Friction and wear are unavoidable in mechanical power transmission. They not only cause energy dissipation but also reduce the operating accuracy and service life of machinery. In my research, I focus on the spur gear, which is one of the most widely used transmission components in modern industry. A spur gear pair is a typical line-contact mechanism, and its meshing surfaces are subjected to very high contact pressure. The reliability of a spur gear transmission depends strongly on the lubrication condition between the tooth flanks. Lubrication is used to separate the rubbing surfaces and to reduce the severity of wear. However, in actual service environments, the lubricating oil is often contaminated by solid particles. These particles may come from external dust, wear debris generated during running-in, or degraded additives. When such particles are mixed with the oil, the fluid is no longer a pure liquid but a liquid-solid two-phase fluid. The presence of solid particles changes the rheological behaviour of the lubricant, modifies the pressure distribution in the oil film, and eventually affects the friction and wear of the spur gear tooth surfaces.
In my work, I aim to investigate the lubrication mechanism and the wear evolution law of spur gears under liquid-solid two-phase lubrication. I selected an involute spur gear pair as the research object. The main variables considered in my study are the particle concentration, particle diameter, surface roughness, entrainment speed, load and equivalent curvature radius. I used two numerical approaches to solve the lubrication problem: the lattice Boltzmann method (LBM) for the microscopic flow field around a single particle in the Hertz contact zone, and the finite difference method (FDM) for the macroscopic elastohydrodynamic lubrication (EHL) analysis. Finally, I combined the lubrication results with the Archard wear model to predict the wear depth distribution along the tooth flank. My goal is to provide a theoretical basis for the condition monitoring and life prediction of spur gear drives working with contaminated lubricants.
The interaction between a solid particle and the oil film is very complex. The particle size in real spur gear systems is often comparable to the oil film thickness. Consequently, particles can pass through the contact zone and may either remain suspended in the oil or become trapped between the asperities. If the particle is larger than the minimum film thickness, it can carry load and generate additional friction. If the particle is smaller, it still influences the effective viscosity of the lubricant. Therefore, the conventional assumption of a pure oil film is not sufficient for a realistic spur gear lubrication analysis. I therefore established a series of models that can describe the liquid-solid two-phase lubrication process in a spur gear contact.
1 Gear Contact and Lubrication Fundamentals
The meshing of a spur gear pair can be idealized as the contact between two equivalent cylinders. According to the Hertz elastic contact theory, the contact region is a narrow rectangle with half-width \(b\). The equivalent radius of curvature \(R\) is defined by the radii of curvature of the two tooth flanks at the meshing point.
\[
\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}
\]
The Hertz contact half-width can be calculated using the load per unit length \(F_n/L\), the elastic moduli \(E_1,E_2\), the Poisson ratios \(\mu_1,\mu_2\), and the equivalent radius \(R\).
\[
b=\sqrt{\frac{4F_n}{\pi L}\cdot
\frac{\frac{1-\mu_1^2}{E_1}+\frac{1-\mu_2^2}{E_2}}{\frac{1}{R_1}+\frac{1}{R_2}}}
\]
The maximum contact pressure follows the elliptic distribution along the contact width. In my calculation, I used the material parameter \(Z_E\) of the spur gear material and expressed the maximum contact stress as
\[
\sigma_H=Z_E\sqrt{\frac{F_n}{bR}}
\]
In tribology, the severity of surface interaction is usually described by the film thickness ratio \(\lambda\). This ratio is defined as the minimum lubricating film thickness divided by the root mean square roughness of the two mating tooth surfaces.
\[
\lambda=\frac{h_{\min}}{\sqrt{\delta_1^2+\delta_2^2}}
\]
According to the value of \(\lambda\), the lubrication state of a spur gear pair can be classified into boundary, mixed and full-film elastohydrodynamic lubrication. Table 1 summarises the classification used in my study.
| Range of \(\lambda\) | Lubrication regime | Typical feature |
|---|---|---|
| \(\lambda \le 0.5\) | Boundary lubrication | Asperities largely in contact, very thin boundary film |
| \(0.5 < \lambda < 3\) | Mixed lubrication | Oil film and asperities share the load |
| \(\lambda \ge 3\) | Full-film EHL | Tooth flanks fully separated by the oil film |
For most industrial spur gear transmissions, especially in low-speed and heavy-load conditions, the value of \(\lambda\) lies in the mixed lubrication region. When solid particles are present in the oil, the effective film thickness ratio is further reduced because particles can interrupt the oil film and promote direct asperity contact. This is why the liquid-solid two-phase lubrication problem is particularly important for spur gear wear prediction.
