基于NSGA⁃II的斜齿轮宏观参数多目标优化

doi:10.13229/j.cnki.jdxbgxb.20210836

摘要/Abstract

摘要：

针对齿轮系统在初始设计阶段存在传递误差波动量大，进而导致的振动噪声突出等问题，提出了一种齿轮副宏观参数多目标优化方法。基于势能法和切片法推导出了斜齿轮传递误差解析计算公式，并以齿轮副总重合度最大、传递误差波动量最小和齿轮副总体积最小为优化目标，以齿轮宏观参数为设计变量，以带精英策略的快速非支配排序遗传算法（NSGA-Ⅱ）为优化算法，建立了齿轮系统宏观参数优化模型。以某二级减速齿轮系统为算例，采用优化模型对齿轮副宏观参数进行了优化设计，基于MASTA软件对优化前、后的动力学指标进行了仿真。结果表明，不同工况下，经过宏观参数优化后的齿轮系统啮合传递误差波动量、系统传递误差波动量和轴承座振动位移幅值均得到不同程度的降低，系统动力学性能得到整体改善。

关键词: 车辆工程, 斜齿轮, 宏观参数, 遗传算法, 优化

Abstract:

In the initial design stage of gear system， there is a large fluctuation of transmission error， which leads to prominent vibration and noise problems of the system. Aiming at that， a multi-objective optimization method of macro parameters of gear pair is presented. Based on the potential energy method and slicing method， the analytical calculation formula of transmission error of helical gear is derived. Taking the maximum total contact ratio， minimum fluctuation of transmission error（TE）， and minimum total volume of gear pair as optimization objectives， the gear macro parameters as design variables， and a fast elitist non-dominated sorting genetic algorithm（NSGA-Ⅱ） as optimization algorithm， the macro parameter optimization model of gear system is established. Taking a two-stage reduction gear system as an example， the macro parameters of the gear pair are optimized by using the optimization model， and the dynamic indexes before and after optimization are simulated based on the software MASTA. The results show that under different working conditions， the fluctuation of meshing TE and system TE as well as the vibration displacement amplitude of bearing seat are reduced in different degrees， and the overall dynamic performance of the gear system is improved.

Key words: vehicle engineering, helical gear, macro parameter, genetic algorithm, optimization

中图分类号:

TH132.41

杨红波,史文库,陈志勇,郭年程,赵燕燕. 基于NSGA⁃II的斜齿轮宏观参数多目标优化[J]. 吉林大学学报(工学版), 2023, 53(4): 1007-1018.

Hong-bo YANG,Wen-ku SHI,Zhi-yong CHEN,Nian-cheng GUO,Yan-yan ZHAO. Multi⁃objective optimization of macro parameters of helical gear based on NSGA⁃Ⅱ[J]. Journal of Jilin University(Engineering and Technology Edition), 2023, 53(4): 1007-1018.

