The study of thermal behavior in gear transmissions is of paramount importance, particularly for high-speed applications. Among the most critical failure modes for spur gear operating at high rotational speeds is scuffing or scoring of the tooth flanks. This phenomenon is intrinsically linked to the thickness of the lubricant film and the temperature at the contacting surfaces. The surface temperature directly influences the viscosity of the lubricant and, consequently, the film thickness separating the mating teeth. Therefore, accurately determining the tooth flank temperature is crucial for predicting and preventing scuffing failures. Furthermore, under high-speed and heavily loaded conditions, the frictional heat generated at the meshing interfaces is substantial. This heat not only elevates the overall temperature of the transmission system but also induces significant thermal deformations. These deformations can severely compromise the performance, operational reliability, and service life of the spur gear drive. Consequently, a detailed investigation into the bulk temperature field within a spur gear tooth is essential for advanced design and reliability assessment.
Traditional analytical methods, such as thermal network approaches, have been employed to estimate average gear body temperatures. However, these methods often fail to provide a comprehensive spatial distribution of the temperature across the tooth volume. Other numerical studies have utilized finite element analysis but frequently incorporated simplifications—such as assuming constant friction coefficients, uniform heat distribution along the contact line, or neglecting load sharing between simultaneous tooth pairs—which can compromise accuracy. Moreover, experimental validation of the predicted bulk temperature fields is often lacking. This work aims to address these gaps by presenting a refined methodology for simulating the steady-state bulk temperature field of a high-speed spur gear. The approach meticulously calculates the frictional heat flux and convective heat transfer coefficients, incorporates detailed load distribution, and is rigorously validated against experimental measurements.
Theoretical Foundation for Bulk Temperature Analysis
According to Blok’s theory, the bulk temperature field of a spur gear tooth can be treated as a steady-state problem once thermal equilibrium is reached. Furthermore, due to the periodic symmetry of the gear, the analysis can be confined to a single spur gear tooth, significantly reducing computational complexity. The three-dimensional heat conduction governing the steady-state temperature distribution within the spur gear tooth is described by Laplace’s equation:
$$
\nabla^2 T_B = \frac{\partial^2 T_B}{\partial x^2} + \frac{\partial^2 T_B}{\partial y^2} + \frac{\partial^2 T_B}{\partial z^2} = 0
$$
where \( T_B \) represents the bulk temperature, and \( x, y, z \) are the spatial coordinates. The solution to this equation requires appropriate boundary conditions that define the heat exchange at the tooth surfaces. For a single spur gear tooth model, the boundaries include the contact flank (\(A_3\)), the two end faces (\(A_5, A_6\)), the tooth root (\(A_2, A_7\)), the tooth tip (\(A_4\)), the non-contact flank (\(A_8\)), the periodic symmetry faces (\(A_1, A_9\)), and the bottom face connected to the gear body (\(A_{10}\)).
The boundary condition on the meshing flank \(A_3\) accounts for both the incoming frictional heat flux and convection to the lubricant:
$$
-\lambda \frac{\partial T_B}{\partial n} = h_{A_3}(T_B – T_o) + q_F \quad \text{on} \quad A_3
$$
The boundary conditions on the end faces \(A_5\) and \(A_6\) consider convection to the surrounding air:
$$
-\lambda \frac{\partial T_B}{\partial n} = h_i(T_B – T_a) \quad \text{on} \quad A_i, \quad i = A_5, A_6
$$
Other surfaces like the root, tip, and non-contact flank experience convective cooling by the oil-air mist/splash:
$$
-\lambda \frac{\partial T_B}{\partial n} = h_j(T_B – T_o) \quad \text{on} \quad A_j, \quad j = A_2, A_4, A_7, A_8
$$
Periodic symmetry is enforced on faces \(A_1\) and \(A_9\):
$$
T_B|_{A_1} = T_B|_{A_9} \quad \text{and} \quad \frac{\partial T_B}{\partial n}|_{A_1} = \frac{\partial T_B}{\partial n}|_{A_9}
$$
Finally, the bottom face \(A_{10}\) is assumed to be adiabatic, representing a deep connection to the massive gear body:
$$
\frac{\partial T_B}{\partial n} = 0 \quad \text{on} \quad A_{10}
$$
In these equations, \( \lambda \) is the thermal conductivity of the spur gear material, \( n \) is the outward normal vector, \( h_{A_3} \), \( h_i \), and \( h_j \) are the convective heat transfer coefficients for the respective surfaces, \( T_o \) is the inlet oil temperature, \( T_a \) is the ambient air temperature inside the gearbox, and \( q_F \) is the average frictional heat flux entering the tooth flank. The accurate determination of \( q_F \) and the various \( h \) coefficients is therefore the cornerstone of a reliable bulk temperature field prediction for the spur gear.
