In recent years, the global demand for sustainable energy solutions has become increasingly prominent. Wind power, as a major representative of renewable energy, has shown a significant growth trend. The gear transmission system is one of the core components of a wind turbine generator set. The design and optimization of gearboxes, especially the improvement of gear meshing performance, have a significant impact on ensuring the stable operation of wind turbines under various working conditions. Due to the large loads, high speeds, and complex service environments of transmission systems in large equipment such as wind turbines, herringbone gears are often employed because of their stronger carrying capacity and better stability. In this research, I focus on the high-speed end herringbone gear transmission of a wind turbine gearbox. The primary subjects include tooth profile modification, parametric modeling, dynamic contact analysis, thermal analysis, thermal-mechanical coupled analysis, and a novel tooth modification method based on the coupled deformation and stress fields.
The herringbone gear is a special form of double-helical gear, which eliminates the axial thrust generated by a single helical gear. This characteristic makes herringbone gears particularly suitable for high-speed and heavy-duty applications. However, the complex geometry and the combined effect of frictional heating and mechanical loading present substantial challenges in predicting the real contact behaviour and thermal deformation. In this thesis, I establish a comprehensive simulation framework that integrates precise gear geometry, friction heat flux, convective heat transfer coefficients, steady-state temperature fields, and sequential thermal-mechanical coupling analysis. Based on the calculated deformation and stress distributions, I then propose a tooth profile modification method specifically tailored for high-speed wind turbine herringbone gears. The methodology and findings are presented in detail.
1. Introduction and Background
The installed capacity of wind power has expanded rapidly worldwide. According to the Global Wind Report 2023, the newly installed capacity of wind power in 2022 recovered strongly after a short downturn caused by the pandemic. China has become a global leader in wind power, with the cumulative installed capacity reaching enormous values. Nevertheless, the technical research on wind turbine gearboxes is still developing, and there is a need for deeper investigations into gear reliability, thermal behaviour, and tooth modification. Gearboxes in wind turbines usually use high-speed gears to convert the low rotational speed of the rotor into the high speed required by the generator. These gears operate under varying loads and speeds, resulting in significant frictional heat generation on tooth surfaces. The resulting temperature rise can cause thermal deformation, which may alter the tooth contact pattern and induce additional dynamic loads, vibration, and noise. Therefore, understanding the thermal-mechanical coupled behaviour of herringbone gears is essential for improving their performance and service life.
Researchers have studied gear contact mechanics, elastohydrodynamic lubrication, and thermal analysis for many years. Early work established the theoretical basis for flash temperature and contact temperature. Later studies used finite element methods to predict the temperature field and stress field of spur and helical gears. However, only a limited number of studies have addressed herringbone gears under combined thermal and mechanical loads. The herringbone gear has a central groove and two opposite helical sections, leading to a more complex load distribution and heat transfer process. Moreover, the tooth profile modification of herringbone gears requires accurate knowledge of the coupled deformation, which has rarely been investigated. In this research, I aim to fill this gap by proposing a modified tooth profile design that accounts for both thermal expansion and elastic deflection.
2. Tooth Profile Modification and Parametric Modeling
Tooth profile modification is an effective way to reduce meshing impact, improve load distribution, and reduce vibration and noise. The fundamental concept is to remove a small amount of material from the tooth surface near the tip or root so that the actual contact under load approaches the ideal conjugate action. In this section, I discuss the principles of meshing impact, the main parameters of profile modification, and the parametric modeling method for herringbone gears.
