Introduction

In the field of mechanical engineering, gear transmission systems play a crucial role in various applications, from automotive engines to industrial machinery. Their performance directly impacts the efficiency, reliability, and lifespan of the overall equipment. Gear transmission system primarily rely on the meshing tooth surfaces to transmit motion and power. Due to poor lubrication, gear overload operation and tooth surface defects, tooth surface wear has emerged as a common failure form in gear transmission system. Wear is one of the most direct causes of vibration, noise and even failure of gear systems due to the destruction of tooth integrity. Tooth surface wear is a dynamic and cumulative process of damage and loss of tooth profile material. Slight wear will change the geometry of the tooth surface and load distribution, increase the transmission error and reduce the gear transmission accuracy. Excessive wear will cause vibration, noise and temperature rise, reduce transmission efficiency, induce tooth pitting, spalling and even broken teeth and other forms of damage, seriously affecting the service life of gear transmission1,2. With the development of gear transmission towards high speed, heavy load, precision, high efficiency, etc., the requirements for the anti-wear performance of gear transmission system also gradually increased. The gear transmission system is usually in the state of mixed elastohydrodynamic lubrication or boundary lubrication, and tooth surface wear runs through the whole life cycle of gear transmission, resulting in a complex coupling relationship between TSW characteristics and lubrication characteristics. When a certain amount of wear occurs on the gear tooth surface, the contact state between the meshing tooth surfaces changes, changing the lubricant film thickness distribution state and bearing ratio, affecting the gear lubrication performance. The lubrication state between the meshing tooth surfaces will affect the tooth surface wear process through the distribution of oil film thickness and oil film temperature rise. The analysis of a model alone does not fully reflect the actual situation. Therefore, in practical production and application, fully taking into account the influence of accumulated tooth surface wear on the lubrication performance is of great significance for improving the lubrication station, reducing frictional wear, prolonging service life, and enhancing transmission performance.

The mechanism of tooth surface wear is complex, with numerous influencing factors and complex relationships among them. A review of the existing literature reveals a variety of research efforts in this area. In order to improve the dynamic performance of hypoid gears and reduce TSW, Jiang et al.3 proposed a tooth surface anti-wear modification design and analysis method by considering the coupling of tooth surface modification, wear and dynamic response. Huangfu et al.4 proposed a life-cycle wear degradation model and GA-LTCA model to incorporate the macro-geometry updating in the wear process, and verified by finite element analysis and run-to-failure tests. Shen et al.5,6 established the torsional dynamics model of planetary gear transmission system considering tooth surface wear by Archard wear formula, and studied the vibration response of planetary gear transmission system under TSW. Chen et al.7,8 proposed a computational model for tooth wear of spur gears in vehicle transmission systems under low speed conditions, and considered the effect of tooth wear on the time-varying meshing stiffness of the system during the evolution of tooth wear. According to the enhanced coordinate transformation and generalized sliding distance model, a adhesive wear prediction model of double helical gears is proposed by Zhou et al. combined with Archard wear Eq.9.Wang et al.10 established a rough tooth surface contact numerical model with high calculation accuracy based on the correlation between tooth surface wear and rough tooth surface contact, and applied it to the tooth surface wear prediction of spur gear. The lubrication state between meshing tooth surfaces is an important factor affecting tooth surface wear11, and most of the above-mentioned research processes are based on Archard’s general formula for tooth surface wear, and the effect of oil film lubrication on tooth surface wear is not considered.