2 Lattice Boltzmann Simulation of the Particle-Laden Hertz Zone
In the first part of my numerical study, I used the lattice Boltzmann method to resolve the oil flow in the Hertz contact zone of a spur gear. The LBM is a mesoscopic approach that simulates the motion of fluid particles on a discrete lattice. Compared with the traditional finite difference method, LBM is advantageous in handling complex particle boundaries and multiphase flow. I used the D2Q9 model, which is the standard two-dimensional nine-velocity lattice model.
The macroscopic density and velocity in the LBM are obtained from the velocity distribution function. The evolution equation of the distribution function in the multiple-relaxation-time (MRT) formulation is expressed as
\[
f_i(\mathbf{x}+\mathbf{c}_i\delta_t,t+\delta_t)-f_i(\mathbf{x},t)
=
\mathbf{M}^{-1}\mathbf{s}\left[\mathbf{m}(\mathbf{x},t)-\mathbf{m}^{eq}(\mathbf{x},t)\right]
+
\mathbf{M}^{-1}\mathbf{F}
\]
In the above equation, \(\mathbf{M}\) is the transformation matrix, \(\mathbf{s}\) is the diagonal relaxation matrix, \(\mathbf{m}\) is the moment vector, \(\mathbf{m}^{eq}\) is the equilibrium moment vector, and \(\mathbf{F}\) is the forcing term caused by the solid particle. The equilibrium distribution function of the D2Q9 model is given by
\[
f_i^{eq}=\rho \omega_i
\left[
1+\frac{\mathbf{c}_i\cdot\mathbf{u}}{c_s^2}
+\frac{(\mathbf{c}_i\cdot\mathbf{u})^2}{2c_s^4}
-\frac{\mathbf{u}\cdot\mathbf{u}}{2c_s^2}
\right]
\]
The macroscopic pressure \(p\) is related to the fluid density through the lattice sound speed \(c_s\),
\[
p=\rho c_s^2
\]
In my simulation, the upper tooth surface and the lower tooth surface were modelled as moving walls with velocities \(u_1\) and \(u_2\). The lubricant was trapped in a narrow channel whose height was equal to the local film thickness. I applied the bounce-back boundary condition on the stationary solid surfaces, the non-equilibrium extrapolation scheme at the inlet, and a pressure outlet condition at the exit.
Before carrying out the full spur gear simulation, I validated the LBM code against the analytical Couette flow solution. The velocity profile in a simple Couette flow is linear:
\[
u(y)=u_{wall}\frac{y-y_0}{y_t-y_0}
\]
The simulated velocity profile agreed very well with the analytical solution. This validation confirmed that the LBM code could correctly reproduce the shear-driven flow in the gear contact region. I then inserted a single circular particle into the computational domain. The particle was permitted to rotate or remain stationary. The physical parameters used in the LBM simulation are listed in Table 2.
| Physical parameter | Value | Lattice parameter | Value |
|---|---|---|---|
| Lubricant density | 870 kg/m³ | Contact length in lattice units | 22000 |
| Lubricant viscosity | 0.13086 Pa·s | Film thickness in lattice units | 20 |
| Minimum film thickness | 2×10⁻⁷ m | Lattice viscosity | 0.1 |
| Contact width | 2.2×10⁻⁴ m | Upper wall velocity | 2.32×10⁻⁶ |
| Relative surface velocity | 0.232 m/s | Relaxation factor | 0.8 |
I investigated six particle motion states: no particle, stationary particle, clockwise rotation, anticlockwise rotation, and two speed ratios. The particle speed ratio was defined as \(\phi = u/u_R\), where \(u\) is the particle surface velocity and \(u_R\) is the reference tooth surface velocity. The velocity distributions around the particle showed that a stationary particle blocks the oil flow and reduces the local velocity because of viscosity. A moving particle, however, induces vortices near its surface. These vortices increase the Reynolds number of the local flow and make the lubricant motion more irregular. The friction between lubricant layers then changes, and this is reflected in the pressure distribution.
