The Wear Behaviour of a New Eccentric Meshing Reducer with Small Teeth Difference

Abstract

Eccentric meshing reducers are widely used in agriculture, industrial robots, and other fields due to their ability to achieve a high reduction ratio within a compact volume. However, the contact wear problem seriously affects the service performance of the eccentric meshing reducer, thereby limiting their range of applications. To effectively address this issue, this study involved a stress analysis of the contact pairs and a surface wear analysis of a new eccentric meshing reducer. The wear equation for the contact pairs was derived using Archard’s wear theory, incorporating geometric and material parameters from both the reducer gear contact pair and the spline contact pair. In parallel, a wear simulation was conducted by integrating the UMESHMOTION subprogram with ALE adaptive grids. Additionally, the effects of load amplitudes on contact pair stress and surface wear were systematically investigated. It is revealed that the contact pair stress of the reducer gear was higher than that of the spline contact pair. Furthermore, the internal spline exhibited the highest wear rate, followed by the output shaft gear, external spline, and input shaft gear, in that order. This work provides a comprehensive and in-depth understanding of the wear behaviors of general reducers with small teeth differences and offers valuable scientific references for design optimization, fault diagnosis, and maintenance strategy formulation.

1. Introduction

In contemporary mechanical transmission systems, planetary reducers have been extensively applied to various mechanical devices, especially where small volume, light weight, and high transmission efficiency are required, due to their unique design characteristics, such as high reduction ratio, compact structure, and operational stability. However, wear issues, which can easily develop in existing RV reducers, harmonic reducers, and ordinary involute planetary reducers, become increasingly prominent after long-term use in industrial robots, agricultural machines, aerospace, and other complex working environments. These wear problems are key factors affecting the system’s reliability and service life. Gear wear not only affects transmission accuracy and efficiency, but also intensifies noises and vibration, and even may cause system failure [1,2]. Due to the arc gear engagement and multi-tooth meshing effect of small teeth differences under high loading conditions, the universal eccentric meshing reducer with an eccentric meshing pair has obvious advantages under high loading conditions, such as agricultural machines. With the extensive application of reducers under complex working conditions, gear component wear is considered the primary factor limiting their performance and service life. Consequently, many studies have examined this wear problem.

Hu et al. [3] pointed out that gear surface wear not only changes the shape of the tooth, resulting in a decrease in transmission accuracy and efficiency, but also may induce vibration and noise problems in the systems. Bai et al. [4] and Shen et al. [5] reported that, with the tooth surface wear of the gear, changes in time-varying meshing stiffness (TVMS) will directly affect the dynamic characteristics of the gear system, including vibration signals and transmission error. Chen et al. [6] focused on the impact of tooth surface wear on the dynamic properties of the system, such as vibration response and load distribution characteristics. On this basis, the coupling relationship between the wear and dynamic performances was disclosed. Shen et al. [7] and Bajpai et al. [8] realized a dynamic simulation of the tooth surface wear process of the gear by introducing advanced finite-element-analysis technology. Feng et al. [9] developed a cyclostationarity-based wear monitoring system to analyze the wear behavior of spur gears.

In the gear wear simulation, Sudhagar et al. [10] developed a mathematical model for predicting the wear pattern in spur gear. The prediction is based on the wear-rate equation, as well as the load shared by the tooth in a single-pair contact zone and a double-pair contact zone. In addition, the distribution of the contact pressure was determined using the Hertzian cylindrical contact theory. Grabovic et al. [11] proposed a wear simulation model to investigate the effect of wear conditions on the tribological evolution of the gear. Chen et al. [12] and Ignatijev et al. [13] discussed the wearing depth prediction based on the Archard wear method, as well as exploring the quantitative influences of wearing depth on the dynamic responses of the gear. Walker et al. [14] proposed a multi-physics transient wear model to predict the wear behavior of helical gear pairs. Masjedi and Khonsari [15] developed a procedure to predict the traction coefficient and wear rate, accounting for the interactions between the surface asperities of the gear teeth at each point along the line of action. Pathak et al. [16] presented linear-complementarity-based formulations for contact analysis and wear prediction in gear trains. The developed formulations are used for the contact analysis of two examples of gear trains.

