In mechanical transmission systems, spur gears are fundamental components that often operate under heavy loads. In such conditions, the lubrication supply may become insufficient, leading to starved or dry friction states. This can result in severe wear and temperature rise on the gear tooth surfaces, potentially causing failures like scuffing, plastic deformation, or even catastrophic breakdown. Therefore, understanding the wear characteristics of spur gears under dry friction is crucial for predicting and preventing such failures. Traditional studies have relied heavily on experimental approaches, but with advancements in computational methods, numerical simulation has emerged as a powerful tool for investigating wear distribution and thermal behavior. In this article, we present a comprehensive numerical analysis of dry friction wear in spur gears, combining theoretical derivations, MATLAB-based simulations, and ANSYS finite element modeling. We focus on the effects of operational parameters like rotational speed and torque load on friction coefficients, wear loss, and temperature fields, aiming to provide insights for gear design and maintenance under harsh lubrication conditions.
Our study begins by establishing the conditions for dry friction in spur gear pairs. Based on lubrication theory, the formation of an elastohydrodynamic (EHD) oil film is essential to prevent direct metal-to-metal contact. For spur gears, the lubricant supply at the meshing zone—particularly from the driven gear—plays a critical role. Using Cameron’s theory, we derive the minimum oil film thickness required to avoid dry friction. The criterion for maintaining an EHD film is given by the following relationship: the oil quantity must exceed a critical value. Specifically, for a spur gear pair, the condition to prevent dry friction is expressed as:
$$ \frac{\eta_0 U}{E R} > K $$
where $\eta_0$ is the dynamic viscosity of the lubricant, $U$ is the entrainment velocity at the meshing point, $E$ is the composite elastic modulus, $R$ is the equivalent radius of curvature, and $K$ is a constant derived from lubrication parameters. When this condition is not met, dry friction occurs, leading to increased wear and heat generation. For spur and pinion gears, the entrainment velocity varies along the path of contact, making it necessary to evaluate this condition at multiple points. We consider the pinion (smaller gear) as more susceptible to wear due to higher cyclic stresses, and thus our analysis primarily targets the pinion tooth surface.
To quantify wear under dry friction, we employ fatigue wear theory, which accounts for the cyclic loading and sliding in gear meshing. The wear process is modeled based on the relative motion between the spur gear and pinion teeth. During a small time interval $dt$, the pinion tooth moves along an involute path from point $C_{11}$ to $C_{12}$, while the gear tooth moves from $C_{21}$ to $C_{22}$. The instantaneous sliding velocity at the contact point is given by:
$$ v_1 = \omega_1 R_{E1}, \quad v_2 = \omega_2 R_{E2} $$
where $\omega_1$ and $\omega_2$ are the angular velocities of the pinion and gear, respectively, and $R_{E1}$ and $R_{E2}$ are their effective radii of curvature. The sliding coefficient, which influences wear rate, is defined as the ratio of sliding velocity to entrainment velocity. For the pinion, the cumulative wear volume over operating time $t$ can be expressed as:
$$ W = \int_0^t \frac{I_h \mu_f F_n v_s}{H} \, dt $$
where $I_h$ is a dimensionless wear coefficient, $\mu_f$ is the friction coefficient, $F_n$ is the normal load, $v_s$ is the sliding velocity, and $H$ is the material hardness. However, for spur gears, a more specific formulation based on contact mechanics is used. Assuming a friction coefficient of 0.1 for dry contact, the wear depth on the pinion tooth is derived as:
$$ h_w = \frac{K_w \mu_f^4 F_n^{3/2}}{E^* \sqrt{R}} \cdot N $$
where $K_w$ is a wear constant incorporating material properties, $E^*$ is the equivalent elastic modulus, $R$ is the contact radius, and $N$ is the number of cycles. This equation highlights the strong dependence of wear on friction coefficient and load. In our simulations, we vary these parameters to analyze their impact.
For temperature field analysis under dry friction, we use the finite element method to solve the heat conduction equation. The gear tooth contact generates frictional heat, which is treated as a surface heat flux. The general heat balance equation for a differential element in the gear body is:
$$ \rho C \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} \left( k_x \frac{\partial T}{\partial x} \right) + \frac{\partial}{\partial y} \left( k_y \frac{\partial T}{\partial y} \right) + \frac{\partial}{\partial z} \left( k_z \frac{\partial T}{\partial z} \right) + q”’ $$
where $\rho$ is density, $C$ is specific heat, $T$ is temperature, $k_x, k_y, k_z$ are thermal conductivities in coordinate directions, and $q”’$ is internal heat generation rate. In dry friction of spur gears, there is no internal heat source within the bulk material, so $q”’ = 0$. For steady-state conditions, $\frac{\partial T}{\partial t} = 0$, simplifying the equation to:
$$ \nabla \cdot (k \nabla T) = 0 $$
with boundary conditions accounting for frictional heat flux at the contact surface and convection elsewhere. Using variational principles, this partial differential equation is transformed into a finite element formulation:
$$ [K] \{T\} = \{P\} $$
where $[K]$ is the thermal conductivity matrix, $\{T\}$ is the nodal temperature vector, and $\{P\}$ is the thermal load vector from frictional heating. This approach allows us to compute temperature distributions in the spur gear and pinion teeth accurately.
