Analysis and Optimization of Meshing Performance of Straight Bevel Gears Machined by Dual Interlocking Circular Cutters

doi:10.3969/j.issn.1004-132X.2025.08.004

Abstract

Abstract:

A method for milling/grinding straight-toothed bevel gears using high-efficiency double-arc cutting tools was proposed to improve their meshing performance. A double arc tool was used to simulate a crown-producing wheel， and tool trimming coefficients， the average radius of the cutter disk and the cutting edge angle were introduced to establish a straight bevel gear tooth face model that considered tooth profile and tooth direction trimming. According to the conditions of continuous tangency of two meshing gears， a tooth contact analysis model was established， and combined with tooth clearance， normal flexibility matrix and mathematical planning， a wheel tooth bearing contact analysis model was established. Then with the goal of achieving symmetry in geometric transmission errors and minimising fluctuations in load transmission errors， the tool profiling origin positions was adjusted and tool parameters were optimised. The geometric transmission errors and load transmission errors of the gear were compared before and after optimization. The example results show that the position of the tool reshaping origin affects the symmetry of the geometric transmission errors； the larger the tool reshaping coefficient is， the larger the amount of tooth profile reshaping is， the amplitude of the geometric transmission errors and the amplitude of the load bearing transmission errors will increase； with the decrease of the average radius of the knife holder and the increase of the blade angle， the amount of tooth profile reshaping is increased， and then the area of the tooth surface mark will be reduced. After optimization， the amount of bearing transmission error fluctuation is reduced by 56.54% under the working load， which may effectively reduce the vibration excitation of the gear pairs.

Key words: dual interlocking circular cutter, straight bevel gear, tooth surface modification, meshing simulation, genetic optimization

摘要：

提出了高效双圆弧刀具铣/磨直齿锥齿轮方法，以改善其啮合性能。采用双圆弧刀具模拟冠状产形轮，引入刀具修形系数、刀盘平均半径和刀刃角，建立考虑齿廓和齿向修形的直齿锥齿轮齿面模型。根据两啮合齿轮连续相切条件建立齿面接触分析模型，结合齿面间隙、法向柔度矩阵和数学规划建立轮齿承载接触分析模型。在此基础上，以几何传动误差对称性和承载传动误差波动量最小为目标，调整刀具修形原点位置和优化刀具参数，对比优化前后齿轮几何传动误差和承载传动误差。算例结果表明，刀具修形原点位置会影响几何传动误差的对称性，刀具修形系数越大，齿廓修形量越大，则几何传动误差幅值和承载传动误差幅值越大；随着刀盘平均半径的减小和刀刃角的增大，齿向修形量增大，则齿面印痕区域减小。优化后在工作载荷下，承载传动误差波动量减小56.54%，有效降低了齿轮副的振动激励。

关键词: 双圆弧刀具, 直齿锥齿轮, 齿面修形, 啮合仿真, 遗传算法

CLC Number:

TH132

Jinzhan SU, Yaoke FENG, Bin LIU, Xinlong CAO, Linlin SUN. Analysis and Optimization of Meshing Performance of Straight Bevel Gears Machined by Dual Interlocking Circular Cutters[J]. China Mechanical Engineering, 2025, 36(8): 1683-1690.

苏进展, 冯要克, 刘镔, 曹新龙, 孙霖霖. 双圆弧刀具加工直齿锥齿轮的啮合性能分析及优化[J]. 中国机械工程, 2025, 36(8): 1683-1690.

Figures/Tables 16

Fig.1 Tool coordinate system

Fig.2 Relative position of the crown generating wheel and the double arc cutter

Fig. 3 Blade angle δ

Fig.4 Expanded coordinate system for straight bevel gears

Fig.5 Transformation of meshing coordinates

Fig.6 Optimize processes

Tab.1 Basic gear parameters

参数	小轮	大轮
齿数N_i	25.0	36.0
模数m/mm	5.0
压力角α/（°）	25.0
轴交角∑/（°）	90.0
齿宽F_w/mm	29.2
齿顶高系数h_a	1.0
齿根高系数h_f	1.25

