Gear tooth surface modification analysis by support vector machine regression

Abstract

A modification to the gear tooth surface is crucial for reducing whistling noise generated by gearboxes of electric vehicles. Using the orthogonal test method, this paper studies geometry parameters of gear pair and the effect of load conditions on modification parameters of tooth surface. To achieve this, the paper develops various test models of gear pairs based on MASTA and then corrects the gear profiles according to load distribution state. The influential factors of gear transmission to modification parameters of tooth surface are thereby intuitively analyzed.

Finally, a prediction model of the result of gear modification is built, based on test parameters, by method of support regression vector machine, providing a valuable reference to determine the tooth-surface modification parameters of the transmission gear of electric vehicles.

Keywords

Gearbox noise tooth-surface modification support vector machine prediction model

1. Introduction

With the implementation of “New Energy Vehicle Industry Development Plan (2021–2035),” the number of electric-powered new energy vehicles has burst. New energy vehicles have gradually won the favor of OEMs and consumers due to their superior qualities, such as energy efficiency and environmental protection. Due to the absence of engine noise in electric vehicles and the significantly higher transmission speed compared to conventional fuel vehicles, the gearbox whistle has become one of the most significant noise sources in electric vehicles [1].

During the working process, the gears transmit power via alternating meshing. However, due to manufacturing error, assembly error, and the deformation of gear pairs during loading process, the force on the tooth surfaces of the gear pair is uneven, resulting in the phenomenon of load impact and whistling [2]. In general, the micro-geometric modification of gears is used to improve meshing status and load distribution on tooth surface, thereby enhancing the gears’ transmission stability, transmission quality, and service life [3]. According to Wu Junjie [4], the reducer is the primary source of noise in an electric vehicle. By modifying the gear tooth surface, the contact state is enhanced, the transmission error is decreased, and the product’s overall NVH is enhanced. Wei Guoxing [5] developed an optimized study on spur tooth surface using energy method and genetic algorithm. Zhao Min et al. [6] took the specific secondary reducer of a new energy vehicle as an example, analyzed the load per unit length, transmission error, and NVH, and then proposed an alternative diagonal modification method. Variable approximation replaces diagonal modification slope variation, which simplifies parameter estimation. The authors of Xu Zhongsi et al. [7] suggested an optimization method for modification parameters that considered both transmission error and tooth surface contact stress. Jia Chao et al. [8] examined the effect of coincidence degree on the parameters of tooth surface alterations and optimized the parameters. Therefore, a good strategy for modifying the tooth surface can effectively enhance load distribution during gear meshing, thereby reducing gear vibration and suppressing gear whistling noise. Unfortunately, none of these preceding studies included a quantitative analysis on the parameters of tooth surface modification.

Throughout the research, the orthogonal test design method is utilized to establish a test plan based on gear design parameters, the position of the gear pair, and the applied load. MASTA is employed to manually trim the gear sets following the same standard in each test plan, and then analyzes the gear sets with trimming parameters and influencing variables. Finally, the prediction model of the support regression vector machine of gear pair parameters and modification parameters is developed to accurately forecast the modification parameters.

2. Theoretical analysis

2.1 Gear mesh

As the gears mesh, they enter the meshing line; the initial contact is SAP, and the final contact is EAP.

In the process of transmitting power, the meshing stiffness of the teeth changes periodically, so the actual meshing point is not always on the meshing line. The instantaneous speed difference will cause meshing interference and impact, resulting in vibration and noise.

The transmission error is the main source of excitation, which can be defined as the angular difference between the theoretical meshing position and the actual meshing position of the gear pair [9]:

$\displaystyle TE_{\textit{mesh}}=u_{in}\cdot i-u_{\textit{out}}$

In particular, $u_{in}$ , $u_{\textit{out}}$ respectively indicate the gear meshing position of the input end and that of the output end.

For reducing meshing interference and impact, improving the contact state of the tooth surface, and decreasing the TE gears need to be modified.

Figure 1.

Process diagram of gear meshing.

