Quality Evaluation of Effective Abrasive Grains Micro-Edge Honing Based on Trapezoidal Fuzzy Analytic Hierarchy Process and Set Pair Analysis

Abstract

Studying the factors affecting machining accuracy, surface quality, and machining efficiency in the powerful honing machining process system, analyzing the basic law between various errors and machining quality, exploring the method of evaluating the quality of honing, and improving the machining quality and transmission performance of hardened gears has important engineering application value. Firstly, this paper establishes an effective abrasive grains micro-edge honing quality evaluation model, proposes a method based on the Trapezoidal Fuzzy Analytic Hierarchy Process (Tra-FAHP) and Set Pair Analysis (SPA) to comprehensively evaluate the quality of the honing process, and obtains the influence weights of each factor on the quality of honing. Secondly, the paper analyzes the influence rules of three types of abrasive grain sizes on helix error, tooth pitch error, tooth profile error, surface roughness, and honing efficiency. Finally, the correctness of the established comprehensive evaluation model of honing quality was verified with the threshold method and weights. The research results show that the model can correctly evaluate the quality of hardened gear honing and can be applied to studying the influence of abrasive grain micro-edge honing on machining characteristics.

1. Introduction

Gear honing, a key finishing process in gear manufacturing, has the advantages of high tooth precision, high load-carrying capacity, low transmission noise, and regular tooth texture, etc. [1,2], These characteristics make it particularly suitable for gear processing in the production of new energy vehicle drivetrains [3,4,5]. The quality of honed teeth plays a crucial role in gear life, transmission noise, and load-carrying capacity. However, the factors affecting the quality of honed teeth are characterized by randomness, complexity, and unpredictability, making a detailed analysis of honing machining characteristics challenging. Compared to traditional grinding methods, the quality of honing is further improved [6].

Honing as a gear finishing method, which uses a pair of staggered-axis cylindrical gears for forced meshing into a movement through the honing wheel on the abrasive grains sliding friction workpiece tooth surface, removes a thin layer of material. Some scholars evaluated the quality improvement of gears with grinding wheels from different perspectives. Sandro Pereira da Silva [7] established the relationship between five input parameters such as spindle speed, machining time, X- and Z-direction feed rate, and spark out time with total profile deviation (fα), total helix deviation (fβ), and total cumulative pitch deviation (fp) by using a grain swarm algorithm. Yuan Bin [8] has optimized and actively controlled the spindle speed, axial feed speed, and radial feed speed of the honing process parameters at a constant cutting speed to minimize the total tooth shape error. The above literature focuses on the effect of various factors on the accuracy of the tooth surface.

Xin Li [9] compared five gear finishing processes and comprehensively evaluated the tooth surface quality with parameters such as surface roughness, surface morphology, surface residual stress, surface texture roughness, and peak radius of curvature after machining, The Author highlighted the synergistic optimization of factors such as abrasive grain size and shape as an effective method to improve both machining efficiency and surface quality. W. Liu [10] designed an orthogonal test of a diamond abrasive grain grinding wheel on silicon carbide grinding to analyze the effects of the processes such as grain size, grinding wheel speed, workpiece speed, and grinding depth on surface roughness, surface topography, and grinding force. The grinding quality was comprehensively analyzed, and the process parameters were optimized based on these factors. The above literature focuses on the effects of various factors on surface quality.

Silva, SP [11] studied the effect of honing process parameters such as speed of pinion, speed of the grinding wheel, feed rate, spark out, and cycle time. The experimental results showed that the helix form error and the evolvent profile error were affected by the cycle time and feed rate. Sergey Riabchenko [12] evaluated the working efficiency of disc circles through parameters such as grinding power, surface roughness, involute tooth shape accuracy, and wear of the disc circles. The experimental results show that the dimensional wear of the grinding wheel is the main factor affecting the accuracy of grinding processing, and the grinding efficiency is correlated with the depth of cut and the grinding time of a single tooth. Fuwei Wang [13] proposed a specific energy model for quantitatively evaluating the material removal rate of honed teeth. Experimentally, it was observed that a larger honing depth resulted in an increase in material removal rate. In addition, at the same workpiece speed and honing depth, an increase in both the cross-axis angle and axial feed rate resulted in an increase in specific honing energy and a decrease in material removal rate. The above studies focus on the effects of various factors on processing efficiency.

Based on controlling systematic errors in the honing process and external disturbances, the honing process and the parameters that characterize the properties of the honing wheel are considered key elements affecting honing quality. Based on the above literature and existing research foundation [13,14,15,16], the honing process parameters mainly include honing speed, honing depth, feed rate, and cross-axis angle, and the main factors characterizing the surface morphology of the honing wheel are abrasive grain size, shape of abrasive grain, effective number of abrasive grain, clearance between abrasive grains, and grinding cutting angle. Some studies showed that the dimensional wear of grinding wheels is the main factor affecting the accuracy of gear machining. Honing of teeth is a generative machining process that requires high dimensional accuracy of honing wheel surfaces. The critical factors affecting the dimensional wear [17,18] of grinding wheels include abrasive grain shedding, abrasive grain fracturing, and abrasive grain abrasion. The 12 parameters listed above were selected as key factors affecting the quality of honed teeth.

At present, research on honing quality is primarily evaluated from the aspects of tooth shape accuracy, tooth surface quality, and honing efficiency. Tooth shape accuracy includes helix error, tooth pitch error, and tooth profile error. Surface roughness affects tooth wear resistance, fit stability, corrosion resistance, etc., which is often used to evaluate surface quality. Compared with other gear finishing processes, the unique surface texture of honing can reduce transmission noise and vibration [19], and the surface residual stresses can significantly increase service life [20]. Therefore, the research on the quality of honed tooth surfaces focuses on three factors: surface roughness, residual stresses, and tooth texture. Honing efficiency is characterized by the ratio of material removal rate to machining time. Existing studies on the quality of honed teeth mostly analyze individual factors without considering a comprehensive evaluation of multiple factors. Based on the above shortcomings, the study introduces a comprehensive evaluation method for multiple factors.