Analysis of Key Parameters: Heat Flux and Convection
Frictional Heat Flux Density
The primary heat source in a spur gear mesh is sliding friction between the tooth flanks. The average frictional heat flux density \( q_F \) at a given meshing position is calculated as:
$$
q_F = \frac{t_F}{t} \beta \gamma \mu p v
$$
where:
\( t_F \) is the time during which heat flux acts on the contact width of the driving spur gear tooth.
\( t \) is the time for one complete revolution of the driving spur gear.
\( \beta \) is the heat partition coefficient.
\( \gamma \) is the thermal conversion efficiency (typically 0.90–0.95).
\( \mu \) is the coefficient of friction.
\( p \) is the average contact pressure.
\( v \) is the relative sliding velocity at the contact line.
The heat partition coefficient \( \beta \), which determines the fraction of frictional heat entering the driving spur gear, depends on the thermophysical properties and tangential speeds of both gears:
$$
\beta = \frac{\sqrt{\lambda_1 \rho_1 c_1 v_1}}{\sqrt{\lambda_1 \rho_1 c_1 v_1} + \sqrt{\lambda_2 \rho_2 c_2 v_2}}
$$
Subscripts 1 and 2 denote the driving and driven spur gears, respectively, with \( \lambda \), \( \rho \), and \( c \) representing thermal conductivity, density, and specific heat capacity.
The coefficient of friction \( \mu \) is not constant but varies along the path of contact. For a spur gear, it can be estimated using an empirical formula that considers load, speed, surface roughness, and lubricant properties:
$$
\mu = 0.048 \left( \frac{F_{NC1}/b}{(v_1 + v_2) R_E} \right)^{0.2} \eta^{-0.05} \left( \frac{R_{a1} + R_{a2}}{2} \right)^{0.25} X_L
$$
Here, \( F_{NC1} \) is the normal load on the driving spur gear tooth flank, \( b \) is the face width, \( R_E \) is the equivalent radius of curvature at the contact point, \( \eta \) is the dynamic viscosity of the lubricant, \( R_a \) is the surface roughness, and \( X_L \) is a surface roughness factor.
The average contact pressure \( p \) is derived from Hertzian contact theory:
$$
p = \frac{\pi}{4} \sqrt{\frac{F_{NC1} E}{2 \pi R_E b (1 – \nu^2)}}
$$
where \( E \) and \( \nu \) are Young’s modulus and Poisson’s ratio of the spur gear material.
The normal load \( F_{NC1} \) is related to the output torque \( T_2 \). Crucially, for an unmodified spur gear, the load is shared between two pairs of teeth in the double-contact regions and carried by a single pair in the single-contact region. The load-sharing is governed by the load distribution factor \( K \):
$$
\begin{aligned}
&K = 1 & & \text{for} & S_1 + S_0 < S < S_2 – S_0 \\
&K = \frac{1}{3} \left(1 + \frac{S – S_1}{S_0}\right) & & \text{for} & S_1 < S < S_1 + S_0 \\
&K = \frac{1}{3} \left(1 – \frac{S – S_2}{S_0}\right) & & \text{for} & S_2 – S_0 < S < S_2
\end{aligned}
$$
with \( S_0 = \pi m (\epsilon – 1) \cos \alpha \), where \( m \) is the module, \( \epsilon \) is the contact ratio, \( \alpha \) is the pressure angle, and \( S \), \( S_1 \), \( S_2 \) define the contact position, start, and end points, respectively.
Therefore, the normal load is:
$$
F_{NC1} = K \frac{T_2 z_1}{z_2 r_c \cos \alpha}
$$
where \( z \) is the number of teeth and \( r_c \) is the radius to the contact point on the driving spur gear.
Convective Heat Transfer Coefficients
Meshing Flank (\(h_{A_3}\)): The convection between the spur gear tooth flank and the lubricant in the contact zone is analogous to that for bevel gears. The coefficient is calculated based on Reynolds number (\(Re\)), Prandtl number (\(Pr\)), and thermal conductivity (\(k\)) of the oil:
$$
h_{A_3} = \frac{0.228 Re^{0.731} Pr^{1/3} k}{d}
$$
where \( d \) is the pitch diameter of the spur gear.
