Herringbone gears are widely adopted in high-speed and heavy-load transmission systems such as aerospace, marine propulsion, and heavy machinery due to their excellent load-carrying capacity, smooth meshing performance, and high transmission efficiency. In these demanding operating conditions, the friction between tooth surfaces generates a substantial amount of heat, which elevates the thermal load on gear teeth. Excessive temperature can reduce transmission efficiency, degrade lubrication performance, and cause surface deformation, ultimately leading to tooth surface scuffing or seizure. Therefore, investigating the contact flash temperature of herringbone gears is of paramount importance for preventing thermal failure and improving service life.
In this study, I focus on the herringbone gear transmission system and conduct a comprehensive analysis combining tooth contact analysis (TCA), loaded tooth contact analysis (LTCA), thermal elastohydrodynamic lubrication (TEHL) theory, and the Blok flash temperature theory. The goal is to establish a precise and reliable method for predicting flash temperature distribution and evaluating scuffing risk. The following sections detail my research methodology, theoretical derivations, numerical results, and validation efforts.
1. Tooth Contact Analysis of Herringbone Gears
Herringbone gears consist of two mirrored helical gear halves with opposite helix angles. The basic parameters of the herringbone gear pair used in this study are presented in Table 1. The active gear (pinion) has 21 teeth, while the driven gear (wheel) has 37 teeth. The normal module is 20 mm, the normal pressure angle is 20°, and the helix angle is 18°.
| Parameter | Driven Wheel (P) | Active Pinion (s) |
|---|---|---|
| Hand of helix | Left (L) / Right (R) | Left (L) / Right (R) |
| Number of teeth | 37 | 21 |
| Normal module / mm | 20 | 20 |
| Normal pressure angle / ° | 20 | 20 |
| Helix angle / ° | 18 | 18 |
| Input speed / rpm | — | 1000 |
| Input torque / N·m | — | 5000 |
For the tooth surface meshing analysis, I established the basic meshing equations in reference coordinate systems. The position vectors and unit normal vectors of the two tooth surfaces can be expressed in their respective coordinate systems as:
$$ \mathbf{r}_i = \mathbf{r}_i(u_i, \theta_i) \in C^2 \quad (i=1,2) $$
$$ \mathbf{n}_i = \frac{\partial \mathbf{r}_i}{\partial u_i} \times \frac{\partial \mathbf{r}_i}{\partial \theta_i} \Big/ \left| \frac{\partial \mathbf{r}_i}{\partial u_i} \times \frac{\partial \mathbf{r}_i}{\partial \theta_i} \right| \quad (i=1,2) $$
At any meshing instant, the two tooth surfaces must maintain continuous tangency. Therefore, the position vectors and unit normal vectors in the inertial reference frame must satisfy:
$$ \mathbf{r}^{(f)}_1(u_1, \theta_1, \phi_1) = \mathbf{r}^{(f)}_2(u_2, \theta_2, \phi_2) $$
$$ \mathbf{n}^{(f)}_1(u_1, \theta_1, \phi_1) = \mathbf{n}^{(f)}_2(u_2, \theta_2, \phi_2) $$
where \(\phi_1\) and \(\phi_2\) are the meshing rotation angles of the pinion and wheel, respectively. These equations constitute five independent scalar equations with six unknowns, which can be solved as nonlinear functions of the pinion rotation angle \(\phi_1\).
1.1 Actual Meshing Line Length and Contact Ratio
For the herringbone gear, the actual length of the meshing line on one side can be calculated using the following geometric relationship:
$$ B_1 B_2 = \sqrt{r_{a1}^2 – r_{b1}^2} + \sqrt{r_{a2}^2 – r_{b2}^2} – a \sin \alpha_t $$
where \(r_{a1}\), \(r_{b1}\) are the tip radius and base radius of the pinion; \(r_{a2}\), \(r_{b2}\) are the corresponding radii of the wheel; \(a\) is the center distance; and \(\alpha_t\) is the transverse pressure angle, which can be obtained from:
$$ \tan \alpha_t = \frac{\tan \alpha_n}{\cos \beta} $$
The base pitch is given by:
$$ P_b = \frac{2\pi r_{b1}}{n_1} $$
The transverse contact ratio and overlap ratio are:
$$ \varepsilon_\alpha = \frac{B_1 B_2}{P_b} $$
$$ \varepsilon_\beta = \frac{m_n \sin \beta}{\pi} $$
Therefore, the total contact ratio for one side of the herringbone gear is:
$$ \varepsilon = \varepsilon_\alpha + \varepsilon_\beta $$
For the gear pair studied here, I calculated \(P_b = 46.7\ \text{mm}\) and the total contact ratio \(\varepsilon = 2.88\). This indicates that at least two pairs of teeth are always in mesh, and at most three pairs simultaneously engage during the meshing cycle.