The application of a wear model under practical working conditions proposed by Li et al. [17] and Lehmann et al. [18] provides beneficial references to wear investigations. Based on the Archard wear method, Li et al. [19] built a wear prediction model of an eccentric meshing pair, taking into account the impact of eccentric movement and dynamic loads. Zhao et al. [20], Janakiraman et al. [21], and Zhang et al. [22]. analyzed the wear characteristics of the planetary gear under different engagement states thoroughly. The authors emphasized the significant influences of wear on the nonlinear responses and dynamic performances of the gear system. In particular, the research on wear between the sun and planet gears of the planetary gear has concluded that the impact of wear on the dynamic characteristics of the system is complicated and varies with the working conditions. With respect to wear monitoring technology, Shen et al. [23] proposed a wear monitoring method based on vibration signals. They realized the real-time updating of the wear parameters by analyzing the vibration and transmission error of the gear, which can provide effective ways to monitor the gear wear online. Chang et al [24], Zhao et al. [25], and Lin et al. [26] discussed the evolution process of wear by combining surface reproduction technology and image analysis from the perspective of surface microstructure, thus providing high-resolution wear information for wear monitoring.

According to the above-mentioned analysis, the gear wear issue has been examined, in terms of developing wear models to perform a comprehensive analysis considering various factors. Although there are established existing works in the literature on gear wear, the assessment of the underlying wear mechanism and the performance evaluation of the universal eccentric meshing reducer with small teeth difference still faces many challenges. This is mainly attributed to the asymmetric tooth-form design and the complicated meshing characteristics. To address this problem, the wear behaviors of a new type of reducer were thoroughly investigated, and theoretical guidance was provided to improve the wear resistance and prolong the service life by building a refined wear performance analysis model. The acquired research results are expected to advance the applications and development of reducers with eccentric meshing pairs in the high-end equipment manufacturing industry, thereby enhancing the technological level and market competitiveness of the entire industry.

2. Modeling and Methods

2.1. Finite Element Model of the Eccentric Meshing Reducer

The overall structure of the new eccentric meshing reducer is shown in Figure 1. The two-dimensional diagram of the eccentric meshing reducer is depicted in Figure 1a. The major components, including the input shaft with eccentric blocks, the spline shell, the intermediate gear, the output shaft, the bearings, and the end cover, are shown in Figure 1b. The power is introduced through the keyway on the input shaft, and the eccentric wheel on the input shaft is connected with the bearing inner race. Thus, power is transmitted into the intermediate gear, which is connected to the shell and the output shaft through the spline and internal gear. Finally, the power is output successfully.

Figure 2 shows the finite element model of the meshing pair in the eccentric meshing reducer. The outer spline is fully constrained. The degree of freedom (DOF) of the input shaft gear along the x and y direction is released. The output shaft gear can only rotate around the z-axis (Figure 2a). The input shaft gear rotates around the z-axis. The meshing schemes of the local areas of the spline contact pair and the gear contact pair are shown in Figure 2b,c. The mesh size in these areas was set as 0.05 mm. The rest of the areas used larger mesh sizes to decrease the computing time. The unit type of the three-dimensional meshing model was C3D8R. The total number of units was 833,412 and the total number of nodes was 746,876.

The relevant geometric parameters of the gear contact pair and spline contact pair are listed in

Table 1. Basic geometry parameters for reducer.

	Input Gear	Output Gear	External Spline	Internal Spline
Teeth number	22	20	14	14
Module (mm)	2	2	-	-
Tooth width (mm)	7	6	19	19
Pressure angle (°)	20	20	-	-

. The teeth number and modules for the input shaft gear were 22 and 2 mm, respectively, with a tooth width of 7 mm. The teeth number and modules for the output shaft gear were 20 and 2 mm, respectively, with a tooth width of 6 mm. The reduction ratio of the reducer was 10. The component materials of the reducer are listed in

Table 2. Chemical composition of the reducer.

