Abstract
Hypoid gear pairs serve as critical power transmission components in rear-drive vehicle driveline systems, and their lubrication conditions directly govern the transmission performance and vibration levels of automotive rear axles. Surface micro-texturing, as an effective technical approach for improving the lubrication performance of contact surfaces, has achieved considerable research progress and engineering applications in mechanical seals, bearings, and related fields. This thesis focuses on the hypoid gear pair of an automotive main reducer, applying surface micro-texturing to enhance lubrication performance and achieve friction reduction and vibration attenuation. Following the design of tooth surface micro-dimples as the main research thread, this work systematically investigates gear tooth contact parameters, two-dimensional micro-dimple shape parameters, three-dimensional micro-dimple arrangement parameters, and the meshing dynamic performance of micro-dimpled tooth surfaces, thereby providing novel design concepts and theoretical support for engineering practice.
First, a mathematical model of hypoid gear pair meshing is established to determine tooth surface motion parameters. A CAE simulation model is constructed based on the finite element method, combined with contact pattern experiments, to characterize gear tooth contact behavior and locate contact imprint positions. Second, the gear meshing lubrication problem is simplified into a two-dimensional plane sliding lubrication model. Tooth surface micro-textures are introduced to improve lubrication and vibration characteristics. Based on experimental data and boundary layer theory, micro-dimples are identified as the appropriate texture type, and parametric simulation studies of their shape parameters are conducted. Furthermore, considering cavitation effects, the influence of sliding velocity on the lubrication performance of square and trapezoidal micro-dimples with equivalent width and depth values is examined. Third, the lubrication performance of single three-dimensional cylindrical and cuboid micro-dimples is compared. Under various sliding velocity conditions, parametric studies of micro-dimple spacing and arrangement patterns are conducted to determine optimal array configurations. Finally, a three-dimensional micro-dimple array with 1600 dimples is arranged on the contact imprint region of the driven gear convex surface, and dynamic simulation software is employed to investigate the influence of micro-dimpled tooth surfaces on hypoid gear pair meshing dynamics.
1. Introduction
Statistical data indicates that 33% to 50% of the world’s energy is consumed by friction, and among every 100 machine component failures, approximately 60 are caused by component wear, while approximately 50 mechanical equipment accidents occur due to lubrication failure. As a common means of daily transportation, automobiles consist of thousands of mechanical components, and friction between contacting components inevitably reduces fuel efficiency. Studies estimate that approximately 10% of automotive energy consumption results from friction and wear phenomena, with friction reducing fuel efficiency by approximately 3% to 5%.
Hypoid gears are key components in automotive rear axle systems, and their operational stability directly affects the vehicle’s Noise, Vibration, and Harshness (NVH) performance. The lubrication method for the main reducer’s driving and driven gears is splash lubrication. Lubricating oil stored in the rear axle housing is carried upward by the rotating driven gear and sprayed onto tooth surfaces under centrifugal force, thereby lubricating the meshing tooth pairs, reducing friction, and improving the NVH performance of the gear system. Traditional research on improving gear lubrication mainly focuses on external conditions, including the physical properties of lubricating oil and oil immersion height, while neglecting research on improving gear lubrication through modification of gear surface structure itself. Therefore, to further improve the NVH performance of the main reducer, it is necessary to conduct research from the perspective of tooth surface structure improvement while considering the improvement of the external lubrication environment.
1.1 Research Status of Surface Micro-textures
Surface micro-texturing, as a method for improving friction performance, has been extensively developed both domestically and internationally. It has been particularly well-established in the fields of mechanical seals, gas seals, parallel thrust bearings, piston rings, and automotive engines. The research team led by scientist Etsion first achieved theoretical and experimental breakthroughs. From 1996 to 2004, Etsion and colleagues developed mathematical models and performed experiments to predict the performance of full-film non-contact mechanical seals based on regular hemispherical micro-textures. In 2002 and 2004, in-depth studies on friction reduction of laser-textured mechanical seal surfaces and bearing surfaces were conducted, demonstrating that surface micro-textures can form convergent gaps from the high-pressure side to the low-pressure side under hydrostatic lubrication conditions, utilizing hydrostatic effects between friction pairs to reduce friction. In 2004, Ravinder B. Siripuram investigated the influence of seven cross-sectional shapes of convex and concave micro-textures, texture dimensions, and area ratios on the pressure distribution, oil film thickness, friction coefficient, and leakage rate of sealing rings. In China, Wang Jiadao applied regular dimples of different dimensions to tripod ring surfaces, and experimental results demonstrated the significant effect of dimples in improving lubrication performance.
Since then, numerous research achievements on the application of surface micro-texture topography to improve tribological properties have emerged, with research focus concentrated on the influence of geometric shape, size, distribution, area density, and depth parameters of surface micro-textures on contact surface friction performance. For spherical point contact models, researchers including U. Pettersson, Lü Wenfei, and U. Sudeep experimentally studied the lubrication performance of point contact models formed by micro-textured planes and smooth spheres. Zhang Shengguang established a starved lubrication model considering surface textures, determining the lubrication state at each point in the computational domain and solving corresponding friction coefficients. The research indicated that the friction reduction effect of textured surfaces is relatively poor under fluid lubrication conditions, but significant under boundary lubrication conditions. S. Li investigated the influence of micro-dimple textures on friction and surface temperature performance under typical gear application speed and load conditions, establishing a thermal elastohydrodynamic coupled point-contact model. For plane contact models, researchers including Liu Hongbin, Wang Xiao, H. Yu, Haiwu Yu, Jinghu Ji, and M. S. Uddin used single micro-texture units as analysis objects to investigate the lubrication performance of different groove shapes and geometric parameter optimization. M. S. Uddin introduced different micro-texture bottom shapes, determining that square micro-textures with single wedge shapes exhibit optimal lubrication performance. Liu Hongbin considered the mutual relationships among micro-units in micro-texture arrays and their influence on lubrication performance.
