基于Ease off的准双曲面齿轮多目标优化

doi:10.13229/j.cnki.jdxbgxb.20220251

Abstract

Abstract:

In order to realize the multi-objective optimization of hypoid gears， a neural network surrogate model was established to describe the relationship between Ease off modification parameters and transfer error， tooth root stress and meshing loss power. Firstly， the dynamics software MASTA was used to establish the hypoid gear drive axle model. Based on the sensitivity coefficient matrix， the machine tool modification parameters corresponding to the second-order Taylor expansion of tooth surface deviation were derived， and the modified gear model was established. Secondly， the transmission error， tooth root stress and meshing loss power of the modified gear model were calculated through MASTA's loading tooth surface contact analysis function， and the neural network agent model was finally established. NSGA-II multi-objective optimization algorithm was used to optimize the surrogate model for comparative verification. The results show that the proposed multi-objective optimization method can effectively reduce transmission error， tooth root stress and meshing power loss of hypoid gears.

Key words: automotive engineering, Ease off, hypoid gear, NSGA-Ⅱ, multi-objective optimization

CLC Number:

U463.2

Xiao WU,Wen-ku SHI,Nian-cheng GUO,Yan-yan ZHAO,Zhi-yong CHEN,Xin-peng LI,Zhuo SUN,Jian LIU. Multi-objective optimization of hypoid gears based on Ease off[J].Journal of Jilin University(Engineering and Technology Edition), 2024, 54(1): 76-85.

Figures/Tables 28

Fig.1

Fig.2

Fig.3

Table 1

Table 2

Fig.4

Fig.5

Fig.6

Table 3

Fig.7

Fig.8

Fig.9

Fig.10

Fig.11

Table 4

Peak-to-peak TE results of 1000 N·m"

参数

数值

主动齿轮载荷/（N·m）

被动齿轮载荷/（N·m）

主动齿轮峰峰值误差/ $μ r a d$

从动齿轮峰峰值误差/ $μ r a d$

1000.00

4100.00

340.8482

83.1337

Table 4

Table 5

Maximum root stress results of 1000 N·m"

参数

主动齿轮

被动齿轮

最大米氏应力（拉伸）/MPa

最大米氏应力（压缩）/MPa

最大主应力（拉伸）/ $μ r a d$

最大主应力（压缩）/ $μ r a d$

147.0513

130.5012

166.4879

89.5532

91.5480

124.0980

111.8853

74.3783

Table 5

Fig.12

Table 6

Parameters of pinion gear transmission efficiency"

参数	单位	数值
中点螺旋角 $β m$	rad	0.61
润滑油运动粘度v	cst	50
接触面宽度 $b w$	mm	10.23
小轮中点分度圆半径 $r m 1$	mm	54.88
端面压力角 $α t m$	rad	0.39
中点锥距 $R m$	mm	197.81
外锥距 $R e$	mm	235.15
节圆半径r	mm	56.30
分锥角 $δ$	rad	0.29
齿顶高系数 $h a e$	mm	18.12
齿面角 $δ a$	rad	0.35
齿宽b	mm	79.05

Table 6

Table 7

Sensitivity analysis"

项目	组合（1）	组合（2）	组合（3）	组合（4）	组合（5）
螺旋角误差系数 $a 0$	0.01	0	0	0	0
压力角误差系数 $a 1$	0	0.01	0	0	0
齿长曲率系数 $a 2$	0	0	0.001	0	0
齿廓曲率系数系数 $a 3$	0	0	0	0.01	0
齿面挠率系数 $a 4$	0	0	0	0	0.001
传递误差平均值	1602.6	499.1	1981.1	3674.4	2021.1

Table 7

Table 8

Parameters of surrogate model"

模型	厚度取值/mm
螺旋角误差系数 $a 0$	0，0.0002
压力角误差系数 $a 1$	0，0.0002
齿长曲率系数 $a 2$	0，0.0001，0.0002，0.0003
齿廓曲率系数系数 $a 3$	0，0.0005，0.001
齿面挠率系数 $a 4$	0，0.0001，0.0002，0.0003，0.0004，0.0005

Table 8

Table 9

Verify parameter of model"