One important observation from my LBM simulations is that the pressure in the film-thickness direction is not constant when a particle is present. In pure-oil lubrication, the pressure gradient across the thin oil film is negligible. However, when a solid particle enters the spur gear contact zone, the pressure around the particle varies along the film-thickness direction. The clockwise rotation of the particle enlarges the pressure difference across the film, while the anticlockwise rotation reduces this difference. The oil film pressure behind the particle is generally higher than the pressure for the particle-free case. This result indicates that the classical assumption of constant pressure across the film must be reconsidered when the spur gear is lubricated by a liquid-solid two-phase fluid.
I also used the LBM to study the influence of the equivalent curvature radius on the contact pressure at the spur gear meshing point. The curvature radius affects the deformed contact shape and the film thickness. I tested three values of \(R\), namely 0.009 m, 0.010 m and 0.012 m, with the particle located at the centre of the contact zone. The maximum contact pressure decreased with increasing curvature radius. Furthermore, the pressure near the particle changed abruptly because of the swirl generated by the particle. This phenomenon is more pronounced when the curvature radius is small and the oil film is thinner.
3 Finite-Difference Elastohydrodynamic Analysis of Particle-Laden Spur Gears
In order to obtain a more complete and quantitative description of the liquid-solid EHL problem for a real spur gear, I also used the finite difference method to solve the modified Reynolds equation. The physical model treats the gear tooth contact as an equivalent cylinder contacting a plane. The modified Reynolds equation for a lubricant containing solid particles can be written as
\[
\frac{d}{dx}\left(\frac{\rho \phi(N,l,h)h^3}{\eta}\frac{dp}{dx}\right)
=
12u_s\frac{d(\rho h)}{dx}
\]
where \(\rho\) is the density of the lubricant, \(\eta\) is the effective viscosity, \(h\) is the film thickness, \(l\) is the particle diameter, \(N\) is the coupling number that expresses the degree of coupling between particle translation and rotation, and \(\phi(N,l,h)\) is a flow factor that adjusts for the influence of particles on the velocity profile. For pure oil, \(\phi=1\) and the equation reduces to the classical Reynolds equation. In the present spur gear simulation, the flow factor was evaluated using the particle-flow model for rigid particles.
The viscosity of the liquid-solid two-phase lubricant changes with pressure and particle concentration. I used the following effective viscosity relation:
\[
\eta_{\rm eff}
=
\eta_0\left(1+2.5\lambda_p\right)
\exp(\alpha p)
\]
where \(\eta_0\) is the viscosity of the base oil, \(\lambda_p\) is the particle mass concentration, and \(\alpha\) is the pressure-viscosity coefficient. The density-pressure relationship was
\[
\rho=\rho_0\left(1+\frac{0.6p}{1+1.7p}\right)
\]
In the finite difference solution, I discretized the Reynolds equation on a uniform grid. The pressure was updated with the Gauss-Seidel iteration in lightly loaded zones and with the Jacobi iteration in heavily loaded zones. The film thickness equation included the gap shape and the elastic deformation:
\[
h(x)=h_0+\frac{x^2}{2R}+v(x)
\]
where \(v(x)\) is the normal elastic deformation of the gear tooth surface. The deformation was calculated with the influence coefficient method. The load balance condition was enforced so that the integral of the pressure over the contact zone was equal to the applied external load per unit length.
\[
\int_{x_0}^{x_e}p(s)ds=w
\]
Using this finite difference solver, I first studied the effect of load on the minimum film thickness and the pressure distribution. The load values were \(1\times10^4\), \(2\times10^4\), \(4\times10^4\), \(6\times10^4\), \(8\times10^4\), and \(1\times10^5\) N. The results showed that an increase in load reduces the minimum film thickness of the spur gear contact. The pressure in the inlet region decreases with load, while the pressure in the central contact region increases. The secondary pressure peak also becomes smaller and moves toward the outlet when the load is increased. This load dependence explains why overloading is harmful for spur gear service life.