2.1 Meshing Impact and Load Distribution
During gear operation, the meshing process repeats periodically. At the instant when one pair of teeth enters or leaves mesh, the load is suddenly transferred from one pair to another. Due to elastic deformation and manufacturing errors, the actual base pitch differs from the theoretical value, causing impact at the meshing-in and meshing-out positions. For the meshing-in impact, when the driven gear tooth tip enters the mesh, the load is still carried by the previous pair. The deformation causes the actual normal base pitch of the driven gear to increase while that of the driving gear decreases. Consequently, the next pair enters mesh earlier than expected, and the instantaneous transmission ratio decreases slightly. This produces a sliding motion at the tooth tip of the driven gear and the tooth root of the driving gear, generating impact. Similarly, at the meshing-out position, the driven gear tooth pair tends to remain in contact longer, resulting in a sudden angular acceleration of the driving gear and impact at the tooth tip of the driving gear.
The load distribution along the line of action in a meshing cycle can be divided into two double-tooth-pair zones and one single-tooth-pair zone. In the double-tooth-pair zones, the load is shared by two pairs of teeth, while in the single-tooth-pair zone, only one pair carries the full load. The transition between these zones is sudden and causes a step change in the load, which is the main source of gear excitation. Tooth profile modification smooths this transition by removing material near the tip so that the actual load curve becomes more gradual.
2.2 Parameters of Tooth Profile Modification
The main parameters of tooth profile modification are the maximum modification amount, the modification length, and the modification curve. These parameters determine the shape and extent of the removed material.
Maximum modification amount is the largest deviation between the modified profile and the theoretical involute profile. Several empirical formulas exist in the literature. In my work, I determine the maximum modification amount from the calculated thermal-mechanical coupled deformation. The idea is that the maximum modification amount should compensate for the combined elastic deflection and thermal expansion at the beginning or end of meshing. Mathematically, the maximum modification amount can be expressed as:
$$ \Delta_{\max} = \delta_{mech} + \delta_{therm} + \delta_{manu} $$
where $\delta_{mech}$ is the elastic deflection due to the normal load, $\delta_{therm}$ is the thermal expansion caused by the temperature rise, and $\delta_{manu}$ is the manufacturing error allowance. In this study, I obtain $\delta_{mech}$ and $\delta_{therm}$ from finite element simulations of the herringbone gear pair under the rated torque and temperature field.
Modification length is the distance from the tooth tip to the starting point of modification along the tooth profile or along the line of action. For high-speed and heavily loaded herringbone gears, long profile modification is recommended, where the modification starts at the lowest point of single-tooth-pair contact on the pinion and extends to the tip. The modification length is related to the base pitch and the transverse contact ratio. In my analysis, the modification length corresponds to the distance from the meshing-in or meshing-out point to the alternating contact point, as indicated by the load distribution diagram.
Modification curve defines how the modification amount varies from zero at the starting point to the maximum value at the tooth tip. The general form is:
$$ \Delta(x) = \Delta_{\max} \left( \frac{x}{L} \right)^p $$
where $x$ is the distance from the starting point, $L$ is the modification length, and $p$ is the power exponent. Typical values are $p=1$ for a linear profile, $p=1.22$, $p=1.5$, and special forms such as the one recommended by some researchers:
$$ \Delta(x) = \Delta_{\max} \left[ 0.44 \left( \frac{x}{L} \right) + 0.56 \left( \frac{x}{L} \right)^2 \right] $$
This particular curve provides a smaller modification amount near the starting point and a larger amount near the tip, which often gives a better compromise between load sharing and transmission error.
2.3 Parametric Modeling of Herringbone Gears
Accurate three-dimensional modeling is essential for reliable finite element analysis. I derive the tooth profile equations using the rack cutter generation principle and coordinate transformation. The tooth profile of an involute gear consists of the addendum circle, involute curve, transition curve, and dedendum circle. The transition curve is generated by the tip radius of the rack cutter, while the involute curve is generated by the straight side of the rack cutter. Using the coordinate transformation between the rack cutter and the gear blank, I obtain the parametric equations for the involute and transition curves.
