60-63. WANG Yanzhong，LIU Yipu，ZHANG Zuzhi，et al. Analysis of load-bearing characteristics of heavy-duty spiral bevel gears[J]. New Technology & New Process，2014(1)：60-63.
[45] 王小林，王延忠，饶芳，等. 准双曲面齿轮齿面参数化方法研究[J]. 机械传动，2005，29(6)：23-26. WANG Xiaolin，WANG Yanzhong，RAO Fang，et al. Study on the parametric method of tooth surface for hypoid gears[J]. Mechanical Transmission，2005， 29(6)：23-26.
[46] 樊奇，让·德福. 格里森专家制造系统(GEMS)开创弧齿锥齿轮及双曲面齿轮数字化制造新纪元[J]. 世界制造技术与装备市场，2005(4)：87-93. FAN Qi，JEAN D. Grisson expert manufacturing system (GEMS) ushers in a new era of digital manufacturing for arc bevel gears and hypoid gears[J]. World Manufacturing Technology & Equipment Market，2005(4)：87-93.
[47] GRANGER C. Ricardo invests in high-performance transmission manufacturing[J]. Machinery Market，2018(1)：TN.6158.
[48] 佚名. 中大创远开展等高齿技术与软件培训服务[J]. 世界制造技术与装备市场，2016(3)：18. Anonymous. Zhongda Chuangyuan launches training services for isocline gear technology and software[J]. World Manufacturing Technology & Equipment Market，2016(3)：18.
[49] LITVIN F L，GONZALEZ-PEREZ I，FUENTES-ANZAR A，et al. Design，generation and stress analysis of face-gear drive with helical pinion[J]. Computer Methods in Applied Mechanics and Engineering，2005，36：3870-3901.
[50] 刘思远，宋朝省，朱才朝. 小交错角面铣准双曲面齿轮副的几何设计与啮合特性[J]. 中国机械工程，2022，33(13)：1521-1528. LIU Siyuan，SONG Chaosheng，ZHU Caichao. Geometric design and meshing characteristics of small crossed angle face-milled hypoid gear pairs[J]. China Mechanical Engineering，2022，33(13)：1521-1528.
[51] LIU S，ZHU C，FUENTES-AZNAR A，et al. Computerized determination of the qualified region of main design parameters of face-milled hypoid gears with low shaft angle[J]. Mechanism and Machine Theory，2021，159：104259.
[52] 董红涛. 弧齿锥齿轮技术研究的现状和发展趋势[J]. 机械传动，2012，36(10)：115-118. DONG Hongtao. Current status and development trends of spiral bevel gear technology research[J]. Mechanical Transmission，2012，36(10)：115-118.
[53] LIU S，SONG C，ZHU C，et al. Investigation on contact and bending stress of face-hobbed and face-milled hypoid gear[J]. Mechanism and Machine Theory，2020，150：103873.
[54] 董学朱. 弧齿锥齿轮变性全展成切齿调整计算新方法[J]. 齿轮，1988(6)：3-7. DONG Xuezhu. A new method for the adjustment calculation of modified full-generating tooth-cutting of spiral bevel gears[J]. Gear，1988(6)：3-7.
[55] 董学朱. 准双曲面齿轮刀倾半展成切齿调整计算新方法[J]. 齿轮，1988(4)：3-9. DONG Xuezhu. A new method for the adjustment calculation of cutter tilt half-generating tooth-cutting of hypoid gears[J]. Gear，1988(4)：3-9.
[56] 董学朱. 准双曲面齿轮刀倾全展成切齿调整计算方法[J]. 齿轮，1988(5)：1-6. DONG Xuezhu. Adjustment calculation method for cutter tilt full-generating tooth-cutting of hypoid gears[J]. Gear，1988(5)：1-6.
[57] 吴序堂. 刀倾半展成法加工弧齿锥齿轮及准双曲面齿轮的机床调整[J]. 机床，1981(11)：6-9. WU Xutang. Machine tool adjustment for cutter tilt half-generating method processing of spiral bevel gears and hypoid gears[J]. Machine Tool，1981(11)：6-9.
[58] 吴序堂. 准双曲面齿轮啮合原理及其在刀倾半展成加工中的应用[J]. 西安交通大学学报，1981(1)：11-26. WU Xutang. The meshing principle of hypoid gears and its application in cutter tilt half-generating processing[J]. Journal of Xi’an Jiaotong University，1981(1)：11-26.
[59] 董学朱，孙殿柱. 通过等距曲面进行真实齿面的啮合分析[J]. 中国农业大学学报，1999(4)：61-63. DONG Xuezhu，SUN Dianzhu. Meshing analysis of real tooth surfaces through equidistant surfaces[J]. Journal of China Agricultural University，1999(4)：61-63.
