**1. Introduction**

Gear transmission is the most important and widely used transmission in mechanical transmission. In particular, long-life precision gears are widely used in mechanical transmission systems with high reliability requirements, such as wind turbine gearboxes [1] and aero engines [2]. As the requirements for long-life precision gears are increasing, the requirements of machining accuracy, assembly accuracy

and gear tooth modification are also growing, and these factors have a grea<sup>t</sup> influence on the tooth surface contact stress (TSCS) of gears.

At present, the main calculation methods of the TSCS include certain international standards such as ISO [3,4]. Most of these standard calculations are based on experience and some experimental results. However, when there are di fferent tooth profile deviations (TPD), meshing errors (ME) and lead crowning modifications (LCM) on the tooth surface, since the influence coe fficient in the calculation formula is derived from research results and field experience, and the determination of some coe fficients is a ffected by many factors, it is di fficult to determine the value of the influence coe fficients. In addition, the interaction between each influence coe fficient is not reflected in the formula. Therefore, the above methods cannot perform accurate calculations [5]. Therefore, it is very important to accurately analyze and calculate the TSCS of gears with the above influence factors, especially when the three factors exist at the same time.

At present, there are a lot of studies on the influence of TPD, ME and LCM on gear performance. Sennba [6] studied the influence of LCM gears and gear shaft misalignment on TSCS and tooth root bending stress through a large number of tests as early as 1974. Ottewill et al. [7] derived an equation of motion incorporating an error function and losses at the mesh interface to study the e ffect of TPD on the rattling of idling gears. They compared the theoretical and experimental trajectories by way of time domain plots as well as via contour plots, and for most profile error functions, good agreemen<sup>t</sup> was achieved between the model and experimental data. The results showed that the TPD weakened the contact strength and bending strength of gears, and intensified the vibration and noise of gears [7]. Ma et al. [8] introduced a method to analyze the dynamic load of the planetary gears in the gearbox of a PT6 turboprop engine, and a rigorous dynamic analysis, which included the e ffects of nonlinear tooth sti ffnesses, ring gear flexibility, gear errors and misalignments, was necessary to determine dynamic tooth loads and the load sharing among the planets. The results were presented from sample calculations for a typical gear stage [8]. Bodas et al. [9] used an advanced contact mechanics model of a planetary gear set to study the e ffects of a series of manufacturing and assembly errors on the load distribution between planetary gears. Three di fferent groups of errors were considered: (i) time-invariant, assembly-independent errors, (ii) time-invariant, assembly-dependent errors, and (iii) time-varying, assembly-dependent errors. With such errors present, the planet load sharing characteristics of an *n*-planet system (*n* = 3 to 6) were investigated for di fferent piloting configurations under both static and dynamic conditions [9]. In view of the shortcomings of traditional analysis methods (the complicated integral equations make it di fficult to instantly obtain proper results for some tooth flanks distorted by heat treatment, and the repetition calculation may not converge, especially in light load conditions), Miyosh et al. [10] proposed a new composite analysis method which can quickly calculate the contact load distribution on the tooth surface of helical gears under any load condition. Although a large number of scholars have carried out experimental research and theoretical research, these studies have not put forward a method or model that can e fficiently and accurately calculate the tooth stress when the above three factors exist at the same time, especially for the influence of these factors on the TSCS when there are di fferent levels of these factors.

In recent years, more and more scholars have used the finite element method (FEM) to model gears, and then conduct the simulation and analysis of the gear performance, and confirm that the FEM is a very e ffective method for the simulation analysis of gears. Li [5] has used the FEM to calculate the TSCS and the tooth root stress with manufacturing errors, tooth profile modifications and assembly errors. By comparing the test results of other scholars and their own test results, it was found that the calculation results of the FEM including ME and LCM are relatively consistent with the test results, which indicates the accuracy of the FEM. It also was found that TPD, ME and LCM exert grea<sup>t</sup> e ffects on the TSCS of the gears. The TSCS and the tooth root stress of the same pair of gears were also calculated by ISO (International Organization for Standardization) and JGMA (Japan Gear Manufacture Association) standards for comparing with the FEM results, and the accuracy and e fficiency of the finite element model have been verified [5]. However, the scholar only studied the influence of a

single influence factor on TSCS under a single torque load through the finite element method, and did not study the influence of di fferent levels of influence factors on the TSCS under di fferent torques. In addition, although the scholar conducted a simple finite element analysis when all factors existed at the same time, the influence degree of each factor on the TSCS and the interaction between the factors have not been studied and analyzed. Wang et al. [11] used the FEM to analyze the TSCS and transmission error of the gears with tooth profile errors, and the results showed that mesh sti ffness decreases and loaded transmission error and the maximum tooth contact stress grow when increasing the tooth lead crown relief. Lin et al. [12] used the FEM to model the machining errors, assembly errors and tooth modifications of the gear transmission system to obtain the static transmission error, so as to analyze the coupled transmission error of a helical gear system. Although there are a lot of FEM studies on the performance of gears with di fferent influencing factors, the main highlight is the influence of one single factor on TSCS, and this does not consider the interaction between various factors, especially the influence of these factors on the gear performance when they exist at the same time.

In addition, most studies often only study the influence of one single precision grade of TPD on gear performance, or assume that the distribution form of TPD on the tooth surface is a sine function, which does not represent the influence of the actual TPD of di fferent precision grades on the gear performance.

When researching the interaction of various factors and the influence degree of various factors on TSCS when they exist at the same time, this paper uses the Taguchi method (TM). The TM has been widely used for the design and analysis of various engineering fields, such as gear manufacturing processes [13], chemical industry [14], soil [15] and energy [16], and it has been demonstrated to be a powerful tool due to its simplicity and robustness [17]. Through the TM, within the specified level range, the optimal level combination of influence factors can also be obtained to determine the optimal TPD grade, ME and LCM quantity in order to obtain a relatively minimum TSCS.

In summary, compared with the experimental method and theoretical method, the FEM can obtain the TSCS more e ffectively, accurately and intuitively, such that a 3D FEM model of one pair of engaged teeth is modeled and the mesh of the contact area is refined by FEM software in this paper. Then, by inputting the values of di fferent levels of one single factor into the FEM model, especially using the real measurement data of TPD, and conducting the TSCS analysis under di fferent torques, the influence degree of one single factor on TSCS is discussed by comparing the ideal model. In view of the fact that other scholars rarely study the degree of influence of each factor on TSCS and the interaction between them when all factors exist at the same time, in this paper, through the TM, the interactions between the influence factors and the influence degree of the factors on the TSCS are studied when the three factors exist on the gear at the same time, and the optimal combination of factor levels can also be obtained.

The general analysis steps of this paper are shown in Figure 1.

**Figure 1.** The general analysis steps.