2.2 Regression of SVMs

Support Vector Machine is a two-class classification method based on statistical learning theory, and its principle is to map nonlinear inputs into high-dimensional feature space through kernel functions, and then to build a linear model in high-dimensional feature space [10, 11]. It is especially suitable for small and medium-size samples, nonlinear, high-dimensional classification, and regression problems.

Suppose there is a training sample set in a feature space:

$\displaystyle T=\{(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{n},y_{n})\}\quad x_{i% }\in R^{n}\quad y_{i}\in\{{+1,-1}\},i=1,2,\ldots N$

In this example, $x_{i}$ is the i-th feature vector and $y_{i}$ is the i-th class label.

It is the separation hypoplane $w\cdot x+b=0$ that correctly separates the training dataset and ensures the largest geometric interval. The optimization problem of correctly dividing the two classes of samples and maximizing the classification interval can be decomposed as follows:

$\displaystyle\min\frac{1}{2}||w||^{2}\quad\text{s.t }y_{i}({w\cdot x_{i}+b})% \geqslant 1,i=1,2,\ldots,N$

Since completely linearly separable data is almost non-existent, a relaxation factor $\xi_{i}\geqslant 0$ is introduced, and the constraint can be expressed as follows:

$\displaystyle y_{i}({w\cdot x_{i}+b})\geqslant 1-\xi_{i}$

To determine the separation hyperplane that maximizes the classification interval, the optimization problem must be considered:

$\displaystyle\min\frac{1}{2}||w||^{2}+C\mathop{\sum}\limits_{i=1}^{m}\xi_{i}$

Where $C$ represents the penalty factor.

For the nonlinear classification problem in the input space, transform it into a linear classification problem in a certain dimensional feature space via nonlinear transformation, learn a linear support vector machine in the high-dimensional feature space, and replace the inner product with the kernel function:

$\displaystyle K({x_{i},x_{j}})=\emptyset({x_{i}})\cdot\emptyset({x_{j}})$

In the dual problem, the kernel function $K({x,z})$ is used to replace the inner product, and the solution is a nonlinear support vector machine:

$\displaystyle f(x)=\textit{sign}\left({\mathop{\sum}\limits_{i=1}^{N}\alpha_{i% }^{\ast}y_{i}K({x_{i},x_{j}})+b^{\ast}}\right)$

A support vector machine is employed in regression problems, which uses a constant $\epsilon>0$ to describe the error between $f(x)$ and $y$ , and then the loss function metric in the regression function can be expressed as:

$\displaystyle\textit{err}({x_{i},y_{i}})=\left\{{{\begin{array}[]{*{20}c}0\\ {|{y_{i}-w\cdot\emptyset({x_{i}})-b}|-\epsilon}\\ \end{array}}}\right.$

The objective function of the regression model can be defined as follows:

$\displaystyle\min\frac{1}{2}||w||_{2}^{2}\quad\text{s.t}|{y_{i}-w\cdot% \emptyset({x_{i}})-b}|\leqslant\epsilon$

Relaxation factors $\xi_{i}^{\vee},\xi_{i}^{\wedge}$ on both sides of the inequality are introduced into the regression problem to maximize the target interval, and the loss function of the regression model becomes:

$\displaystyle\min\frac{1}{2}||w||_{2}^{2}+C\mathop{\sum}\limits_{i=1}^{m}({\xi% _{i}^{\vee}+\xi_{i}^{\wedge}})$

In summary, support vector machine regression is a supervised learning algorithm, to perform discrete value prediction by finding the optimized fitting hyperplane of multidimensional data. Its superiority is the calculation complexity increases insignificantly when data dimension expands and meanwhile keeps a high forecast precision. There are researches that have applied support vector machine regression in the design area of mechanical parts which achieve good results [12, 13].

3. Model construction of involute cylindrical gear pairs

MASTA is a professional software for the design and manufacturing of transmission systems. Functions include modeling of transmission system, analyzing, optimizing and so forth. It is a commonly used CAE software for the analysis of gear tooth modification [14, 15]. To examine how gear parameters and the state of the system where the gear pair locates effect on surface modification parameters, a simple gear pair transmission system is established within MASTA software. The diagram is in Fig. 2 as follows.