Many research achievements have been obtained on the study of the EHL characteristics of gear transmission systems. The effects of non-Newtonian characteristics of fluid, transient extrusion effect, micro tooth surface morphology and thermal effect have been fully considered. The forward solution method, quasi solution method, Newton-Raphson method and multiple grid method have also been introduced into the numerical solution of EHL, and a relatively complete theoretical system of gear EHL has been gradually formed12,13. The gear transmission system is usually in the mixed EHL state, the tooth surface load is shared by the lubricant film and the asperity, and the wear process is different from the tooth surface wear in the dry contact condition14. Zhang et al.15,16 proposed a fast prediction method of tooth wear rate under MTEHL conditions, established a mathematical model of frictional wear of meshing tooth surfaces, gave the method of determining each parameter in the computational model, and discussed the scope of its applicability in practical engineering. Xiao et al.17 applied the transient non-Newtonian thermal EHL point contact model and adhesive wear model to the steady-state operation stage and start-up and stop stage of heavy-duty herringbone gear transmission, and investigated the influence of different factors on its lubrication performance and tooth surface wear depth. Combining elasticity theory, gear meshing theory, EHL theory and dynamical theory, Wang18 established a model for calculating the depth of tooth wear of gears under different working conditions, and analysed the influence of gear design parameters and shape modification parameters on the depth of tooth wear to provide a theoretical basis for the anti-wear design of gear transmission systems and analysis of vibration and noise reduction. TSW is essentially a gradual material removal process, and the gear transmission system will show different contact behavior whether the tooth surface wear is considered or not19,20. Each meshing cycle of the gear transmission will have a certain surface wear, which will change the tooth profile. The aforementioned studies have demonstrated the existence of a reciprocal influence relationship between TSW and the lubrication state of tooth surfaces. However, it should be noted that the mechanism underlying TSW is relatively intricate, and most lubrication models often confine themselves to dynamic updates of gear geometry models and global models during the process of TSW. This limitation adversely affects the accuracy of calculating the lubrication performance between meshing tooth surfaces.

DIG is a novel gear type with unique features. It combine the advantages of involute gears and double arc gears. The tooth profile of this kind of gear is composed of two staggered involutes, which are connected by a transition curve, and the involutes at both ends of tooth tip and tooth root are arranged in a stepwise manner (Fig. 1). Previous studies have proved that this kind of gear has high bearing capacity and good dynamic performance21,22. At present, certain achievements have been made in the research on the lubrication characteristics of the meshing tooth surfaces of DIGs23,24,25. Meanwhile, a tooth surface wear prediction model under the mixed EHL state has also been established, and the accuracy of the established model was verified by comparing its calculation results with those of the classical Archard wear calculation26. However, during the research process, the influence of tooth surface wear on the meshing characteristics and lubrication characteristics has not been clearly expounded. For this reason, this paper breaks through the limitation of traditional research on tooth surface wear within a single field and deeply considers the influence of tooth surface wear on the amount of tooth profile deformation. Under the state of mixed EHL lubrication, this research established a close correlation among the meshing characteristics of DIG tooth profiles, lubrication characteristics, and tooth surface wear characteristics. It aims to conduct a comprehensive and in-depth exploration of the distribution law of tooth surface wear and its influence on the lubrication performance between meshing tooth surfaces. This research has great significance for improving the computational accuracy of the meshing characteristics and lubrication characteristics of DIG and for developing effective techniques for suppressing tooth surface wear.

Fig. 1
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Double involute gear: (a) physical drawing, (b) basic tooth profile.

In the figure, αaand αd are the tooth angle of tooth tip and root, l*a and l*dare the height coefficient of tooth tip and root, y*a and y*d are the tangential modification coefficients of tooth tip and root22.

The meshing model of digs considering TSW

TSW can causes the actual tooth profile of the gear to deviate from the initial tooth profile, resulting in a change in its meshing characteristics. In the first tooth wear calculation of DIGs transmission, the gear tooth parameters are the design parameters, and for the second and subsequent meshing cycles, the relevant parameters are constantly changing with the wear process. Assuming that the pinion maintains a constant velocity and the gear applies a certain torque, the gear tooth engagement with TSW of DIGs is shown in Fig. 2. Where hlub1,2 are the accumulated wear depths of the pinion and the gear tooth surfaces, respectively, and Tp is the input torque of the gear. In the tooth surface wear area, the curvature radius of the pinion (R1m), the gear (R2m), and the comprehensive radius of curvature (Rm) of DIGs are

$$\left\{ {\begin{array}{*{20}{c}}{{R_{1m}}(i,j) = {R_1}(i,j) - {h_{{\mathop{\rm lub}} 1}}(i,nb)}\\{{R_{2m}}(i,j) = {N_1}{N_2} - {h_{{\mathop{\rm lub}} z}}(i,nb) - {R_{1m}}(i,j)}\\{{R_m}(i,j) = \frac{{{R_{1m}}(i,j){R_{2m}}(i,j)}}{{{R_{1m}}(i,j) + {R_{2m}}(i,j)}}}\end{array}} \right.$$
(1)

where R1, R2, and R are the curvature radius of the pinion, the gear and DIGs without considering the TSW, respectively27. hlub1 represents the wear depth of the pinion tooth surface. hlubz denotes the sum of the accumulated tooth wear depths of the pinion and the gear, and N1N2 is the length of the theoretical meshing line.