1.2 Contact Line Length Analysis
Due to the high contact ratio, I investigated the variation of the contact line length for individual teeth and for the total meshing process. The meshing process of a single tooth can be divided into three stages: (1) the contact line grows from a point to a full line; (2) the contact line length remains constant while both the front and rear faces are in mesh; (3) the contact line shortens from a line back to a point until the tooth exits meshing.
For the first stage, when \(0 \le L_x \le 61\ \text{mm}\), the contact line length is:
$$ L_{T1} = \frac{L_x}{\sin \beta_b} $$
During the second stage, when \(61\ \text{mm} < L_x \le 70.83\ \text{mm}\), the contact line length remains constant. In the third stage, when \(70.83\ \text{mm} < L_x \le 131.83\ \text{mm}\), the contact line length is:
$$ L_{T1} = \frac{\varepsilon_\beta P_b – L_x}{\sin \beta_b} $$
Figure 1 illustrates the variation of the total contact line length over one meshing cycle. The total contact line length exhibits a periodic pattern, with a maximum value of approximately 0.31 m.
2. Loaded Tooth Contact Analysis with Fine Grid Discretization
Based on the tooth contact analysis, I established a local contact region analysis model for the herringbone gear. When the left and right tooth flanks contact at points \(M_L\) and \(M_R\), respectively, the elastic deformation under load transforms the point contacts into elliptical contact regions. According to Hertz contact theory, the semi-major axis \(a\) and semi-minor axis \(b\) of the contact ellipse can be expressed as:
$$ a = k_a \sqrt[3]{\frac{3F}{E_c (A+B)}} $$
$$ b = k_b \sqrt[3]{\frac{3F}{E_c (A+B)}} $$
where \(k_a\) and \(k_b\) are the contact ellipse coefficients, \(E_c\) is the equivalent elastic modulus, \(A\) and \(B\) are coefficients related to the principal curvatures, and \(F\) is the normal load.
For the herringbone gear, the load is simultaneously applied to both the left and right tooth flanks. The load balance equation at a contact point \(M\) is:
$$ \sum_{k=1}^{s} F_{ML}^{(k)} + \sum_{k=1}^{s} F_{MR}^{(k)} = F_n $$
where \(F_{ML}^{(k)}\) and \(F_{MR}^{(k)}\) are the normal loads on the \(k\)-th tooth pair of the left and right flanks, respectively, \(s\) is the number of tooth pairs in simultaneous mesh, and \(F_n\) is the total normal load.
2.1 Contact Ellipse Grid Discretization
To accurately determine the pressure distribution within the contact ellipse, I discretized the local contact region \(S\) into \(m \times n\) small rectangular elements, each with area \(ds = dx \cdot dy\), where \(dx = 2a/m\) and \(dy = 2b/n\). The discretization scheme is illustrated in the model. For each small element, the load is \(F_{ij}\), and the total load is:
$$ F_M = \sum_{i=1}^{m} \sum_{j=1}^{n} F_{ij} $$
The load balance condition for the discretized contact region becomes:
$$ \sum_{k=1}^{s} \sum_{i=1}^{m} \sum_{j=1}^{n} F_{ijL}^{(k)} = F_L, \quad \sum_{k=1}^{s} \sum_{i=1}^{m} \sum_{j=1}^{n} F_{ijR}^{(k)} = F_R $$
where \(F_L\) and \(F_R\) are the total loads on the left and right flanks, respectively, with \(F_L + F_R = F_n\).