For the transition curve, consider a point on the rounded tip of the rack cutter. The coordinates in the cutter coordinate system can be expressed as:
$$ x_0 = x_c + \rho_0 \cos \gamma, \quad y_0 = y_c – \rho_0 \sin \gamma $$
where $(x_c, y_c)$ is the center of the tool tip radius, $\rho_0$ is the radius of the tool tip, and $\gamma$ is the angle parameter. By applying the coordinate transformation to the gear coordinate system, the transition curve equation becomes:
$$ \begin{cases} x = r(\varphi – \gamma) \cos \varphi + (y_0 – r) \sin \varphi \\ y = r(\varphi – \gamma) \sin \varphi – (y_0 – r) \cos \varphi \end{cases} $$
Similarly, for the straight cutting edge, the involute curve can be derived as:
$$ \begin{cases} x = r \left[ \cos \varphi + (\varphi – \alpha) \sin \varphi \right] \\ y = r \left[ \sin \varphi – (\varphi – \alpha) \cos \varphi \right] \end{cases} $$
where $\alpha$ is the pressure angle, $\varphi$ is the rolling angle, and $r$ is the pitch circle radius. These equations allow me to generate precise tooth profile points in MATLAB. The herringbone gear is then created by mirroring a helical gear section about a central plane. In my modeling procedure, I first generate the 2D tooth profile points, then use a spline to create the profile curve, and finally sweep the profile along a helix to form the 3D tooth surface. The resulting gear models are imported into HyperMesh for meshing and into Abaqus for subsequent thermal and structural analyses.
| Parameter | Symbol | Pinion (Z1) | Gear (Z2) |
|---|---|---|---|
| Number of teeth | $Z$ | 31 | 102 |
| Normal module (mm) | $m_n$ | 4.5 | 4.5 |
| Normal pressure angle (°) | $\alpha_n$ | 20 | 20 |
| Helix angle (°) | $\beta$ | 25 | 25 |
| Face width (mm) | $B$ | 45 | 45 |
| Groove width (mm) | $B_1$ | 10 | 10 |
| Addendum modification coefficient | $x$ | 0 | 0 |
| Input torque (N·m) | $T$ | 4000 | — |
| Rotational speed (r/min) | $n$ | 5000 | 1519.6 |
3. Dynamic Contact and Thermal Analysis of High-Speed Herringbone Gears
The frictional heat generated at the tooth interface is one of the primary heat sources in high-speed gear transmission. To predict the temperature field of herringbone gears accurately, it is necessary to calculate the relative sliding velocity, contact pressure, friction coefficient, and the resulting frictional heat flux. In addition, the convective heat transfer coefficients on the gear surfaces must be determined. In this section, I describe the calculation procedure and present the numerical results for the studied gear pair.
3.1 Relative Sliding Velocity
For two involute gears in mesh, the instantaneous contact point moves along the line of action. The radius of curvature of each tooth profile at a given mesh point can be obtained from the involute geometry. For the driving gear, the radius of curvature at a point $k$ is:
$$ \rho_{1k} = r_{b1} \tan \alpha_{1k} = \frac{r_{b1} \sin \alpha_k}{\cos \beta_b} $$
where $r_{b1}$ is the base circle radius of the driving gear, $\alpha_{1k}$ is the pressure angle at point $k$, and $\beta_b$ is the base helix angle. Similarly, for the driven gear:
$$ \rho_{2k} = r_{b2} \tan \alpha_{2k} = \frac{r_{b2} \sin \alpha_k}{\cos \beta_b} $$
The combined radius of curvature is:
$$ \frac{1}{\rho_{12}} = \frac{1}{\rho_{1k}} + \frac{1}{\rho_{2k}} $$
The tangential velocities of the two tooth surfaces at the contact point are different due to the different radii of curvature. The relative sliding velocity $v_g$ is along the tangent direction and can be expressed as:
$$ v_g = v_{1k} – v_{2k} = \omega_1 \rho_{1k} – \omega_2 \rho_{2k} $$
Since the angular velocities $\omega_1$ and $\omega_2$ are proportional to the number of teeth, the relative sliding velocity reaches its maximum at the tooth tip and tooth root, and becomes zero at the pitch point. This behavior is critical because the frictional heat generation is proportional to the product of relative sliding velocity and contact pressure.
