1. Introduction
Spur gears are used in a multitude of engineering applications, including but not limited to automotive transmissions. There is a growing volume of legislation, stringent directives, and regulated targets to reduce harmful emissions from all machinery, particularly from road transport, thus requiring ever-improved energy-efficient transmissions. It is also important to improve noise, vibration, and harshness (NVH) refinement of all gearing systems, as this is another customer-demanded attribute. Additionally, the structural integrity and durability of all mechanical components such as gearing and bearings is of primary concern, particularly with the ever-harsher operating conditions (increased loading, faster contact kinematics, and elevated temperatures). It has been shown that operating conditions for transmissions affect their energy consumption and NVH performance, which are interlinked and often require contradictory methods of palliation [
1]. The current investigation focuses on the transmission efficiency and structural integrity of spur gears of high-performance vehicles.
It is now well-established that some degree of traction is needed to transmit power through gear interactions. However, beyond a certain limit, excess traction can lead to power loss throughout a typical meshing cycle [
2,
3,
4]. Therefore, it is important to develop techniques to predict the generated friction, thus accurately assessing the power losses of a transmission system. Petry-Johnson et al. [
5] have shown the influence of surface roughness and a formed thin lubricant film in gear teeth pair conjunctions upon generated friction and eventually the efficiency of a gear box operating at speeds up to 10,000 rpm. Li and Kahraman [
6] developed a tribodynamic model to predict friction throughout a meshing cycle of a spur gear pair. Using theoretical tooth profiles, the authors showed the essential coupling between gear dynamics and tribology of contact throughout the meshing cycle. There have been other contributions with a focus on spur gear pairs [
7,
8,
9,
10].
Two key factors influence the structural integrity of teeth–pair contacts. These are the induced root stresses and contact surface defects caused by the generated sub-surface stresses. The latter form part of the current investigation, as they are more of concern for high-performance transmissions based upon spur gears. It has been shown that the onset of fatigue spalling is a crucial problem in spur gears and is generally induced by generated high sub-surface shear stresses [
11,
12,
13,
14]. However, an important issue is the proper consideration for the determination of real contact surface geometry and kinematics, as well as an accurate lubrication model, which are the main contributions of the current paper.
Traditionally, the contact parameters used for the tribological studies are obtained using Tooth Contact Analysis (TCA) [
1,
15,
16,
17,
18]. However, the traditional TCA approach assumes perfect geometry of the contacting teeth under dry contact conditions. Consequently, surface defects from manufacturing processes or those caused during operation are not taken into account. Surface defects result in some changes in surface geometry and thus the contact conditions. Furthermore, gear teeth contacts are lubricated. These issues were recently taken into account by Oglieve et al. [
19] in the reported Lubricated Loaded Tooth Contact Analysis (LLTCA) approach, which is adopted here. Details and procedures in this regard are highlighted in
Section 2.1.
In order to evaluate the effect of LLTCA alone, the current study uses a standard one-dimensional non-Newtonian thermoelastohydrodynamic lubrication (TEHL) model of spur gear teeth pair with the instantaneous contact geometry and kinematics determined using the LLTCA, which is compared with the traditional TCA method. In this manner, the effect of LLTCA in a more accurate prediction of contact conditions becomes clear. Such an approach has not hitherto been reported in the literature.
5. Sub-Surface Stress Field
Fatigue spalling is a determining factor for the structural integrity of gearing systems. In a mixed regime of lubrication, the asperities on the opposing contacting surfaces can interact and cause surface wear and/or fatigue. However, in a pure EHL contact, which is the case assumed here, the root cause of fatigue emanates through sub-surface stresses reaching certain limits in the presence of sub-surface flaws. Therefore, it is important to predict the likelihood of onset of fatigue spalling in such cases. This is usually caused by the coupling of high generated contact pressures and shear [
11,
34,
46]. In the absence of any surface coating, the approach adopted by Johnson [
47] is commonly used to calculate the sub-surface stresses induced by pressure and shear under EHL conditions:
where
is the intermediate parameter along the
x-coordinate on the computational domain and
is the Cartesian coordinate into the depth of the contacting solids. Viscous shear stress in the
x-direction,
, on the lower surface is obtained as:
It must be noted that a 2D sub-surface analysis approach should consider both shear stress in the direction of entrainment and pressure orthogonal to the surface; thus, the approach expounded here is quite suitable for the idealized 1D infinite line contact EHL. However, a more comprehensive approach would require a full 3D analysis, particularly for the case of finite line contacts. The computational domain for the sub-surface stress field is the same as that used for the evaluation of the generated pressures. The dimension into the depth of contacting solids uses a length of and 1000 elements are used along the z-axis. The sub-surface stress solver utilizes a trapezoidal numerical integrator.