In the field of conformal contact, Ning Ren and Toshikazu Nanbu investigated the influence of micro-texture distribution patterns, bottom shapes, and surface relative motion on concentrated conformal contact lubrication. D. Zhu subsequently used a mixed elastohydrodynamic lubrication model to study the lubrication performance of micro-textures in concentrated conformal contacts, with results indicating that sinusoidal micro-textures with relatively small wavelength amplitudes perpendicular to the sliding direction exhibit optimal lubrication performance. Zhu Hua experimentally studied the friction-reducing effect of variable-density micro-dimple surface textures on reciprocating piston-cylinder liners, and Wang Hongtao subsequently used numerical simulation to analyze the influence of cylindrical micro-dimple arrangement, depth, radius, and texture unit position offset on friction force and hydrodynamic load-carrying capacity.
1.2 Research Status of Elastohydrodynamic Lubrication Numerical Methods
Elastohydrodynamic lubrication (EHL) problems are inherently difficult to solve due to their involvement of nonlinear equation systems composed of the Reynolds equation, load equation, and lubricant density-viscosity equations. Although various solution methods exist including direct iteration, inverse methods, Newton’s method, and multigrid methods, obtaining complete numerical solutions is challenging. The multigrid method has been increasingly refined for solving coupled EHL problems incorporating thermal effects and roughness. However, due to its algorithmic complexity, its practical application has limitations. In recent years, researchers have increasingly utilized computational fluid dynamics (CFD) software FLUENT for solving lubrication problems involving surface micro-textures. Tae-Jo Park applied three-dimensional micro-dimples to parallel thrust bearings for friction reduction, discovering that lubricant pressure, velocity, and density distributions are affected by micro-dimple position and number. Ramesh simulated friction characteristics of textured surfaces with width ranges of 20–100 µm, depth ranges of 1–100 µm, and density ranges of 4%–63%, with maximum friction reduction of 80%. Giovanni Caramia solved the two-dimensional dimpled sliding plane N-S equations using FLUENT, investigating the influence of different micro-dimple depths, widths, and plane spacings on resistance. In domestic research, Yin Huiying used FLUENT to analyze the gap flow field distribution composed of micro-dimple annular surfaces, approximately simulating the flow condition and pressure distribution of lubricating oil films between micro-dimpled cylinder liners and pistons. Qian Yanqiang established three-dimensional lubricating oil film models for floating ring bearing clearances with smooth, internally textured, and externally textured configurations, demonstrating that both internal and external textures improve oil film load-carrying capacity, with internal textures providing greater improvement.
Traditionally, Reynolds boundary conditions are employed for surface micro-texture EHL solutions with cavitation pressure set to zero. Research incorporating cavitation effects has also progressed. Zavos Anastasios divided the ring-bore contact lubrication region in internal combustion engines into full oil lubrication zones, cavitation zones, and oil film regeneration zones, studying the influence of micro-dimple shape and depth on friction force and minimum oil film thickness. Shipra Aggarwal incorporated cavitation and thermal effects to investigate the influence of circular, square, trapezoidal, and triangular cross-sectional micro-grooves on the friction and load-carrying characteristics of sector-shaped pad thrust bearings. In domestic research, Jinyu Zhang directly observed cavitation regions in micro-textured thrust bearing ring-ring contacts through experiments and compared them with Reynolds model and JFO model predictions, with experimental results aligning more closely with JFO model solutions. Meng Xiangkai used the JFO model to study the viscosity wedge effect of dimpled surfaces, demonstrating that wedge effects have greater influence on oil film load-carrying and friction coefficients compared to cavitation effects.
2. Loaded Tooth Contact Analysis of Hypoid Gear Pairs
2.1 Friction and Lubrication Overview of Hypoid Gears
Under high-frequency operating conditions, the friction-induced heat generated by high-speed rolling and sliding between tooth surfaces of automotive hypoid gear systems is particularly pronounced, leading to gear failures including scuffing, pitting, and tooth fracture. These failures directly impact gear meshing performance, strength, stability, and generate NVH problems. The lubrication oil film formation between hypoid gear tooth surfaces is influenced by multiple factors including tooth surface geometry, machining accuracy, and operating conditions. Under adequate oil supply conditions, stable elastohydrodynamic lubrication is established; under insufficient oil supply, the oil film thins and the gear pair enters starved lubrication conditions; in severe cases, the oil film is destroyed resulting in direct metal-to-metal contact and adhesive wear.
2.2 Mathematical Model of Hypoid Gear Meshing
For the hypoid gear pair with a shaft angle of $90^\circ$, the axes are perpendicular and offset in space, causing relative sliding between pitch surfaces. Any instantaneous relative motion can be decomposed into axial sliding along the relative sliding axis and relative rotational motion about that axis. The relative sliding axis necessarily lies on the instantaneous rotation axis within the pitch plane.
During hypoid gear transmission, when gear material and lubrication environment are determined, the lubrication state is governed by the geometric curvature of meshing surfaces, relative sliding velocity, and meshing load. The sliding velocity between teeth is particularly critical due to its high magnitude and direct association with scuffing and pitting failures. The theoretical meshing mode of hypoid gears is point contact; however, due to finite tooth surface stiffness and elastic deformation under load, the actual meshing state forms an elongated elliptical contact zone inclined to the tooth length direction.