2. Related Work

Fuzzy Hierarchical Analysis (FHA) is a multi-objective decision-making method that is widely used for comprehensive evaluation in the field of machining. In order to solve the complex problem of machinability of titanium alloy, which has low density, high hardness, low thermal conductivity, and low modulus of elasticity. K. Fuse [21] applied a combination of FHA and ideal similarity ranking preferred fuzzy technology for multi-objective decision-making on the wire-cutting machining process, combined with experiments to optimize a set of optimal process parameters. For the complexity of laser melting coating, there is no practical method to evaluate the quality of the coating. W.X. Liang [21] proposed a fuzzy comprehensive model for evaluating the quality of plating from the macroscopic profile, microstructure, and mechanical properties, using FHA to rank the evaluation indexes in terms of weights and optimizing the process parameters. A.S. Patil [22] modeled and evaluated the milling quality of Ti6AI4V in terms of Surface integrity, Flank, and Crater wear. FHA and prioritization were used to determine the better process parameters. Trapezoidal fuzzy numbers, a type of fuzzy mathematics, can also be used for multi-factor evaluation of honing quality.

However, a single evaluation method has a large deviation between the theoretical calculation results and the actual value, unable to address the ambiguities, uncertainties, and issues related to quantification among multiple factors. SPA is a mathematical method for dealing with the interaction of determinism and uncertainty in a system. Y.H. Liu [23] combined FAH and SPA to carry out a comprehensive evaluation of regional comprehensive energy consumption, through the measured data, the reliability of the comprehensive evaluation method and results was verified.

G.Z. Huang [24] combined the fuzzy comprehensive analysis method and SPA for construction safety evaluation and found that combining the two methods could simultaneously handle the uncertainty and fuzziness of information in construction safety and achieve better evaluation results. J.W. Wang [25] compared the evaluation results of traditional FAH, entropy weighting, fuzzy comprehensive evaluation, gray correlation analysis, and ideal solution similarity preference ranking method and proposed an environmental risk synthesis method based on the combination of intuitive FAH and SPA. According to the results, the evaluation model is more objective and reasonable for addressing the ambiguities and uncertainties in environmental risks.

In this paper, for the factors affecting the quality of honed teeth, which are characterized by randomness, complexity, and unpredictability, an effective abrasive grains micro-edge honing characteristic evaluation model is established to solve the problem of unsystematic and incomplete evaluation of honing quality and provide a decision-making basis for the manufacture of high-performance hard gears.

3. Evaluation Methodology and Factor Analysis

3.1. Evaluation Methodology

In this study, a combination of Tra-FAHP and SPA was used to evaluate the quality of the Cubic Boron Nitride (CBN) grit micro-edge honing process, and the evaluation process is shown in Figure 1.

Firstly, analyze and identify the honing quality indexes and establish an effective abrasive grains micro-edge honing quality evaluation system based on AHP. Then, obtain the consistency matrix. Secondly, the trapezoidal fuzzy evaluation matrix is obtained by trapezoidal fuzzification of the consistency matrix elements through the trapezoidal fuzzy center inverse formula. Finally, the transformation of fuzzy numbers to linkage numbers is achieved through ean-variance, which introduces SPA to determine the final quality evaluation composite weights.

3.2. Establishing a Three-Stage and Four-Level Indicator System

Based on the analysis of the factors affecting the honing process quality and the relationships between the indicators mentioned above, a three-level and a four-level indicator system, as illustrated in Figure 2, has been established to comprehensively evaluate the honing quality.

4. Evaluation Procedure

4.1. AHP Determines the Consistency Matrix

AHP is a subjective assignment method that can clearly identify evaluation models for complex, cross-cutting decision-making problems. By making pairwise comparisons between two factors at the same level, it can derive the importance of each factor relative to the elements of the previous level. The specific process is as follows:

(1)

Build hierarchy

In Figure 2, a logical hierarchy is constructed based on the factors associated with the enhancement of quality. “It’s an indicator system composed of three tiers and four levels, with the top-level objective (A) serving as the overarching evaluation. This is then followed by the criterion level (B), which encompasses observation B for objective A. The sub-criterion level (C) then follows, with observation C for B forming its basis. Finally, the element level (D) addresses the factors affecting the top-level objective (A).

(2)

Constructing the Pairwise Comparison Matrix

The relative importance of the two elements is evaluated using the “1–9 point scale”, as illustrated in

Table 1. Satie’s 1–9 point scale and corresponding definitions.

The Comparison of the Importance of Element ai and aj	Digital Scale	Inversion
Equally important	1	1
Moderately important	3	1/3
Strongly important	5	1/5
Extremely important	7	1/7
Completely important	9	1/9
Intermediate value of the above importance	2, 4, 6, 8	1/2, 1/4, 1/6, 1/8

[26]. An $n\ast n$ pairwise comparison judgment matrix regarding $Q$ is established.

According to the hierarchical structure of the evaluation model, created $\mathrm{A}={\left(B_{ij}\right)}_{3\times 3}$, $B_1={\left(C_{ij}\right)}_{3\times 3}$, $B_2={\left(C_{ij}\right)}_{2\times 2}$, …, $C_8={\left(D_{ij}\right)}_{12\times 12}$ among other paired comparison matrices totaling twelve.

$$Q_P={\left(a_{ij}\right)}_{n\times n}=\left[\begin{array}{ccc}\begin{array}{c}\begin{array}{cc}{\displaystyle \frac{a_1}{a_1}}& {\displaystyle \frac{a_1}{a_2}}\end{array}\\ \begin{array}{cc}{\displaystyle \frac{a_2}{a_1}}& {\displaystyle \frac{a_2}{a_2}}\end{array}\end{array}& \begin{array}{c}\dots \\ \cdots \end{array}& \begin{array}{c}{\displaystyle \frac{a_1}{a_n}}\\ {\displaystyle \frac{a_2}{a_n}}\end{array}\\ \begin{array}{cc}⋮& ⋮\end{array}& \ddots & ⋮\\ \begin{array}{cc}{\displaystyle \frac{a_n}{a_1}}& {\displaystyle \frac{a_n}{a_2}}\end{array}& \dots & {\displaystyle \frac{a_n}{a_n}}\end{array}\right]$$

(1)

In the formula, $Q_P$ represents the judgment matrix, where n denotes the total number of elements. The value of element $a_{ij}$ indicates the degree of importance of the i-th element $a_i$ relative to the j-th element $a_j$ within the same level in terms of their influence on the upper-level element. It holds that $a_{ij}$ and $a_{ji}$ are reciprocals, that is, $a_{ij}={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$a_{ji}$}\right.}$, and $a_{ii}=1$.

(3)

Check consistency

Consistency testing CR is calculated according to Equation (2), and a CR value of less than 0.1 indicates that the consistency test has been passed. The consistency index RI is shown in

Table 2. Random index values for different scales.

n	1	2	3	4	5	6	7	8	9
RI	0	0	0.52	0.89	1.12	1.26	1.36	1.41	1.46

, and the consistency index (CI) is computed as per Equation (3).

$$CR={\displaystyle \frac{CI}{RI}}$$

(2)

$$CR={\displaystyle \frac{CI}{RI}}$$

(3)

In this context, ${\lambda }_{max}$ refers to the maximum eigenvalue of the judgment matrix, and n represents the number of columns or rows of the judgment matrix.