The tangential velocities of the pinion and the gear at a certain engagement position after considering the TSW are

$$\left\{ {\begin{array}{*{20}{c}}{{u_{1m}}(i,j) = {\omega _1}{R_{1m}}(i,j)\cos {\beta _b}}\\{{u_{2m}}(i,j) = {\omega _2}{R_{2m}}(i,j)\cos {\beta _b}}\end{array}} \right.$$
(2)

where ω1,2 are the angular velocities of the pinion and the gear, respectively. and βb is the base cylindrical helix angle.

The relative sliding veolocity, entertainment velocity and slide-roll ratio of DIGs considering the TSW can be expressed as

$$\left\{ {\begin{array}{*{20}{c}}{{u_{sm}}(i,j) = \left| {{u_{1m}}(i,j) - {u_{2m}}(i,j)} \right|}\\{{u_{rm}}(i,j) = [{u_{1m}}(i,j) + {u_{2m}}(i,j)]/2}\\{{S_{rm}}(i,j) = {u_{sm}}(i,j)/{u_{rm}}(i,j)}\end{array}} \right.$$
(3)

According to the point-to-point sliding distance calculation method proposed by Flodin et al.28,29,30. After considering the TSW, the sliding distances of a certain meshing point on the tooth profile of the pinion and the gear can be expressed as

$$\left\{ {\begin{array}{*{20}{c}}{{S_{1m}}(i,j) = 2{b_m}\left| {{u_{1m}}(i,j) - {u_{2m}}(i,j)} \right|/{u_{1m}}(i,j)}\\{{S_{2m}}(i,j) = 2{b_m}\left| {{u_{1m}}(i,j) - {u_{2m}}(i,j)} \right|/{u_{2m}}(i,j)}\end{array}} \right.$$
(4)

where bm is the contact half-width of the engagement point along the sliding direction, \({b_m}=\sqrt {4w{R_m}/(\pi {E^\prime })}\).

Fig. 2
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Schematic diagram of DIGs engagement with TSW.

The mixed EHL model of digs considering the TSW

Mixed EHL model of the gear system

In the mixed EHL condition, the tooth surface load of the gear system is shared by the lubricant film and the asperity31. In the calculation process, the parameters of the meshing characteristics considering the TSW of DIGs are substituted into the EHL model proposed by Masjedi et al.32,33, which can obtain the center oil film thickness hcm, the minimum oil film thickness hmm, the asperity load ratio Lam, and the film thickness ratio \(\Lambda _m\), which are calculated as

$$\left\{ {\begin{array}{*{20}{c}} {{h_{cm}}=2.691{R_m}W_{m}^{{ - 0.135}}U_{m}^{{0.705}}{G^{0.556}}(1+0.2{{\bar {\sigma }}^{1.222}}{{\bar {H}}^{0.223}}W_{m}^{{ - 0.229}}U_{m}^{{ - 0.748}}{G^{ - 0.842}})} \\ {{h_{mm}}=1.652{R_m}W_{m}^{{ - 0.077}}U_{m}^{{0.716}}{G^{0.695}}(1+0.026{{\bar {\sigma }}^{1.12}}{{\bar {H}}^{0.185}}W_{m}^{{ - 0.312}}U_{m}^{{ - 0.809}}{G^{ - 0.977}})} \\ {{L_{am}}=0.005W_{m}^{{ - 0.408}}U_{m}^{{ - 0.088}}{G^{0.103}} \times \ln (1+4470{{\bar {\sigma }}^{6.015}}{{\bar {H}}^{1.168}}W_{m}^{{0.485}}U_{m}^{{ - 3.741}}{G^{ - 2.898}})} \\ {{\Lambda _m}={h_{mm}}/\sigma } \end{array}} \right.$$
(5)

where Wm, Um, \(\bar {\sigma }\), σ, G, \(\bar {H}\)are the dimensionless load, dimensionless velocity, dimensionless tooth surface roughness, dimensional tooth surface roughness, dimensionless material parameters and dimensionless tooth surface hardness considering the effect of the TSW, respectively. The specific calculation formula is