[60] 董学朱. 延伸外摆线齿准双曲面齿轮几何设计和切齿调整计算新方法[J]. 机械传动，1999，23(4)：16-19. DONG Xuezhu. A new method for the geometric design and tooth-cutting adjustment calculation of extended hypocycloid hypoid gears[J]. Mechanical Transmission，1999，23(4)：16-19.
[61] 郭晓东，郑昌启，林超. 锥齿轮设计制造现代应用技术的研究[J]. 重庆大学学报(自然科学版)，1993(1)：37-44. GUO Xiaodong，ZHENG Changqi，LIN Chao. Research on modern application technology of bevel gear design and manufacturing[J]. Journal of Chongqing University (Natural Science Edition)，1993(1)：37-44.
[62] 王星. HGT准双曲面齿轮的齿面设计及啮合特性分析[D]. 西安：西北工业大学，2016. WANG Xing. Tooth surface design and meshing characteristics analysis of HGT hypoid gears[D]. Xi’an：Northwestern Polytechnical University，2016.
[63] LITVIN F L，GUTMAN Y E. Methods of synthesis and analysis for hypoid gear-drives of “formate” and “helixform”—Part 1. Calculations for machine settings for member gear manufacture of the formate and helixform hypoid gears[J]. Journal of Mechanical Design，1981(1)：83-88.
[64] LITVIN F L，GUTMAN Y. Methods of synthesis and analysis for hypoid gear-drives of “formate” and “helixform”—Part 2. Machine setting calculations for the pinions of formate and helixform gears[J]. Journal of Mechanical Design，1981(1)，89-101.
[65] LITVIN F L，GUTMAN Y. Methods of synthesis and analysis for hypoid gear-drives of “formate” and “helixform”—Part 3. Analysis and optimal synthesis methods for mismatch gearing and its application for hypoid gears of “formate” and “helixform”[J]. Journal of Mechanical Design，1981(1)，122-125.
[66] TSAY C，LIN J. A mathematical model for the tooth geometry of hypoid gears[J]. Mathematical and Computer Modelling，1993，18(2)：23-34.
[67] LIN C，TSAY C. Computer-aided manufacturing of spiral bevel and hypoid gears with minimum surface deviation[J]. Mechanism and Machine Theory，1998，33(6)：785-803.
[68] FUENTES-AZNAR A，RUIZ-ORZAEZ R，GONZALEZ-PEREZ I. Numerical approach for determination of rough-cutting machine-tool settings for fixed-setting face-milled spiral bevel gears[J]. Mechanism and Machine Theory，2017，112：22-42.
[69] LIN C Y，TSAY C B，FONG Z H. Mathematical model of spiral bevel and hypoid gears manufactured by the modified roll method[J]. Mechanism and Machine Theory，1997，32(2)：121-136.
[70] LIN C Y，TSAY C B，FONG Z H. Computer-aided manufacturing of spiral bevel and hypoid gears by applying optimization techniques[J]. Journal of Materials Processing Technology，2001，114(1)：22-35.
[71] 续鲁宁. 摆线齿锥齿轮刀具制造及检测方法[D]. 重庆：重庆理工大学，2010. XU Luning. Manufacture and inspection methods of cycloidal tooth bevel gear cutters[D]. Chongqing：Chongqing University of Technology，2010.
[72] ZHANG W，CHEN B. A motion control method for face hobbing on CNC hypoid generator[J]. Mechanism & Machine Theory，2015，92：127-143.
[73] 梁成成，宋朝省，朱才朝，等. 基于刀具法向基准的奥利康准双曲面齿轮精确建模与验证[J]. 重庆大学学报，2020，43(2)：1-11. LIANG Chengcheng，SONG Chaosheng，ZHU Caichao，et al. Accurate modeling and verification of Orlikon hypoid gears based on tool normal benchmark[J]. Journal of Chongqing University，2020，43(2)：1-11.
[74] RONG K，DING H，CHEN S，et al. Top-Rem grinding tool modification considering loaded edge contact for spiral bevel gears[J]. Advanced Engineering Informatics，2022，53：101697.
[75] LIU S，SONG C，ZHU C，et al. Investigation on the influence of work holding equipment errors on contact characteristics of face-hobbed hypoid gear[J]. Mechanism and Machine Theory，2019，138：95-111.
[76] LIANG C，SONG C，ZHU C，et al. Investigation of tool errors and their influences on tooth surface topography for face-hobbed hypoid gears[J]. Journal of Mechanical Design，2020，142(4)：044501.
