1 Introduction

Gears play an integral role in many industrial segments, as they supply one of the basic mechanical components for transmitting motion and/or power to ensure the proper functioning of machines, instruments, and equipment [1]. Especially in the field of new energy vehicles, gears need to withstand higher rotational speeds and greater loads, which require gear machining precision to reach 4-5 levels (ISO 1328-1:2013). Insufficient gear machining precision will produce slight vibrations during meshing, which affect the vehicle’s noise, stability, and service life [2, 3]. Therefore, ensuring the gear machining precision that includes tooth surface deviations, tooth profile deviations, tooth lead deviations, and tooth pitch deviations is important [4, 5]. Previous studies have shown that gear machining precision can be improved by reducing the tooth surface deviations, tooth profile deviations, tooth lead deviations, and tooth pitch deviations of machined gears [6, 7].

Gear grinding is usually the last step in the finishing gear manufacturing process, which can remove defects such as uneven materials and pores on the gear surface, thereby meeting the requirements for gear machining precision in most fields [8]. However, the gear grinding mechanism is too complex to ensure gear machining precision [9]. Therefore, it is urgent to study methods for improving gear machining precision [10]. Gear grinding machine is gear machine tools that use the grinding wheel as a cutting tool to process workpiece gears to realize the grinding process. Since the machining process is affected by geometric, thermal, and force errors that can cause a decrease in gear machining precision [11], many researchers have modeled errors and compensated for gear machine tools to achieve an increase in gear machining precision. For instance, CHEN et al [12] developed the geometric error model and a NUM error compensation system for gear grinding machines, which can reduce tooth profile deviations, tooth lead deviations, and tooth pitch deviations. XIA et al [13] proposed a geometric error modeling and compensation for gear grinding machines based on single-axis kinematic measurements and the actual inverse kinematic model, and experiments showed that the tooth profile deviation of the machined gear was reduced. TANG et al [14] proposed an innovative geometric error modeling and compensation for gear grinding machine with the non-rotary cutter, and experiments verified through theoretical calculations and actual machining to reduce the tooth surface deviation of machined gears. LIU et al [15] introduced the effects of geometric and thermal errors on gear grinding machines, proposed an analytical model for rolling guide rails to calculate the geometric errors of the X-axis and Z-axis, and the thermal error models of the spindle and C-axis are established based on transfer learning models. Experimental results showed that this method can significantly reduce tooth profile deviation. GUI et al [16, 17] proposed a new mist-edge-fog-cloud computing system for geometric and thermal error prediction and compensation for gear grinding machines, and the results showed that the tooth surface deviation was reduced. However, the error must be re-modeled when facing different gear machine tools, resulting in the poor universality of this method. Gear machine tools are also affected by other errors from force and motion control errors during machining, and there is a coupling relationship between the errors [18], which makes it difficult to describe precisely the error model for gear machine tools.

As highlighted earlier, modeling and compensating errors for gear machine tools can improve gear machining precision. However, this method has poor universality, and it is difficult to describe precisely the error model for gear machine tools. Moreover, the correlation between machining process parameters and cutting force/heat, changing machining process parameters, will cause changes in cutting force/heat and affect machining precision [19, 20]. Hence, it is meaningful to study the relationship between machining process parameters and machining precision in response to the difficulty of precisely modeling gear machine tools. GUERRINI et al [21] studied the relationship between the machining precision and the process parameters and optimized the process parameters experimentally to improve the gear machining precision. YADAV et al [22] studied and optimized the effect of different process parameter variables on gear roughness for the magnetorheological gear profile finishing process and improved the gear machining precision using optimized process parameters. LIN et al [23] proposed an improved PSO-BP-PSO algorithm for predicting process parameters to establish a prediction model between turning process parameters and gear machining precision. The experimental results show that the optimized process parameters have improved the gear machining precision by 2-3 levels. HAN et al [24] studied the influence of hobbing process parameters on gear machining precision. The experimental results showed that the lower the cutting depth and feed speed, the higher the gear machining precision. SUN et al [6] proposed a neural network based on improved PSO and BP methods to predict the geometric deviation of hobbing gears and optimize process parameters to achieve higher gear machining precision. However, these studies focus on proposing a method to optimize process parameters and improve gear machining precision.