$${W_m}=w/{R_m}E^\prime,{U_m}={\mu _0}{u_{rm}}/{R_m}E^\prime,\bar {\sigma }=\sigma /{R_m},G=\alpha E^\prime,\bar {H}=H/E^\prime$$
(6)

where w is the unit line load, α is the viscous pressure coefficient, µ0 is the initial lubricating oil viscosity, H is the Vickers hardness, and E’ is the integrated modulus of elasticity, which is calculated as

$$\frac{1}{{E^\prime}}=\frac{{1 - \upsilon _{1}^{2}}}{{{E_1}}}+\frac{{1 - \upsilon _{2}^{2}}}{{{E_2}}}$$
(7)

where \({E_{1,2}}\)and \({\upsilon _{1,2}}\)are the elasticity modulus and Poisson’s ratio of the two gears, respectively.

Modified Archard tooth surface wear model

Masjedi and Khonsari34 investigated the TSW in the mixed EHL condition based on the classical Archard tooth surface wear formulation, proposed a modified Archard tooth surface wear model, and deduced the tooth surface wear volume Vlubm per unit time in this condition.

$${V_{\operatorname{lub} m}}=K{\psi _m}(\frac{{{L_{am}}}}{{100}})\frac{{F{u_{sm}}}}{H}$$
(8)

where K is the dimensionless wear coefficient, and K = 5 × 10− 435. ψm is the fractional film defect, which is calculated as

$${\psi _m}=1 - \exp [ - \frac{X}{{{u_{sm}}{t_0}}}\exp ( - \frac{{{E_a}}}{{{R_g}{T_s}}})]$$
(9)

where X is the lubricant molecular diameter,t0 is the basic time of molecular vibration in the adsorbed state, Ea is the adsorption heat in interfacial lubricant, Rg is the molar gas constant, Tsm is the interfacial flash temperature, and Tsm = T0 + ΔTm. In which, T0 is the bulk temperature and ΔTm is the surface temperature rise in the contact area, which can be expressed as36

$$\Delta {T_m}=\frac{{2{b_m}{q_m}}}{{\sqrt \pi ({K_1}\sqrt {1+{P_{e1m}}} +{K_2}\sqrt {1+{P_{e2m}}} )}}$$
(10)

where K1 and K2 are the thermal conductivity of the two contact tooth surfaces, respectively. Pe1 and Pe2 are the Peclet coefficients of the two contact surfaces, respectively, Peim=bmusm/2ki. ki is the thermal diffusivity of the two meshing interfaces. qm is the average heat of the tooth surface in the mixed EHL condition, and consisting of the heat generated by the asperity contact qam and the lubricant shear qhm, which are calculated as follows.

$$\left\{ {\begin{array}{*{20}{c}} {{q_m}={q_{am}}+{q_{hm}}} \\ {{q_{am}}={f_c} \cdot {u_{sm}} \cdot {p_{am}}={f_c} \cdot {u_{sm}} \cdot {p_m} \cdot ({L_{am}}/100)} \\ {{q_{hm}}={\Lambda _{\lim }} \cdot {u_s} \cdot {p_h}={\Lambda _{\lim }} \cdot {u_{sm}} \cdot {p_m} \cdot (1 - {L_{am}}/100)} \end{array}} \right.$$
(11)

where fc is the asperity contact friction coefficient. \(\Lambda _{\lim }\) is the ultimate shear stress coefficient, which is an inherent characteristic of the lubricant. The average contact pressure pm in the mixed EHL condition is consists of the asperity contact pressure pam and the oil film pressure phm, which are calculated as37

$$\left\{ {\begin{array}{*{20}{c}}{{p_m} = {p_{am}} + {p_{hm}} = \frac{{4F(i,t)}}{{3\pi {b_m}l(i,t)}}}\\{{p_{am}} = {p_m} \cdot \frac{{{L_{am}}}}{{100}}}\\{{p_{hm}} = {p_m} \cdot (1 - \frac{{{L_{am}}}}{{100}})}\end{array}} \right.$$
(12)

Divide both sides of Eq. (8) by the sliding velocity usm and the contact area A at the same time, and the wear depth per unit sliding distance of the tooth surface can be obtained.

$$\frac{{{V_{\operatorname{lub} m}}}}{{A{u_{sm}}}}=K{\psi _m}(\frac{{{L_{am}}}}{{100}})\frac{{F{u_{sm}}}}{{HA{u_{sm}}}} \Leftrightarrow \frac{{{h_{\operatorname{lub} m}}}}{{{u_{sm}}}}=K{\psi _m}(\frac{{{L_{am}}}}{{100}})\frac{{{p_m}}}{H}$$
(13)

where Vlubm is the wear volume, and Vlubm=A·hlubm. Pm is the average contact pressure, and Pm=F/A.