At contact point $M_{ij}$, the surface velocity vectors of the driving and driven gears are expressed as:
$$v_{ij}^{p} = \omega_p \cdot (a_{ij}^p \times r_{ij}^p), \quad v_{ij}^{g} = \omega_g \cdot (a_{ij}^g \times r_{ij}^g) \tag{2-1}$$
where $\omega_p$, $\omega_g$ are angular velocities, $a_{ij}^p$, $a_{ij}^g$ are axis vectors, and $r_{ij}^p$, $r_{ij}^g$ are pitch cone distances for the driving and driven gears respectively. The transmission ratio relation is:
$$\omega_p = R \cdot \omega_g \tag{2-2}$$
The total surface velocity $v_{ij}^{h}$ can be decomposed into a component along the common normal direction $w_{ij}^{h}$ and a component within the tangent plane $u_{ij}^{h}$. Due to the contact constraint:
$$w_{ij}^{p} = w_{ij}^{g} \tag{2-3}$$
$$(v_{ij}^{p} – v_{ij}^{g}) \cdot n_{ij} = (u_{ij}^{p} – u_{ij}^{g}) \cdot n_{ij} = 0 \tag{2-4}$$
Decomposing $u_{ij}^{h}$ in the tangent plane along two perpendicular directions $t$ and $t’$, the sliding velocity components are:
$$(v^{s})_{ij\,t} = (u^{p})_{ij\,t} – (u^{g})_{ij\,t} \tag{2-5}$$
$$(v^{s})_{ij\,t’} = (u^{p})_{ij\,t’} – (u^{g})_{ij\,t’} \tag{2-6}$$
The total sliding velocity is:
$$v_{ij}^{s} = \sqrt{[(v^{s})_{ij\,t}]^{2} + [(v^{s})_{ij\,t’}]^{2}} \tag{2-7}$$
The hypoid gear pair parameters used in this study are presented in Table 1.
| Parameter | Driving gear | Driven gear |
|---|---|---|
| Number of teeth | 10 | 43 |
| Mean pressure angle | 19° | |
| Pitch circle diameter | 63 mm | 180 mm |
| Mean spiral angle | 46°48′ | 28°37′ |
| Pitch cone angle | 18°15′ | 70°42′ |
| Spiral direction | Left | Right |
| Offset distance | 25 mm | |
| Shaft angle | 90° | |
| Face width | / | 34 mm |
Given a driving gear speed of 2000 r/min, MATLAB was used to compute the sliding velocity at meshing points as a function of driving gear rotation angle. The relative sliding velocity varied within the range of 2 to 8 m/s, increasing with the driving gear rotation angle. From tooth root to tooth tip, the sliding velocity increased by approximately 5 m/s.
2.3 Finite Element Model for Loaded Tooth Contact Analysis
Due to the complex spatial curved surfaces and small features such as chamfers and holes in hypoid gears, non-structural hexahedral mesh generation presents significant challenges. The gear pair was extracted from the main reducer and simplified by removing chamfers, fillets, and threaded holes. The Hypermesh solid cutting function was used to create five-tooth models for both the driving and driven gears. The driving gear model comprised 83,720 nodes and 73,980 elements, while the driven gear model comprised 54,351 nodes and 47,392 elements.
The gear material is 20CrMnTi with properties listed in Table 2.
| Material property | Value |
|---|---|
| Elastic modulus (GPa) | 207 |
| Poisson’s ratio | 0.25 |
| Density (kg/m³) | 7800 |
For the contact settings, the driving gear concave surface was designated as the target surface and the driven gear convex surface as the contact surface using asymmetric contact with a friction coefficient of 0.08. The augmented Lagrange algorithm was selected. Only rotational degrees of freedom about the respective central axes were released. A driving speed of 2000 r/min was applied to the driving gear rotational joint, and a resistive torque was applied to the driven gear rotational joint.
The driven gear load torque $M$ was calculated as:
$$M = \frac{G \cdot r \cdot (f_r + f_h + f_j)}{i_0 \cdot n \cdot \eta} \tag{2-8}$$
where $G$ is the vehicle fully loaded weight, $r$ is the rolling radius, $f_r$ is the road rolling coefficient (0.018), $f_h$ is the climbing coefficient (0.08), $i_0$ is the transmission ratio, $n$ is the number of drive axles, and $\eta$ is the transmission efficiency (0.9). The performance coefficient $f_j$ is:
$$f_j = \frac{\max[16(0.195G/M_{max})]}{100} \tag{2-9}$$
Substituting the vehicle parameters yielded a driven gear load of 250 N·m. The simulation time span was 0.02 s with initial, minimum, and maximum time steps of 1e-5 s, 1e-7 s, and 1e-3 s, respectively.
2.4 Contact Pattern Verification and Analysis
The contact pattern was experimentally measured by applying red lead oxide paste mixed with engine oil to the driven gear tooth surfaces and rotating the gear set at 2000 r/min using a main reducer vibration testing system. The resulting elliptical contact imprint on the driven gear convex surface showed good agreement with the finite element simulation results, thereby validating the simulation model.
The finite element analysis revealed that meshing begins at the driven gear large end and ends at the small end. The maximum equivalent meshing stress exhibited significant fluctuation, with stress values at four detected peaks ranging from 1216.2 MPa to 1364.2 MPa. The time interval between peaks was approximately 0.003 s, corresponding to the theoretical time for the driven gear to rotate through one tooth pitch $\left(8.37^\circ \times 1/(360 \times 465/60) = 0.003\ \text{s}\right)$.
3. Lubrication Characteristics Analysis of 2D Micro-dimpled Tooth Surfaces
3.1 Hydrodynamic Lubrication Mechanisms of Surface Micro-textures
The hydrodynamic pressure generation of surface micro-textures operates through three primary mechanisms:
(1) Cavitation mechanism: The classical Reynolds equation yields antisymmetric pressure distributions for micro-textured surfaces, which cannot generate net load-carrying capacity. By applying half-Sommerfeld cavitation boundary conditions, the pressure distribution becomes asymmetric, thereby generating hydrodynamic pressure and net oil film load-carrying capacity.