The matrices that pass the consistency test are deemed to exhibit satisfactory consistency and can be used for trapezoidal fuzzy transformation.

4.2. Determine the Trapezoidal Fuzzy Evaluation Matrix

The concept of trapezoidal fuzzy numbers was proposed by Zadeh in 1965. It utilizes trapezoidal membership functions to address problems in uncertain environments.

(1)

Constructing a fuzzy judgment matrix

Assuming $U=\left(u_1,u_2,u_3,u_4\right)$, where $u_1,u_2,u_3,u_4$ are real numbers in the set $U$, and $0<u_1<u_2<u_3<u_4$. Here, $u_1$ and $u_4$ represent the lower and upper bounds of the trapezoidal fuzzy number $U$, respectively, while $u_2$ and $u_3$ are the median values of the trapezoidal fuzzy number. The formula for the trapezoidal membership function is given as (4) [27].

$$\mu \left(x\right)=\left\{\begin{array}{c}\begin{array}{c}\begin{array}{c}0,\left(x\le u_1\right)\\ {\displaystyle \frac{x-u_1}{u_2-u_1}},\left(u_1<x<u_2\right)\end{array}\\ 1,\left(u_2\le x\le u_3\right)\\ {\displaystyle \frac{u_4-x}{u_4-u_3}},\left(u_3<x<u_4\right)\end{array}\\ 0,\left(x\ge u_4\right)\end{array}\right.$$

(4)

Considering the subjectivity, ambiguity, and uncertainty in the scoring provided by experts in the direction of honing, the trapezoidal fuzzy numbers are calculated using the trapezoidal centroid formula. This process transforms the general judgment matrix into a fuzzy judgment matrix. In order to better reflect the experts’ thought processes, as shown in

Table 3. ‘1–9 point scale‘ method and improved trapezoidal fuzzy number.

The Comparison of the Importance of Element ai and aj	Traditional aij Assignment	Improving aij Assignment	Improved aij Trapezoidal Fuzzy Assignment
equality	1	5/5	(1, 1, 1, 1)
A little important	3	6/4	(1, 11/9, 13/7, 7/3)
Obviously important	5	7/3	(3/2, 13/7, 3, 4)
Strongly important	7	8/2	(7/3, 3, 17/3, 9)
Extremely important	9	9/1	(4, 17/3, 9, 9)

, an improved valuation method is applied to the traditional “1–9 point scale” method [28].

$$U={\displaystyle \frac{{\int }_u\mu \left(x\right)\cdot xdx}{{\int }_u\mu \left(x\right)dx}}{\displaystyle \frac{{(u}_3^2+u_3·u_4+u_4^2)-{(u}_1^2+u_1·u_2+u_2^2)}{3·\left(u_4{+u}_3-u_1{-u}_2\right)}}$$

(5)

In order to facilitate subsequent calculations, a brief introduction to the arithmetic rules of trapezoidal fuzzy numbers is provided. Let two trapezoidal fuzzy numbers be $U=(u_1,u_2,u_3,u_4)$ and $M=(m_1,m_2,m_3,m_4)$, and let l be any constant. The arithmetic rules are as follows in

Table 4. Trapezoidal Fuzzy Number Arithmetic Rules.

Algorithm	Operational Formula
addition	$U+M=(u_1+m_1,u_2+m_2,u_3+m_3,u_4+m_4)$
subtraction	$U-M=(u_1-m_4,u_2-m_3,u_3-m_2,u_4-m_1)$
multiplication	$U\times M=(u_1{m}_1,u_2{m}_2,u_3{m}_3,u_4{m}_4)$
division method	${\displaystyle \frac{U}{M}}=({\displaystyle \frac{u_1}{m_4}},{\displaystyle \frac{u_2}{m_3}},{\displaystyle \frac{u_3}{m_2}},{\displaystyle \frac{u_4}{m_1}})$

:

By using the data from Table 4 to replace the elements in the simplified matrix with trapezoidal fuzzy numbers, ensuring consistency, the fuzzy judgment matrix U expressed in terms of trapezoidal fuzzy numbers can be obtained. The simplified matrix must satisfy the consistency requirement, and the improved trapezoidal fuzzy number judgment matrix also meets the consistency requirement [28].

Assuming that n (n > 1) experts with equal status are invited to score at the same time, the reduced matrix after the consistency test is performed, and trapezoidal fuzzy processing is performed. The matrix $U^k$ after fuzzification of the nth expert data is recorded as:

$$U^k={\left(u_{ij}^k\right)}_{n\times n}$$

(6)

In Formula (6), $u_{ij}^k=({u_1}_{ij}^k,{u_2}_{ij}^k,{u_3}_{ij}^k,{u_4}_{ij}^k)$, K = 1, 2, ⋯, n.

Based on the preferences of n experts, calculate the average of $u_{ij}^k$ as the evaluation fuzzy matrix $\overline{U}$ using the formula:

$$\overline{U}=\left({{\overline{u}}_1}_{ij},{{\overline{u}}_2}_{ij},{{\overline{u}}_3}_{ij},{{\overline{u}}_4}_{ij}\right)={\displaystyle \frac{1}{\mathrm{n}}}\otimes \left({\sum }_{L=1}^n{u_1}_{ij}^k,{\sum }_{L=1}^n{u_2}_{ij}^k,{\sum }_{L=1}^n{u_3}_{ij}^k,{\sum }_{L=1}^n{u_4}_{ij}^k\right)$$

(7)

(2)

Index weight calculation.