Therefore, the tooth wear depth at a certain meshing position of the pinion and the gear after a single engagement under the mixed EHL condition is

$${h_{{\mathop{\rm lub}} 1,2m}}(i,j) = K{\psi _m}(\frac{{{L_{am}}}}{{100}})\frac{{{p_m}(i,j)}}{H}{S_{1,2m}}(i,j)$$
(14)

After N times of meshing cycles, the accumulated wear depth of the tooth surface at any engagement position of the pinion and the gear is

$${h_{Z{\mathop{\rm lub}} 1,2m}}(i,j) = \sum\limits_{n = 1}^N {{h_{{\mathop{\rm lub}} 1,2m}}(i,j)}$$
(15)

Friction coefficient of the tooth surfaces

For DIGs considering the TSW under the mixed EHL condition, the friction coefficient between the meshing tooth surfaces is composed of the asperity contact friction coefficient and the fluid shear friction coefficient according to the load bearing ratio34, that is:

$${f_m}=({L_{am}}/100) \cdot {f_c}+\frac{{2{b_m}l(i,t){\tau _{\lim }}}}{F}\left[ {1 - \exp (\frac{{ - {\mu _{avgm}} \cdot {u_{sm}}}}{{{\tau _{\lim }} \cdot {h_{cm}}}})} \right]$$
(16)

where µavgm is the average viscosity of the lubricant considering the TSW, and its calculation formula is

$${\mu _{avgm}}={\mu _0}\exp \left\{ {(\ln {\mu _0}+9.67)\left[ { - 1+{{(1+5.1 \times {{10}^{ - 9}}{p_{hm}})}^Z}} \right] - {K_T}\Delta {T_m}} \right\}$$
(17)

where KT is the viscosity temperature coefficient, and KT=0.03K− 1. Z is the viscosity pressure index, and \(Z=\alpha /[(\ln {\mu _0}+9.67) \times 5.1 \times {10^{ - 9}}]\), in which α is the pressure viscosity coefficient.

Results and discussion

The double involute gear drive geometry parameters, lubricant parameters and working parameters are shown in Table 1. Figure 3 presents the calculation flowchart for the wear depth on the tooth surface of double involute gears under the mixed EHL condition. In the solving process, first of all, the meshing characteristics of DIGs are calculated based on the gear geometric parameters. Then, the relevant meshing parameters are substituted into the mixed EHL model, and thereby the lubrication characteristics between the meshing tooth surfaces are calculated. Next, the lubrication parameters between the meshing tooth surfaces are substituted into the modified Archard wear model, from which the tooth surface wear depth under a single meshing cycle is obtained. After that, the tooth profile after tooth surface wear is updated in real time, and the above calculation process is repeated based on the new tooth profile parameters. This cycle is repeated repeatedly until the tooth surface wear depth and lubrication state under the specified number of meshing cycles are obtained.

Table 1 Geometric parameters, lubricant parameters and working conditions of digs.
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Fig. 3
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Calculation flowchart for the wear depth of DIG tooth surface.

Distribution pattern of tooth surface wear depth

The wear of the tooth surface occurs throughout the entire service life of the gear transmission. During the cumulative wear process, the thickness of the gear teeth gradually decreases, resulting in changes in meshing characteristics and subsequently affecting lubrication conditions between meshing tooth surfaces, such as variations in oil film thickness, contact pressure distribution, tooth surface temperature distribution, and friction coefficient. These changes consequently impact the wear process of the gear pair. Therefore, when studying lubrication characteristics considering tooth surface wear, it is necessary to consider the geometrical updating of the gear.