(2) Inertia mechanism: In conventional lubrication problems without micro-textures, the oil film dimension in the flow direction far exceeds that in the thickness direction, allowing inertia terms to be neglected. However, with surface micro-textures, the directional dimensions become comparable in scale, requiring the full Navier-Stokes equations. The inclusion of inertia or convection terms breaks the antisymmetry of pressure distribution, shifting the pressure profile downstream toward the positive pressure region, thereby generating positive oil film pressure.
(3) Wedge mechanism: When the upper wall moves at velocity $U$ in the $x$-direction, lubricant flowing into the micro-dimple experiences a sudden gap increase forming a divergent wedge, causing pressure reduction. Conversely, lubricant flowing out of the micro-dimple experiences a sudden gap decrease forming a convergent wedge, causing pressure increase. If the positive and negative pressure peaks do not completely cancel, a net hydrodynamic effect is generated.
3.2 Governing Equations and Non-dimensionalization
For the two-dimensional micro-dimple lubrication problem, assuming an incompressible Newtonian fluid, isothermal conditions, constant viscosity and density, steady laminar flow, and negligible body forces, the N-S equations in component form are:
$$\rho\left(u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}\right) = -\frac{\partial p}{\partial x} + \eta\left(\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}\right) \tag{3-1}$$
$$\rho\left(u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y}\right) = -\frac{\partial p}{\partial y} + \eta\left(\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}}\right) \tag{3-2}$$
The continuity equation is:
$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \tag{3-3}$$
Introducing the following non-dimensional parameters:
$$X = \frac{x}{L}, \quad Y = \frac{y}{h_0}, \quad U = \frac{u}{u_0}, \quad V = \frac{v}{u_0}, \quad P = \frac{p}{p_0}, \quad \rho^{*} = \frac{\rho}{\rho_0}, \quad \eta^{*} = \frac{\eta}{\eta_0} \tag{3-4}$$
with characteristic pressure $p_0 = \eta_0 u_0 / L$ and Reynolds number $\mathrm{Re} = \rho_0 u_0 L / \eta_0$, the non-dimensional N-S equations become:
$$\rho^{*}\left(U\frac{\partial U}{\partial X} + V\frac{\partial U}{\partial Y}\right) = -\frac{\partial P}{\partial X} + \frac{1}{\mathrm{Re}}\left(\frac{\partial^{2} U}{\partial X^{2}} + \frac{\partial^{2} U}{\partial Y^{2}}\right) \tag{3-5}$$
$$\rho^{*}\left(U\frac{\partial V}{\partial X} + V\frac{\partial V}{\partial Y}\right) = -\frac{\partial P}{\partial Y} + \frac{1}{\mathrm{Re}}\left(\frac{\partial^{2} V}{\partial X^{2}} + \frac{\partial^{2} V}{\partial Y^{2}}\right) \tag{3-6}$$
$$\frac{\partial U}{\partial X} + \frac{\partial V}{\partial Y} = 0 \tag{3-7}$$
3.3 Micro-dimple Geometry and Design Parameters
Four micro-dimple cross-sectional shapes were considered: rectangular (R), triangular (T), spherical segment (S), and trapezoidal (L). The key geometric parameters are the micro-dimple width $w$, maximum depth $h_{max}$, friction pair gap $h_0$, and micro-dimple spacing. The boundary layer thickness was estimated using flat-plate boundary layer theory:
$$\delta = 0.035 x \mathrm{Re}_x^{-1/7} \tag{3-8}$$
With lubricant kinematic viscosity of $5 \times 10^{-5}\ \text{m}^2/\text{s}$, sliding velocity of 5 m/s, and characteristic dimension of 200 µm, the Reynolds number was 66.7, yielding a boundary layer thickness of 3.84 µm. Therefore, micro-dimple depths were designed in the range of 1.5–7.5 µm, consistent with the boundary layer thickness order of magnitude. Micro-dimple widths were selected in the range of 40–180 µm. The dimensionless width $W$ and depth $H$ were defined as:
$$W = \frac{w}{L_d}, \quad H = \frac{h_{max}}{h_0} \tag{3-9}$$
The complete simulation design is summarized in Table 3.
| Shape | Width (µm) | Depth (µm) | Spacing (µm) | Slip velocity (m/s) | Simulation count |
|---|---|---|---|---|---|
| Rectangular (R) | 40, 70, 100, 130, 160, 180 | 1.5, 2.5, 3.5, 4.5, 5.0, 6.5, 7.5 | 200 | 5 | 168 |
| Triangular (T) | 40, 70, 100, 130, 160, 180 | 168 | |||
| Spherical (S) | 40, 70, 100, 130, 160, 180 | 168 | |||
| Trapezoidal (L) | 40, 70, 100, 130, 160, 180 | 168 |
3.4 CFD Model Setup
The computational domain consisted of a rectangular region bounded by the upper moving wall, the lower stationary wall containing the micro-dimple, and periodic boundary conditions on the left and right sides to simulate periodic micro-dimple distribution. The lubricating oil was 75W90 grade with dynamic viscosity of 0.0135 kg/(m·s) and density of 900 kg/m³ at 90°C. The SIMPLE algorithm was employed for pressure-velocity coupling with second-order pressure discretization and QUICK momentum discretization. Convergence criteria were set to 1e-15.