I also investigated the effect of the equivalent curvature radius on the spur gear EHL performance. I used the same entrainment speed of 0.8 m/s and particle concentration of 0.5%. The tested curvature radii were 0.01, 0.015, 0.02, 0.025 and 0.03 m. The minimum film thickness increased with the curvature radius. The oil film pressure in the contact zone decreased as the curvature radius increased, whereas the inlet pressure showed the opposite tendency. The secondary pressure peak shifted to the left when the radius was increased. Table 3 summarises the qualitative trends obtained from the finite difference calculations.
| Parameter increased | Minimum film thickness \(h_{\min}\) | Central oil film pressure | Secondary pressure peak |
|---|---|---|---|
| Load \(w\) | Decreases | Increases | Decreases and moves to outlet |
| Curvature radius \(R\) | Increases | Decreases | Decreases and moves to inlet |
4 Combined Effects of Surface Roughness and Solid Particles on Spur Gear Lubrication
In a real spur gear, the tooth flank is not perfectly smooth. The surface roughness amplitude is usually of the same order as the lubricating film thickness. Therefore, surface roughness cannot be ignored when the lubrication state is mixed. In my model, I generated a Gaussian random roughness profile and added it to the film thickness equation. The roughness height \(r(x)\) was produced using the standard normal random distribution in the numerical code, with a root mean square value of 0.6 μm. The total film thickness became
\[
h(x)=h_0+\frac{x^2}{2R}+v(x)+r(x)
\]
The random roughness profile creates local fluctuations in the oil film pressure and causes many pressure spikes. When solid particles are also present, the roughness asperities can trap particles and intensify the interaction between the two surfaces. I therefore solved the modified Reynolds equation with both particle-induced viscosity and roughness terms. This combined model represents a three-body contact problem, in which the two rough gear tooth surfaces and the solid particle all interact with the lubricant.
I first studied the influence of the entrainment speed. The entrainment speed is the average velocity of the two tooth surfaces at the meshing point. I varied the entrainment speed from 0.6 m/s to 1.4 m/s. The results showed that the film thickness becomes wavy because of the roughness profile. As the entrainment speed increased, the overall film thickness increased. A thicker film reduces the number of direct asperity contacts and therefore reduces the wear of the spur gear tooth flank. The oil film pressure peak in the contact zone decreased with increasing entrainment speed, while the inlet pressure increased. The secondary pressure peak became weaker at higher speeds.
Next, I studied the effect of the curvature radius on the mixed lubrication of the spur gear. With the same load and speed, the film thickness increased with the curvature radius. The pressure fluctuations became smaller when the curvature radius was large. This is because a larger curvature radius produces a wider Hertz contact area and a smaller local pressure gradient. From a practical point of view, the spur gear contact points near the pitch circle and the tooth root have different curvature radii. These changes directly affect the lubrication state and wear distribution along the tooth profile.
I then analysed the effect of particle concentration. I used particle mass concentrations of 0, 0.5%, 2%, 5% and 7%. The particle diameter was 0.6 μm. The results showed that the minimum film thickness changes only slightly when the particle concentration is increased. However, the pressure distribution is significantly altered. Compared with the particle-free lubricant, the oil containing particles produces a larger contact area and a higher pressure peak. As the particle concentration increased, the secondary pressure peak became smaller and moved to the inlet side. The reason for this behaviour is that particles increase the effective viscosity of the lubricant and enhance the hydrodynamic action in the inlet region, which changes the location of the maximum pressure.
Finally, I examined the particle size effect on the friction generated by the particles. In mixed lubrication, the total friction consists of the fluid shear resistance, the micro-asperity contact friction, and the particle shear friction:
\[
F_{\rm total}=F_{\rm fluid}+F_{\rm asperity}+F_{\rm particle}
\]
When the particle diameter is larger than the local film thickness, the particle is squeezed between the two surfaces and must bear a portion of the normal load. The particle then undergoes shear deformation during the sliding motion. I calculated the particle-induced friction for particle diameters of 0.5, 0.6, 0.7, 0.8, 0.9 and 1.0 μm. The particle friction increased with particle diameter, but the rate of increase gradually slowed down. Larger particles are more likely to be trapped in the inlet zone and therefore create more resistance to the sliding motion. This is an important reason why large debris particles in the lubricant should be removed.