Figure 2.

Schematic diagram of a gear pair transmission system.

A simple transmission system consists of a pair of gear sets, an input shaft, an output shaft, and concept bearings that are used on both ends of the input shaft and output shaft. The adopted concept of bearing does not consider the influence of bearing assembly relationship and bearing structure on the system. The input shaft and output shaft are both solid shafts with diameter of 25 mm and length of 150 mm.

A gear pair is characterized by the following parameters: gear module, helix angle, pressure angle, number of teeth, addendum coefficient, dedendum coefficient, modification coefficient, tooth width and center distance between gear pairs. There is a certain correlation between these main parameters of gears. For the same center distance, transmission ratio, modulus, helix angle and pressure angle are the primary parameters used to design gear pairs. The modification coefficient is usually calculated according to ISO 21771 based upon the above parameters. Addendum coefficient and dedendum coefficient are mainly used to get the heights of the tooth tips and tooth roots of the calculated gear pair. Therefore, this paper only focuses on the center distance, transmission ratio, modulus, helix angle, pressure angle, and tooth width. These five parameters are the key factors to study tooth structure. Moreover, influences of the load on the transmission system and the axial position of the gear pair in transmission system are also considered.

Orthogonal design is a statistical approach to solve multifactorial issues. It is commonly employed when the result is influenced by multiple factors and is required to clarify the effect of each factor or find the best parameter combination. Its advantages are fewer analysis times, high efficiency, and easy implementation. Now it has been widely applied in the design area of mechanical parts, to effectively analyze the influence of gear tooth modification parameters on meshing results [16, 17].

Based on above analysis and settings, the orthogonal method is used to design 8 factors, and each factor has 5 levels, yielding to a total of 50 groups of experiments. The orthogonal design experiments are described as follows.

Table 1

Orthogonal design table

Centre distance	Ratio of gears	Modulus	Angle of helix	Pressure angle	Teeth width	Dimensional parameter	Torque N.m
70	1	2	5	14.5	15	25	100
85	1.5	2.25	10	17.5	20	50	200
100	2	2.5	15	20	25	75	300
115	2.5	2.75	20	22.5	30	100	400
130	3	3	25	25	35	125	500

4. Calculation of the tooth surface modification model

The modification methods shown in ISO21771 can be divided into three categories, namely, helix modifications, transverse profile modifications, and triangular modifications. Among these, transverse profile modifications and helix modifications are the most frequently adopted modification methods. The helix modification methods include helix slope modification, helix crowning and helix end relief. The transverse profile modification methods include transverse profile slope modification, profile crowning and tip and root relief, amongst which tip and root relief is not commonly employed [18]. Helix end and tip relief are designed to avoid the crush of tooth edge due to increasing stress distribution but are not crucial to the stress distribution on tooth surface.

Figure 3.

Modification method diagram.

This test follows the common methods of transverse profile modifications and helix modifications, including 4 methods: helix slope modification, helix crowning, transverse profile slope modification and profile crowning.

A pre-finish at the distance of 10% from tooth ends is conducted, with the amount of 10 $\mu$ m cut. In MASTA, the modification target is that stress distribution in more than 60% of the tooth surface is greater than 70% of the maximum stress, and the design experiment group is modified in relation to the optimization value of total energy. The stress distribution results before and after modification of the 10th group are illustrated in Fig. 4. For the other groups, the modification refers to the unified standard, and the results after modification are similar. For all 50 test sets, the area percent of stress over 70% of the maximum value is controlled within the range 62.8%–78.3% and the average value of that of the 50 sets is 71.4%. The detailed parameters for modification are displayed in Table 2.

Figure 4.

Comparison of trimming results.