End Faces (\(h_i\)): For spur gears cooled by oil jets, the end faces are primarily exposed to air. The convection here is modeled as a rotating disk problem. For laminar flow, the coefficient is:
$$
h_i = 0.308 k_a (q + 2)^{0.5} Pr_a^{0.5} \left( \frac{\omega}{\eta_a} \right)^{0.5}
$$
where \( k_a \), \( Pr_a \), and \( \eta_a \) are the thermal conductivity, Prandtl number, and kinematic viscosity of air, respectively; \( \omega \) is the angular velocity; and \( q \) is a constant defining the radial temperature distribution on the disk (typically \( q=2 \)).
Other Surfaces (\(h_j\)): There is no universally accepted formula for convection on the root, tip, and non-contact flanks of a spur gear. Based on established practice for similar analyses, the coefficient for these surfaces is taken as a fraction (between 1/3 and 1/2) of the end face convection coefficient \( h_i \).
Analytical and Experimental Methodology
Finite Element Modeling
A three-dimensional finite element model of a single spur gear tooth was developed using ANSYS parametric design language (APDL) to allow for easy modification of geometry and operating conditions. The model was discretized using SOLID70 elements, which are 8-node hexahedral elements suitable for 3-D thermal analysis. The mesh was refined in the region near the contact surface to capture the steep temperature gradients accurately.
The application of boundary conditions required special attention. Since the same surface (the contact flank) is subjected to both an inward heat flux (\( q_F \)) and convective cooling (\( h_{A_3} \)), surface effect elements (SURF152) were overlaid on the relevant faces. The frictional heat flux \( q_F \), calculated as a function of meshing position, was applied as a boundary load, while the convective heat transfer coefficients were applied to the surface effect elements.
Experimental Setup
The simulation results were validated using a mechanical power-circulating gear test rig. This closed-loop system consists of a test gearbox and a slave gearbox connected via a drive motor, load coupling, and torsion shaft. This configuration allows high torque testing with relatively low motor power input, as energy circulates within the system and the motor only overcomes frictional losses.
The test spur gear pair was mounted in the test gearbox. The operating conditions, including input torque, input speed, and lubricant inlet temperature, could be precisely controlled. An inspection window and a temperature measurement port were machined into the test gearbox housing to allow optical access and temperature measurement. The surface temperature of the spur gear teeth was measured using a Fluke Ti32 infrared thermal imager with an accuracy of ±2°C or 2% of the reading. After running the test rig for approximately 30 minutes to reach thermal steady-state (indicated by stabilized oil return temperature), thermal images of the gear flank were captured through the measurement port. The emissivity and transmissivity settings of the thermal imager were calibrated according to the test conditions.
The geometric parameters of the driving spur gear used for both simulation and experiment are summarized in the table below:
| Parameter | Value |
|---|---|
| Number of Teeth (Driving) | 16 |
| Number of Teeth (Driven) | 24 |
| Module (mm) | 4.5 |
| Pressure Angle (°) | 22.43 |
| Face Width (mm) | 20 |
| Driving Gear Profile Shift Coefficient | 0.85 |
| Driven Gear Profile Shift Coefficient | -0.5 |
| Center Distance (mm) | 91.5 |
| Surface Roughness, \(R_a\) (μm) | 0.8 |
The baseline operating conditions for analysis were: Torque \( T = 70 \, \text{N·m} \), Rotational Speed \( n = 1500 \, \text{r/min} \), Lubricant Inlet Temperature \( T_o = 40^\circ \text{C} \), Ambient Air Temperature \( T_a = 40^\circ \text{C} \). The spur gear material was steel with typical properties: \( E = 210 \, \text{GPa} \), \( \nu = 0.3 \), \( \rho = 7800 \, \text{kg/m}^3 \), \( c = 500 \, \text{J/(kg·K)} \), \( \lambda = 40 \, \text{W/(m·K)} \).
Results and Discussion
Boundary Condition Calculations and Temperature Distribution
Using the formulas described, the average frictional heat flux \( q_F \) was calculated for different meshing positions along the path of contact for the spur gear. The results show a characteristic profile: \( q_F \) is higher near the start and end of contact (root and tip regions) where sliding velocity is significant, and it approaches a minimum near the pitch point where the sliding velocity is nearly zero. This non-uniform heat input is a critical factor determining the final temperature distribution on the spur gear tooth flank.