Component	Material	Chemical Composition
		C	Si	Mn	Cr	P	S	Ni	Ti
Input gear	20CrMnTi	0.18	0.26	1.20	1.20	0.01	0.004	0.01	0.07
Output gear	40Cr	0.40	0.27	0.60	1.0	0.01	0.004	0.01
External spline	20CrMnTi	0.18	0.26	1.20	1.20	0.01	0.004	0.01	0.07
Internal spline	45	0.45	0.30	0.6	0.20	0.01	0.004	0.01

. More specifically, the internal spline was made of 45# steel, the input shaft gear and external spline were integrated parts made of 20CrMnTi, and the output shaft gear was made of 40Cr. The friction coefficient for the gear and spline contact pair is set to 0.08 [27].

2.2. Wearing Techniques

Based on the Archard wear theory, the wearing behaviors of the contact pairs in the eccentric meshing reducer were studied. According to this theory, the wear of the contact surface is mainly affected by the contact stress and relative sliding distance. It can be expressed as follows [28]:

$$V=K{\displaystyle \frac{W}{H}}s$$

(1)

where $W$ and $s$ are the contact normal force and the relative sliding distance, respectively. $V$ refers to the wear loss. $H$ and $K$ represent the material surface hardness and dimensionless coefficient of the wear, respectively. With respect to the wearing condition of the local contact area of the reducer, the local wear equation is as follows [28]:

$$h=kps$$

(2)

where $h$ is the wearing depth (mm) and $p$ is the contact pressure (MPa). $k$ is the dimensional coefficient of wear (mm²/N). The wear coefficients of the different materials in the reducer are listed in

Table 3. Wear coefficient for different materials.

Material	20CrMnTi	40Cr	45
$k$ (mm2/N)	$1.9\times {10}^{-13}$	$2.5\times {10}^{-11}$	$1.35\times {10}^{-10}$

. 20CrMnTi has the lowest coefficient of wear ($1.9\times {10}^{-13}{\mathrm{mm}}^2/\mathrm{N}$), while the 45# steel shows the highest coefficient of wear ($1.35\times {10}^{-10}{\mathrm{mm}}^2/\mathrm{N}$). The wear resistance of the internal spline is the lowest. However, the spline connection area bears relatively low loads considering that the tooth width in this area is 19 mm and the gear contact pair is 6 mm.

This eccentric meshing reducer is equipped for high-cycle fatigue working conditions. It is impossible to simulate wear under each loading cycle. Hence, the “jump-in cycle” algorithm [29] accelerates the solving speed of the tooth surface wear. This algorithm assumes that the contact pressure and sliding distance of the tooth surface remain the same in the given fixed loading cycle ($\mathrm{\Delta }N$). Therefore, the wearing depth of the meshing surface can be calculated as follows [30]:

$$h_j^i=h_j^{i-1}+\mathrm{\Delta }{h}_j^i$$

(3)

where $h_j^i$ and $h_j^{i-1}$ are the cumulative abrasion losses of the node $j$ in the $i$th and $i-1$th loading modules, respectively. $\mathrm{\Delta }{h}_j^i$ is the wear increment of the $i$th loading module, and it can be calculated as follows [30]:

$$\mathrm{\Delta }{h}_j^i=k{p}_j^i{s}_j^i\mathrm{\Delta }N$$

(4)

where $\mathrm{\Delta }N$ represents the number of the loading cycles in the $i$th loading module.

The wear process of the reducer was simulated by combining the UMESHMOTION wear subprogram of the commercial finite element software ABAQUS (2022) and ALE adaptive mesh technology. The wear simulation process of the meshing pairs is shown in Figure 3. First, a finite element model of the meshing pair was built according to material properties, boundary conditions, the geometric model, etc. Later, the sliding distance of the nodes on the wear surface, normal pressure, and other information were solved in the increment step based on the ABAQUS solver. Secondly, wear increment was calculated according to the Archard wear model. Meanwhile, the displacement of the nodes on the wear surface was controlled through the UMESHMOTION (2022) subprogram. In this process, remeshing was carried out through ALE adaptive meshing technology to prevent the excessive deformation of grids in the wear region, thus interrupting the calculation. After a wear cycle was calculated, it was evaluated whether the model reached the presetting wear cycle. If yes, the calculation was ended; otherwise, the finite element analysis of wear was repeated.