Table 4 gives a summary of the influence of the main parameters on the lubricating film and friction of a spur gear.
| Parameter increased | Minimum film thickness | Contact pressure | Friction |
|---|---|---|---|
| Entrainment speed | Increases | Decreases | Decreases |
| Curvature radius | Increases | Decreases | — |
| Particle concentration | Nearly unchanged | Pressure peak increases | Increases in root-side zone |
| Particle diameter | — | — | Increases |
5 Wear Prediction of Spur Gear Tooth Flank
After obtaining the lubrication characteristics of the liquid-solid two-phase flow, I combined the lubrication model with the Archard wear law to predict the wear of the spur gear tooth flank. The Archard equation describes the wear volume \(V\) as a function of the normal load \(W\), the sliding distance \(S\), the material hardness \(H\) and the dimensionless wear coefficient \(K\).
\[
\frac{V}{S}=K\frac{W}{H}
\]
For local wear calculation, the wear depth \(h_w\) at a point on the tooth flank can be expressed as
\[
\frac{dh_w}{dt}=k\,p\,v
\]
where \(k\) is the local wear coefficient, \(p\) is the local contact pressure, and \(v\) is the relative sliding velocity. Because the meshing of a spur gear is periodic, I calculated the accumulated wear depth after a given number of meshing cycles. The accumulated wear depth at a discrete point \(Q\) after \(n\) wear cycles is
\[
h_{Q,n}=h_{Q,n-1}+\Delta t\sum_{i=1}^{M}k_i\,N_i\,p_i\,v_i
\]
where \(N_i\) is the number of meshing events in the current interval, \(M\) is the number of discrete time steps per cycle, and \(p_i\) and \(v_i\) are the contact pressure and sliding velocity at each time step.
5.1 Gear and particle parameters used in wear calculation
I selected a standard involute spur gear pair with the parameters listed in Table 5. The tooth profile was discretised into 100 points from the tooth root to the tooth tip. The dynamic load, sliding velocity and contact pressure were computed for each meshing point along the line of action.
| Parameter | Value |
|---|---|
| Number of teeth of pinion \(Z_1\) | 29 |
| Number of teeth of gear \(Z_2\) | 45 |
| Module \(m\) | 4 mm |
| Face width \(B\) | 30 mm |
| Pressure angle | 20° |
| Elastic modulus \(E_1,E_2\) | 2.21×10¹¹ Pa |
| Poisson ratio \(\mu_1,\mu_2\) | 0.3 |
| Roughness \(R_a\) | 0.6 μm |
| Particle density | 2250 kg/m³ |
| Particle diameter | 0.6 μm |
| Particle shear stress | 4×10⁷ N/m² |
5.2 Film thickness ratio and lubrication state analysis
I first calculated the minimum film thickness and the film thickness ratio along the full tooth profile. The results for different rotational speeds are shown qualitatively in Figure 1. The minimum film thickness increases along the meshing line from the tooth root to the tooth tip, but there is a clear reduction near the single-tooth contact zone. This is because the load is transferred from two pairs of teeth to only one pair in the single-tooth zone. The sudden increase in load reduces the film thickness and creates a discontinuous change in the film thickness ratio.
For the operating conditions with a rotational speed of 450 r/min and a load of 900 N·m, the film thickness ratio is always in the mixed lubrication range. Therefore, I could not assume a pure full-film lubrication condition in my spur gear wear analysis. The mixed lubrication assumption is also closer to the real industrial situation, especially when the lubricating oil contains solid particles.