We implemented the wear calculation model in MATLAB to simulate dry friction behavior. The program computes friction coefficients and wear depth along the tooth profile for different operating conditions. Key parameters include rotational speeds of 3000, 6000, and 10000 rpm, and torque loads of 150, 250, and 350 N·m. The spur gear pair has standard involute teeth with module 3 mm, pressure angle 20°, and 24 teeth on the pinion and 48 on the gear. Material properties are typical for steel: elastic modulus 210 GPa, Poisson’s ratio 0.3, density 7850 kg/m³, and hardness 200 HB. The friction coefficient in dry contact is initially assumed as 0.1 but varies with sliding conditions in the simulation.
The MATLAB simulation results for friction coefficient distribution are summarized in Table 1. These values are averaged over the meshing cycle for the pinion tooth flank.
| Rotational Speed (rpm) | Torque Load (N·m) | Average Friction Coefficient | Maximum Friction Coefficient |
|---|---|---|---|
| 3000 | 150 | 0.098 | 0.105 |
| 3000 | 250 | 0.102 | 0.112 |
| 3000 | 350 | 0.108 | 0.120 |
| 6000 | 150 | 0.092 | 0.100 |
| 6000 | 250 | 0.096 | 0.106 |
| 6000 | 350 | 0.101 | 0.115 |
| 10000 | 150 | 0.085 | 0.095 |
| 10000 | 250 | 0.090 | 0.102 |
| 10000 | 350 | 0.095 | 0.110 |
As observed, the friction coefficient decreases with increasing rotational speed but increases with higher torque loads. This trend is consistent across all speeds, indicating that higher sliding velocities may reduce adhesion effects, while greater normal pressures enhance frictional resistance. The distribution pattern along the tooth profile remains similar, with peaks near the pitch point where sliding velocity is minimal.
For wear depth, the simulation outputs are shown in Table 2. Wear is calculated as maximum depth on the pinion tooth after 10^5 cycles, normalized to microns.
| Rotational Speed (rpm) | Torque Load (N·m) | Maximum Wear Depth (µm) | Location on Tooth |
|---|---|---|---|
| 3000 | 150 | 12.5 | Near root, single-double contact point |
| 3000 | 250 | 18.7 | Near root, single-double contact point |
| 3000 | 350 | 25.3 | Near root, single-double contact point |
| 6000 | 150 | 9.8 | Near root, single-double contact point |
| 6000 | 250 | 14.2 | Near root, single-double contact point |
| 6000 | 350 | 19.5 | Near root, single-double contact point |
| 10000 | 150 | 7.6 | Near root, single-double contact point |
| 10000 | 250 | 11.1 | Near root, single-double contact point |
| 10000 | 350 | 15.8 | Near root, single-double contact point |
The wear depth is highest at the transition region between single and double tooth contact, close to the root of the pinion tooth. This is due to stress concentration and higher sliding in that zone. Notably, wear decreases with increasing speed but increases significantly with torque load. At 350 N·m and 3000 rpm, wear depth reaches 25.3 µm, indicating severe material loss. This underscores the critical impact of load on dry friction wear in spur gears.
To complement the wear analysis, we performed finite element thermal simulations using ANSYS. The model includes a three-dimensional tooth geometry of the spur pinion and gear, with fine meshing at the contact region. Steady-state temperature fields are computed for dry friction conditions with a constant heat flux derived from frictional power. The heat flux $q$ is calculated as:
$$ q = \mu_f F_n v_s / A_c $$
where $A_c$ is the contact area. For the spur and pinion pair, the contact area changes along the meshing path, so we use an average value based on Hertzian contact theory. The steady-state temperature distribution for the pinion tooth at 3000 rpm and 250 N·m is shown in Figure 1 (refer to the inserted image for visualization). The highest temperatures are localized along the contact line, with values exceeding 150°C in some regions, while the gear bulk remains near ambient temperature. This confirms that frictional heating is confined to a thin surface layer, but can lead to thermal softening or oxidation over time.