The distribution patterns of tooth surface wear depth under two different states are compared in Fig. 4, taking into account whether the gear geometric model is updated or not. As depicted in the figure, the distribution pattern of tooth surface wear depth remains consistent for both the pinion and gear under these two distinct working conditions. Notably, greater wear depth is observed at the tooth root and tooth tip meshing positions, while lesser wear depth occurs near the tooth waist order positions. The pinion exhibits more severe wear at its tooth root meshing position, with significantly higher tooth surface wear depth compared to that of the gear. The distribution law of tooth surface wear depth obtained from this research is basically consistent with the calculation results in Reference 26, which effectively verifies the accuracy of the model established in this paper. When a small number of wear cycles are considered, the tooth surface wear depth of the pinion and the gear are both small, resulting in similar operating conditions. However, as the number of wear cycles increases, there is a gradual increase in differences between pinion and driven gear’s teeth surfaces’ wear depths under these two working conditions, particularly when considering geometrical updating leading to larger values for tooth surface wear depths. The maximum difference in tooth surface wear depth for a given working condition between these two scenarios reaches 287 μm. The above studies show that different running cycles will cause different degrees of wear of the tooth profile, the meshing characteristics of the gear pair will also change accordingly, and the wear tooth profile curves of DIGs is shown in Fig. 5.

Fig. 4
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Tooth surface wear distribution of DIGs with or without geometrical updating.

Fig. 5
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The variation of tooth profile due to the TSW.

Effect of the TSW on the meshing characteristics of digs

From the study in "Distribution pattern of tooth surface wear depth" Sect., it reveals that when the tooth surface wear depth accumulates to a certain degree, the tooth profile of DIGs will change, and the variation in tooth profile will inevitably have an impact on their meshing characteristics. Figure 5 shows the effect of the TSW on the meshing characteristics of DIGs after 1 × 1012 meshing cycles. As shown in Fig. 6(a)-Fig. 6(d), after considering the influence of the TSW, the comprehensive curvature radius and entertainment velocity decreases, the slide-roll ratio and relative sliding velocity increase. TSW has a greater influence on the meshing characteristics of DIGs near the meshing in and meshing out positions, which is mainly related to the distribution of the tooth surface wear depth. As can be seen from Eq. (4), the relative sliding distance between the meshing tooth surfaces is mainly related to the relative sliding velocity. The sliding distance of the pinion and the gear all increases after considering the TSW, the sliding distance of the pinion changes greatly, and the maximum sliding distance difference appears at the meshing in position of the pinion, while the changes at other meshing positions is smaller, as shown in Fig. 6(e)-Fig. 6(f). It can be seen that in the study of the lubrication performance of DIGs, the influence of tooth surface wear on their meshing characteristics cannot be ignored.

Fig. 6
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Effect of the TSW on the meshing characteristics of DIGs.

Effect of the TSW on the time-varying excitation of digs

According to Eq. (5) and Eq. (11), dimensionless load, dimensionless velocity and dimensionless roughness are important excitation parameters affecting the lubrication characteristics of the tooth surfaces, and average contact pressure, surface flash temperature and fractional film defect are the excitation parameters affecting the TSW characteristics. Figure 7 reveals the effect of the tooth surface wear on the time-varying excitation parameters of DIGs after 1 × 1012 meshing cycles under the mixed EHL condition. The figure shows that the variation trend of the above relevant excitation parameters under the two different working conditions are basically the same whether the TSW is considered or not, and the excitation parameters all increase after considering the influence of the TSW, especially near the meshing in position.

Fig. 7
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Effect of the TSW on the excitation parameters of DIGs (N = 1012).