In order to reduce the influence of extreme data on the overall weight, the paper uses the geometric average method to ascertain the fuzzy evaluation value of the factor layer $D_i$, and the weight of the evaluation index ${\tilde{V}}_i$ is:

$${\tilde{V}}_i=\left(v_1,v_2,v_3,v_4\right)=({\displaystyle \frac{{\alpha }_i}{\delta }},{\displaystyle \frac{{\beta }_i}{\gamma }},{\displaystyle \frac{{\gamma }_i}{\beta }},{\displaystyle \frac{{\delta }_i}{\alpha }})$$

(8)

In Formula (8), $v_j$ is the weight of the index $D_i$, and the values of ${\alpha }_i$, ${\beta }_i$, ${\gamma }_i$ and ${\delta }_i$ are calculated by Formula (9), and $\alpha ={\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\alpha }_i$, $\beta ={\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\beta }_i$, $\gamma ={\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\gamma }_i$, $\delta ={\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\delta }_i$.

$$\left\{\begin{array}{c}\begin{array}{c}{\alpha }_i={\left[{\prod }_{j=1}^n{\overline{u}}_1\right]}^{1/n}\\ {\beta }_i={\left[{\prod }_{j=1}^n{\overline{u}}_2\right]}^{1/n}\\ {\gamma }_i={\left[{\prod }_{j=1}^n{\overline{u}}_3\right]}^{1/n}\end{array}\\ {\delta }_i={\left[{\prod }_{j=1}^n{\overline{u}}_4\right]}^{1/n}\end{array}\right.$$

(9)

Normalize the calculation of fuzzy evaluation index weights $V_i$.

$$V_i={\displaystyle \frac{E_{vi}}{{\sum }_{i=1}^m{E}_{vi}}}$$

(10)

In Formula (10), $E_v={\displaystyle \frac{v_1+v_2+v_3+v_4}{4}}$ is the expectation of the weight of the evaluation index. By combining Formulas (5)–(7), the Tra-FAHP weight and normalized weight of each evaluation index can be obtained.

4.3. SPA Evaluation Model

SPA is a systematic method proposed by scholar Zhao Keqin in 1989, which utilizes the dialectical unity of opposites to analyze the relationships between determinacy and indeterminacy, with a focus on the connectivity function [29]. Based on the dialectical unity thought, it analyzes the identity, opposition, and differences of set pairs, reflecting the interconnection between the determinacy and indeterminacy of things.

4.3.1. Establishment of a SPA Evaluation Model

Let the sub-criteria layer C of the criterion layer $D_k(k=1,2,\cdots ,n)$ have the attribute weights relative to the target layer A denoted by $w_t$, and the attribute values of the criterion layer be represented as $p_{kt}(t=\mathrm{1,2},\cdots n)$. The weighted comprehensive value of uncertain multi-attribute decision-making $M\left(D_k\right)(k=1,2,\cdots ,n)$ is given by:

$$M\left(D_k\right)={\sum }_{t=1}^n{w}_t{p}_{kt}={\left[M\left(S_1\right),M\left(S_2\right),M\left(S_3\right),M\left(S_4\right)\right]}^t$$

(11)

Equation (11) represents the weighted comprehensive optimization decision value for the quality evaluation scheme of effective abrasive grains micro-edge honing process using CBN coated internal gear honing wheels.

The SPA evaluation matrix is constituted as a 12-row by 8-column matrix, with the factor layer D represented as row vectors and the sub-criterion layer C as column vectors, where the composite value $M\left(D_k\right)$ serves as the individual elements.

4.3.2. SPA Analysis

Let the set pair H be constituted by sets A and B, denoted as $H=(A, B)$. The relationship between set A and set B, as well as the degree of their superiority and inferiority, can be represented by the connectivity degree ${\mu }_{A\leftrightarrow B}$.

$${\mu }_{A\leftrightarrow B}=a+bi+cj$$

(12)

In Equation (12), a, b and c are identity degree, difference degree, and opposition degree, respectively, $a,b,c\in \left[\mathrm{0,1}\right]$, and $a+b+c=1.$ In reference [30], it was proposed that the mean and variance of sample characteristic parameters in statistics can be represented by trapezoidal fuzzy number characteristic parameters based on the uncertainty system theory of SPA. The characteristics of the ‘mean + variance’ combination are represented by the connective $A+Bi$. The utilization of this feature, the connection number $A+Bi$, establishes a coupling between trapezoidal fuzzy number and SPA.

For trapezoidal fuzzy numbers $U=\left(u_1,u_2,u_3,u_4\right)$, if $u_1, u_2, u_3, u_4$ are considered as four observations of the same object U, then the average value $\overline{u}$ of $U$ is:

$$\overline{u}={\displaystyle \frac{1}{n}}{\sum }_{j=1}^n{u}_j={\displaystyle \frac{u_1+u_2+u_3+u_4}{4}}$$

(13)

The variance s of trapezoidal fuzzy number is:

$$s=\sqrt{{\displaystyle \frac{{\sum }_{j=1}^n\left(u_j-\overline{u}\right)}{n-1}}}=\sqrt{{\displaystyle \frac{{\left(u_1-\overline{u}\right)}^2+{\left(u_2-\overline{u}\right)}^2+{\left(u_3-\overline{u}\right)}^2+{\left(u_4-\overline{u}\right)}^2}{3}}}$$

(14)

$\overline{u}$ represents the certainty of fuzzy numbers, and the variances are the description of the uncertainty of fuzzy numbers, illustrating the centralized (certainty) and discrete (un-certainty) nature of the trapezoidal fuzzy numbers in a statistical context.

Then the mean-variance connection number $u\left(\overline{u},s\right)$ is:

$$u\left(\overline{u},s\right)=A+Bi=\overline{u}+si$$

(15)

The mean $\overline{u}$ and variance s can be mapped to the two-dimensional deterministic-uncertain plane of SPA, and the mean $\overline{u}$ and variance s are expressed as the deterministic-uncertain weight in the “modulus” value of the set pair plane vector, which is calculated by Formula (16):

$$r=\sqrt{{\overline{u}}^2+s^2}$$

(16)

The overall weight is obtained by normalizing the “model”.

$${\mathrm{E}}_{\mathrm{i}}={\displaystyle \frac{{\mathrm{r}}_{\mathrm{i}}}{{\sum }_{i=1}^t{\mathrm{r}}_{\mathrm{i}}}}\times 100\%$$

(17)

5. Verification of Comprehensive Evaluation Model for Honing Processing Quality

5.1. Determining the Weight of Influencing Factors of Honing Machining Quality

According to the gear honing quality evaluation hierarchy shown in Figure 2, the questionnaire was designed to hire five experts of equal status with relevant industry backgrounds to score the factors affecting machining quality. After consistency adjustment and testing, an improved trapezoidal fuzzy number assignment calculation method was applied. It determines the quality of the internal gear honing wheel’s CBN coating, the effectiveness of abrasive grains, and the micro-edge honing processing quality. The hierarchical structure of the factors under the relative evaluation index and the corresponding Tra-FAHP weights are shown in

Table 5. The Tra-FAHP weights of the relative evaluation indicators of factors.