3.5 Evaluation Parameters
The oil film load-carrying capacity $F_y$, wall friction force $F_x$, and comprehensive evaluation coefficient $f$ were defined as:
$$F_y = \int p \, dx \tag{3-10}$$
$$F_x = \int \tau \, dx, \quad \tau = \eta \frac{\partial v}{\partial y} \tag{3-11}$$
$$f = \frac{F_x}{F_y} \tag{3-12}$$
3.6 Results of 2D Parametric Studies
Compared to smooth surfaces where oil film pressure remains minimal, the introduction of micro-dimples generates a negative pressure zone at the dimple entrance and a positive pressure zone at the exit. The square and trapezoidal micro-dimples exhibit abrupt pressure transitions with concentrated low- and high-pressure regions, while triangular and spherical micro-dimples show gradual transitions with expanded pressure zones.
From the parametric analysis, the following conclusions were drawn regarding micro-dimple shape effects:
Triangular micro-dimples produced the smallest oil film pressures across all widths, with pressures gradually increasing with depth but plateauing after $H = 0.9$. Spherical segment micro-dimples showed a peak in maximum pressure at small widths, followed by gradual increases at larger widths. Square and trapezoidal micro-dimples exhibited nearly coincident pressure curves, with maximum pressure first increasing then decreasing with depth, and the optimal depth shifting rightward with increasing width. The overall pressure magnitude relationship was: $R \approx L > S > T$.
The optimal parameter combinations for each shape are presented in Table 4.
| Shape | Optimal W | Optimal H | Max. pressure (dimensionless) |
|---|---|---|---|
| Rectangular (R) | 0.8 | 0.9 | 0.3225 |
| Triangular (T) | 0.8 | 1.5 | 0.2758 |
| Spherical (S) | 0.8 | 1.3 | 0.2930 |
| Trapezoidal (L) | 0.8 | 0.9 | 0.3228 |
Streamline analysis revealed that square micro-dimples generate large triangular vortex zones at the entrance and exit, storing more lubricant without weakening wedge effects. Trapezoidal micro-dimples, with their gentler entrance and exit slopes, guide lubricant flow, forming more concentrated vortex zones. Triangular micro-dimples create large vortex zones at their sharp bottom corners, weakening wedge effects, while spherical segment micro-dimples form elongated elliptical vortex zones along the flow direction.
Friction performance analysis showed that smooth surfaces yield negative load-carrying capacity, whereas all optimized micro-dimples generate positive load-carrying capacity. The trapezoidal micro-dimple produced the largest oil film load-carrying capacity (only 1.06% greater than square), while square micro-dimples produced the smallest wall friction force, with trapezoidal, spherical, triangular, and smooth surfaces being 0.013%, 2.29%, 3.99%, and 23.63% larger, respectively. The comprehensive evaluation coefficient ordering was: $L > R > S > T > \text{Smooth}$.
3.7 Cavitation Effects
Cavitation occurs when local pressure falls below the saturated vapor pressure of the lubricant. For lubricating oils, this pressure is approximately 5000 Pa. Cavitation in micro-dimples can generate beneficial pressure spikes that enhance load-carrying capacity. The Mixture multiphase model with the Schnerr-Sauer cavitation model was employed to describe vapor-liquid phase transformation.
Six slip velocities (1, 3, 5, 7, 9, and 12 m/s) were investigated. The results showed that no cavitation occurred at 1 m/s and 3 m/s. The critical slip velocity for cavitation inception lies between 3 and 5 m/s. At velocities of 5 m/s and above, cavitation bubbles form in the micro-dimple entrance region, expand with increasing velocity, and follow the lubricant flow direction.
Cavitation significantly enhances oil film pressure and reduces wall shear stress. At a slip velocity of 5 m/s, cavitation increased the oil film load-carrying capacity of square and trapezoidal micro-dimples by 61% and 39.4%, respectively. After cavitation, the square micro-dimple exhibited 14.2% greater load-carrying capacity than the trapezoidal micro-dimple. Wall friction forces decreased by 0.46% and 0.22% for square and trapezoidal micro-dimples due to cavitation. The comprehensive evaluation coefficient increased by 61.6% and 39.8% for square and trapezoidal micro-dimples, respectively.
The vapor volume fraction contours demonstrated that square micro-dimples produced larger cavitation regions, with more disturbed velocity fields. The pressure and shear stress comparisons at 5 m/s are summarized in Table 5.
| Parameter | Square (non-cav.) | Square (cav.) | Trapezoidal (non-cav.) | Trapezoidal (cav.) |
|---|---|---|---|---|
| Load capacity increase | — | +61% | — | +39.4% |
| Friction force decrease | — | −0.46% | — | −0.22% |
| Comprehensive coefficient increase | — | +61.6% | — | +39.8% |
Based on these findings, the square micro-dimple with $W = 0.8$ and $H = 0.9$ (width 160 µm and depth 4.5 µm) was identified as the optimal shape parameter combination.
4. Influence of Micro-dimple Array Layout on 3D Tooth Surface Lubrication
4.1 Comparison of Single 3D Cylindrical and Cuboid Micro-dimples
Extending the optimal 2D square micro-dimple design to three dimensions, single cylindrical and cuboid micro-dimple units were compared. The cylindrical micro-dimple had a diameter of 160 µm and height of 4.5 µm, while the cuboid micro-dimple had side lengths of 160 µm and height of 4.5 µm. The smooth wall region extended 200 µm in both directions. The computational domain used a mixed tetrahedral and triangular prism non-structural mesh with 5 boundary layers, a first-layer thickness of 0.12 µm, and growth rate of 1.2, totaling approximately 540,000 mesh elements.