3. Results and Discussions

3.1. Tooth Surface Stress Analysis of the Reducer

The von Mises stress contour of a spline contact pair in the initial meshing moment is illustrated in Figure 4. As can be found that three spline contact pairs participate in meshing, hence the contact ratio ranges from 2 to 3. In the gear contact pair, only two pairs of gears participate in meshing and the contact ratio ranges between 1 and 2. Moreover, the tooth width of the spline contact pair is 19 mm and the gear pair is 6 mm. Hence, the tooth surface stress of the gear pair is significantly higher than that of the spline contact pair.

The von Mises stress contours of the gear contact pair and the spline contact pair of the reducer at different meshing positions are shown in Figure 5 and Figure 6. The stress responses in two different contact pairs are highly concentrated near the contact area. On one hand, in gear contact pairs, the contact stress of the output shaft gear is considerably higher than that of the input shaft gear throughout the meshing process. The contact stress for the output shaft gear increases from 153 MPa at the moment of engagement to a maximum of 699 MPa near the nodal line, then decreases to 147 MPa at disengagement. In comparison, the contact stress of the input shaft gear rises from 90 MPa at engagement to a peak of 480 MPa near the nodal line, before falling to 117 MPa at disengagement. On the other hand, in spline contact pairs, the contact stress of the external spline increases from 92 MPa at engagement to a maximum of 366 MPa near the nodal line, then decreases to 20 MPa at disengagement. Similarly, the contact stress of the internal spline rises from 125 MPa at engagement to a peak of 433 MPa near the nodal line, and then decreases to 10 MPa at disengagement.

The stress variation trends in the input and output shaft gears along the roots of the teeth are shown in Figure 7. Clearly, the stresses at the roots of teeth of the input and output shaft gears first increase and then decrease with the increase in the distance. The stress reaches the maximum (145 MPa) on the input shaft gear about 0.75 mm above the root of the teeth, while the maximum stress (130 MPa) on the output shaft gear was achieved at 0.50 mm above the root of the teeth. Therefore, the contact stress of the gear contact pair was far higher than that at the roots of the teeth.

The stress variation trends along the tooth width at the maximum meshing stress for the input and output shaft gears are illustrated in Figure 8. For the input shaft gear, the stress increases from the outer end towards the tooth width center, initially rising to a peak of 480 MPa and then stabilizing as it gradually decreases to 450 MPa. In contrast, for the output shaft gear, the stress decreases gradually from a maximum of 699 MPa at the outer end to 510 MPa, remaining relatively constant within 2 mm of the gear center. Although the gear stresses are theoretically uniform along the tooth width, fluctuations can occur due to stress concentration on the gear surface and variations in tooth widths between the input and output shaft gears.

The evolution of stress along the tooth profile at the maximum meshing stresses for the input and output shaft gears is illustrated in Figure 9. Obviously, the two-gear meshing stress reaches the maximum when the output shaft gear rotates from the initial position by 28.4° and the stress of the output shaft gear along the tooth profile is higher than that of the input shaft gear. According to the data, on the 2 mm path from the top to the root of the teeth, their stress responses all gradually increased and then decreased. The input shaft gear achieved the maximum stress (506 MPa) at the position 0.72 mm away from the initial point, and the output shaft gear achieved the maximum stress (699 MPa) at the position 1.10 mm away from the initial point.

The evolutionary laws of maximum von Mises stress of the output shaft gear with the input torque are shown in Figure 10. It can be observed that there is an approximately linear relationship between the stress response and the input torque. As the input torque increases gradually from 30 Nm to 120 Nm, the stress rises linearly from 699 MPa to 1285 MPa. This is because the gear is made of 20CrMnTi and the yield limit exceeds 1300 MPa after surface hardening. The gear is still in the elastic deformation stage under the torque as high as 120 Nm. Hence, the stress presents an approximate linear growth in the torque range.