2.2 Relative Sliding Velocity and Comprehensive Curvature Radius
The relative sliding velocity between the pinion and wheel at any contact point is crucial for flash temperature calculation. The absolute velocities of the pinion and wheel at point \(M\) are:
$$ \mathbf{v}_{M1} = \boldsymbol{\omega}_1 \times \mathbf{r}_{M1} $$
$$ \mathbf{v}_{M2} = \boldsymbol{\omega}_2 \times \mathbf{r}_{M2} $$
where \(\boldsymbol{\omega}_1\) and \(\boldsymbol{\omega}_2\) are the angular velocity vectors of the pinion and wheel. The tangential components of these velocities along the contact line are:
$$ \mathbf{v}_{tM1} = \mathbf{v}_{M1} – (\mathbf{v}_{M1} \cdot \mathbf{n}_{M1}) \times \mathbf{n}_{M1} $$
$$ \mathbf{v}_{tM2} = \mathbf{v}_{M2} – (\mathbf{v}_{M2} \cdot \mathbf{n}_{M2}) \times \mathbf{n}_{M2} $$
The relative sliding velocity is then:
$$ \mathbf{v}_c = \mathbf{v}_{tM1} – \mathbf{v}_{tM2} $$
My calculations show that the relative sliding velocity first increases and then decreases along the meshing path, reaching a maximum at the tooth tip and root regions, and zero at the pitch point.
The comprehensive curvature radius at a contact point is calculated as:
$$ \rho_{red} = \frac{1}{\rho_1^{-1} + \rho_2^{-1}} $$
where \(\rho_1\) and \(\rho_2\) are the curvature radii of the two tooth surfaces at the contact point. The comprehensive curvature radius is smallest at the initial meshing point, which corresponds to the highest Hertzian contact stress.
2.3 Load Distribution Coefficient
Based on the loaded tooth contact analysis model, I established the displacement compatibility equations for both the left and right tooth flanks:
$$ \sum_{i=1}^{m} \sum_{j=1}^{n} \lambda_{ijL} F_{ijL} + w_{ijL} = u(x,y)_{L} + d_{ijL} $$
$$ \sum_{i=1}^{m} \sum_{j=1}^{n} \lambda_{ijR} F_{ijR} + w_{ijR} = u(x,y)_{R} + d_{ijR} $$
where \(\lambda_{ijL,R}\) is the total bending-shear compliance, \(w_{ijL,R}\) is the initial gap, \(u(x,y)\) is the normal displacement, and \(d_{ijL,R}\) is the residual gap. When the surfaces are in contact, \(d_{ij} = 0\) and \(F_{ij} > 0\); when separated, \(d_{ij} > 0\) and \(F_{ij} = 0\).
The load distribution coefficient for the left and right flanks can be expressed as:
$$ L_{ML} = \frac{\sum_{i=1}^{m} \sum_{j=1}^{n} F_{ijL}}{F_n}, \quad L_{MR} = \frac{\sum_{i=1}^{m} \sum_{j=1}^{n} F_{ijR}}{F_n} $$
When shaft deformation is considered, the horizontal shaft angle error \(\Delta\theta\) causes asymmetric load distribution between the left and right tooth flanks. This leads to an axial force difference \(\Delta F_z\):
$$ \Delta F_z = \sum_{k=1}^{s} \sum_{i=1}^{m} \sum_{j=1}^{n} F_{ijL}^{(k)} \cos \alpha_j – \sum_{k=1}^{s} \sum_{i=1}^{m} \sum_{j=1}^{n} F_{ijR}^{(k)} \cos \alpha_j $$
To mitigate the eccentric load, I adopted an axially floating installation method for the pinion. The axial floating displacement \(\varepsilon\) is introduced into the calculation as an initial gap correction:
$$ w_{ijL,R}’ = w_{ijL,R} + \varepsilon_n $$
where \(\varepsilon_n\) is the axial displacement projected onto the tooth normal direction. The convergence criterion is:
$$ \Delta = \frac{\Delta F_z}{F_n} \le 0.01\% $$
Table 2 summarizes the load distribution coefficients under different shaft angle errors with and without axial floating.