From the review of past works on improving gear machining precision by various methods, there has been relatively little research on the effect of process parameters optimized by different methods on gear machining precision. This study analyzes and discusses the effect of process parameters obtained from statistical analysis and intelligent optimization methods on machining precision, providing guidance for researchers who use process parameters to improve machining precision, thereby saving machining costs to a certain extent. The present work has established a multi-objective optimization model for the cutting speed, feed rate, and cutting depth of the worm wheel gear grinding machine, as well as the relationship between tooth surface deviations, tooth profile deviations, and tooth lead deviations. Based on experimental data, four methods, response surface method (RSM), gray relational analysis-principal component analysis (GRA-PCA), particle swarm optimization (PSO), and genetic algorithm-particle swarm optimization (GA-PSO), are introduced to explore the effect of different process parameters on machining precision. The experimental results show that intelligent optimization is slightly better than statistical analysis in improving machining precision. In addition, the tooth surface deviations, tooth profile deviations, and tooth lead deviations obtained using the GA-PSO are superior to other methods in this paper.

2 Theoretical analysis of gear machining precision

The actual position of the grinding wheel is affected by the various errors when the grinding wheel moves along the gear tooth surface, which causes theoretical and actual contact lines mismatch, thus resulting in the pose error. The resulting pose error will produce tooth surface deviations. In this study, tooth surface deviations are defined as the projection of the position vector deviation between the theoretical tooth surface and the actual tooth surface at the same meshing point in the normal direction of the theoretical tooth surface through meshing point division, as shown in Figure 1. Specifically, the difference is made between the position vector of the theoretical tooth surface and the position vector of the actual tooth surface at the same meshing point and multiplied by the normal vector of the theoretical tooth surface at that point, as shown in Eq. (1):

Figure 1全屏查看

Definition of tooth surface deviations

(1)

where is tooth surface deviations; is the position vector of the theoretical tooth surface; is the position vector of the actual tooth surface; is the normal vector of the theoretical tooth surface.

The gear grinding machine grinds gear by point contact, and the grinding wheel will leave a grinding trace line consisting of grinding points on the gear tooth surface when the gear rotates one tooth. However, the conventional meshing point division method uses equidistant division of tooth profile and tooth lead to obtain crossing points, which has the problem of mismatch between crossing points and grinding points. Therefore, the new meshing point division method is proposed to select the meshing points on the gear tooth surface.

The new meshing point division process is shown in Figure 2. The cyan surface represents the gear tooth surface. Dividing the gear tooth surface equally along the tooth profile and lead can obtain the crossing points. The orange and green arrows represent the tooth lead and profile, respectively. A band area with width is formed with as the center line based on the maximum distance between neighboring grinding locations along the tooth width direction on the same grinding trace line. The upper and lower boundaries of the area are end-face tooth profiles, and the crossing points all fall on the middle line. The width interval of the banded area is calculated as follows:

Figure 2全屏查看

New meshing point division method

(2)

where is the value range of the banded area; is the sequence number of the banded area; is the tooth width of the gear; is the number of equal divisions along the tooth width of the gear; is the maximum distance along the tooth width direction between adjacent grinding points on the same grinding trace line.

In the banded area, the distance between the grinding point and the crossing point is expressed by constructing the radius difference between radius in the grinding point and radius in the crossing point, calculated as follows:

(3)

where is the radius difference; and are the coordinates of the grinding point; is the radius of the crossing point.