For gears and bearings made of ductile material, failure is usually governed by cyclic reversing orthogonal shear stresses,
, as shown by various authors [
34,
48,
49,
50].
6. Results and Discussion
The current investigation focuses on the high-performance transmissions of race vehicles. Therefore, the material and lubricant properties are chosen accordingly. An important point to note is that the surface topography is very smooth (of the order of 0.05–0.1 μm). Therefore, there is no need to consider boundary friction as well as surface fatigue, which would otherwise be required for the direct contact of rough surfaces in more conventional gearing systems.
Table 1 lists the surface material and lubricant rheological data. The lubricant is considered to have similar shear thinning characteristics as those described by Paouris et al. [
50]; thus,
,
and
have been chosen accordingly and are included in
Table 2.
Figure 8 shows a comparison of the predicted minimum film thickness during a meshing cycle, using the geometrical and kinematic data obtained through use of LLTCA and the conventional TCA. Despite the difference in the speed of lubricant entrainment and the contact radii of curvature, the lubricant film thickness remains almost unaltered between the two cases. However, at the start and the end of a meshing cycle, use of the TCA data appears to show a thicker film thickness value compared with that using the LLTCA data. The thinner film thickness predicted by the LLTCA case suggests that the inlet would move closer to the contact center, resulting in starvation during the early and later stages of the meshing cycle, which explains the reason for the insensitivity to lubricant entraining speed.
The introduction of shear thinning originated from the non-Newtonian behavior of the lubricant has a significant effect on the lubricant film thickness.
Figure 8 clearly illustrates the transition between Newtonian and non-Newtonian lubricant shear behavior with a sudden reduction in lubricant film thickness. Up to a 50% difference between Newtonian and non-Newtonian behavior for LLTCA and 48% for the case of TCA are noted. At the pitch point, there is no relative velocity between the contacting surfaces (i.e., pure rolling). With no sliding speed, the shear thinning effect is reduced, thus resulting in an increase in the thickness. However, even at low sliding speeds, the shear thinning effect has a significant influence on film thickness. With the LLTCA input conditions, at the pitch point, there is a reduction of 13%.
Figure 9 shows the change in pressure and film thickness between Newtonian and non-Newtonian behavior, including the effect of shear thinning. These results correspond to isothermal conditions at 40 °C. Clearly, the change in film thickness is the result of non-Newtonian shear thinning, reducing the lubricant film availability for entrainment into the contact. The pressure distribution remains unaltered as EHL films are insensitive to load, which primarily determines the generated pressures.
As shown earlier, the contact appears more starved when taking shear thinning into account. This is shown by the reduced pressure spike at the contact exit for the non-Newtonian response, as well as the reduced film thickness.
It is noteworthy that since the gears considered in this study are for a high-performance application, the typical composite surface roughness values are well below 0.1 µm. For the LLTCA meshing cycle, including lubricant shear thinning, the Stribeck film ratio remains around 6.5. Thus, with the film thickness values encountered in this study, it is evident that the probability of asperity contact is negligible. Therefore, rough surface interactions are neglected in the current analysis.