图/表 18

图1

图2

图3

图4

图5

图6

表1

表2

表3

图7

图8

图9

图10

图11

图12

图13

表4

表5

参考文献 36

1	齿轮手册编委会. 齿轮手册(上册) [M]. 北京:机械工业出版社, 1990.
2	李润方, 王建军. 齿轮系统动力学—振动·冲击·噪声 [M]. 北京: 科学出版社, 1997.
3	刘艳芳, 赖俊斌, 岳会军, 等. 斜齿轮振动噪声分析方法[J]. 振动、测试与诊断, 2016, 36(5): 960-966, 1027-1028.
	Liu Yan-fang, Lai Jun-bin, Yue Hui-jun, et al. Research on the noise and vibration of the helical gear[J]. Journal of Vibration, Measurement & Diagnosis, 2016, 36(5): 960-966, 1027-1028.
4	Cai Y, Hayashi T. The optimum modification of tooth profile for a pair of spur gears to make its rotational vibration equal zero[C]∥Proceedings of International Power Transmission and Gearing Conference,Tokyo,Japan, 1992, 57: 3957-3963.
5	付学中, 方宗德, 贾超, 等. 面齿轮传动啮合刚度分析与修形减振优化[J]. 振动与冲击, 2019, 38(5): 265-272.
	Fu Xue-zhong, Fang Zong-de, Jia Chao, et al. Meshing stiffness analysis and optimization of vibration reduction and modification for face-gear drives[J]. Journal of Vibration and Shock, 2019, 38(5): 265-272.
6	张容川, 周云山, 胡哓岚, 等. 双排行星齿轮系统的齿轮优化修形[J]. 汽车工程, 2018, 40(9): 1118-1124.
	Zhang Rong-chuan, Zhou Yun-shan, Hu Xiao-lan, et al. The optimized gear modification for double-row planetary gear train[J]. Automotive Engineering, 2018, 40(9): 1118-1124.
7	赵宁, 康士朋, 郭辉, 等. 基于遗传算法的弧齿锥齿轮动态特性优化设计[J]. 航空动力学报, 2010, 25(10): 2396-2402.
	Zhao Ning, Kang Shi-peng, Guo Hui, et al. Optimization design of spiral bevel gear's dynamic characteristics based on genetic algorithm[J]. Journal of Aerospace Power, 2010, 25(10): 2396-2402.
8	赵宁, 秋朋园, 刘贵立. 高重合度人字齿轮传动动态性能优化设计[J]. 国防科技大学学报, 2015, 37(2): 166-174.
	Zhao Ning, Qiu Peng-yuan, Liu Gui-li. Optimized design of dynamic behavior of double helical gears with high contact ratio[J]. Journal of National University of Defense Technology, 2015, 37(2): 166-174.
9	杨硕文, 唐进元. 一种新的直齿轮复合修形设计方法 [J]. 中南大学学报: 自然科学版, 2019, 50(5): 1082-1088.
	Yang Shuo-wen, Tang Jin-yuan. A new design method for compound modification of spur gear[J]. Journal of Central South University (Science and technology), 2019, 50(5): 1082-1088.
10	贾超, 方宗德. 高速齿轮传递误差和啮入冲击的激励模拟及齿面优化修形[J]. 振动与冲击, 2019, 38(23): 103-109, 138.
	Jia Chao, Fang Zong-de. Simulation for transmission error and mesh-in impact excitation of high speed gears and their tooth surface optimal modification[J]. Journal of Vibration and Shock, 2019, 38(23): 103-109, 138.
11	赵宁, 秋朋园, 刘贵立. 高重合度人字齿轮轮齿最佳修形优化设计[J]. 国防科技大学学报, 2015, 37(1): 165-170.
	Zhao Ning, Qiu Peng-yuan, Liu Gui-li. Modification optimization of double helical gears with high contact ratio [J]. Journal of National University of Defense Technology, 2015, 37(1): 165-170.
12	王峰, 方宗德, 李声晋, 等. 人字齿轮系统振动传递分析优化与试验验证[J]. 机械工程学报, 2015, 51(1): 34-42.
	Wang Feng, Fang Zong-de, Li Sheng-jin, et al. Analysis optimization and experimental verification of herringbone gear transmission system[J]. Journal of Mechanical Engineering, 2015, 51(1): 34-42.
13	贾超, 方宗德, 张永振. 高速内啮合人字齿轮多目标优化修形[J]. 哈尔滨工业大学学报, 2017, 49(1): 166-172.
	Jia Chao, Fang Zong-de, Zhang Yong-zhen. Multi-objective optimal modification for internal double helical gears with high speed[J]. Journal of Harbin Institute of Technology, 2017, 49(1): 166-172.
14	刘玄, 方宗德, 赵宁. 人字齿轮小轮轴向窜动的多目标复合修形优化[J]. 西安交通大学学报, 2021, 55(1): 118-126.
	Liu Xuan, Fang Zong-de, Zhao Ning. Multi-objective compound modification optimization of pinion axial floating for the double helical gears[J]. Journal of Xi'an Jiaotong University, 2021, 55(1): 118-126.
15	蒋春明, 阮米庆. 汽车机械式变速器多目标可靠性优化设计[J]. 汽车工程, 2007, 29(12): 1090-1093.
	Jiang Chun-ming, Ruan Mi-qing. Multi-objective reliability optimal design of automotive mechanical transmission [J]. Automotive Engineering, 2007, 29(12): 1090-1093.
16	Chen X, Song C S, Zhu C C, et al. Effect of macro-parameters on vibration and radiation noise for high speed wheel gear transmission in electric vehicles[J]. Journal of Mechanical Science and Technology, 2018, 32(9): 4153-4164.
17	宗长富, 任明辉, 万滢, 等. 变速器斜齿轮宏观参数减振优化设[J]. 吉林大学学报: 工学版, 2016, 46(6): 1772-1779.