According to the machining law of the gear grinding machine, the gear tooth surface will form a large number of grinding points. When dividing the meshing points on the gear tooth surface, if the crossing points of the conventional meshing point division method and the grinding points do not coincide, the radius difference between the crossing points and grinding points can be calculated to determine the meshing points. Based on the radius difference, the crossing points can always match the corresponding grinding points, and the distance between the two points is tiny. Therefore, this study selected the grinding points closest to the crossing points as the meshing points based on the minimum radius difference.

The tooth profile deviations and tooth lead deviations are shown in Figure 3. Tooth profile deviations are the difference between the two design tooth profiles adapted to the actual tooth profile trajectory within the working part of the tooth profile at the gear end [25]. Tooth lead deviations are the deviations between the actual helix and the theoretical helix measured in the tangential direction of the base circle of the gear end face.

Figure 3全屏查看

Tooth profile and tooth lead deviations

Tooth surface deviations contain information about tooth profile and tooth lead deviations, as shown in Figure 4. The gear machining precision is only affected by the tooth surface deviations when machining gears with only tooth surface deviations, as shown in Figure 5(a). However, there may not only be tooth surface deviations in the machining process but also tooth profile and tooth lead deviations. According to Figure 5(a), it can be seen that the tooth surface deviations are affected by the tooth profile and tooth lead. If there are tooth profile and tooth lead deviations, the tooth profile and tooth lead of the tooth surface will deviate from the theoretical position, causing an offset of the meshing points on the actual tooth surface. The meshing points on the theoretical tooth surface always remain unchanged, so the tooth surface deviations changes at this point. Similarly, if the tooth surface deviations change, the corresponding tooth profile and tooth lead deviations will also change. The effect of tooth profile deviations, tooth lead deviations, and these two deviation terms on tooth surface deviations are shown in Figures 5(b)-(d). Therefore, the method to improve the gear machining precision in this study is to consider reducing the tooth surface deviations, tooth profile deviations, and tooth lead deviations.

Figure 4全屏查看

Tooth surface deviations contain information about tooth profile and tooth lead deviations

Figure 5全屏查看

Effect of different deviations on tooth surface deviations: (a) Normal; (b) Tooth profile deviations; (c) Tooth lead deviations; (d) Tooth profile deviations and tooth lead deviations

3 Experimental procedures

The experiment is carried out on the YW7232 worm wheel gear grinding machine. The tool used is the grinding wheel and the workpiece is helical gear with the parameters shown in Table 1. Through literature research [26, 27], the main process parameters that affect the machining accuracy performance for gear machine tools are cutting speed, cutting depth, and feed rate. Accordingly, the process parameters of cutting speed, cutting depth, and feed rate are selected for this experiment, and other process parameters are kept constant. The lubrication conditions used in the experiment are to spray HG68 lubricating oil with a cleanliness level of NAS1638-9 onto the contact surface. The experimental design method chosen is the central composite design of the RSM, and the selection of three process parameters is shown in Table 2. The tooth surface deviations, tooth profile deviations, and tooth lead deviations of machined gear are obtained by experiments, as shown in Table 3.

In order to improve the gear machining precision, it is necessary to reduce tooth surface deviations, tooth profile deviations, and tooth lead deviations. However, reducing tooth surface deviations, tooth profile deviations, and tooth lead deviations can be regarded as a multi-objective optimization problem. The two commonly used methods for multi-objective optimization problems are intelligent optimization algorithms and statistical analysis. In this paper, the intelligent optimization methods used are PSO and GA, and the statistical analysis methods used are RSM and GRA-PCA. The flowchart of the adopted method is shown in Figure 6.

Figure 6全屏查看

Flowchart of adopted method

4 Process parameter optimization for different methods

4.1 RSM method

The effect of process parameters of cutting depth, feed rate, and cutting speed on the response surface of tooth surface deviations, tooth profile deviations, and tooth lead deviations are shown in Figures 7-9. The darker the response surface, the more significant the effect of process parameters.