| Condition | Shaft angle error / ° | Left flank load coefficient | Right flank load coefficient |
|---|---|---|---|
| Fixed pinion | 0 | 0.500 | 0.500 |
| Fixed pinion | 0.00094 | 0.523 | 0.477 |
| Fixed pinion | 0.00188 | 0.549 | 0.451 |
| Fixed pinion | 0.00376 | 0.587 | 0.413 |
| Floating pinion | 0.00094 | 0.501 | 0.499 |
| Floating pinion | 0.00376 | 0.502 | 0.498 |
These results clearly demonstrate that the axial floating method effectively equalizes the load distribution between the left and right flanks, preventing eccentric loading and improving meshing stability. This uniform load distribution is essential for accurate flash temperature prediction.
3. Thermal Elastohydrodynamic Lubrication Analysis
Lubrication plays a critical role in the performance and durability of herringbone gears. When the oil film ruptures, direct metal-to-metal contact occurs, leading to increased friction, elevated temperatures, and ultimately scuffing failure. I established a point-contact TEHL model based on the local contact ellipse to analyze the lubrication characteristics of herringbone gears.
3.1 Governing Equations
For the point-contact TEHL analysis, I used the Reynolds equation to describe the pressure distribution in the oil film. Considering the entrainment velocity coinciding with the minor axis of the contact ellipse, the steady-state Reynolds equation is:
$$ \frac{\partial}{\partial x} \left( \frac{\rho h^3}{\eta} \frac{\partial p}{\partial x} \right) + \frac{\partial}{\partial y} \left( \frac{\rho h^3}{\eta} \frac{\partial p}{\partial y} \right) = 12 u_e \frac{\partial (\rho h)}{\partial x} $$
The boundary conditions are:
$$ p(x_{in}, y) = p(x_{out}, y) = 0, \quad p(x, y) \ge 0, \quad \frac{\partial p(x_{out}, y)}{\partial x} = 0 $$
The oil film thickness equation is:
$$ h(x, y) = h_0 + h_g(x, y) + V_e(x, y) $$
The geometry term is:
$$ h_g(x, y) = \frac{x^2 – x_0^2}{2R_x} + \frac{y^2 – y_0^2}{2R_y} $$
The elastic deformation is:
$$ V_e(x, y) = \frac{2}{\pi E’} \iint_{\Omega} \frac{p(\xi, \zeta)}{\sqrt{(\xi – x)^2 + (\zeta – y)^2}} \, d\xi \, d\zeta $$
The density-pressure-temperature relationship (Dowson-Higginson):
$$ \rho = \rho_0 \left( 1 + \frac{0.6 \times 10^{-9} p}{1 + 1.7 \times 10^{-9} p} \right) – 0.00065 (T – T_0) $$
The viscosity-pressure-temperature relationship (Roelands):
$$ \eta = \eta_0 \exp \left\{ (\ln \eta_0 + 9.67) \left[ \left( \frac{T – 138}{T_0 – 138} \right)^{-S_0} \left( 1 + \frac{p}{p_0} \right)^{Z_0} – 1 \right] \right\} $$
The load balance equation is:
$$ w = \iint_{\Omega} p(x, y) \, dx \, dy $$
3.2 Energy Equation and Dimensionless Form
The energy equation for the oil film is:
$$ \rho_f c_f \left( u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} \right) – \frac{\partial}{\partial z} \left( k_f \frac{\partial T}{\partial z} \right) = \frac{u T}{\rho} \frac{\partial \rho}{\partial T} \left( u \frac{\partial p}{\partial x} + v \frac{\partial p}{\partial y} \right) + \eta \left[ \left( \frac{\partial u}{\partial z} \right)^2 + \left( \frac{\partial v}{\partial z} \right)^2 \right] $$
For the solid gear bodies, the heat conduction equation is:
$$ \rho_s c_s u_s \frac{\partial T}{\partial x} = k_s \frac{\partial^2 T}{\partial z^2} $$
At the solid-liquid interface, the heat flux continuity condition is:
$$ k_s \left. \frac{\partial T}{\partial z} \right|_{z=0} = k_f \left. \frac{\partial T}{\partial z} \right|_{z=0} $$
All governing equations were normalized using appropriate dimensionless parameters. The dimensionless variables are defined in Table 3.