	Zong Chang-fu, Ren Ming-hui, Wan Ying, et al. Optimization of macro-geometric parameters of helical gears of transmissions to reduce vibration[J]. Journal of Jilin University (Engineering and Technology Edition), 2016, 46(6): 1772-1779.
18	赵宁, 杨杰. 高重合度圆柱齿轮传动多目标优化设计 [J]. 机械传动, 2012, 36(7): 43-46.
	Zhao Ning, Yang Jie. Multi-objective optimization design of high contact ratio cylindrical gear drive [J]. Journal of Mechanical Transmission, 2012, 36(7): 43-46.
19	何畅然, 贺敬良, 何渠. 基于MASTA的齿轮变速箱啸叫研究[J]. 北京信息科技大学学报, 2014, 29(5): 87-91.
	He Chang-ran, He Jing-liang, He Qu. Research on geared transmission whine based on MASTA[J]. Journal of Beijing Information Science and Technology University, 2014, 29(5): 87-91.
20	Garambois P, Perret-Liaudet J, Rigaud E. NVH robust optimization of gear macro and microgeometries using an efficient tooth contact model[J]. Mechanism and Machine Theory, 2017, 117: 78-95.
21	Garambois P, Rigaud E, Perret-Liaudet J. Robust optimization of gear macro and micro-geometries in order to minimize static transmission error and gear stiffness fluctuations[J]. International Conference on Gears, 2017, 2294(1/2): 927-938.
22	蔡文奇. 纯电动汽车两挡变速箱齿轮振动噪声仿真分析与优化设计[D]. 长春: 吉林大学机械与航空航天工程学院, 2019.
	Cai Wen-qi. Simulation analysis of gear vibration noise and optimization design of pure electic vehicle two-speed gearbox[D]. Changchun: College of Mechanical and Aerospace Engineering, Jilin University, 2019.
23	包英豪. 纯电动汽车两挡变速箱噪声与效率分析及参数优化[D]. 长春: 吉林大学汽车工程学院, 2020.
	Bao Ying-hao. Noise and efficiency analysis and parameter optimization of two-speed transmission of pure electric vehicle[D]. Changchun: College of Automotive Engineering, Jilin University, 2020.
24	张利, 黄筱调, 谢杰, 等. 基于Romax的斜齿圆柱齿轮减速器参数优化[J]. 机械设计与制造, 2020, 57(6): 215-218.
	Zhang Li, Huang Xiao-diao, Xie Jie, et al. Parameter optimization of helical cylindrical gear reducer based on Romax[J]. Machinery Design and Manufacture, 2020, 57(6): 215-218.
25	赵宁, 秋朋园, 刘贵立. 高重合度人字齿轮轮齿最佳修形优化设计[J]. 国防科技大学学报, 2015, 37(1): 165-170.
	Zhao Ning, Qiu Peng-yuan, Liu Gui-li. Modification optimization of double helical gears with high contact ratio [J]. Journal of National University of Defense Technology, 2015, 37(1): 165-170.
26	王峰, 方宗德, 李声晋, 等. 人字齿轮系统振动传递分析优化与试验验证[J]. 机械工程学报, 2015, 51(1): 34-42.
	Wang Feng, Fang Zong-de, Li Sheng-jin, et al. Analysis optimization and experimental verification of herringbone gear transmission system[J]. Journal of Mechanical Engineering, 2015, 51(1): 34-42.
27	刘宗晟. 变速器动力学建模与振动噪声分析[D]. 长春: 吉林大学机械与航空航天工程学院, 2016.
	Liu Zong-sheng. Dynamics modeling and vibro-acoustic characteristics analysis of transmission [D]. Changchun: College of Mechanical and Aerospace Engineering, Jilin University, 2016.
28	Yang D, Lin J Y. Hertzian damping, tooth friction and bending elasticity in gear impact dynamics[J]. Journal of Mechanisms, Transmissions, and Automation in Design, 1987, 109(2): 189-196.
29	Tian X. Dynamic simulation for system response of gearbox including localized gear faults[D]. Alberta, Canada: University of Alberta, 2004.
30	Sainsot P, Velex P, Duverger O. Contribution of gear body to tooth deflections——a new bidimensional analytical formula[J]. Journal of Mechanical Design, 2004, 126(4): 748-752.
31	Wan Z G, Cao H R, Zi Y Y, et al. An improved time-varying mesh stiffness algorithm and dynamic modeling of gear-rotor system with tooth root crack[J]. Engineering Failure Analysis, 2014, 42: 157-177.
32	成大先. 机械设计手册[M].6版. 北京: 化学工业出版社, 2016.
33	Deb K, Member A, Pratap A, et al. A fast and elitist multiobjective genetic algorithm: NSGA-Ⅱ[J]. Transactions on Evolutionary Computation, 2002, 6(2): 182-197.
34	Ahmed F, Deb K. Multi-objective optimal path planning using elitist non-dominated sorting genetic algorithms[J]. Soft Computing, 2013, 17(7): 1283-1299.
35	付学中, 方宗德, 关亚彬, 等. 基于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 Xue-zhong, Fang Zong-de, Guan Ya-bin, 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.
36	Yao Q Z. Multi-objective optimization design of spur gear based on NSGA-Ⅱ and decision making[J]. Advances in Mechanical Engineering, 2019, 11(3): 1-8.