输入集/10^-5					MASTA模型计算结果			代理模型计算结果			相对误差/%
$a 0$	$a 1$	$a 2$	$a 3$	$a 4$	TE/μRad	RS/MPa	PMi/W	TE/μrad	RS/MPa	PMi/W	TE	RS	PMi
10	10	5	5	5	202.492	268.010	199.561	199.761	261.973	204.972	-1.349	-2.253	+2.712
10	10	15	15	15	248.317	378.304	213.967	240.194	384.523	207.173	-3.271	+1.644	-3.175
5	5	5	5	5	200.561	375.798	198.673	219.913	368.846	204.691	+9.659	-1.850	+3.029
5	15	25	25	25	435.081	354.052	238.767	428.970	349.637	250.111	-1.405	-1.248	+4.751
5	5	15	15	15	245.919	319.348	210.820	261.183	330.652	216.814	+6.207	+3.540	+2.843

Table 9

Fig.13

Table 10

Model optimal solution and verification"

最优解集/10^-5					MASTA模型计算结果			代理模型计算结果			相对误差/%
$a 0$	$a 1$	$a 2$	$a 3$	$a 4$	TE/μrad	RS/MPa	PMi/W	TE/μrad	RS/MPa	PMi/W	TE	RS	PMi
0	0	16.7	0	38.9	112.681	292.985	185.378	107.947	297.218	182.961	-4.201	+1.445	-1.304
0	0	16.7	22.2	16.7	107.935	300.674	183.095	115.236	289.028	179.898	+6.764	-3.873	-1.746
0	4.4	16.7	22.2	16.7	121.356	245.574	190.279	115.263	261.931	185.196	-5.021	+6.661	-2.671
4.4	20	16.7	0	33.3	119.601	253.077	189.586	111.027	259.501	182.787	-7.169	+2.538	-3.586
2.2	0	16.7	11.1	33.3	151.667	275.755	192.056	145.458	288.334	193.534	-4.094	+4.562	+0.770