Tab.2 Tool parameters

参数	小轮	大轮
刀盘平均半径ρ_m/mm	80.0/160.0/200.0	200.0
刀顶圆角半径ρ_f/mm	0.8	0.8
刀刃角δ/（°）	1.5/2.0/2.5	2.0
修形系数a_t	0.0001/0.0002/0.0003	0.0

Fig.7 TCA and LTCA results for different blade mean radii

Fig.8 TCA and LTCA results for different knife edge angles

Fig.9 TCA and LTCA results with different trimming coefficients

Tab.3 Optimizing the boundaries of the parameters

	a_t	δ/（°）	ρ_m/ mm
上界	0.0	0.0	100.0
下界	0.0008	2.5	300.0

Tab.4 Parameter optimization results

a_t1	δ₁/（°）	ρ_m1/ mm
0.000 56	1.7832	181.6523

Fig.10 TCA and LTCA results

Fig.11 Tooth surface deviation

Fig.12 Comparison of transmission error amplitude for different loads

References 23

[1]	STADTFELD H J. Straight Bevel Gears on Phoenix Machines Using Coniflex Tools ［J］. Gear Solutions， 2007， 2007（10）： 32-39.
[2]	STADTFELD H J. Straight Bevel Gear Cutting and Grinding on CNC Free Form Machines［C］∥AGMA Fall Technical Meeting. Detroit，2007： 7-9.
[3]	FUENTES-AZNAR A， GONZALEZ-PEREZ I. Mathematical Definition and Computerized Modeling of Spherical Involute and Octoidal Bevel Gears Generated by Crown Gear［J］. Mechanism and Machine Theory， 2016， 106： 94-114.
[4]	FUENTES-AZNAR A， YAGUE-MARTINEZ E， GONZALEZ-PEREZ I. Computerized Generation and Gear Mesh Simulation of Straight Bevel Gears Manufactured by Dual Interlocking Circular Cutters［J］. Mechanism and Machine Theory， 2018， 122： 160-176.
[5]	SHIH Y P， HSIEH H Y. Straight Bevel Gear Generation Using the Dual Interlocking Circular Cutter Cutting Method on a Computer Numerical Control Bevel Gear-cutting Machine［J］. Journal of Manufacturing Science and Engineering， 2016， 138（2）： 021007.
[6]	SHIH Y P. A Lengthwise Modification for Face-Hobbed Straight Bevel Gears［C］∥ 25th International Conference on Design Theory and Methodology； ASME 2013 Power Transmission and Gearing Conference. Portland， 2013： V005T11A002.
[7]	SHIH Y P. Mathematical Model for Face-hobbed Straight Bevel Gears［J］. Journal of Mechanical Design， 2012， 134（9）： 091006.
[8]	SHIH Y P， HUANG Yachuan， LEE Yihui， et al. Manufacture of Face-hobbed Straight Bevel Gears Using a Six-axis CNC Bevel Gear Cutting Machine［J］. The International Journal of Advanced Manufacturing Technology， 2013， 68（9）： 2499-2515.
[9]	王志永，李文强，张宇.双刀盘铣齿机床与Free-form型机床的加工运动等效转换［J］.机械工程学报，2024，60（21）：312-319.
	WANG Zhiyong， LI Wenqiang， ZHANG Yu. Equivalent Conversion of Machining Motion between Double Cutter Disks Gear Milling Machine Tool and Free-form Machine Tool［J］. Journal of Mechanical Engineering， 2024，60（21）：312-319.
[10]	王志永，张竹，段志宏，等. 内凹刀盘加工的直齿锥齿轮齿面接触分析［J］. 机械传动， 2024， 48（2）： 104-109.
	WANG Zhiyong， ZHANG Zhu， DUAN Zhihong， et al. Tooth Contact Analysis of Straight Bevel Gears Machined with Concave Cutters［J］. Journal of Mechanical Transmission， 2024， 48（2）： 104-109.
[11]	蒋进科，刘钊，刘红梅. 齿面磨损最小直齿锥齿轮Ease-off修形设计与分析［J］. 西安交通大学学报， 2020， 54（6）： 99-106.