Figure 10 shows the pressure distribution and the corresponding film thickness throughout a meshing cycle using the LLTCA input to the analysis with non-Newtonian shear taken into account under isothermal conditions. The equivalent Hertzian pressure distribution, which is applicable to dry elastostatic contact conditions, is also included on all the figures. This shows that the pressure distribution closely follows the Hertzian pressure profile except for the inlet hydrodynamic trail and the secondary pressure peak (i.e., the pressure spike or pip) in the vicinity of contact exit. As the inlet trail shrinks toward the edge of the Hertzian domain, contact starvation grows, leading to the diminution of the exit pressure spike. With increased loading and reduced contact kinematics as well as shear thinning, the exit pressure spike tends to the edge of Hertzian region with a reduced magnitude. In the extreme case of starvation, the pressure distribution tends to the Hertzian condition. This trend can be observed in the results of
Figure 10, and it has been observed by many authors, including with experimental verification [
51,
52,
53].
For high-performance transmissions, it is important to be able to accurately predict the frictional power loss. Due to high lubricant shear strain rates in gear teeth meshing, the shear stress of the lubricant occurs mostly beyond the Eyring shear stress (i.e., non-Newtonian traction). Under these conditions, shear stress is no longer dependent on the lubricant film thickness, but it is dependent on the generated contact pressure. Hence, friction with input from both LLTCA and TCA are quite similar as shown in
Figure 11. However, one should note that in practice, as the gear teeth go through numerous meshing cycles, the average temperature of the lubricant would rise as a result. In turn, this would reduce the local viscosity, resulting in an overall reduction in the generated viscous friction.
Figure 12 illustrates the resulting generated contact temperature of the meshing surfaces throughout the meshing cycle with LLTCA input.
Given the assumed initial low bulk lubricant temperature for this analysis, the predicted flash contact temperature is quite reasonable. The greatest contact temperatures occur in regions with highest Deborah number, which is associated with thinnest lubricant film under non-Newtonian traction. With reduced film thickness caused by cases of lubricant shear thinning, the flash temperature of solid surfaces increases accordingly.
Frictional power loss follows the same trend reported in the literature for gearing systems [
18,
22].
Figure 13 shows that for spur gears, the power loss diminishes at the pitch point, since there is a relative sliding of the meshing gear teeth pair (i.e., pure rolling contact). For both cases of LLTCA and TCA inputs, the power loss is quite similar from the start of meshing to the position of the pitch point. Thereafter, LLTCA input results in greater power loss prediction because of higher predicted sliding velocities. In fact, an increase of 22% is observed at some of the meshing points. Therefore, it is clear that for a more accurate evaluation of transmission efficiency, one should consider measured geometry as in the case of LLTCA. In particular, considering the number of meshing cycles in real applications, such relatively small differences can amount to a large deviation in the prediction of gear efficiency in the long term.
For high-speed transmissions, which are based mostly on spur and bevel gear pairs, operational reliability is the paramount requirement, as their meshing teeth pairs are subjected to high pressures, shear, and generated temperature. Thin films can result in the direct contact of surfaces and wear. However, one of the main causes of failure is often sub-surface shear stresses of a cyclic nature, potentially causing fatigue spalling [
11,
49,
52].
Figure 14 shows the axisymmetric nature of the orthogonal shearing stresses,
at the beginning and the end of a meshing cycle. The pressure spike in the vicinity of contact exit induces a localized sub-surface field of its own, which is similar to the observations of Paouris et al. [
50].
Sub-surface stresses are dependent on the generated contact pressure and friction. Since both LLTCA and TCA predict similar friction, henceforth, the analysis focuses on the difference between Newtonian and non-Newtonian LLTCA cases only.
The equivalent stress with the maximum alternating shear stress hypothesis, which is shown to be the closer representation of fatigue failures of bearings and gears [
46,
48], is given as:
where
and
are the alternating maximum and minimum shear stresses as in
Figure 14a,b. As the equivalent stress approaches the yield of the material, failure due to fatigue spalling can occur in the presence of any sub-surface flaws such as pores or inclusions. Therefore, the larger the value of equivalent stress, the greater the chance of inelastic deformation due to sub-surface alternating shear stresses.
Figure 15 shows that non-Newtonian shear yields higher orthogonal shear stresses.
Most gear meshing is subjected to the shear thinning of lubricant and non-Newtonian regime of traction under thermal EHL. Therefore, the prevalent sub-surface stress conditions are best represented by alternating shear stresses under non-Newtonian conditions.