The three-dimensional N-S equations and continuity equation in non-dimensional form are:
$$\rho^{*}\left(U\frac{\partial U}{\partial X} + V\frac{\partial U}{\partial Y} + S\frac{\partial U}{\partial Z}\right) = -\frac{\partial P}{\partial X} + \frac{1}{\mathrm{Re}}\nabla^{2}U \tag{4-1}$$
$$\rho^{*}\left(U\frac{\partial V}{\partial X} + V\frac{\partial V}{\partial Y} + S\frac{\partial V}{\partial Z}\right) = -\frac{\partial P}{\partial Y} + \frac{1}{\mathrm{Re}}\nabla^{2}V \tag{4-2}$$
$$\rho^{*}\left(U\frac{\partial S}{\partial X} + V\frac{\partial S}{\partial Y} + S\frac{\partial S}{\partial Z}\right) = -\frac{\partial P}{\partial Z} + \frac{1}{\mathrm{Re}}\nabla^{2}S \tag{4-3}$$
$$\frac{\partial U}{\partial X} + \frac{\partial V}{\partial Y} + \frac{\partial S}{\partial Z} = 0 \tag{4-4}$$
The results showed that the cuboid micro-dimple generates larger positive-negative pressure differences (3769 Pa) compared to the cylindrical micro-dimple (770 Pa). The cuboid micro-dimple also produced lower maximum wall shear stress (16,786.5 Pa) compared to the cylindrical type (17,169.8 Pa). Quantitative comparisons of friction performance are presented in Table 6.
| Parameter | Cylindrical | Cuboid | Difference |
|---|---|---|---|
| Load-carrying capacity (relative) | 1.00 | 2.686 | +168.6% |
| Wall friction force (relative) | 1.058 | 1.00 | −5.8% |
| Comprehensive coefficient (relative) | 1.00 | 2.844 | +184.4% |
4.2 Micro-dimple Array Design
Based on the contact analysis and experimental imprint results, the contact imprint on the driven gear convex surface is located within the region of 50%–70% along the tooth length and 55%–75% along the tooth height. This elliptical region has a major axis of 16–19 mm and minor axis of 3–4 mm. Micro-dimples were designed within this region to enhance hydrodynamic lubrication effects.
Based on prior research, the optimal center-to-center spacing for adjacent micro-dimples typically ranges from 1.2 to 1.8 times the dimple width. Since the micro-dimple width was 160 µm, the spacing range was 192–288 µm. The longitudinal spacing was varied at 200, 225, 250, and 275 µm, while the lateral spacing was fixed at 200 µm. Square arrangement and staggered (cross) arrangement patterns were compared. The computational domain comprised a $4 \times 4$ micro-dimple array, corresponding to lateral dimensions of 800 µm and longitudinal dimensions from 800–1100 µm, allowing curvature effects to be neglected.
Table 7 summarizes the simulation matrix.
| Arrangement | Lateral spacing (µm) | Longitudinal spacing (µm) | Slip velocity (m/s) | Total cases |
|---|---|---|---|---|
| Square | 200 | 200, 225, 250, 275 | 1, 3, 5, 7, 9, 12 | 48 |
| Staggered | 48 |
4.3 Results of Micro-dimple Array Analysis
Section pressure analysis: For the square arrangement, pressure curves along three sections parallel to the velocity direction exhibited periodic variation with decreasing amplitude from H1 to H3. The longitudinal spacing was identified as the key parameter affecting the continuity of oil film pressure between adjacent rows. For the staggered arrangement, the offset relationship between adjacent rows caused the high-pressure zone of the next row to closely oppose the low-pressure zone of the previous row, resulting in smaller pressure periods with complete sub-cycles within each main cycle.
Load-carrying capacity: The square arrangement produced positive dimensionless mean pressures, indicating effective oil film load-carrying capacity. Mean pressures decreased with increasing longitudinal spacing, with 250 µm and 275 µm yielding nearly identical values. For spacings of 200 µm and 225 µm, pressure increased significantly with slip velocity, while for 250 µm and 275 µm, the pressure increase was more gradual. The staggered arrangement produced negative mean pressures, indicating that the pump-back effect of negative pressure zones dominated without generating effective positive pressure.
Wall friction: For the square arrangement, the minimum average shear stress occurred at 225 µm spacing. For the staggered arrangement, the minimum occurred at 200 µm spacing. Both arrangements exhibited linear increases in average shear stress with slip velocity, and both arrangements produced comparable average wall friction force levels.
Comprehensive evaluation: For the square arrangement, the comprehensive evaluation coefficient decreased with increasing longitudinal spacing, and higher slip velocities improved the coefficient, particularly at smaller spacings. For the staggered arrangement, the coefficient increased with longitudinal spacing but decreased with slip velocity.
The optimal configuration was identified as the cuboid micro-dimple (160 µm × 160 µm × 4.5 µm) arranged in a square pattern with lateral and longitudinal spacing of 200 µm. The key performance metrics of this configuration are summarized in Table 8.
| Parameter | Optimal configuration / value |
|---|---|
| Micro-dimple shape | Cuboid |
| Length × Width | 160 µm × 160 µm |
| Depth | 4.5 µm |
| Lateral spacing | 200 µm |
| Longitudinal spacing | 200 µm |
| Arrangement pattern | Square |
| Load-carrying capacity | Positive (effective) |
5. Meshing Dynamics Analysis of Hypoid Gear Pairs with Micro-dimpled Tooth Surfaces
5.1 Effects of Micro-dimple Arrays on Tooth Contact Conditions
The spatial meshing constraint for the teeth generated by the cradle-type cutting process is expressed as:
$$n \cdot v = 0 \tag{5-1}$$
where $n$ is the radius vector of a point on the tooth surface in the machine coordinate system and $v$ is the relative velocity between the workpiece gear and the cutting tool at that point. For every contact point on both meshing gear surfaces, this vector relationship must be satisfied.