Figure 7全屏查看

Response surface of tooth surface deviations: (a) Cutting speed and feed rate interaction; (b) Cutting speed and cutting depth interaction; (c) Feed rate and cutting depth interaction

Figure 8全屏查看

Response surface of tooth profile deviations: (a) Cutting speed and feed rate interaction; (b) Cutting speed and cutting depth interaction; (c) Feed rate and cutting depth interaction

Figure 9全屏查看

Response surface of tooth lead deviations: (a) Cutting speed and feed rate interaction; (b) Cutting speed and cutting depth interaction; (c) Feed rate and cutting depth interaction

The interaction of the feed rate and the cutting speed on the tooth surface deviations is shown in Figure 7(a). When the feed rate is given, there is a significant change in surface color as the cutting speed increases. When the cutting speed is given, there is a greater change in surface color than the former as the feed rate increases. The results indicate that the feed rate has more effect on tooth deviations than the cutting speed.

The interaction of the cutting depth and the cutting speed on the tooth surface deviations is shown in Figure 7(b). When the cutting depth is given, the surface color changes insignificantly as the cutting speed increases. When the cutting speed is given, there is a significant change in surface color as the cutting depth increases. The results indicate that the cutting depth has more effect on tooth deviations than the cutting speed.

The interaction of the cutting depth and the feed rate on the tooth surface deviations is shown in Figure 7(c). When the cutting depth is given, the surface color is obvious with the increase of the feed rate. When given the feed rate, the surface color changes significantly more than the former as the cutting depth increases. The results indicate that the cutting depth has more effect on tooth deviations than the feed rate.

Through the above analysis, the effect of process parameters on tooth surface deviations is ranked as cutting depth, feed rate, and cutting speed in order of importance.

Furthermore, the interaction of process parameters on tooth profile deviations and tooth lead deviations is shown in Figures 8 and 9. Similarly, the following conclusions are available. The degree of effect of process parameters on tooth profile deviations is arranged according to cutting depth, feed rate, and cutting speed. The degree of effect of process parameters on tooth lead deviations is ranked as feed rate, cutting depth, and cutting speed.

4.3 PSO and GA-PSO methods

The initialization parameters of PSO are initialization weight set to 1, learning factors set to 2, population size set to 50, the number of iterations set to 200, particle position range consistent with process parameter levels, and particle velocity range of -0.5 to 0.5 [28]. The initialization parameters of GA are chromosome coding length set to 3, crossover probability set to 0.65, and mutation probability set to 0.005.

The GA performs crossover and mutation operations on the particle positions updated by PSO to obtain new particle positions and then calculates the fitness corresponding to new particle positions [29]. As a result, the GA-PSO has a global search ability and prevents the population from falling into a local optimality dilemma. For multi-objective problems, there is usually a coupling between the solutions, which makes it hard to obtain an optimal solution. Pareto optimality is an effective method for solving multi-objective problems [30]. An optimal set can be obtained by Pareto, which is a set consisting of mutual non-domination among solutions. The Pareto optimal front is the set of objective values corresponding to the Pareto optimal set. Trade-offs or compromises are made between the objective values of the Pareto optimal front to obtain the experimentally optimal process parameters. The process parameters selected using PSO and GA-PSO are determined based on the Pareto optimal front, as shown in Figure 10.

Figure 10全屏查看

Pareto front of PSO and GA-PSO

5 Results and discussion

According to Figures 7-9, the process parameters obtained using RSM are that the cutting speed is 35 m/s, the feed rate is 180 mm/min, and the cutting depth is 0.4 mm. The experimental results are analyzed using GRA-PCA, and the results are shown in Table 5. The process parameters of cutting speed, feed rate, and cutting depth obtained are 40 m/s, 180 mm/min, and 0.332 mm, respectively. The process parameters of PSO and GA-PSO can be obtained from Figure 10. The process parameters obtained by PSO are that the cutting speed is 36.516 m/s, the feed rate is 177.483 mm/min, and the cutting depth is 0.332 mm. Similarly, the process parameters of cutting speed, feed rate, and cutting depth obtained using GA-PSO are 37.331 m/s, 180.673 mm/min, and 0.332 mm, respectively.