[77] CAI K，SONG C，LIANG C，et al. Investigation of the effect of multi-segment cutter section on meshing characteristics for face-hobbed hypoid gears[J]. Journal of Advanced Mechanical Design Systems and Manufacturing，2023，17(6)：0080.
[78] LIANG C，SONG C，ZHU C，et al. Investigation of the effects with linear，circular and polynomial blades on contact characteristics for face-hobbed hypoid gears[J]. Mechanism and Machine Theory，2020，146：103739.
[79] LIU S，ZHU C，SONG C，et al. Compensation of errors of alignment and contact pattern repositioning in hypoid gears with low crossing shaft angle[J]. Meccanica，2021(1)：1-15.
[80] 梁成成. 基于三面刀盘的端面滚齿准双曲面齿轮啮合性能分析与实验研究[D]. 重庆：重庆大学，2021. LIANG Chengcheng. Analysis and experimental study of the meshing performance of end-face hobbing hypoid gears based on three-face cutter disk[D]. Chongqing：Chongqing University，2021.
[81] 刘赣华，邹洋，钱锦年. 压力淬火对弧齿锥齿轮齿形影响仿真研究[J]. 机械设计与制造，2018(5)：141-147. LIU Ganhua，ZOU Yang，QIAN Jinnian. Simulation study on the influence of pressure quenching on the tooth profile of spiral bevel gears[J]. Mechanical Design and Manufacture，2018(5)：141-147.
[82] 刘赣华，唐乃夫，汪啟. 等距螺旋锥齿轮的精确建模与金属粉末注射成型工艺试制[J]. 粉末冶金技术，2024，42(2)：207-214. LIU Ganhua，TANG Naifu，WANG Qi. Precise modeling and metal powder injection molding process trial of equidistant spiral bevel gears[J]. Powder Metallurgy Technology，2024，42(2)：207-214.
[83] 邓诗义. 基于热流固耦合模型的螺旋锥齿轮淬火冷却过程数值模拟与实验研究[D]. 赣州：江西理工大学，2023. DENG Shiyi. Numerical simulation and experimental study of the quenching cooling process of spiral bevel gears based on thermo-fluid-solid coupling model[D]. Ganzhou：Jiangxi University of Science and Technology，2023.
[84] 刘付洋，刘赣华，谢显峰，等. 弧齿锥齿轮动态热边界渗碳淬火工艺优化设计[J]. 机械传动，2018(3)：45-48. LIU Fuyang，LIU Ganhua，XIE Xianfeng，et al. Dynamic thermal boundary carburizing quenching process optimization design of spiral bevel gears[J]. Mechanical Transmission，2018(3)：45-48.
[85] FAN Q. Computerized modeling and simulation of spiral bevel and hypoid gears manufactured by Gleason face hobbing process[J]. Journal of Mechanical Design，2006(1)：1315-1327.
[86] SHEVELEVA G I，VOLKOV A E，MEDVEDEV V I. Algorithms for analysis of meshing and contact of spiral bevel gears[J]. Mechanism and Machine Theory，2007，42(2)：198-215.
[87] 李家琦，郭玉梁，魏冰阳，等. 曲面综合法复杂齿面啮合仿真分析方法研究[J]. 机械传动，2022，46(11)：16-20. LI Jiaqi，GUO Yuliang，WEI Bingyang，et al. Research on the simulation analysis method of complex tooth surface meshing using the surface integration method[J]. Mechanical Transmission，2022，46(11)：16-20.
[88] 古德万，魏冰阳，任明辉，等. EASE-OFF修形高减速比准双曲面齿轮接触仿真与动态性能试验[J]. 机械传动，2023，47(5)：128-133. GU Dewan，WEI Bingyang，REN Minghui，et al. EASE-OFF shaping high reduction ratio hypoid gear contact simulation and dynamic performance testing[J]. Mechanical Transmission，2023，47(5)：128-133.
[89] POWELL D，BARTON H R. Analytical study of surface loading and sliding velocity of automotive hypoid gears[J]. ASLE Transactions，1959，2(2)：173-183.
[90] WILCOX L，COLEMAN W. Application of finite elements to the analysis of gear tooth stresses[J]. Journal of Manufacturing Science & Engineering，1973，95(4)：1139-1148.
[91] 汪中厚，周晓玲. 螺旋锥齿轮动力学研究方法及进展[J]. 中国机械工程，2006，17(11)：1203-1208. WANG Zhonghou，ZHOU Xiaoling. Research methods and progress in the dynamics of spiral bevel gears[J]. China Mechanical Engineering，2006，17(11)：1203-1208.