Introducing micro-dimple arrays on the driven gear convex surface has minimal influence on the spatial curvature characteristics of the tooth surface because the micro-dimple dimensions are several orders of magnitude smaller than the radius of curvature at the meshing points. Additionally, the simulated and measured contact imprints form inclined elliptical zones with the major axis at an angle to the tooth length direction, and arranging the micro-dimple array along the tooth length direction avoids alignment with the meshing line direction, thereby preserving the original contact conditions.
5.2 Establishment of Virtual Prototype Model
A 20 × 80 micro-dimple array was created on the driven gear convex surface contact region of 18 consecutive gear teeth using the following procedure. A datum plane was established through a point-and-direction method, and the convex surface tooth tip and root boundary curves were projected onto this plane. The midpoint of the line connecting the midpoints of the upper and lower boundary curves served as the sketch origin. Twenty micro-dimples were arranged along the tooth height direction and eighty along the tooth length direction. The array was created by extruding the sketch along the normal direction of the convex surface boundary curve, with the extrusion terminating at a surface offset by 4.5 µm from the convex surface. Boolean subtraction with the driven gear solid body produced the tooth surface micro-dimple array.
The simplified driving and driven gear models were assembled and exported to ADAMS for dynamic simulation. The material properties of 20CrMnTi were assigned, and revolute joints were created between ground and both gears. Contact forces were defined using the impact function method with the following parameters:
The contact stiffness coefficient was calculated as:
$$K = \frac{4}{3} R^{1/2} E^{*} \tag{5-2}$$
$$R = \frac{R_1 R_2}{R_1 + R_2}, \quad E^{*} = \left(\frac{1-\mu_1^{2}}{E_1} + \frac{1-\mu_2^{2}}{E_2}\right)^{-1} \tag{5-3}$$
With equivalent radii of 27.28 mm and 81.21 mm for the driving and driven gears respectively, the contact stiffness coefficient was computed as $5.66517 \times 10^{5}\ \text{N/mm}$. The force deformation characteristic exponent was set to 1.5 for metal-metal contact, the boundary penetration depth was 0.1 mm, and the damping coefficient was 100 N·s/mm. Coulomb friction was defined with static friction coefficient of 0.08, dynamic friction coefficient of 0.05, static slip velocity of 0.1 mm/s, and dynamic slip velocity of 10 mm/s.
The driving speed and driven gear load were applied using Step functions:
$$v(t) = \mathrm{Step}(t, 0, 0^{\circ}, 0.2, 360^{\circ} \times 2000/60) \tag{5-4}$$
$$T(t) = \mathrm{Step}(t, 0, 0, 0.2, 250000) \tag{5-5}$$
The simulation duration was 0.8 s with a time step of 0.0001 s.
5.3 Validation of the Virtual Prototype Model
The simulation results showed that the driven gear load torque increased smoothly from zero to 2.5e5 N·mm over 0–0.2 s, consistent with the Step function definition. The driving gear angular velocity reached 12,000 deg/s, while the driven gear theoretical output angular velocity was 2790.70 deg/s (based on the 10/43 tooth ratio). The simulation yielded an average driven gear angular velocity of 2790.6975 deg/s over 0.2–0.8 s, showing excellent agreement with the theoretical value and validating the simulation model.
5.4 Transmission Error Analysis
The transmission error was defined as the ratio of the difference between theoretical and simulated average output angular velocities to the theoretical value. The angular velocity curves for both smooth and micro-dimpled tooth surfaces are shown in Figure 5-8. The micro-dimpled tooth surface caused periodic small-amplitude oscillations in the driven gear angular velocity after 0.2 s, with oscillations occurring every approximately 0.05 s, corresponding to the time required for the driven gear to rotate through 18 teeth (the region where micro-dimpled teeth actively participate in meshing).
The average angular velocities over the oscillation window from 0.4866 to 0.5378 s were 2790.6597 deg/s for the smooth surface and 2790.6379 deg/s for the micro-dimpled surface. This corresponds to transmission errors of 0.0014% and 0.0022% respectively. The introduction of micro-dimpled tooth surfaces had a negligible effect on transmission error, differing by only 0.0008 percentage points (Table 9).
| Parameter | Smooth surface | Micro-dimpled surface | Difference |
|---|---|---|---|
| Average angular velocity (deg/s) | 2790.6597 | 2790.6379 | 0.0218 |
| Transmission error | 0.0014% | 0.0022% | +0.0008% |
5.5 Meshing Force Analysis
The meshing force curves for both configurations were extracted and compared. The smooth tooth surface gear pair exhibited relatively small meshing force fluctuations, essentially symmetric about approximately 3200 N after 0.2 s. The micro-dimpled tooth surface gear pair exhibited larger fluctuations when the micro-dimpled teeth entered meshing, with notable force spikes.
Over the window from 0.4866 to 0.5378 s, the average meshing forces were 3188.0196 N and 3208.5083 N for the smooth and micro-dimpled configurations respectively, representing an increase of 20.4887 N (0.64%) due to the micro-dimpled tooth surfaces (Table 10).
| Parameter | Smooth gear pair | Micro-dimpled gear pair | Difference |
|---|---|---|---|
| Average meshing force (N) | 3188.0196 | 3208.5083 | +20.4887 N (+0.64%) |
Frequency domain analysis was conducted using FFT on the meshing force signals. The rotational frequencies of the driving and driven gears are:
$$f_1 = \frac{n_1}{60}, \quad f_2 = \frac{n_2}{60} \tag{5-6}$$
$$f_1 = 33.3\ \text{Hz}, \quad f_2 = 7.75\ \text{Hz}$$
The meshing frequency is:
$$f_c = \frac{n_1 z_1}{60} = \frac{2000 \times 10}{60} = 333\ \text{Hz} \tag{5-7}$$
For the smooth gear pair, meshing force peaks appeared at frequencies of 333.36 Hz, 666.67 Hz, 1000.01 Hz, 1333.34 Hz, 1666.68 Hz, and 2000.01 Hz, corresponding to integer multiples of the meshing frequency. The maximum meshing force amplitude was 133.12 N at 2000.01 Hz.