5.3 Friction coefficient and friction force
The friction coefficient depends strongly on the lubrication state. For the mixed lubrication of a spur gear, I used the following composite expression:
\[
f_{\rm mix}=f_{\lambda}f_e+\left(1-f_{\lambda}\right)f_b
\]
where \(f_e\) is the friction coefficient in the full-film EHL zone, \(f_b\) is the friction coefficient in the boundary lubrication zone, and \(f_{\lambda}\) is the load sharing factor of the full-film zone. The parameter \(f_{\lambda}\) was expressed as a function of the film thickness ratio:
\[
f_{\lambda}
=
\frac{1.2\lambda^{0.64}}{1+0.37\lambda^{1.26}}
\]
Table 6 lists the coefficients used in the full-film friction coefficient expression, which are the result of a regression analysis of the spur gear EHL friction data.
| Coefficient | Value |
|---|---|
| \(b_1\) | -8.91 |
| \(b_2\) | 1.03 |
| \(b_3\) | 1.03 |
| \(b_4\) | 0.35 |
| \(b_5\) | 2.81 |
| \(b_6\) | -0.1 |
| \(b_7\) | 0.75 |
| \(b_8\) | -0.39 |
| \(b_9\) | 0.62 |
My calculations showed that the friction coefficient along the line of action first increases and then decreases. The maximum friction coefficient occurs at the pitch point, where the sliding direction reverses. With increasing rotational speed, the friction coefficient decreases. With increasing load, the friction coefficient increases. The presence of particles creates a small disturbance in the friction coefficient curve in the meshing zone close to the tooth root on the pinion side. This is because the local film thickness in that area is smaller than the particle diameter, so the particles can directly contact the rough tooth surface.
The friction force follows a similar trend. The maximum friction force is observed at the pitch point. A sudden change appears at the transition from double-tooth contact to single-tooth contact. The friction force decreases with increasing speed and increases with increasing load. I also studied the effect of particle concentration on friction. When the particle concentration increases, the friction force in the tooth-root side near the pitch point increases. In the tooth-tip side, the film is thicker than the particle diameter, so the particle concentration has negligible effect. This result confirms that the particles mainly affect the spur gear friction in zones where the film is relatively thin.
5.4 Wear coefficient and wear depth
In my wear model, the wear coefficient is not a constant. It depends on the lubrication state through the film thickness ratio. For boundary lubrication, the wear coefficient is equal to the boundary value \(k_0\). When the film thickness ratio increases and the spur gear enters the mixed lubrication zone, the wear coefficient decreases because the oil film carries a larger share of the load. In the full-film region, the wear coefficient tends to zero because the gear tooth surfaces are completely separated.
I adopted the following piecewise function for the dynamic wear coefficient:
\[
k =
\begin{cases}
k_0, & \lambda \le 0.5 \\[4pt]
k_0\left(4\lambda-2\right), & 0.5 < \lambda < 3 \\[4pt]
0, & \lambda \ge 3
\end{cases}
\]
The boundary wear coefficient \(k_0\) was selected from the published spur gear wear experiments. The dynamic wear coefficient was then used in the wear depth calculation. The sliding distance \(S\) at each meshing point was obtained from the sliding coefficient of the spur gear pair. The sliding coefficient of the pinion and the gear is defined as
\[
\lambda_{\rm slip}=\frac{S_1-S_2}{S_1}
\]
where \(S_1\) and \(S_2\) are the distances travelled by the two mating tooth surfaces during the same time interval. The actual sliding distance in one mesh cycle is approximately equal to twice the Hertz contact half-width times the sliding coefficient.
\[
S=2b\lambda_{\rm slip}
\]
The sliding coefficient of the pinion and the gear is zero at the pitch point. From the tooth root to the pitch point, the sliding coefficient of the pinion is larger than that of the gear. From the pitch point to the tooth tip, the opposite trend is observed. The sliding distance shows the same distribution because it is proportional to the sliding coefficient.
Using the Archard wear equation, I computed the accumulated wear depth after 3 million, 6 million and 9 million meshing cycles. The load was 900 N·m, the rotational speed was 450 r/min, the particle concentration was 0.5%, and the particle diameter was 0.6 μm. The main conclusions from the wear calculation are as follows:
First, the wear depth distribution along the tooth profile of the pinion and the gear has the same trend. The wear depth first decreases and then increases along the line of action. At the pitch point, the wear depth is zero because there is no relative sliding. The maximum wear depth occurs in the meshing zone near the tooth root side of the pitch point. In this zone, the equivalent curvature radius is small, the contact pressure is high, and the oil film is relatively thin. Solid particles therefore have a stronger influence in this region.