The calculated convective heat transfer coefficients for the baseline case are presented below:
| Surface | Convective Heat Transfer Coefficient (W/(m²·K)) |
|---|---|
| Meshing Flank (\(h_{A_3}\)) | 662.14 |
| End Face (\(h_i\)) | 42.90 |
| Other Surfaces (\(h_j\), taken as \(h_i/2.4\)) | 17.88 |
The finite element analysis using these boundary conditions yielded the steady-state bulk temperature field for the spur gear tooth. The simulation predicted a maximum temperature of 76.23°C on the contact surface. The corresponding experimental measurement from the thermal imager was 77.78°C. The close agreement (within ~2%) validates the accuracy of the proposed simulation methodology for analyzing the spur gear bulk temperature field.
The temperature distribution shows that the highest temperature zone is located at the center of the contact surface along the face width. The temperature decreases from the contact surface towards the interior of the spur gear tooth body, exhibiting a gradient. Analyzing the temperature along the tooth flank reveals two key trends:
1. Along the face width: The temperature distribution is approximately parabolic, with the minimum temperatures at the two ends. This is a direct consequence of enhanced convective cooling at the end faces of the spur gear.
2. Along the tooth profile (from root to tip): The temperature profile mirrors the heat flux input. It shows a decrease, followed by an increase, and then another decrease, with a local minimum near the pitch point where the heat generation is lowest. This pattern is consistent for both simulation and experimental data.
Influence of Operational Parameters
The validated model was used to investigate the influence of key operational parameters on the maximum spur gear tooth temperature.
Effect of Torque: With speed and oil temperature held constant, increasing the input torque leads to a clear increase in the maximum spur gear flank temperature. This is because higher torque increases the normal load \( F_{NC1} \), which in turn raises the average contact pressure \( p \) and the frictional heat flux \( q_F \), as seen from the equations \( p \propto \sqrt{F_{NC1}} \) and \( q_F \propto \mu p v \). The increased heat input outweighs any minor changes in convection, resulting in a higher steady-state temperature.
Effect of Rotational Speed: Holding torque and oil temperature constant, an increase in rotational speed also causes the maximum spur gear temperature to rise. The relationship is complex: speed increases the sliding velocity \( v \), directly raising \( q_F \). It also increases the convective coefficients (\( h_{A_3} \) and \( h_i \)), which promotes cooling. However, the net effect is dominated by the increased heat generation, leading to a higher overall temperature. The increase in \( h_{A_3} \) with speed, given by \( h_{A_3} \propto Re^{0.731} \), is significant but not sufficient to counteract the rise in \( q_F \).
Effect of Lubricant Inlet Temperature: Increasing the inlet oil temperature \( T_o \), while keeping torque and speed constant, results in a higher maximum spur gear temperature. The primary mechanism here is the reduction in the convective cooling potential. The temperature difference \( (T_B – T_o) \) that drives heat transfer is reduced. Furthermore, a higher \( T_o \) typically lowers the lubricant viscosity \( \eta \), which can slightly increase the friction coefficient \( \mu \) according to the formula \( \mu \propto \eta^{-0.05} \), contributing marginally to more heat generation.
In all parametric studies, the simulation results showed consistent trends with the experimental measurements. The experimental values were systematically slightly higher than the simulation predictions. This consistent offset is likely attributable to additional heat sources present in the physical test rig—such as bearing losses and windage—that were not modeled in the single-tooth spur gear FEM analysis, which assumes an adiabatic connection to the gear body.
Conclusion
This study presents a comprehensive methodology for simulating the bulk temperature field of high-speed spur gears. The approach integrates principles from gear meshing, tribology, and heat transfer to accurately calculate the position-dependent frictional heat flux, accounting for load sharing and variable friction coefficients, and the convective heat transfer coefficients for all spur gear tooth surfaces. A three-dimensional finite element model of a single spur gear tooth was developed and solved under steady-state conditions.
The key findings are:
1. The maximum temperature on a spur gear tooth is located at the center of the contact surface along the face width. The temperature distribution from the flank to the tooth interior shows a distinct gradient.
2. On the contact surface itself, the temperature follows an approximate parabolic distribution along the face width due to end cooling, and it varies along the profile according to the heat flux input, with a local minimum near the pitch line.
3. The maximum operating temperature of the spur gear increases with rising input torque, rotational speed, and lubricant inlet temperature. The underlying physical mechanisms for each trend have been explained through the analytical formulas governing heat generation and dissipation.
4. The close correlation between the simulated bulk temperature field and experimental measurements obtained from a gear test rig validates the proposed analytical and numerical methodology. This confirms that the approach is reliable for the thermal analysis and design assessment of spur gear transmissions operating under high-speed conditions.
The validated model serves as a useful tool for predicting thermal behavior, optimizing cooling strategies, and assessing the risk of scuffing failure in spur gear design, ultimately contributing to the development of more reliable and efficient gear systems.