Evaluation Objective	Factor of Evaluation	Tra-FAHP Weight
Analysis of Honing Characteristics A	Honing accuracy B1	(0.278, 0.373, 0.712, 0.964)
	Processing efficiency B2	(0.097, 0.136, 0.254, 0.341)
	Surface quality B3	(0.166, 0.222, 0.403, 0.542)
Honing accuracy B1	Tooth pitch error C1	(0.250, 0.331, 0.577, 0.774)
	Tooth profile error C2	(0.181, 0.237, 0.398, 0.518)
	Helix error C3	(0.151, 0.198, 0.331, 0.426)
Processing efficiency B2	Material removal amount C4	(0.188, 0.241, 0.395, 0.515)
	Processing time C5	(0.164, 0.212, 0.350, 0.452)
Tooth pitch error C1	Abrasive grain abrasion D1	(0.032, 0.047, 0.098, 0.140)
	Abrasive grain fracture D2	(0.043, 0.064, 0.137, 0.202)
	Abrasive grain detachment D3	(0.064, 0.096, 0.217, 0.348)
	Grinding cutting angle D4	(0.024, 0.037, 0.077, 0.111)
	Shape of abrasive grain D5	(0.031, 0.046, 0.098, 0.147)
	Abrasive grain size D6	(0.036, 0.053, 0.110, 0.159)
	Effective number of abrasive grain D7	(0.044, 0.066, 0.143, 0.219)
	Clearance between abrasive grains D8	(0.028, 0.041, 0.081, 0.112)
	Honing speed D9	(0.026, 0.040, 0.087, 0.125)
	Honing depth D10	(0.049, 0.073, 0.158, 0.241)
	Honing speed D11	(0.026, 0.039, 0.080, 0.114)
	Cross axis angle D12	(0.054, 0.082, 0.180, 0.280)
Tooth profile error C2	Abrasive grain abrasion D1	(0.031, 0.046, 0.095, 0.136)
	Abrasive grain fracture D2	(0.048, 0.070, 0.145, 0.214)
	Abrasive grain detachment D3	(0.062, 0.093, 0.210, 0.331)
	Grinding cutting angle D4	(0.031, 0.045, 0.090, 0.126)
	Shape of abrasive grain D5	(0.024, 0.035, 0.070, 0.097)
	Abrasive grain size D6	(0.036, 0.053, 0.109, 0.156)
	Effective number of abrasive grain D7	(0.033, 0.049, 0.101, 0.145)
	Clearance between abrasive grains D8	(0.031, 0.045, 0.092, 0.131)
	Honing speed D9	(0.046, 0.067, 0.141, 0.211)
	Honing depth D10	(0.047, 0.068, 0.139, 0.204)
	Honing speed D11	(0.034, 0.050, 0.101, 0.143)
	Cross axis angle D12	(0.049, 0.072, 0.150, 0.223)
Helix error C3	Abrasive grain abrasion D1	(0.032, 0.046, 0.090, 0.126)
	Abrasive grain fracture D2	(0.043, 0.064, 0.136, 0.201)
	Abrasive grain detachment D3	(0.059, 0.089, 0.205, 0.326)
	Grinding cutting angle D4	(0.027, 0.039, 0.079, 0.110)
	Shape of abrasive grain D5	(0.023, 0.034, 0.069, 0.097)
	Abrasive grain size D6	(0.034, 0.049, 0.100, 0.142)
	Effective number of abrasive grain D7	(0.033, 0.049, 0.101, 0.146)
	Clearance between abrasive grains D8	(0.025, 0.037, 0.074, 0.104)
	Honing speed D9	(0.046, 0.068, 0.147, 0.223)
	Honing depth D10	(0.045, 0.067, 0.141, 0.211)
	Honing speed D11	(0.044, 0.065, 0.137, 0.203)
	Cross axis angle D12	(0.054, 0.080, 0.173, 0.262)
Material removal amount C4	Abrasive grain abrasion D1	(0.028, 0.041, 0.083, 0.117)
	Abrasive grain fracture D2	(0.034, 0.049, 0.095, 0.133)
	Abrasive grain detachment D3	(0.047, 0.067, 0.135, 0.195)
	Shape of abrasive grain D5	(0.024, 0.036, 0.075, 0.106)
	Abrasive grain size D6	(0.046, 0.064, 0.122, 0.168)
	Effective number of abrasive grain D7	(0.055, 0.078, 0.153, 0.220)
	Clearance between abrasive grains D8	(0.035, 0.050, 0.098, 0.137)
	Honing speed D9	(0.059, 0.083, 0.166, 0.241)
	Honing depth D10	(0.062, 0.088, 0.180, 0.266)
	Honing speed D11	(0.051, 0.072, 0.143, 0.205)
	Cross axis angle D12	(0.054, 0.078, 0.161, 0.237)
Processing time C5	Honing speed D9	(0.188, 0.241, 0.395, 0.515)
	Honing depth D10	(0.164, 0.212, 0.350, 0.452)
	Honing speed D11	(0.137, 0.172, 0.268, 0.334)
	Cross axis angle D12	(0.122, 0.159, 0.262, 0.336)
Surface roughness C6	Abrasive grain abrasion D1	(0.027, 0.040, 0.082, 0.118)
	Abrasive grain fracture D2	(0.037, 0.053, 0.106, 0.151)
	Abrasive grain detachment D3	(0.049, 0.071, 0.149, 0.221)
	Grinding cutting angle D4	(0.026, 0.039, 0.081, 0.116)
	Shape of abrasive grain D5	(0.025, 0.037, 0.078, 0.113)
	Abrasive grain size D6	(0.047, 0.069, 0.144, 0.213)
	Effective number of abrasive grain D7	(0.044, 0.065, 0.138, 0.205)
	Clearance between abrasive grains D8	(0.036, 0.052, 0.106, 0.151)
	Honing speed D9	(0.042, 0.063, 0.136, 0.199)
	Honing depth D10	(0.058, 0.085, 0.180, 0.273)
	Honing speed D11	(0.038, 0.056, 0.119, 0.174)
	Cross axis angle D12	(0.043, 0.062, 0.129, 0.191)
Surface texture C7	Grinding cutting angle D4	(0.078, 0.103, 0.176, 0.230)
	Abrasive grain size D6	(0.094, 0.129, 0.237, 0.324)
	Honing speed D9	(0.095, 0.130, 0.239, 0.327)
	Honing depth D10	(0.095, 0.128, 0.229, 0.308)
	Honing speed D11	(0.081, 0.111, 0.199, 0.268)
	Cross axis angle D12	(0.107, 0.144, 0.262, 0.361)
Residual stress C8	Abrasive grain abrasion D1	(0.027, 0.040, 0.084, 0.121)
	Abrasive grain fracture D2	(0.037, 0.055, 0.113, 0.163)
	Abrasive grain detachment D3	(0.052, 0.076, 0.162, 0.238)
	Grinding cutting angle D4	(0.030, 0.044, 0.089, 0.126)
	Shape of abrasive grain D5	(0.027, 0.040, 0.081, 0.115)
	Abrasive grain size D6	(0.039, 0.057, 0.115, 0.165)
	Effective number of abrasive grain D7	(0.037, 0.053, 0.109, 0.158)
	Clearance between abrasive grains D8	(0.026, 0.038, 0.074, 0.102)
	Honing speed D9	(0.043, 0.063, 0.131, 0.191)
	Honing depth D10	(0.056, 0.081, 0.171, 0.256)
	Honing speed D11	(0.049, 0.072, 0.153, 0.229)
	Cross axis angle D12	(0.052, 0.076, 0.158, 0.235)

.