Table 10

Fig.14

Fig.15

Fig.16

Fig.17

Table 11

References 15

1	Park Nogill. A generalized hypoid gear synthesized with common crown rack positioned between pinion and gear blanks[J]. Journal of Mechanical Design, 2017, 139(8): No. 085001.
2	魏冰阳, 杨建军, 仝昂鑫, 等. 基于等距Ease-off曲面的轮齿啮合仿真分析[J]. 航空动力学报, 2017, 32(5): 1259-1265.
	Wei Bing-yang, Yang Jian-jun, Tong Ang-xin, et al. Simulation analysis of gear tooth meshing based on isometric ease-off surface[J]. Journal of Aerospace Power, 2017, 32(5): 1259-1265.
3	Li G, Zhu W. An active ease-off topography modification approach for hypoid pinions based on a modified error sensitivity analysis method[J]. Journal of Mechanical Design, 2019, 141(9): 1-10.
4	Wang Q, Zhou C, Gui L, et al. An ease-off based approach to designing a high-order transmission motion for face-milled spiral bevel gears[J]. International Journal of Vehicle Design, 2019, 81(3/4): 241-252.
5	Shih Y P. A novel ease-off flank modification methodology for spiral bevel and hypoid gears[J]. Mechanism & Machine Theory, 2010, 45(8): 1108-1124.
6	Peng X L, Zhang L, Fang Z D. Manufacturing process for a face gear drive with local bearing contact and controllable unloaded meshing performance based on ease-off surface modification[J]. Journal of Mechanical Design, 2016, 138(4): No.043302.
7	Fan Q. Ease-Off and Application in Tooth Contact Analysis for Face-Milled and Face-Hobbed Spiral Bevel and Hypoid Gears[M]. Berlin: Springer International Publishing, 2016.
8	聂少武, 唐思成, 黄菊花, 等. 基于刀倾法的弧齿锥齿轮齿面失配啮合分析[J]. 机械传动, 2019, 43(9): 96-102.
	Nie Shao-wu, Tang Si-cheng, Huang Ju-hua, et al. Analysis of tooth surface mismatch meshing of spiral bevel gear based on cutter tilt method[J]. Mechanical Transmission, 2019, 43(9): 96-102.
9	聂少武, 邓静, 邓效忠, 等. 基于齿面Ease off拓扑的弧齿锥齿轮齿面偏差等效修正方法[J]. 中国机械工程, 2017, 28(20): 2434-2440.
	Nie Shao-wu, Deng Jing, Deng Xiao-zhong, et al. Equivalent correction method for tooth surface deviation of spiral bevel gears based on ease off topology[J]. China Mechanical Engineering, 2017, 28(20): 2434-2440.
10	曹雪梅, 邓效忠, 聂少武. 基于共轭齿面修正的航空弧齿锥齿轮高阶传动误差齿面拓扑结构设计[J]. 航空动力学报, 2015, 30(1): 195-200.
	Cao Xue-mei, Deng Xiao-zhong, Nie Shao-wu. Topological structure design of aeronautical spiral bevel gear high order transmission error based on conjugate tooth surface modification[J]. Journal of Aerospace Power, 2015, 30(1): 195-200.
11	聂少武, 邓静, 邓效忠, 等. 弧齿锥齿轮齿面拓扑修形及加工参数计算[J]. 航空动力学报, 2017, 32(08): 2009-2016.
	Nie Shao-wu, Deng Jing, Deng Xiao-zhong, et al. Topological modification and machining parameters calculation of spiral bevel gears[J]. Journal of Aerospace Power, 2017, 32(8): 2009-2016.
12	聂少武, 蒋闯, 邓效忠, 等. Ease-off拓扑修正的准双曲面齿轮齿面修形方法[J]. 中国机械工程, 2019, 30(22): 2709-2715, 2740.
	Nie Shao-wu, Jiang Chuang, Deng Xiao-zhong, et al. Tooth surface modification method for hypoid gears with ease-off topology modification[J]. China Mechanical Engineering, 2019, 30(22): 2709-2715, 2740.
13	Cinelli C, Hazlett C. Making sense of sensitivity: extending omitted variable bias[J]. Journal of the Royal Statistical Society Series B, 2020, 82(1): 39-67.
14	Deb K, Pratap A, Agarwal S, et al. A fast and elitist multi-objective genetic algorithm: NSGA-II[J]. IEEE Transactions on Evolutionary Computation, 2002, 6(2): 182-197.
15	Srinivas N, Deb K. Multi-objective function optimization using non-dominated sorting genetic algorithms[J]. Evolutionary Computation, 1994, 2(3): 1301-1308.

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参数	数值
主减速比	41
桥壳网格单元最大尺寸/mm	10
桥壳网格单元最小尺寸/mm	4
桥壳网格单元坍塌比最小值	0.2
桥壳网格单元总数	3 071 501
桥壳密度/（kg·m^-3）	7 850
桥壳弹性模量/MPa	206 800
桥壳泊松比	0.29
轴系密度/（kg·m^-3）	7 850
轴系弹性模量/MPa	212 000
轴系泊松比	0.28
齿轮密度/（kg·m^-3）	7 800
齿轮弹性模量/MPa	207 000
齿轮泊松比	0.29

参数/单位	主动齿轮	被动齿轮
齿数	10	41
旋向	左	右
轴夹角/（°）	90	90
节圆模数/mm	10.98	10.98
中点螺旋角/（°）	35	35
平均压力角/（°）	22.50	22.50
齿宽/mm	79.02	72.00
全齿高/mm	20.45	20.69
齿顶高/mm	18.12	18.35
平均节圆直径/mm	112.26	380.90
节锥角/（°）	16.48	73.10
面锥角/（°）	20.24	73.89
根锥角/（°）	15.71	69.36

参数	主动齿轮	被动齿轮
网格大小/mm	5.95	27.14
轮廓网格密度	4	4
表面网格数	8	8
圆角网格数	8	8
径向网格数	4	4

模型	TE/μrad	RS/MPa	PMi/W
性能提升百分比/%	39.600	22.395	19.964
原始齿面	178.701	387.444	228.767
Ease off最优面	107.935	300.674	183.095

Multi-objective optimization of hypoid gears based on Ease off

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