Table 2

Shows the modification results as follows

Sequence no.	Offset amount	Stress percentage after trimming	Transmission error		Maximum stress		Trimming amount
			Original set	Set after trimming	Original set	Set after trimming	Helix angle	Helix crowning	Transverse profile slope angle	Profile crowning
1	5.25	81%	0.855	2.01	1401	1187	5	8	$-$ 12	15
2	9	74%	4.5	0.9	1147	1050	9	5	0	20
3	0.5	76%	1.8	0.5	841	870	0	3	3	15
4	$-$ 10	82%	1.2	0.3	797	733	$-$ 14	3	0	10
5	$-$ 9.6	76%	1.08	0.2	681.7	600	$-$ 12	5	0	4
6	17	77%	1.9	1.089	2061	1363	17	10	$-$ 10	20
7	24.59	83%	11.93	3.7	1831	1407	26	10	$-$ 10	30
8	1	74%	2	2.3	1160	1127	0	5	0	8
9	$-$ 12	76%	1.3	0.4	679	574	$-$ 11	5	1	0
10	$-$ 20	89%	11.6	1.4	1688	1634	$-$ 19	10	0	30
11	29.06	82%	9.39	2.9	2252	1977	29	10	$-$ 10	56
12	18	81%	2.5	0.6	979	910	20	5	0	8
13		78%	4.8	5	1037	1047	0	10	0	0
14	$-$ 51	75%	14	1.5	2390	1919	$-$ 65	10	1	55
15	$-$ 45	82%	2.68	1	1781	1424	$-$ 60	10	$-$ 10	20
16	22	89%	7.4	1.3	1177	1147	27	5	10	27
17	36	82%	0.42	0.9	1367	1152	45	15	$-$ 5	20
18	5.8	81%	0.6	0.5	2295	1910	0	9	$-$ 20	20
19	$-$ 51	73%	45	9	3002	3020	$-$ 51	36	0	110
20	$-$ 27	79%	14	2.6	1459	1374	$-$ 30	8	0	40
21	35	75%	5.29	5.6	1496	1312	35	25	$-$ 10	5
22	119	73%	20.46	5.88	5447	2541	150	20	$-$ 70	35
23	0	81%	5	0.5	1929	1421	0	17	$-$ 10	20
24	$-$ 37	83%	0.8	1	1741	1565	$-$ 32	5	$-$ 5	10
25	$-$ 41	75%	13.5	7.8	1723	1693	$-$ 45	8	0	46
26	12.06	88%	5.09	1.58	6154	1533	12	15	$-$ 25	$-$ 20
27	20	82%	2	0.8	1409	1188	22	5	$-$ 5	20
28	0	83%	3	1.3	601	634	0	5	0	8
29	$-$ 7.6	79%	1.4	0.8	652	645	$-$ 12	3	$-$ 1	10
30	$-$ 5.7	74%	5.8	1.3	876	895	$-$ 7	5	9	20
31	13	88%	1.02	0.78	1071	926	13	5	2	8
32	19.9	81%	1.88	0.4	974	820	27	5	0	8
33	2.4	72%	7.5	1.1	1156	1185	1	5	1	25
34	$-$ 18	83%	9.6	5	1160	1077	$-$ 20	10	$-$ 9	15
35	$-$ 42	85%	9.4	0.75	3679	2185	$-$ 57	6	$-$ 2	62
36	18.18	75%	2.47	1.69	1129	1073	20	10	0	12
37	26	76%	2.9	2.6	2065	1834	30	16	$-$ 8	0
38	5	72%	2.8	1.4	1641	1606	0	5	$-$ 5	60
39	$-$ 33	82%	3	0.7	1993	1481	$-$ 33	12	$-$ 15	12
40	$-$ 35	86%	12	3.4	1511	1320	$-$ 40	12	0	26
41	26	75%	10	2.17	1851	1574	30	10	$-$ 5	48
42	16.8	78%	11	2.64	1249	1292	17	10	0	40
43	$-$ 0.9	81%	4.8	1.7	1780	2015	0	10	0	50
44	$-$ 49	78%	5	1	1744	1490	$-$ 55	13	$-$ 10	12
45	$-$ 46	82%	3.3	1.6	1953	1485	$-$ 53	10	$-$ 20	30
46	25	79%	2.16	1.25	1052	1004	27	20	0	5
47	80	82%	25	3.75	3072	2256	110	20	$-$ 20	60
48	2	81%	47	6.8	3093	2900	5	20	$-$ 70	120
49	$-$ 60	80%	14	2.6	4001	2733	$-$ 70	0	0	130
50	$-$ 31	76%	1.13	0.36	1195	1189	$-$ 35	8	0	30