[92] 李特文，国楷，叶凌云，等. 齿轮几何学与应用理论[M]. 上海：上海科学技术出版社，2008. LITVIN F L，GUO Kai，YE Lingyun，et al. Gear geometry and applied theory[M]. Shanghai：Shanghai Scientific & Technical Publishers，2008.
[93] VIJAYAKAR S. A combined surface integral and finite contact problem element solution for a three-dimensional[J]. Int. J. Numer. Methods Eng，1991，31：525-545.
[94] ABAQUS/Standard User’s Manual[M]. Rhode Island，2018.
[95] CHEN P，WANG S，ZOU H. Efficient semi-analytic method for single tooth contact analysis of loaded spiral bevel gears[J]. Applied Mathematical Modelling，2024，129：754-779.
[96] LIU Z. Semi-analytical loaded tooth contact analysis method for spiral bevel gears[J]. International Journal of Mechanical Sciences，2023，253：108329.
[97] QU W，DING H，TANG J. An innovative semi- analytical determination approach to numerical loaded tooth contact analysis (NLTCA) for spiral bevel and hypoid gears[J]. Advances in Engineering Software，2020，149：102892.
[98] HU Z. Numerical determination to loaded tooth contact performances in consideration of misalignment for the spiral bevel gears[J]. International Journal of Mechanical Sciences，2019，151：343-355.
[99] WEI B Y，LI J Q，CAO X M，et al. Calculation of gear mesh stiffness and loaded tooth contact analysis based on EASE-OFF surface topology[J]. Advances in Mechanical Engineering，2022，14(11)：1-11.
[100] PENG S，DING H，TANG J. Accurate numerical computation of loaded tooth surface contact pressure and stress distributions for spiral bevel gears by considering time-varying meshing characteristics[J]. Advances in Engineering Software，2019，135：102683.
[101] KOLIVAND M，KAHRAMAN A. An EASE-OFF based method for loaded tooth contact analysis of hypoid gears having local and global surface deviations[J]. Journal of Mechanical Design，2009，132(7)：245-254.
[102] KOLIVAND M，KAHRAMAN A. A load distribution model for hypoid gears using EASE-OFF topography and shell theory[J]. Mechanism and Machine Theory，2009，44(10)：1848-1865.
[103] ARTONI A，GABICCINI M，KOLIVAND M. EASE-OFF based compensation of tooth surface deviations for spiral bevel and hypoid gears：Only the pinion needs corrections[J]. Mechanism and Machine Theory，2013，61：84-101.
[104] DING H，WAN Z，ZHOU Y，et al. A data-driven programming of the human-computer interactions for modeling a collaborative manufacturing system of hypoid gears by considering both geometric and physical performances[J]. Robotics and Computer-Integrated Manufacturing，2018，51：121-138.
[105] HE W，SONG C，LIU S，et al. Targeted tooth surface predesign and machine-tool settings identification methodology of spiral bevel gear considering EASE- OFF[J]. Journal of Advanced Mechanical Design，Systems，and Manufacturing，2024，18(5)：0059.
[106] CHEN X，DING H，WANG Y，et al. Sensitive data-driven tooth surface collaborative grinding model for aerospace spiral bevel gears[J]. Simulation Modelling Practice and Theory，2022，119：102566.
[107] DING H，Li Y，ZHANG Y，et al. High-performance tooth flank collaborative optimization model for spiral bevel and hypoid gears[J]. Advanced Engineering Informatics，2023，57：102059.
[108] 严宏志，肖蒙，胡志安，等. 基于EASE-OFF的螺旋锥齿轮齿面分区修形方法[J]. 中南大学学报，2018，49(4)：824-830. YAN Hongzhi，XIAO Meng，HU Zhian，et al. EASE-OFF based tooth surface zone modification method for spiral bevel gears[J]. Journal of Central South University，2018，49(4)：824-830.
[109] 陈义忠，严宏志，艾伍轶，等. 圆弧刀廓加工螺旋锥齿轮的全齿面分区修形[J]. 制造业自动化，2019，41(11)：136-142. CHEN Yizhong，YAN Hongzhi，AI Wuyi，et al. Full tooth surface zone modification of spiral bevel gears using arc-profile cutter[J]. Manufacturing Automation，2019，41(11)：136-142.
[110] MU Y，FANG Z. Design and analysis of high contact ratio spiral bevel gears by modified curvature motion method[J]. Proceedings of the Institution of Mechanical Engineers，Part C：Journal of Mechanical Engineering Science，2018，232(19)：3396-3409.