Effect of the TSW on the lubrication characteristics of digs

To study the influence of TSW on the lubrication characteristics between the meshing tooth surfaces of DIGs, Fig. 8 presents a comparative analysis of relevant lubrication characteristics under two different working conditions, with and without considering TSW, under the given working condition. As can be seen from Fig. 8(a) and Fig. 8(b), after considering TSW, both the oil film pressure and the asperity load ratio increase. This is because the average contact pressure in the mixed EHL state is composed of the oil film pressure and the asperity contact pressure. The increase in the average contact pressure after TSW (Fig. 7(d)) causes both the oil film pressure and the asperity contact pressure to increase. The increase in the asperity load ratio indicates that the increase in the asperity contact pressure after wear is more significant relative to the oil film pressure. Considering that the tooth surface flash temperature increases after TSW (Fig. 7(e)), the viscosity of the lubricating oil decreases, which reduces the film forming ability of the lubricating oil, resulting in a further decrease in the oil film thickness and the film thickness ratio between the meshing tooth surfaces, as shown in Fig. 8(c) - Fig. 8(e). The decrease in the oil film thickness and the increase in the asperity load ratio after TSW cause more asperities on the meshing tooth surfaces to participate in collisions and friction, generating higher frictional forces, thus leading to an increase in the tooth surface wear coefficient, as shown in Fig. 8(f). From the above mentioned research, it can be seen that TSW has an adverse impact on the lubrication performance between the meshing tooth surfaces of DIGs. Moreover, TSW has a more significant impact on the lubrication performance near the meshing-in and meshing-out positions, while having a relatively minor impact on the lubrication performance near the tooth waist.

Fig. 8
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Influence of the TSW on the lubrication characteristics of DIGs (N = 1012).

  1. (1)

    Influence of the TSW on the lubrication characteristics under different tooth waist order parameters.

The tooth waist order parameters (l*a、l*d、y*a、y*d) are crucial geometric parameters that distinguish double involute gears from common involute gears36. In this paper, the tooth waist height coefficient is denoted as l*a=l*d= l*, and the tooth waist tangential modification coefficient is denoted as y*a=y*d= y*. To investigate the impact of tooth surface wear on the lubrication characteristics of DIGs with varying tooth waist order parameters, Fig. 8 illustrates the numerical differences of the oil film pressure, center film thickness and friction coefficient at the meshing in position of DIGs for the variation of l* and y* in two different states considering whether the tooth surface is worn or not. As depicted in Fig. 9, the changing trend of the differences in oil film pressure, center film thickness, and friction coefficient between meshing tooth surfaces under two working conditions is consistent with the changing trend of the tooth waist height coefficient, but opposite to the changing trend of the tangential modification coefficient. This is mainly due to the fact that the change of the tooth waist order parameters alters the contact line length, which in turn leads to the change of the unit line load. This research shows that the influence of tooth surface wear on the lubrication characteristics between the meshing tooth surfaces of DIG transmission will increase with the increase of the tooth waist height coefficient and gradually weaken with the increase of the tangential modification coefficient. However, it should be noted that although there is some influence from tooth surface wear on the lubrication characteristics of DIGs with varying tooth waist order parameters, this influence remains relatively small.

Fig. 9
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Effect of the TSW on the lubrication characteristics at the meshing in position of DIGs under different tooth waist order parameters (N = 1012).

(2) Influence of the TSW on the lubrication characteristics under different velocity and torque conditions.

The effect of the TSW on the oil film pressure, center oil film thickness, and friction coefficient at the meshing in position of DIGs under different velocity and torque conditions is illustrated in Fig. 10. In Fig. 10(a)-Fig. 10(c), both the oil film pressure and center oil film thickness gradually increase with increasing velocity, while the friction coefficient gradually decreases. Once the velocity reaches 5000r/min, both the oil film pressure and friction coefficient tend to stabilize gradually. This is because under low rotational velocity conditions, the amount of oil involved between the meshing tooth surfaces is relatively small, the lubrication state between the meshing tooth surfaces is relatively poor, and a relatively large amount of wear will occur on the meshing tooth surfaces, which in turn has a greater impact on the lubrication state. Under high rotational velocity conditions, the inhibitory effect of the heat generated by the tooth surface contact on the increase in oil film thickness is enhanced, causing the influence of tooth surface wear on the lubrication state to gradually weaken. Moreover, under a small number of meshing cycles, tooth surface wear has minimal impact on relevant lubrication characteristics, and this is due to the fact that the wear depth is minor when the meshing cycles is low. As shown in Fig. 10(d)-Fig. 10(f), as input torque increases, both the oil film pressure and friction coefficient become larger while the center oil film thickness gradually decreases. The influence of TSW on lubrication characteristics between meshing tooth surfaces is enhanced by an increasing number of meshing cycles, particularly under large torque conditions. This study demonstrates that TSW significantly affects DIGs’ lubrication characteristics under low velocity and high torque conditions. Furthermore, greater tooth surface wear depth will leads to a more pronounced effect on these characteristics.