5. Result analysis of tooth surface modification

Generally, transmission error of gear pairs and maximum stress on tooth surface are effectively reduced after the tooth surface modification. In addition, meshing state on the tooth surface is greatly improved. Above all, this paper identifies the relationship among the geometry, torque, position in system and trim amount of each modification method of gear pairs through intuitive analysis.

5.1 Intuitive analysis

5.1.1 Analysis of helix angle modification parameters

The size and direction of the helix angle modification varies under the influence of different levels of various factors, as shown in Fig. 5. The most important factor affecting the modification of helix angle is axial position of the transmission system where the gear pair resides. Furthermore, it was discovered that when the gear pair is in the middle of the shaft system, the helix angle modification is the smallest, and when one-fourth of the distance away from the edge of the shaft system, the modification amount is the largest. The difference in the position of the gear pair also results a significant change in the modification direction.

Figure 5.

Results of helix angle modification.

5.1.2 Analysis of helix crowning modification parameters

Figure 6 depicts the modification of the helix crowning under the influence of different levels of various factors. As the torque is continuously increased, the crowning amount shows a noticeable increasing trend. Then within modulo range, the crowning amount increases and then decreases as the modulo rises. The peak value is reached when the modulo is 2.5. Meanwhile, changing pressure angle causes a significant change in crowing amount, but no clear changing pattern is observed.

Figure 6.

Results of helix crowning modification.

5.1.3 Analysis of transverse profile slope modification parameters

As shown in Fig. 7, all parameters present no clear linearly changing pattern for transverse profile slope modification. Changes in helix angle have little effect on the amount of profile pressure angle modification. Nonetheless, an increase in pressure angle and torque will cause the amount of transverse profile slope modification to increase within a broader parameter range. It is also found that modification amount of transverse profile angle drops within a wide parameter range when the center distance gets larger.

Figure 7.

Results of transverse profile slope modification.

5.1.4 Analyses of profile crowning modification parameters

As depicted in Fig. 8, under the influence of different levels of various factors, the amount of profile crowning modification rises along with the increment of the transmission ratio and the torque, as well as the modulus. However, the amount of profile crowning modification grows down when tooth width, helix angle, and center distance grow up. The change in position causes an initial increase in profile crowning modification, followed by a decrease, but the overall variation range is relatively small.

Figure 8.

Results of profile crowning modification.

To explore the relationship between modification results and variables, it is proposed to develop a regression model based on support vector machine to reasonably predict the modification parameter results under the influence of these design variables.

5.2 Training SVM regression prediction model

Using “libsvm” toolbox in MATLAB to predict parameters of 4 types of tooth surface modification Three optimization parameters are available in the SVR module, namely: $\sigma$ in radial basis kernel function, $\varepsilon$ in insensitive loss function and penalty factor c.

Gaussian kernel function is employed in support vector machine regression analysis. Let e-SVR’s loss function $P=\varepsilon$ and the value of $\varepsilon$ be 0.01.

The training set randomly selects data from the 40 sets of orthogonal design experimental results above, and the test set has the remaining 10 sets. The training test and prediction results of the 4 modification methods are as follows:

5.2.1 Prediction model for helix angle modification parameters

As shown in Fig. 9, after training with 40 data sets, the squared correlation coefficient of the training set is 0.99947 and that of the test set is 0.83452. The final prediction model is then obtained, with optimized parameters c $=$ 2.8284 and $\sigma=$ 0.5946.

Figure 9.

Modification model for helix angle prediction.