The data in Table 5 are passed through the algorithm in Table 3, which could obtain the weights of each attribute of the sub-criteria layer C with respect to the top layer A, as shown in

Table 6. The weight of sub-criterion C relative to top-level A.

Sub Criterion Layer C	The Weight of Sub Criterion C Relative to Top-Level A	$\mathbf{Normalized} \mathbf{Weights} {\mathit{E}}_{\mathit{i}}$
Tooth pitch error C1	(0.070, 0.123, 0.411, 0.746)	0.475
Tooth profile error C2	(0.050, 0.088, 0.283, 0.499)	0.312
Helix error C3	(0.042, 0.074, 0.236, 0.411)	0.284
Material removal amount C4	(0.018, 0.033, 0.100, 0.176)	0.120
Processing time C5	(0.016, 0.029, 0.089, 0.154)	0.216
Surface roughness C6	(0.045, 0.083, 0.285, 0.546)	0.371
Surface texture C7	(0.019, 0.036, 0.118, 0.213)	0.134
Residual stress C8	(0.024, 0.044, 0.145, 0.261)	0.211

The weights are normalized to obtain $w_t$.

Firstly, the SPA matrix is processed using “mean + variance”. Secondly, the “mode” of the plane vector set is calculated and normalized, and finally, the integral weights of elements at each layer are obtained, as shown in Figure 3.

From the data in Figure 3, it can be concluded that the honing speed, honing depth, feed rate, and cross-axis angle are controllable parameters in the actual machining process. The honing speed has a weight value of 45.4% in the overall evaluation of micro-edge honing characteristics, which significantly impacts the quality of honing operations. Additionally, abrasive grain fracturing (7.5%), abrasive grain shedding (11.5%), and abrasive grain size (7.3%) are affected by the honing wheel life and the hardness of the workpiece. In different honing conditions, only by choosing reasonable characteristic process parameters can the honing ability of the honing wheel be brought into full play and good honing effect be obtained. To further confirm the significance of the factors, sensitivity analyses were performed through the trapezoidal fuzzy set-pair analysis method.

5.2. Analyze the Sensitivity of the Key Factor of the Machining Quality

In order to clarify the research objectives, it is first assumed that the importance of criterion layer B will change, followed by the calculation of the trapezoidal fuzzy set for a single criterion. The sensitivity results of key factors are shown in Figure 4. As can be seen from the size of the weighting value, the effect of abrasive grain shedding on honing accuracy and machining efficiency is the most significant, while the effect of abrasive grain size on surface quality is also considerable.

On the one hand, the hardness of abrasive particles present on the CBN honing wheel is much higher than the hardness of the workpiece tooth surface. In addition, the cutting force and cutting heat of abrasive particles in the gear honing process are low, so the abrasive particles rarely break off, maintaining the integrity of the honing wheel. On the other hand, because the honing wheel’s characteristics wear very little during the initial and normal wear stages, the impact of abrasive breakage and falling off on the machining quality of the workpiece can be negligible.

In the stage of rapid wear, the honing force and temperature rise sharply, resulting in grinding wheel blockage and insufficient coating retention. As a result, the abrasive particles break or fall off, forming a self-sharpening phenomenon and keeping the grinding wheel continuously honing. At the same time, a large number of axial shedding is easy to cause helix errors and pitch errors. A large number of radial drops are easy to cause tooth profile errors and pitch errors. The falling off and breaking of abrasive particles affect the integrity of the polishing wheel and reduce the quality of the honing process. To sum up, we have analyzed the whole cutting process, as shown in Figure 4, and the analysis results are consistent with reality.

5.3. CBN Abrasive Grain Micro-Edge Honing Processing Experiment

In order to verify the accuracy of the comprehensive evaluation model, taking the key factor of abrasive grain size as an example, the influence experiment of different abrasive mesh sizes on honing quality is designed.

5.3.1. Specimen Requirements

Take the high-speed gear in the gearbox of a new energy vehicle as an example. The design parameters of the process requirements adopted are shown in

Table 7. Transmission gear parameters.

Category	Modulus	Pressure Angle	Tooth Number	Cumulative Total Pitch Deviation	Cumulative Total Deviation of Tooth Profile	Total Deviation of Helix	Accuracy Class
code	m	αn	Z	FP	Fα	Fβ	GB/T10095.1-2008 [31]
numerical value	4	20°	44	36 μm	15 μm	13 μm	6

, and the processing quality of the gear is evaluated and analyzed.

5.3.2. Processing Conditions

According to GB/T10095.1-2008 cylindrical gear precision system, GB/T 131-2006 [32] product geometry series specifications and design parameters requirements, select YK4825 power gear honing machine, and use the experimental parameters shown in

Table 8. Honing gear experimental parameters.

Parameter	Mesh Size	Coating Thickness	Impact Toughness T1	Workpiece	Honing Speed vc	Honing Depth ap	Feed Speed f	Time t	Cutting Conditions
Numerical value	100# 300# 540#	0.25 0.075 0.025	41% 48% 50%	20CrMnTi (HRC58-60)	4 m/s	10 μm	3 m/s	120 s	No Cutting fluid

, and then honing the gear.

In the Table 8, # denotes the mesh size, and 100# denotes the abrasive mesh size of 100.

As shown in Figure 5. The gear honing accuracy test was completed in the 3903T gear measuring Center of Harbin Measuring Tool Factory. The screw error, tooth profile error, and tooth spacing error can be measured simultaneously in this experiment. The VHX-7000 Ultra depth of field microscope from KEYENCE (Osaka, Japan) was used to measure the roughness of the top of the tooth surface, the indexing circle, and the root of the tooth, and then the average roughness was calculated. In addition, the material removal rate is calculated by the weight change of the gear before and after the honing process.