[111] 苏进展，魏刚，杨羽，等. 大重合度弧齿锥齿轮设计与分析[J]. 西安交通大学学报，2021，55(3)：117-125. SU Jinzhang，WEI Gang，YANG Yu，et al. Design and analysis of high contact ratio spiral bevel gears[J]. Journal of Xi'an Jiaotong University，2021，55(3)：117-125.
[112] 苏进展，许鹏，刘镔，等. 弧齿锥齿轮复合传动误差设计与分析[J]. 西安交通大学学报，2022，56(11)：41-48. SU Jinzhang，XU Peng，LIU Bin，et al. Design and analysis of compound transmission error in spiral bevel gears[J]. Journal of Xi’an Jiaotong University，2022，56(11)：41-48.
[113] ARTONI A，KOLIVAND M，KAHRAMAN A. An EASE-OFF based optimization of the loaded transmission error of hypoid gears[J]. Journal of Mechanical Design，2010，132(1)：011010.
[114] ARTONI A，GABICCINI M，GUIGGIANI M，et al. Multi-objective EASE-OFF optimization of hypoid gears for their efficiency，noise，and durability performances[J]. Journal of Mechanical Design，2011，133(12)：121007.
[115] SIMON V. Generation of hypoid gears on CNC hypoid generator[J]. Journal of Mechanical Design，2011，133(12)：121003.
[116] SIMON V. Optimization of face-hobbed hypoid gears[J]. Mechanism and Machine Theory，2014，77：164-181.
[117] SIMON V. FEM stress analysis in hypoid gears[J]. Mechanism and Machine Theory，2000，35(9)：1197-1220.
[118] ZHOU Z，PANG J，GAO F，et al. Digital visualization of time-varying local bearing contact recognition for non-orthogonal aviation spiral bevel gears[J]. Advanced Engineering Informatics，2023，57：102098.
[119] LEI D，RONG K，SONG B，et al. Digital twin modeling for tooth surface grinding considering low-risk transmission performance of non-orthogonal aviation spiral bevel gears[J]. ISA Transactions，2022，128：646-663.
[120] ARTONI A，GABICCINI M，GUIGGIANI M. Grinding face-hobbed hypoid gears through full exploitation of 6-axis hypoid generators[C]// International Gear Conference 2014-Conference Proceedings Volume I. Woodhead Publishing，2014：107-117.
[121] ZHANG W，GUO X，WANG Y，et al. A CNC tooth grinding method for formate face hobbed hypoid gears[J]. Mechanism and Machine Theory，2020，144：103628.
[122] ZHANG W，WEI X，GUO X，et al. A novel continuous indexing method for face-hobbed hypoid gear tooth grinding[J]. Mechanism and Machine Theory，2022，173：104826.
[123] FAN Q. Computerized modeling and simulation of spiral bevel and hypoid gears manufactured by Gleason face hobbing process[J]. Journal of Mechanical Design，2006，128(6)：1315-1327.
[124] FAN Q. Enhanced algorithms of contact simulation for hypoid gear drives produced by face-milling and face-hobbing processes[J]. ASME Journal of Mechanical Design，2006，128：1315-1327.
[125] FAN Q，DAFOE R S，SWANGER J W. Higher-order tooth flank form error correction for face-milled spiral bevel and hypoid gears[J]. Journal of Mechanical Design，2008，130(7)：1029-1033.
[126] SHIH Y. A novel EASE-OFF flank modification methodology for spiral bevel and hypoid gears[J]. Mechanism & Machine Theory，2010，45(8)：1108-1124.
[127] SHIH Y，FONG Z. Flank correction for spiral bevel and hypoid gears on a six-axis CNC hypoid generator[J]. Journal of Mechanical Design，2008，130(6)：876-877.
[128] ARTONI A，GABICCINI M，KOLIVAND M. EASE- OFF based compensation of tooth surface deviations for spiral bevel and hypoid gears：Only the pinion needs corrections[J]. Mechanism and Machine Theory，2013，61：84-101.
[129] 古德万，魏冰阳，任明辉，等. EASE-OFF修形高减速比准双曲面齿轮接触仿真与动态性能试验[J]. 机械传动，2023，47(5)：128-133. GU Dewan，WEI Bingyang，REN Minghui，et al. Contact simulation and dynamic performance test of ease-off modified high reduction hypoid gears[J]. Journal of Mechanical Transmission，2023，47(5)：128-133.
[130] WANG Qin. Design and analysis for hypoid gears with EASE-OFF flank modification[J]. Applied Sciences，2022，12(2)：822.