| Dimensionless quantity | Definition | Physical meaning |
|---|---|---|
| \(X, Y\) | \(X = x/b, Y = y/b\) | Coordinates scaled by semi-minor axis \(b\) |
| \(P\) | \(P = p/p_H\) | Pressure scaled by Hertzian pressure |
| \(\bar{\eta}\) | \(\bar{\eta} = \eta/\eta_0\) | Dimensionless viscosity |
| \(\bar{\rho}\) | \(\bar{\rho} = \rho/\rho_0\) | Dimensionless density |
| \(\bar{T}\) | \(\bar{T} = T/T_0\) | Dimensionless temperature |
| \(H\) | \(H = h R_x / b^2\) | Dimensionless film thickness |
| \(W\) | \(W = w / (E’ R_x^2)\) | Dimensionless load |
3.3 Lubrication Characteristics of Herringbone Gears
I selected three representative positions along the meshing line—the initial meshing point A, the pitch point B, and the final meshing point C—on both the left and right tooth flanks. Figure 2 shows the oil film pressure and thickness distributions. The following observations were made:
Oil film pressure: For all three positions, the pressure reaches its peak at the center of the contact ellipse, followed by a secondary pressure peak near the outlet region. The pressure at the initial meshing point A is the highest due to the small comprehensive curvature radius and high contact stress.
Oil film thickness: The film thickness decreases rapidly in the inlet region, remains relatively stable in the Hertzian contact zone, decreases near the outlet, and slightly recovers at the exit. The minimum film thickness occurs at the initial meshing point, consistent with the maximum pressure there.
Due to the axial floating installation, the pressure and film thickness distributions on the left and right flanks are nearly identical, confirming the effectiveness of the load equalization method.
3.4 Oil Film Temperature Field
The oil film temperature field was solved using the energy equation. I obtained the mid-layer film temperature and the interface temperatures on the two gear surfaces. The results show that:
The oil film temperature increases gradually in the inlet region, rises sharply in the Hertzian contact zone due to compression and shear heating, and drops rapidly after the secondary pressure peak. The two solid surface temperatures follow the oil film temperature but remain lower than the mid-layer temperature. At the pitch point B, the sliding velocity is zero, so there is minimal viscous shear heating. However, due to tooth surface compression, a slight temperature rise of approximately 0.6°C still occurs.
3.5 Influence of Operating Conditions
Using the single-variable method, I examined the effects of torque and speed on the TEHL characteristics.
Effect of torque: With the speed fixed at 1000 rpm, the input torque was varied as \(T = 500\), 1000, and 2000 N·m. Table 4 summarizes the maximum oil film pressure, central film thickness, and central film temperature at mesh point A.
| Torque / N·m | Max film pressure / GPa | Central film thickness / μm | Central film temperature / °C |
|---|---|---|---|
| 500 | 1.05 | 1.42 | 82 |
| 1000 | 1.42 | 1.18 | 116 |
| 2000 | 1.89 | 0.93 | 165 |
It is clear that as torque increases, the oil film pressure increases significantly, the film thickness decreases, and the central film temperature rises substantially.
Effect of speed: With the torque fixed at 300 N·m, the speed was varied as \(n = 1500\), 2000, and 3000 rpm. Table 5 presents the corresponding lubrication performance parameters.
| Speed / rpm | Max film pressure / GPa | Central film thickness / μm | Central film temperature / °C |
|---|---|---|---|
| 1500 | 0.78 | 1.52 | 88 |
| 2000 | 0.79 | 1.47 | 102 |
| 3000 | 0.80 | 1.38 | 132 |
The results indicate that when the torque remains constant and speed increases, the central pressure remains nearly unchanged, the film thickness decreases slightly, and the central film temperature rises noticeably. This is because higher speed intensifies the shear rate in the oil film, generating more viscous heat.