5.4. Effect of CBN Grain Size on Honing Quality

5.4.1. The Relationship Between Grain Size and Honing Precision

(1)

Total deviation between grain size and helix

As shown in Figure 6, As the abrasive mesh size increases, the size of the abrasive decreases. After honing, the average value of the spiral deviation of 300^# and 540^# abrasive particles decreases successively, and the difference between them is very small. The smaller the abrasive grain size, the more concentrated the helical total deviation distribution of the workpiece tooth surface after honing so that the minimum helical error of 540^# is smaller than the corresponding value of 300^#.

After machining with 100^#, 300^#, and 540^# grinding wheels, the total screw deviation of the workpiece tooth surface is improved in different degrees, and the improvement degree is affected by the error accumulation and error recurrence before honing. Under the machining of the 540^# honing wheel, the maximum error of the tooth line of No. 21 gear did not decrease significantly. In actual production, by increasing the number of honing operations, the error replication coefficient and the maximum error of the helix of the workpiece tooth surface can be effectively reduced. The honing precision can be improved to meet the design requirements.

(2)

Grain size and single pitch deviation.

In the electronic gearbox, the workpiece gear and the honing wheel are forced to engage according to the theoretical transmission ratio, which plays a significant role in correcting the pitch deviation. In Figure 7, with the increase in abrasive mesh size, the algebraic difference between the measured tooth pitch and the theoretical tooth pitch decreases progressively. The distribution of pitch deviation is also gradually approaching the zero line, which indicates that the ability to correct the change of common normal length is enhanced, so the machining accuracy of honing teeth is improved. The pitch error of the workpiece shows an obvious sinusoidal distribution after 540^# abrasive honing, which reflects the circular run-out error between the honing wheel and the workpiece. It is a random error generated by the previous process. After honing, the pitch error is obviously reduced. Due to error accumulation, the pitch error of the workpiece is still approximately sinusoidal. In the subsequent work, the error recurrence coefficient can be effectively reduced by increasing the number of honing operations.

(3)

Cumulative total deviation of grain size and tooth profile

As shown in Figure 8, the average value of the cumulative total deviation of the tooth profile decreases with the increase of abrasive mesh size, which indicates that the accuracy of the workpiece is gradually improved after honing. For the wheel with a large prehoned tooth profile deviation, the tooth profile deviation curve of the honed wheel after precision reshaping is closer to the theoretical value. After the secondary enveloping cutting process of the workpiece tooth profile, the involute tooth profile deviation of the workpiece gear is reduced.

100^# abrasive particles are used in roughing operations to remove process residues with high throughput and efficiency. The use of 300^# and 540^# abrasive particles for finishing can effectively reduce the tooth profile error, installation error, and short cycle error in the machine tool transmission chain. The error between the actual and designed tooth shapes is reduced while the machining precision is improved. It is also found that the small abrasive mesh size leads to the uniform distribution of the spacing, and the high precision of the honing wheel tooth shape reduces the tooth shape deviation, both of which improve the precision of the honing workpiece.

5.4.2. The Relationship Between Surface Roughness

It can be seen from Figure 9 that the average, maximum, and minimum values of workpiece tooth surface roughness decrease with the increase of abrasive mesh size when other process conditions remain unchanged, thus improving the honing tooth surface quality. The reason can be revealed according to the Ra expression of honing surface roughness.

Figure 9. Relationship between abrasive grain size and surface roughness with honing gear.

Figure 9. Relationship between abrasive grain size and surface roughness with honing gear.

$$Ra=0.256(1+c){\left({\displaystyle \frac{1}{n_{w}t}}\right)}^{0.4}{H}_v$$

(18)

In Equation (18) c curl compensation coefficient, $n_w$ workpiece rotational speed, t honing machining time, and the maximum valley height of the workpiece surface after honing $H_v$ are:

$$H_v=0.97w^{1.2}{\left(cot\theta \right)}^{0.4}{\left\{{\displaystyle \frac{v_1}{v_2}}\sqrt{{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{{\rho }_2}}-{\displaystyle \frac{1}{{\rho }_1}}\right)}\right\}}^{0.4}$$

(19)

In Equation (19), $\theta$ abrasive grain top cone angle, $v_1$ workpiece contact point velocity, $v_2$ honing wheel contact point velocity, the radius of curvature of the workpiece tooth surface ${\rho }_1$, the radius of curvature of the abrasive grain wheel tooth surface ${\rho }_2$, and the average abrasive grain spacing $w$ are simplified as:

$$w=137.9G^{-1.4}\sqrt[3]{{\displaystyle \frac{\pi }{32-S}}}$$

(20)

$G$ is the honing wheel grit size, $S$ is the organization number. The combined analysis of the above equations shows the surface roughness $Ra\propto H_v$, and the maximum height of the valley on the surface of the workpiece after honing $H_v\propto w^{1.2}$. And because of the average interval of abrasive grain $w\propto G^{-1.4}$, therefore $Ra\propto H_v\propto w^{1.2}\propto G^{-1.68}$, $Ra\propto {\displaystyle \frac{1}{G^{1.68}}}$. That is, as the grain mesh size increases, the abrasive grain size decreases, and the average spacing of the abrasive grain decreases, resulting in a decrease in the maximum valley height $H_v$, which in turn decreases the surface roughness value $Ra$ and the surface roughness quality improves.

5.4.3. The Relationship Between Grain Size and Processing Efficiency

As shown in Figure 10, with the reduction in grain mesh size of the CBN honing wheel, cutting performance and honing processing efficiency are significantly improved. On the one hand, the size of abrasive particles increases with the decrease of grain mesh size. At the same time, the surface width of the abrasive particles and the nominal width of the abrasive chip cutting layer in front of the tool is increased, resulting in more material being cut off from the workpiece surface per unit time, which improves the honing efficiency. On the other hand, when the grain mesh size is small, the contact area between the abrasive particle and the binder is larger, and the height of the abrasive edge is also larger, which makes the chip space and lubricating cooling effect of the honing wheel better, thus enhancing the ability to resist external intrusion. The abrasive particles are less likely to fall off, and the low wear rate of the honing wheel improves the life and processing efficiency of the honing wheel.