For the micro-dimpled gear pair, meshing force peaks appeared at frequencies of 333.38 Hz, 666.75 Hz, 1000.13 Hz, 1333.51 Hz, 1666.88 Hz, and 2000.26 Hz. The maximum meshing force amplitude was 122.43 N at 1666.88 Hz. The frequency components were essentially consistent between the two configurations, though the micro-dimpled tooth surface produced slightly higher frequencies with reduced peak amplitudes. Notably, the micro-dimpled gear pair exhibited a “three-peak” pattern around the maximum amplitude frequency, indicating somewhat more complex frequency content, whereas the smooth gear pair showed a clean single peak.
The rotational frequency of the driving gear was recalculated as $f_1 = 2000/60 = 33.3$ Hz, and that of the driven gear as $f_2 = 465/60 = 7.75$ Hz, with the meshing frequency confirmed at $f_c = 333.3$ Hz.
5.6 Summary of Dynamic Performance Comparison
The introduction of micro-dimple arrays on the driven gear convex surface produced the following effects on the meshing dynamics of hypoid gear pairs (Table 11):
| Metric | Smooth | Micro-dimpled | Relative change |
|---|---|---|---|
| Transmission error | 0.0014% | 0.0022% | +0.0008 pp |
| Average meshing force | 3188.02 N | 3208.51 N | +0.64% |
| Max meshing force amplitude (FFT) | 133.12 N | 122.43 N | −8.0% |
| Peak frequency | 2000.01 Hz | 1666.88 Hz | −333 Hz |
| Frequency content | Single peaks | Triple peaks at maximum | Slightly more complex |
The meshing force time-domain and frequency-domain analyses demonstrated that the micro-dimple array had minimal influence on the overall meshing frequency characteristics. The slight increase in average meshing force (0.64%) and transmission error (0.0008 percentage points) was considered acceptable given the substantial improvements in lubrication performance achieved by the surface micro-dimples.
6. Conclusions and Outlook
6.1 Main Conclusions
This thesis systematically investigated the lubrication and dynamics characteristics of hypoid gear pairs with micro-dimpled tooth surfaces. The following conclusions were drawn:
(1) Contact parameters. The mathematical model of hypoid gear pair meshing was established, and the relative sliding velocity at meshing points ranged from 2 to 8 m/s at a driving gear speed of 2000 r/min. A five-tooth finite element model for loaded tooth contact analysis was validated against experimental contact pattern measurements. The maximum meshing equivalent stress reached 1364.2 MPa, occurring at the tooth root region as meshing progressed from the large end toward the small end of the driven gear.
(2) Two-dimensional micro-dimple shape optimization. Through 168 parametric CFD simulations, the optimal shapes and parameters were identified. Square and trapezoidal micro-dimples with dimensionless width of 0.8 and dimensionless depth of 0.9 exhibited superior and comparable lubrication performance. Considering cavitation effects, the critical slip velocity for cavitation inception was determined to be between 3 and 5 m/s. At 5 m/s slip velocity, cavitation enhanced the oil film load-carrying capacity of square and trapezoidal micro-dimples by 61% and 39.4%, respectively, while reducing wall friction forces by 0.46% and 0.22%. The square micro-dimple with width of 160 µm and depth of 4.5 µm was identified as the optimal 2D configuration.
(3) Three-dimensional micro-dimple array optimization. The cuboid micro-dimple outperformed the cylindrical micro-dimple, with 168.6% greater load-carrying capacity and 184.4% higher comprehensive evaluation coefficient. Parametric studies of array configuration revealed that square arrangement with lateral and longitudinal spacing of 200 µm produced positive and effective oil film load-carrying capacity, while staggered arrangement generated negative pressures. The optimal 3D configuration was determined as cuboid micro-dimples (160 µm × 160 µm × 4.5 µm) in a square array with 200 µm spacing in both directions.
(4) Dynamics performance. A virtual prototype model of the micro-dimpled tooth surface hypoid gear pair was established and validated. The introduction of a 20 × 80 micro-dimple array on the driven gear convex surface resulted in negligible changes in transmission error (increase of 0.0008 percentage points, from 0.0014% to 0.0022%). The average meshing force increased by 20.4887 N (0.64%), from 3188.0196 N to 3208.5083 N. However, the peak meshing force amplitude in the frequency domain decreased from 133.12 N to 122.43 N. The frequency components remained essentially consistent with integer multiples of the meshing frequency, confirming that the micro-dimple array preserves the smooth transmission characteristics inherent to hypoid gear pairs.
6.2 Limitations and Future Work
Several limitations of this study should be acknowledged. First, the lubrication problem was simplified to a single-phase isothermal hydrodynamic lubrication analysis, neglecting the coupled thermal-elastohydrodynamic effects and multi-phase phenomena that occur in actual hypoid gear operation. Future research should establish more comprehensive coupled mathematical models.
Second, the three-dimensional micro-dimple array models used for parameter optimization were simplified small-area flat-plate sliding contact models that neglected the curvature effects of actual hypoid gear tooth surfaces. Future research should investigate dynamic lubrication processes with micro-dimples arranged on actual tooth surfaces using dynamic mesh techniques.
Third, experimental validation of the optimized micro-dimple design parameters was not completed within the scope of this study due to time and resource constraints. Future work should develop feasible micro-dimple processing methods and experimental testing protocols to verify the simulation results and refine the design parameters for engineering application in hypoid gear systems.