Fig. 10
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Effect of the TSW on the lubrication characteristics at the meshing in position of DIGs under different working conditions.

(3) Influence of the TSW on the lubrication characteristics under different roughness.

Figure 11 shows the comparison of the oil film pressure, central oil film thickness and friction coefficient at the initial meshing position of DIGs under two different working conditions (with and without considering TSW ) for various tooth surface roughness levels. As observed from the figure, an increase in tooth surface roughness leads to a gradual decrease in the oil film pressure between meshing tooth surfaces, while the center oil film thickness and friction coefficient gradually increase. This is because, in the mixed EHL condition, the contact pressure is jointly borne by the asperities and the oil film. When the tooth surface roughness increases, the proportion of contact pressure borne by the asperities increases, resulting in a decrease in oil film pressure. The increase in roughness leads to an increase in the amount of oil stored between the meshing tooth surfaces and impedes the fluidity of the oil, prolonging the residence time of the oil between the tooth surfaces and thus increasing the oil film thickness. Additionally, the increase in roughness causes more asperities to be directly involved in the contact, leading to an increase in the friction coefficient between the meshing tooth surfaces. When the number of load cycles is small and wear depth on the tooth surface is minimal, there are no significant changes in numerical differences of oil film pressure, center oil film thickness, and friction coefficient between meshing tooth surfaces under two different working conditions with increasing tooth surface roughness. However, as the number of load cycles increases, differences between these lubrication parameters for tooth surfaces under two working conditions progressively amplify. Moreover, larger roughness values result in greater disparities among these parameters. These results indicate that as the number of load cycles increases, the tooth surface wear depth of DIG also increases. Meanwhile, the influence of tooth surface wear on the lubrication performance between the meshing tooth surfaces increases with the increase of tooth surface roughness.

Fig. 11
Fig. 11The alternative text for this image may have been generated using AI.
Full size image

Effect of the TSW on the lubrication characteristics at the meshing in position of DIGs under different roughness conditions.

Conclusions

Tooth surface wear occurs throughout the entire life cycle of gear transmission. At present, most of the research on the lubrication characteristics between meshing tooth surfaces does not consider the impact of tooth surface wear, which seriously affects the calculation accuracy of the lubrication characteristics. Aiming at this issue, based on the line contact mixed EHL theory and the Archard adhesive wear model, this paper constructs a tooth surface wear prediction model for DIGs under mixed EHL conditions. It studies the cumulative distribution pattern of tooth surface wear of DIGs and its subsequent influence on the lubrication characteristics between meshing tooth surfaces. It focuses on exploring the impacts of tooth surface wear on lubrication characteristics when working conditions, tooth waist order parameters, and tooth surface roughness change. The main conclusions are as follows:

(1) The wear depth of DIGs is relatively large at the meshing positions of tooth roots and tooth tips, while it is relatively small at the stepped positions of the tooth waists, and the wear depth increases with the increase of meshing cycles. Meanwhile, TSW will reduce the comprehensive curvature radius and entrainment velocity, and increase the slide-roll ratio, relative sliding velocity, and sliding distance, thereby changing the meshing characteristics of the tooth profiles of DIGs.

(2) When the wear depth of DIGs accumulates to a certain extent, it will cause an increase in oil film pressure, asperity load ratio, and friction coefficient, while the minimum oil film thickness, center oil film thickness, and film thickness ratio will decrease, thus having an adverse impact on the lubrication performance of the meshing tooth surfaces, and this impact is more significant at the meshing positions of tooth roots and tooth tips than at the stepped positions of the tooth waists.

(3) Changes in the tooth waist order parameters will cause changes in the wear depth of the tooth surface, which in turn will have different degrees of influence on the lubrication characteristics of DIGs. The degree of this influence increases with the increase of the tooth waist height coefficient and decreases with the increase of the tooth waist tangential modification coefficient, but the degree of influence is relatively small.

(4) For double involute gears, when the meshing cycles are relatively small, the influence of tooth surface wear on the lubrication performance of the meshing tooth surfaces has little correlation with changes in parameters such as rotational velocity, torque, and tooth surface roughness. When the meshing cycle times are relatively large, the influence of TSW on the lubrication characteristics of the meshing tooth surfaces decreases with the increase of rotational velocity and increases with the increase of torque and roughness.