There are differences in the process parameters obtained by different methods, and experiments are needed to verify which is more effective in improving gear machining precision. The tooth surface deviations, tooth profile deviations, and tooth lead deviations obtained by different process parameters are shown in Table 6. The results obtained by the four methods are very similar, which is explained by the fact that the range of process parameters does not fluctuate greatly. The three deviation terms obtained from GRA-PCA are somewhat reduced compared to RSM. The GRA-PCA gives better process parameters than RSM due to an in-depth analysis of potential connections between RSM experimental results. The PSO has a relatively large tooth lead deviations compared to the GRA-PCA. The worse tooth lead deviations of PSO than GRA-PCA may be due to the optimization process falling into a local optimal. Compared with RSM, GRA-PCA, and PSO, the three deviation terms obtained by GA-PSO are all the smallest because GA-PSO solves the PSO local optimal problem.

Overall, intelligent optimization algorithms are better than statistical analysis in solving multi-objective problems. Because the intelligent optimization algorithms explore the optimal solution within the upper and lower bounds of the process parameters. The statistical analysis is limited to a given level of process parameters. However, statistical analysis can explore potential relationships between experimental results independent of the initial solution. Intelligent optimization has different effects depending on the initial solution, which limits the development of intelligent optimization. In this study, the process parameters obtained by intelligent optimization are more effective in improving the gear machining precision of the machined gear.

From the perspective of the effect of process parameters on gear machining precision, gear machining precision decreases as the cutting depth increases, showing a negative correlation. The reason is that increasing the cutting depth will increase the material removal on the gear surface, resulting in uneven distribution of cutting force and accumulation of cutting heat, thus reducing the gear machining precision [31-33]. As the feed rate increases, the gear machining precision first increases and then decreases. Properly increasing the feed rate can increase the relative speed between the grinding wheel and the gear, promote the timely removal of chips, reduce the accumulation of cutting heat, and help improve gear machining precision [34]. However, excessive feed rate increases the contact area between the grinding wheel and the gear, thereby increasing cutting force and leading to a decrease in gear machining precision [35, 36]. As the cutting speed increases, the gear machining precision first increases and then decreases. Appropriately increasing the cutting speed can speed up the material removal speed, causing the chips to take away more heat, thereby reducing the heat transmitted to the gear, which is beneficial to improving the gear machining precision [37]. However, excessive cutting speed may increase the wear of the grinding wheel, accumulate too much heat, and thus reduce the gear machining precision [38, 39].

6 Conclusions

This paper establishes a multi-objective optimization model through tooth surface deviations, tooth profile deviations, and tooth lead deviations to optimize the process parameters with the goal of improving gear machining precision. The optimization process uses different methods to obtain the optimal process parameters. The experimental results show that the optimal process parameters effectively improve the gear machining precision. The contribution of this paper is summarized as follows.

1) The new meshing point division method is proposed to solve the problem of mismatch between crossing points and grinding points, and to realize meshing point division of the gear tooth surface.

2) The coupling relationship between tooth surface deviations, tooth profile deviations, and tooth lead deviations is determined through theoretical analysis, thereby establishing a multi-objective optimization model to improve gear machining precision.

3) The experimental results show that cutting depth has the greatest impact on gear machining precision, followed by feed rate and cutting speed. Increasing the cutting depth will reduce the gear machining precision while increasing the feed rate and cutting speed will lead to a trend of first improving and then decreasing the gear machining precision. Based on this result, how the process parameters affect the gear machining precision is further analyzed.

4) The effect of process parameters on gear machining precision is discussed from the perspectives of statistical analysis (RSM, GRA-PCA) and intelligent optimization (PSO, GA-PSO), which provides guidance for relevant researchers, thereby saving machining costs to a certain extent.