4. Contact Flash Temperature Analysis
The flash temperature is the transient temperature rise at the tooth surface during meshing, superimposed on the bulk tooth temperature. I applied the Blok flash temperature theory combined with the loaded tooth contact analysis to calculate the flash temperature distribution on the herringbone gear tooth flanks.
4.1 Blok Flash Temperature Formula with Discretization
Based on the discretized contact ellipse, the instantaneous flash temperature at the center of each small rectangular element \((i,j)\) of the \(k\)-th meshing tooth pair can be expressed as:
$$ T_{f,ij}^{(k)} = \frac{1.11 \, \mu_{m,ij}^{(k)} w_{ij}^{(k)} \left| V_{t1,ij}^{(k)} – V_{t2,ij}^{(k)} \right|}{b^{(k)} \left[ B_1 \sqrt{V_{t1,ij}^{(k)}} + B_2 \sqrt{V_{t2,ij}^{(k)}} \right]} \cdot \frac{1}{\sqrt{2b^{(k)}}} $$
where \(\mu_{m,ij}^{(k)}\) is the local coefficient of friction, \(w_{ij}^{(k)}\) is the load density, \(V_{t1,ij}^{(k)}\) and \(V_{t2,ij}^{(k)}\) are the tangential velocities of the pinion and wheel surfaces, \(b^{(k)}\) is the contact half-width, and \(B_1\), \(B_2\) are the thermal contact coefficients (taken as 13.6 N/(mm·s^0.5·K) for hardened gears).
The local coefficient of friction is calculated as:
$$ \mu_{m,ij}^{(k)} = 0.12 \left( \frac{w_{ij}^{(k)} R_a}{\eta_a V_{\Sigma,ij}^{(k)} R_x} \right)^{0.25} $$
where \(R_a\) is the surface roughness, \(\eta_a\) is the dynamic viscosity at bulk temperature, and \(V_{\Sigma,ij}^{(k)}\) is the sum of tangential velocities.
4.2 Flash Temperature Distribution
Using the operating conditions of \(n = 1000\) rpm and \(T = 5000\) N·m, I calculated the flash temperature distribution on the left and right tooth flanks. The results show that the flash temperature distributions are nearly identical on both sides, confirming that the axial floating method successfully equalizes the load and hence the temperature. The maximum flash temperature occurs at the initial meshing point, where the comprehensive curvature radius is minimal and the relative sliding velocity is largest. The flash temperature then decreases toward the pitch point, reaching a minimum value of approximately 0.6°C at the pitch point due to zero sliding velocity. After passing the pitch point, the flash temperature increases again as the sliding velocity reverses and increases.
4.3 Validation and Comparative Analysis
To validate my flash temperature prediction method, I compared my theoretical results with three independent approaches: (1) Romax software simulation, (2) TEHL calculation, and (3) the traditional ISO calculation method. The comparison was performed on the left flank at three key positions. Table 6 summarizes the results.
| Position | My method / °C | Romax / °C | TEHL / °C | ISO / °C | Max difference / % |
|---|---|---|---|---|---|
| Meshing entry | 102 | 96 | 108 | 98 | 5.9 |
| Pitch point | 0.6 | 0 | 0.8 | 0 | — |
| Meshing exit | 45 | 43 | 48 | 42 | 5.7 |
The comparative analysis reveals the following findings:
The maximum difference between my method and the other three approaches is less than 6%, which confirms the accuracy and reliability of my flash temperature computation.
At the pitch point, both the Romax simulation and ISO calculation yield zero flash temperature because they neglect elastic deformation effects. In contrast, my method, which accounts for tooth surface deformation, produces a small but non-zero flash temperature of 0.6°C, consistent with the TEHL result. This demonstrates that my method provides a more realistic representation of the actual tooth contact state.
The traditional ISO method treats the load distribution coefficient as constant over certain intervals, whereas my grid-based approach captures the variation at every discrete point, providing a more detailed and accurate flash temperature profile across the entire tooth flank.
4.4 Parametric Influence on Flash Temperature
I also investigated the effects of key parameters on the flash temperature of herringbone gears.