3.2. Wear Analysis of Tooth Surface

3.2.1. Wear Laws of Different Contact Pairs

Figure 11 displays the variation curves of the wear depth of the internal spline at different points within the wear region across various cycle counts. The abrasion loss of the meshing surface of the internal spline from the bottom to the top gradually increased to the maximum and then quickly decreased. Under a different number of loading cycles, the maximum abrasion loss of the internal spline is 5.1 mm away from the bottom of the spline. As the loading cycles increased from 200,000 to 600,000, the maximum abrasion loss gradually increased from 3.60 × 10⁻³ mm to 6.13 × 10⁻³ mm.

Figure 12 shows the wear evolutions for the input gear and the external spline as the loading cycle increases. Although the spline and input shaft gear have the same material properties (20CrMnTi) and the meshing width of the input shaft gear (6 mm) is far lower than that of the external spline (19 mm), the spline exhibits greater abrasion loss under the same number of cycles. For example, given the torque of 30 Nm, the maximum wearing depth on the tooth surface of the input shaft gear increased from 0.707 × 10⁻³ mm to 2.229 × 10⁻³ mm as the loading cycle increased from 2 × 10⁸ to 8 × 10⁸. On the contrary, the maximum wearing depth on the external spline increased from 5.110 × 10⁻³ mm to 10.969 × 10⁻³ mm. This is because the spline pair has a greater sliding distance than the meshing process of the gear and it achieves a greater wearing depth under the calculation of the Archard wear model. Therefore, the wear rate of the internal spline was higher compared to that of the gear, since it had a longer contact width (19 mm) and lower unit loads.

3.2.2. Impact of the Load Amplitude on Wear

Figure 13 and Figure 14 show the wear depths for output gear and input gear under different loading amplitudes. The maximum wear depth for the input shaft gear increased from 0.70 × 10⁻³ mm to 1.75 × 10⁻³ mm when the loads’ amplitude gradually increased from 30 Nm to 120 Nm after 2 × 10⁸ loading cycles, while the maximum wear depth of the output shaft gear increased from 0.91 × 10⁻³ mm to 2.05 × 10⁻³ mm when the loads’ amplitude gradually increased from 30 Nm to 120 Nm after 2 × 10⁶ loading cycles. It is revealed that the wear rate for the output shaft gear was far higher compared to that of the input shaft gear, due to the different materials.

The wear rates of different contact pairs of the reducer under different load amplitudes are shown in

Table 4. The wear rate for different component under different loading amplitudes.

Torque (N·m)	30	60	90	120
Output gear	4.53 × 10−10	6.60 × 10−10	8.50 × 10−10	10.25 × 10−10
Input gear	3.54 × 10−12	6.38 × 10−12	8.88 × 10−12	11.15 × 10−12

. In this reducer, the wear rate of the input gear was lower compared with the output gear under the same loads. For example, the wear rates of the output gear and the input gear were 6.60 × 10⁻¹⁰ mm/cycle and 6.38 × 10⁻¹² mm/cycle under the torque 60 Nm, respectively. In addition, the wear rates of the different parts all show approximately linear changes with the increase in the load amplitudes.

4. Conclusions

In this work, the stress responses and surface wear of the gear contact pair and spline contact pair of a new eccentric meshing reducer were systematically investigated. A three-dimensional finite element analysis model of the contact pairs was built based on the geometric parameters and material parameters. The wear equation of contact pairs was derived based on the theory of Archard wear. A wear simulation analysis was carried out by combining the UMESHMOTION subprogram and ALE adaptive meshing technology. Additionally, the effects of load amplitudes on the contact pair stress and surface wear were investigated. Major research results could be drawn:

(1)

In this eccentric meshing reducer, the stress responses of the gear contact pair were stronger than those of the contact pair. Under the conditions of 30 Nm, the maximum stress of the gear contact pair was 699 MPa and the maximum stress of the corresponding spline contact pair was 550 MPa.

(2)

In this eccentric meshing reducer, the wear rate of the external spline was higher than that of the input shaft gear, which uses the same material. This effect can be attributed to the greater relative sliding distance. In the two contact pairs, the internal spline showed the highest wear rate, followed by the output shaft gear, external spline, and input shaft gear, successively.