Metrics

Viewed

Full text

Abstract

Cited

Shared

Discussed

宏观参数	第一级（小齿 /大齿）	第二级（小齿 /大齿）
齿数z	15/42	13/40
齿宽B/mm	12/12	20/20
齿顶圆直径d_a/mm	27.794/68.289	28.087/75.207
齿根圆直径d_f/mm	21.044/61.539	20.212/67.332
旋向	右旋/左旋	左旋/右旋
法向模数m_n/mm	1.5	1.75
螺旋角β/（°）	15	5
法向压力角α/mm	20	20
齿顶高系数h_a	1.0	1.0
顶隙系数c	0.25	0.25
变位系数x	0.3/0.22	0.4/0.47
中心距s/mm	45	48
端面重合度ε	2.8	3.08
弹性模量E/MPa	2.07×10⁵
剪切模量G/MPa	7.96×10⁴
泊松比ν	0.3
密度ρ/（kg·m^-3）	7.8×10³

宏观参数	策略A1	策略B1	策略C1
法向模数m_n/mm	1.75	1.75	1.75
螺旋角β/（°）	18	18	20
变位系数x₁	0.37	0.38	0.36
变位系数x₂	0.17	0.18	0.16
端面重合度ε	3.14	3.10	3.11

宏观参数	策略A2	策略B2	策略C2
法向模数m_n/mm	2.0	2.0	2.0
螺旋角β/（°）	8	8	10
变位系数x₁	0.34	0.36	0.35
变位系数x₂	0.4	0.42	0.39
端面重合度ε	3.22	3.21	3.19

分析指标	初值	优化	减小量/%
1^st齿轮副啮合传递误差波动/μm	2.05	1.75	14.6
2^nd齿轮副啮合传递误差波动/μm	4.67	2.20	52.9
系统传递误差波动/（m·rad）	0.16	0.12	25.0
轴承座振动位移幅值/μm	4.95	4.35	12.1

分析指标	工况	初值	优化	减小量/%
1^st齿轮副啮合传递误差/μm	工况二	3.10	3.05	1.61
	工况三	4.31	3.93	8.82
	工况四	6.22	5.36	13.83
2^nd齿轮副啮合传递误差/μm	工况二	4.08	3.65	10.54
	工况三	3.57	2.71	24.09
	工况四	18.04	13.34	26.05
系统传递误差/（m·rad）	工况二	0.21	0.15	28.57
	工况三	0.45	0.32	28.89
	工况四	0.61	0.46	24.59
轴承座振动位移幅值/μm	工况二	5.32	5.15	3.20
	工况三	5.73	5.26	8.20
	工况四	6.23	5.78	7.22