Second, the pinion and the gear have almost the same wear depth in the meshing zone between the pitch point and the tooth tip. In the meshing zone between the tooth root and the pitch point, the pinion wears much more than the gear. This asymmetry is caused by the different numbers of teeth. The pinion rotates faster and undergoes more meshing cycles in the same operating period, so its accumulated wear depth is larger. This result is consistent with the common observation that the pinion of a spur gear pair usually fails earlier than the gear in a contaminated lubricant environment.
Third, the wear depth increases with the number of meshing cycles. After 3 million cycles, the wear is still relatively small, and the wear coefficient changes only slightly. After 6 million cycles, the tooth surface profile changes noticeably, and the wear coefficient begins to increase because the surface roughness is altered. After 9 million cycles, the wear depth grows more rapidly. This nonlinear evolution indicates that the spur gear wear process accelerates once the tooth surface topography is modified by initial wear.
Table 7 provides a summary of the wear depth results in a qualitative way. The values are normalised by the maximum wear depth after 9 million cycles so that the distribution can be compared between the pinion and the gear.
| Meshing zone | Pinion | Gear |
|---|---|---|
| Tooth root side before the pitch point | High, maximum at the boundary of single-tooth zone | Moderate, lower than pinion |
| Pitch point | Zero | Zero |
| Between pitch point and tooth tip | Similar to gear | Similar to pinion |
| Single-tooth contact zone | Sudden increase in wear depth | Sudden increase in wear depth |
6 Conclusions
My research investigated the liquid-solid two-phase lubrication and wear law of an involute spur gear pair. I used the lattice Boltzmann method to examine the microscopic flow behaviour around a single solid particle in the Hertz contact zone, and the finite difference method to solve the macroscopic elastohydrodynamic lubrication problem. I then combined the lubrication results with the Archard wear model to predict the wear depth distribution on the tooth flank. The principal conclusions of my study are summarised below.
First, the LBM simulation of the spur gear contact zone demonstrates that a solid particle significantly changes the oil flow. A stationary particle retards the oil flow, whereas a moving particle creates vortices and increases the local turbulence. The oil film pressure in the film-thickness direction is no longer constant when particles are present. The pressure behind the particle is greater than the particle-free case. With increasing curvature radius, the maximum contact pressure decreases. The LBM results agree qualitatively with the finite difference solution, which indicates that the lattice Boltzmann method is applicable to the lubrication analysis of a spur gear.
Second, the finite difference analysis of the spur gear EHL shows that the minimum film thickness decreases with increasing load and increases with increasing curvature radius. The oil film pressure in the contact zone increases with load and decreases with curvature radius. These results are important for the design and operation of spur gear transmissions because they define the operating conditions that favour a protective oil film.
Third, when the surface roughness is taken into account together with solid particles, the film thickness profile becomes wavy, and the pressure distribution contains many random fluctuations. The film thickness increases with entrainment speed and curvature radius. The particle concentration has a small effect on the minimum film thickness, but it increases the pressure peak and modifies the location of the secondary pressure peak. The friction caused by particles increases with particle diameter, but the rate of increase gradually slows down.
Fourth, the wear analysis of the spur gear tooth flank under liquid-solid lubrication shows that the film thickness ratio along the line of action remains in the mixed lubrication regime for the investigated operating conditions. The friction coefficient and friction force reach their maximum values at the pitch point. The wear coefficient decreases with increasing rotational speed and increases with increasing load. The accumulated wear depth increases nonlinearly with the number of meshing cycles. The maximum wear depth of both the pinion and the gear occurs in the meshing zone near the tooth root side of the pitch point. In this zone, the pinion wears more than the gear because it experiences more meshing cycles. In the meshing zone between the pitch point and the tooth tip, the wear depths of the pinion and the gear are comparable.
In conclusion, solid particles in the lubricant have a significant influence on the lubrication and wear behaviour of spur gears. The presence of particles should not be neglected in the life prediction of gear transmissions. My work provides a theoretical basis for understanding the lubrication mechanism of liquid-solid two-phase flow in a spur gear contact and for predicting the wear evolution of the tooth flank. Future research will extend the current model to three-dimensional contacts, thermal effects, and non-circular particles. I believe that the combination of mesoscopic simulation and macroscopic wear theory will become a powerful tool for the tribological design of spur gears.