We also conducted transient thermal analyses to investigate the effect of torque load on peak temperatures. The simulation applies a step load and monitors the maximum surface temperature over time until a steady state is approached. Results for different torque loads at a fixed speed of 6000 rpm are summarized in Table 3.
| Torque Load (N·m) | Peak Temperature (°C) at 0.1s | Peak Temperature (°C) at 1s | Steady-State Peak Temperature (°C) |
|---|---|---|---|
| 150 | 85 | 120 | 125 |
| 250 | 110 | 165 | 175 |
| 350 | 135 | 210 | 225 |
The peak temperature increases linearly with torque load, as higher loads generate more frictional heat. At 350 N·m, the steady-state peak temperature reaches 225°C, which is sufficient to alter material properties and accelerate wear mechanisms. This linear relationship can be expressed as:
$$ T_{max} = T_0 + \alpha \cdot \tau $$
where $T_0$ is a base temperature, $\alpha$ is a proportionality constant, and $\tau$ is the torque load. For our spur gear system, $\alpha$ is approximately 0.5°C per N·m, indicating high thermal sensitivity to load.
Further, we analyzed the influence of rotational speed on temperature rise. Table 4 shows steady-state peak temperatures for different speeds at a constant torque of 250 N·m.
| Rotational Speed (rpm) | Steady-State Peak Temperature (°C) | Heat Flux (W/mm²) |
|---|---|---|
| 3000 | 160 | 0.45 |
| 6000 | 175 | 0.62 |
| 10000 | 195 | 0.85 |
Temperature increases with speed due to higher sliding velocities and thus greater frictional power. However, the rise is less dramatic compared to load effects, suggesting that load control is more critical for temperature management in dry friction of spur gears.
Our numerical models also allow us to explore wear progression over multiple cycles. Using Archard’s wear law integrated with the fatigue wear theory, we simulate cumulative wear depth on the pinion tooth after 10^6 cycles. The wear rate $k$ is derived from:
$$ \frac{dV}{ds} = k \frac{F_n}{H} $$
where $dV$ is wear volume, $ds$ is sliding distance. For spur gears, the sliding distance varies along the profile, so we compute it incrementally. Results show that wear accumulates non-linearly, with accelerated loss after initial run-in. This is particularly evident for high-load scenarios, where the wear coefficient $k$ increases due to thermal effects.
To generalize our findings, we propose a dimensionless parameter $\Pi$ that combines operational variables for spur and pinion wear prediction:
$$ \Pi = \frac{\mu_f^4 F_n^{3/2} N}{E^* \sqrt{R} v^{0.5}} $$
where $v$ is the pitch line velocity. This parameter correlates well with simulated wear depth across all cases, with a correlation coefficient of 0.95. It can serve as a design guideline to estimate wear life under dry friction.
In terms of temperature field dynamics, the transient analysis reveals that the time to reach steady state decreases with higher loads and speeds. For instance, at 350 N·m and 10000 rpm, steady state is achieved within 0.5 seconds, whereas at 150 N·m and 3000 rpm, it takes over 2 seconds. This rapid heating under severe conditions can cause thermal shock, leading to micro-cracking on the tooth surface. The spatial temperature gradient is steepest near the contact line, with values dropping by over 100°C within a millimeter of depth. This gradient induces thermal stresses, which we approximate using:
$$ \sigma_{th} = E \beta \Delta T $$
where $\beta$ is the coefficient of thermal expansion and $\Delta T$ is the temperature difference. For steel gears, $\beta \approx 12 \times 10^{-6}$ /°C, so a $\Delta T$ of 150°C yields thermal stresses around 360 MPa, which is significant for fatigue life.
We also examined the effect of tooth geometry modifications on wear and temperature. By introducing tip relief on the pinion tooth, we reduced the maximum wear depth by 15% and peak temperature by 10% in simulations. This highlights the potential of profile optimization to mitigate dry friction effects.
Our MATLAB and ANSYS simulations are validated against published experimental data for dry friction wear in spur gears. The trends in friction coefficient and wear depth match qualitatively, though absolute values differ due to idealized assumptions. For instance, our model assumes constant friction coefficient, whereas in reality, $\mu_f$ varies with surface roughness and temperature. Future work could incorporate dynamic friction models and consider third-body abrasion.
In conclusion, this numerical study provides a detailed analysis of dry friction wear in spur gears. We derived theoretical conditions for dry friction, implemented wear calculations based on fatigue theory, and performed finite element thermal simulations. Key findings include: (1) Friction coefficient and wear depth decrease with increasing rotational speed but increase with torque load; (2) Wear is most severe near the root transition zone of the pinion tooth; (3) Temperature fields show localized heating along the contact line, with peak temperatures rising linearly with load; (4) A dimensionless parameter is proposed to predict wear under various operating conditions. These insights can aid in designing spur and pinion systems for harsh environments, optimizing lubrication strategies, and predicting service life. The integration of MATLAB and ANSYS tools demonstrates the efficacy of numerical simulation for gear research, offering a cost-effective alternative to extensive testing.
For practical applications, we recommend monitoring load and speed parameters closely to avoid dry friction conditions. If unavoidable, surface treatments like coatings or geometry modifications can reduce wear and thermal damage. Further research could explore the combined effects of wear and temperature on gear dynamics, including vibrations and noise, to develop comprehensive failure prediction models for spur gear transmissions.