Torque effect: With speed fixed at 1000 rpm, the input torque was varied as \(T_1 = 3000\), \(T_2 = 5000\), and \(T_3 = 10000\) N·m. The maximum flash temperatures were 70°C, 102°C, and 171°C, respectively. Torque has a pronounced influence on flash temperature because higher torque directly increases the load density and friction coefficient at the tooth surface.
Speed effect: With torque fixed at 5000 N·m, the speed was varied as \(n_1 = 1000\), \(n_2 = 2000\), and \(n_3 = 3000\) rpm. The maximum flash temperatures were 102°C, 118°C, and 131°C, respectively. Higher speed increases the relative sliding velocity, which intensifies frictional heating, but the effect is less significant than torque.
Surface roughness effect: For \(R_a = 0.3\), 0.5, and 0.8 μm, the flash temperature increased progressively with roughness. This is attributed to the higher friction coefficient associated with rougher surfaces, which generates more heat at the contact interface.
4.5 Scuffing Verification
Finally, I performed scuffing verification using the highest contact temperature criterion:
$$ T_{f,\max} + T_M \le T_S $$
where \(T_M = 60^\circ\text{C}\) is the bulk tooth temperature, and the critical scuffing temperature \(T_S\) is determined from the empirical relationship:
$$ T_S = 26.2 \ln(\nu_{40}) $$
For the selected lubricant FVA345 M320 (ISO VG 320), \(\nu_{40} = 327\ \text{mm}^2/\text{s}\), which gives \(T_S = 152^\circ\text{C}\). Figure 3 shows the contact temperature distribution for three operating conditions.
For condition 1 (\(n_1 = 1000\) rpm, \(T_1 = 3000\) N·m), the maximum contact temperature is \(60 + 70 = 130^\circ\text{C}\), which is below the critical scuffing temperature. For condition 2 (\(n_2 = 1000\) rpm, \(T_2 = 5000\) N·m), the maximum contact temperature is \(60 + 102 = 162^\circ\text{C}\), exceeding the critical value. For condition 3 (\(n_3 = 2000\) rpm, \(T_3 = 5000\) N·m), the maximum contact temperature is \(60 + 118 = 178^\circ\text{C}\), also above the critical limit.
The analysis reveals that the most vulnerable region for scuffing is the initial meshing zone, where the flash temperature is highest. This finding provides a theoretical basis for gear tooth modification and optimization to reduce flash temperature and enhance scuffing resistance.
5. Conclusion
In this study, I conducted a comprehensive investigation of the contact flash temperature of herringbone gears under high-speed and heavy-load conditions. The following conclusions can be drawn:
(1) The total contact ratio of the herringbone gear pair is 2.88, indicating that at least two pairs of teeth are always in mesh. The total contact line length exhibits a periodic variation, with a maximum value of approximately 0.31 m.
(2) A loaded tooth contact analysis model was established with fine grid discretization of the contact ellipse. The results show that shaft deformation causes asymmetric load distribution between the left and right tooth flanks. By adopting the axially floating pinion installation, the load distribution coefficients on both flanks were effectively equalized, preventing eccentric loading.
(3) The point-contact TEHL analysis revealed that the oil film pressure increases with torque, while the film thickness decreases. The oil film temperature rises with both torque and speed, with torque having a more pronounced effect. At high speeds, the risk of scuffing due to elevated oil film temperature becomes significant.
(4) Based on the Blok flash temperature theory combined with loaded tooth contact analysis, I obtained the flash temperature distribution on both tooth flanks. The maximum flash temperature occurs at the initial meshing point. The comparison with Romax simulation, TEHL calculation, and ISO method shows good agreement, with a maximum difference of 6%, validating the accuracy of my approach.
(5) The parametric study indicates that torque, speed, and surface roughness all positively correlate with flash temperature. The scuffing verification using the highest contact temperature criterion shows that the herringbone gear system may experience scuffing at the initial meshing zone when the input torque exceeds 5000 N·m with a speed of 1000 rpm. This finding highlights the critical importance of optimizing tooth profile modifications and selecting appropriate lubricants to mitigate thermal failure in herringbone gears.