5.2.2 Prediction model for helix crowning modification parameters

As shown in Fig. 10, after 40 sets of data training, the squared correlation coefficient of the training set is 0.99973, and that of the test set is 0.83550. The optimized parameters of final prediction model are c $=$ 0.28284 and $\sigma=$ 0.707107.

Figure 10.

Prediction model for helix crowning.

Figure 11.

Prediction model for transverse profile slope.

5.2.3 Predicting model for transverse profile slope modification parameters

Figure 11 illustrates how the final prediction model is obtained after 40 sets of data training: the square correlation coefficient of the training set is R2 $=$ 0.99396 and that of the test set is R2 $=$ 0.83326. Furthermore, the optimized parameters are c $=$ 4 and $\sigma=$ 0.707107.

5.2.4 Prediction model for profile crowning modification parameters

According to Fig. 12, after 40 sets of data training, the squared correlation coefficient of the training set is R2 $=$ 0.99966 and that of the test set is R2 $=$ 0.83287. The final prediction model optimized parameters are c $=$ 2 and $\sigma=$ 0.707107.

Figure 12.

Prediction model for profile crowning modification.

In summary, for the prediction models of 4 modification methods obtained from the design parameters and working state, the squared correlation coefficient of the training set is higher than 0.99, and that of the test set is also higher than 0.83, indicating that this model predicts with good accuracy and high practicality.

6. Test of the prediction model of tooth surface modification

To further verify the engineering applicability of the prediction models of the 4 modification methods, the reducer of an electric vehicle was optimized with predicted tooth profile modification parameters based on SVM. The base samples and the optimized samples were tested on a high-speed motor test bench for comparison. The test bench is as presented in Fig. 13. The driving end of the test bench adopts K&A motor: the maximum test power is 388 kW; the maximum speed is 2000 rpm; the maximum torque is 530 Nm (@7000 rpm). The loading end adopts Siemens motor: the maximum speed is 3300 rpm; the maximum torque is 3700 Nm (@800 rpm); the control accuracy of steady-state torque of the system is $\pm$ 1 Nm; the measurement accuracy of steady-state torque is $\pm$ 0.05%FS; the measurement accuracy of steady-state speed is $\pm$ 1 rpm.

Table 3
Comparison of test data

Name	Base prototype	Optimized prototype
Center distance	90 mm	90 mm
Number of teeth	23/51	23/51
Modulus	2.15	2.15
Pressure angle	20 ${}^{\circ}$	20 ${}^{\circ}$
Helix angle	25 ${}^{\circ}$	25 ${}^{\circ}$
Addendum coefficient	1/1	1/1
Dedendum coefficient	1.25/1.25	1.25/1.25
Modification coefficient	0/0.08	0/0.08
Helix slope modification	0 $\mu$ m	8 $\mu$ m
Helix crowning	2 $\mu$ m	5 $\mu$ m
Transverse profile slope modification	0 $\mu$ m	0 $\mu$ m
Profile crowning	2 $\mu$ m	8 $\mu$ m

Figure 13.

Test of the reducer of an electric vehicle.

Table 3 illustrates the test results, parameters of base sample and optimized samples of the gear pair. Figure 13 shows contact spots after tests. Contacting surface of baseline part(left) is mostly on one side; however, that of optimized part(right) is more even, which indicates that the optimization effectively corrects the contact spots and maintains the center contact of the gear pair. Figure 14 illustrates the comparison data of the testing noise obtained with GRAS microphone. Results show that noise level of optimized part(red) is 3–5 dB lower than that of base part(blue) in the full frequency range, indicating an overall noise reduction for the reducer. The efficacy and efficiency of the modification parameters of the 4 prediction models are also validated by this test.

Figure 14.

Contact spots before and after optimization.

Figure 15.

Experimental noise before and after optimization.