	JIANG Jinke， LIU Zhao， LIU Hongmei. Design and Analysis for Straight Bevel Gears with Ease-off Flank Modification Based on Minimal Wear［J］. Journal of Xi’an Jiaotong University， 2020， 54（6）： 99-106.
[12]	周延泽，吴继泽. 直齿锥齿轮齿根应力的有限元分析［J］. 北京航空航天大学学报， 1996， 22（1）： 88-93.
	ZHOU Yanze， WU Jize. Finite Element Analysis of Tooth Root Stress of Straight Bevel Gear［J］. Journal of Beijing University of Aeronautics and Astronautics， 1996， 22（1）： 88-93.
[13]	曹雪梅，娄佳佳，马战勇. 直齿锥齿轮修形齿面安装误差敏感性分析［J］. 机械传动， 2014， 38（4）： 40-43.
	CAO Xuemei， LOU Jiajia， MA Zhanyong. Sensitivity Analysis of Installation Errors of the Straight Bevel Gear Modification Tooth Surface［J］. Journal of Mechanical Transmission， 2014， 38（4）： 40-43.
[14]	曹雪梅，孙宁，邓效忠. 直齿锥齿轮低安装误差敏感性设计与实验验证［J］. 航空动力学报， 2016， 31（1）： 227-232.
	CAO Xuemei， SUN Ning， DENG Xiaozhong. Design for Straight Bevel Gear Based on Low Installation Error Sensitivity and Experiment Tests［J］. Journal of Aerospace Power， 2016， 31（1）： 227-232.
[15]	KOLIVAND M， LIGATA H， STEYER G， et al. Actual Tooth Contact Analysis of Straight Bevel Gears［J］. Journal of Mechanical Design， 2015， 137（9）： 093302.
[16]	陈霞，夏巨谌，胡国安，等. 直齿锥齿轮齿廓修形［J］. 机械设计， 2007， 24（7）： 42-44.
	CHEN Xia， XIA Juchen， HU Guoan， et al. Tooth Profile Modification of Spur Bevel Gear［J］. Journal of Machine Design， 2007， 24（7）： 42-44.
[17]	陈霞，汪姣，夏巨谌，等. 直齿圆锥齿轮修形仿真［J］. 机械科学与技术， 2009， 28（3）： 386-390.
	CHEN Xia， WANG Jiao， XIA Juchen， et al. Simulation of the Modification of a Straight Bevel Gear Tooth［J］. Mechanical Science and Technology for Aerospace Engineering， 2009， 28（3）： 386-390.
[18]	沈玲莉，陆宝山，季业益，等. 一种具修形的直齿锥齿轮副齿面接触分析方法［J］. 机械设计， 2021， 38（7）： 68-73.
	SHEN Lingli， LU Baoshan， JI Yeyi， et al. Method of the Tooth-contact Analysis for the Straight Bevel Gear’s Pair with Modified Teeth［J］. Journal of Machine Design， 2021， 38（7）： 68-73.
[19]	FUENTES A， ISERTE J L， GONZALEZ-PEREZ I， et al. Computerized Design of Advanced Straight and Skew Bevel Gears Produced by Precision Forging［J］. Computer Methods in Applied Mechanics and Engineering， 2011， 200（29/30/31/32）： 2363-2377.
[20]	方宗德. 齿轮轮齿承载接触分析（LTCA）的模型和方法［J］. 机械传动， 1998， 22（2）： 1-3.
	FANG Zongde. Model and Approach for Loaded Tooth Contact Analysis （LTCA） of Gear Drives［J］. Journal of Mechanical Transmission， 1998， 22（2）： 1-3.
[21]	付学中，方宗德，关亚彬，等. 采用NSGA-Ⅱ算法的面齿轮副小轮拓扑修形多目标优化［J］. 西安交通大学学报， 2017， 51（7）： 98-104.
	FU Xuezhong， FANG Zongde， GUAN Yabin， et al. NSGA-Ⅱ Based Multi-objective Optimization on Topologically Modified Pinions for Face Gear Pairs［J］. Journal of Xi’an Jiaotong University， 2017， 51（7）： 98-104.
[22]	DEB K. Multi-objective OPTIMIZATION USING EVOLUTIONARY ALGORITHMS ［M］. Hoboken：John Wiley & Sons， 2001.
[23]	DEB K， PRATAP A， AGARWAL S， et al. A Fast and Elitist Multiobjective Genetic Algorithm： NSGA-II［J］. IEEE Transactions on Evolutionary Computation， 2002， 6（2）： 182-197.