In addition, it is found that the honing state of clogging and surface passivation is likely easy to occur when the honing wheel is machined with large grain mesh size CBN. This phenomenon leads to a decrease in honing efficiency because the large grain mesh size will cause the pore volume and the nominal cross-sectional area of the debris of the honing wheel to be reduced. In addition, the number of effective abrasive cutting edges and abrasive chips increases. Therefore, in the range of small cut times, a part of the grinding chips is taken away by the cutting fluid, and the other part is retained in the pore volume. This results in a lower actual cutting depth of abrasive particles and a lower effective material removal rate per unit of time compared to the theoretical material removal rate.

With the increase in honing times, the probability of abrasive cutting-edge wear increases. The effective cutting depth of the 100^# honing wheel is greater than that of the 540^# honing wheel. The chip fusion in the pore of the honing wheel increases with the increase in temperature. After a certain number of honing cycles, the amount of abrasive plugging increases with the increase of the grain mesh size of the CBN honing wheel. Therefore, the honing wheel with a large grain mesh size should be selected in semi-fine honing and fine honing. On the contrary, the honing wheel with a small grain mesh size has a large penetration depth of abrasive particles, high honing temperature, and large material removal. At the same time, there are more grinding chips and fused caking materials blocked in the gap between abrasive particles, so the mesh number of small particles is more suitable for coarse honing.

5.5. Analysis of Comprehensive Evaluation Result of Gear Honing Processing Quality

The experiments mainly measured the experimental values of different abrasive grain size D6 on the five factors of sub-criterion layer indexes C1, C2, C3, C4, and C6. Based on the characteristics that the unit, order of magnitude, and trend of each index’s influence on honing quality are not uniform and cannot be measured, the threshold method is adopted to conduct dimensionless processing of experimental data [33] to ensure the reliability of the evaluation.

The greater the indicator value, the better the honing quality. The material removal rate indicator value falls into the category of “the larger, the better”. The formula for dimensionless processing is as follows:

$${\lambda }_{ik}={\displaystyle \frac{x_{ik}-min{x}_k}{ma\mathrm{x}{x}_k-min{x}_k}}$$

(21)

If the index value of the sub-criterion layer increases, the quality of honing teeth will deteriorate. Parameters such as surface roughness, single pitch deviation, total tooth profile deviation, and total helix deviation all follow the principle of “the smaller, the better.” The dimensionless processing formula is as follows:

$${\lambda }_{ik}={\displaystyle \frac{ma\mathrm{x}{x}_k-x_{ik}}{ma\mathrm{x}{x}_k-min{x}_k}}$$

(22)

In this context, ${\lambda }_{ik}$ represents the normalized value of the i-th factor in the k-th group of experiments, while $x_{ik}$ denotes the honed experimental value of the i-th factor in the k-th group.

By incorporating the aforementioned experimental values into the sub-criteria layer C, we can evaluate the quality differences of CBN grinding with varying granularity. The comprehensive evaluation model can be derived from Equation (23).

$$M_z=W_z{R}_z$$

(23)

In the equation, $W_z$ represents the evaluation weight matrix of factors, with specific values obtained for the weight matrix $W_z$ of sub-criteria layer C referenced in Table 6 as $W_z=[0.24 0.18 0.16 0.25 0.17]$. $R_z$ denotes the evaluation weight of experimental data.

$R_{z6}$ is a dimensionless matrix of ${100}^{\#}$,${300}^{\#}$,${540}^{\#}$, The calculation result is:

$$R_{z6}={\left[\begin{array}{ccc}0& 0.42& \begin{array}{ccc}0.64& 0.63& 0.92\end{array}\\ 0.12& 0.68& \begin{array}{ccc}0.76& 0.85& 0.45\end{array}\\ 0.11& 0.51& \begin{array}{ccc}0.39& 0.62& 0.06\end{array}\end{array}\right]}^T$$

Bring the results of $W_z$ and ${ R}_{z6}$ into Formula (23), obtained $M_z=[0.49 0.56 0.35]$. The comprehensive evaluation of the honing quality corresponding to the three-grain sizes was ranked 0.53 > 0.47 > 0.34. This indicates that 300^# CBN grit has the best honing quality, 540^# CBN grit has the worst honing quality, and 100^# is in between. In the ${ R}_{z6}$ matrix, the weights of 100^# and 300^# abrasive grits are in good agreement, and the difference in the evaluation of machining efficiency between the two in the matrix is 0.23, and the difference in the evaluation of surface roughness is 0.52, which is larger than that of the evaluation of the individual tooth pitch deviation, the cumulative total deviation of the tooth profile, and the total deviation of the helix. This indicates that the abrasive grain size has a greater impact on machining efficiency and surface roughness and a smaller impact on honing accuracy. The conclusion is consistent with the conclusion of the comprehensive evaluation sensitivity in Figure 4, which verifies the correctness of the established comprehensive evaluation model of honing machining quality and can provide a decision-making basis for the manufacture of high-performance hardened gears.

6. Conclusions

(1)

Based on Tra-FAHP and SPA, a comprehensive evaluation model for CBN micro-edge honing quality is established, which eliminates the subjective influence of AHP and ensures that the evaluation results are more reasonable and objective. The established evaluation model can quantitatively and effectively identify and evaluate the quality of the honing process with certainty and uncertainty factors, guiding the honing process of high-performance hardened gears.

(2)

With increasing grain size, the mean value of total helix deviation sequentially decreases. The distribution of tooth pitch deviation gradually approaches the zero line; the correction ability of the change of the length of the common normal line increases; the average value of the total profile deviation decreases, making the accuracy of the honed workpiece gradually improve. However, affected by the phenomenon of error reenactment, the maximum error reduction of the helix of the tooth surface after honing with large-sized abrasive grain is not obvious. Increasing the number of honing operations and reducing the error reflection coefficient can effectively reduce the maximum error of the workpiece tooth surface helix and improve the honing machining accuracy.

(3)

With abrasive grain size decreasing, the average spacing of abrasive grain decreases, resulting in decreased maximum valley height ${\mathrm{H}}_{\mathrm{v}}$, which makes the surface roughness value $\mathrm{R}\mathrm{a}$ smaller. Improvement of surface quality after honing.

(4)

With the increase of abrasive grain size, the cutting performance improves, and the honing machining efficiency obviously improves, which is applied to rough honing machining.

The article evaluates the effect of abrasive grain size on the quality of the honing process. The established comprehensive evaluation model can also be applied to the evaluation of other 11 key factors on the quality of the honing process. The study provides a theoretical basis for the comprehensive evaluation of honing quality and the optimization of process parameters. In the future, we will focus on studying the influence of key factors, such as abrasive grain and process parameters, on honing quality by improving the comprehensive evaluation model of honing characteristics and proposing a more scientific control strategy to guide the development of honing wheels and the honing process. of the manuscript.