6.1 Summary

To explore the influence of gear pair design and working state parameters of transmission system on tooth profile modification, this paper employs the orthogonal design test method and considers 8 factors, including center distance, helix angle, pressure angle, tooth width, etc., to design 50 test sets, analyzing with built models based on test parameters and modifying according to unified standards. Via intuitive analysis, the influence of each factor on the 4 modification methods is determined, and prediction models of the gear system parameters based on each modification method are established using the support regression vector machine method. Results of both training sets and test sets demonstrate that the prediction model has a high precision. Test results also shows this prediction model has sufficient engineering applicability and can be used effectively to forecast tooth profile modification parameters.

This paper has not discussed the coupling influence of each modification parameters. This work will be conducted in the next study to present a more precise prediction model.

Footnotes

Project funding

Supported by Natural Science Foundation of Chongqing (CSTC2019JCYJ-BSHX0031); Sponsored by Science and Technology Research Program of Chongqing Education Commission (KJZD-K201803201); and the Doctoral Fund Project of Chongqing Industrial Vocational and Technical College (2022GZYBSZK1-01).

References

Kang

, et al. A research on the evaluation method of high-frequency whining noise in electric drivetrain. Automobile Engineering. 2019; 41(06): 682.

Yuan

. Gear noise and gear modification. Mechanics Research and Application. 2006; 20(5): 7.

Jiang

Yang

, et al. Application study of micro-modification of tooth surfaces in noise reduction of automobile transmissions. Automobile Engineering. 2009; 31(6): 557.

Liu

. Optimization of reducer whistling based on gear modification. Automobile Technology. 2019; (05): 16.

Wei

. Optimization study of spur gear modification based on energy methods and genetic algorithms. Northwestern University; 2015.

Zhao

Chen

Tang

, et al. Research on noise reduction of secondary reducer gears for new energy vehicles based on diagonal modification. Science Technology and Engineering. 2020; 20(23): 9354.

Cheng

Gao

, et al. Dual-objective optimization of howling noise reduction for the reducer of electric vehicles. Automobile Engineering. 2018; 40(1): 76.

Jia

Fang

. Simulation of excitation and tooth surface optimization and modification of high-speed gear transmission errors and meshing impacts. The Vibration and Shock Journal. 2019; 38(23): 103.

Smith

. “Gear Noise and Vibration” second edition, Marcel Dekker Inc. 2003.

10.

Vapnik

. The Nature of Statistic Learning Theory. The Nature of Statistical Learning Theory. 2000.

11.

Boser

Guyon

Vapnik

. A Training Algorithm for Optimal Margin Classifiers. Proceedings of Annual ACM Workshop on Computational Learning Theory. 1996. doi: 10.1145/130385.130401.

12.

Bao

Zheng

. Transverse flux permanent magnet machine optimizationbased on orthogonal design and support vector machines. Journal of Basic Science and Engineering. 2012; 20(05): 912.

13.

Sun

Yang

. Study on para meter design based on orthogonal test and support vector machine. China Mechanical Engineering. 2011; 22(08): 971.

14.

Yang

Dong

. The study on gear micro modification based on MASTA. Journal of Beijing Information Science and Technology University. 2012; 27(04): 89.

15.

Wang

Sun

, et al. Research on low-noise design of gear based on MASTA software. Ship Science and Technology. 2016; 38(10): 89.

16.

Chen

Zhong

Xiao

. Design method of robust optimization based on orthogonal tests and its application. China Mechanical Engineering. 2004; 15(04): 283.

17.

Zhai

Liang

Wang

, et al. Research on the parameters optimum design of polarized magnetic system based on orthogonal design. Proceedings of the CSEE., 2003; 23(10): 158.

18.

Sun

Feng

, et al. Design of addendum modification parameters of spur gear pair. Engineering Mechanics. 2018; 35(7): 243.

Gear tooth surface modification analysis by support vector machine regression

Abstract

Keywords

1. Introduction

2. Theoretical analysis

2.1 Gear mesh

3. Model construction of involute cylindrical gear pairs

5.1 Intuitive analysis

5.1.1 Analysis of helix angle modification parameters

5.2.1 Prediction model for helix angle modification parameters

5.2.4 Prediction model for profile crowning modification parameters

Table 3 Comparison of test data

Footnotes

Project funding

References

Table 3
Comparison of test data