3.2 Contact Pressure and Load Sharing
The normal load on a meshing tooth pair varies with the mesh position. Because herringbone gears have a high overlap ratio, multiple tooth pairs may be in contact simultaneously. The total normal load can be derived from the transmitted torque and the base circle radius. In my analysis, I use the average line load $w_t$ per unit length of the contact line:
$$ w_t = \frac{F_n}{L_c} = \frac{T_2 / Z_2 Z_1}{B r_1 \cos \alpha_n \cos \beta} $$
where $T_2$ is the torque on the driven gear, $Z_1$ and $Z_2$ are the numbers of teeth, $B$ is the face width, $r_1$ is the pitch radius of the driving gear, $\alpha_n$ is the normal pressure angle, and $\beta$ is the helix angle. The contact line length $L_c$ changes during meshing due to the helical geometry. The average contact pressure $p_k$ at a particular mesh position is calculated using the Hertzian contact theory for equivalent cylinders:
$$ p_k = \sqrt{ \frac{w_t E’}{2 \pi \rho_{12} (1 – \nu^2)} } $$
where $E’$ is the equivalent elastic modulus and $\nu$ is Poisson’s ratio. The pressure distribution along the contact line is assumed to be elliptic in the local contact zone, but for thermal calculation, I use the average pressure over the Hertzian contact width.
3.3 Friction Coefficient
The friction coefficient between gear teeth depends on the lubricant viscosity, surface roughness, sliding velocity, and contact pressure. For oil-lubricated gears, a commonly used formula is:
$$ f = 0.12 \left( \frac{w_t R_a}{\eta_l v_g \rho_{12}} \right)^{0.25} $$
where $R_a$ is the average surface roughness, $\eta_l$ is the dynamic viscosity of the lubricant, and $v_g$ is the relative sliding velocity. In my study, I take a constant representative value of $f = 0.06$ based on typical operating conditions. This assumption is acceptable for the steady-state thermal analysis because the variation of the friction coefficient along the tooth profile is much smaller than the variation of the sliding velocity and contact pressure.
3.4 Frictional Heat Flux Distribution
The instantaneous frictional heat flux generated at the contact interface is:
$$ q_k = f \, p_k \, v_g \, \gamma $$
where $\gamma$ is the fraction of frictional energy converted into heat, which I take as 0.95. This heat is distributed between the two gears according to the thermal properties of the materials. The heat partition factor $\xi$ is:
$$ \xi = \frac{\sqrt{\lambda_1 c_1 \rho_1 v_1}}{\sqrt{\lambda_1 c_1 \rho_1 v_1} + \sqrt{\lambda_2 c_2 \rho_2 v_2}} $$
where $\lambda$ is thermal conductivity, $c$ is specific heat, $\rho$ is density, and $v$ is the surface velocity. The frictional heat flux on the driving gear is then:
$$ q_{1k} = \xi q_k, \quad q_{2k} = (1 – \xi) q_k $$
Since the contact at a given point lasts only during the short period when the tooth pair is in mesh, the average heat flux over one rotation cycle is obtained by multiplying the instantaneous heat flux by the contact duration ratio. The average heat flux is:
$$ Q_{1k} = q_{1k} \frac{\tau_1}{T_1}, \quad Q_{2k} = q_{2k} \frac{\tau_2}{T_2} $$
where $\tau_1$ and $\tau_2$ are the contact times and $T_1$ and $T_2$ are the rotational periods. Figure 3.5 shows the distribution of the average friction heat flux along the tooth profile for the pinion and gear. The heat flux has a maximum near the tooth root and tooth tip, and a local minimum at the pitch point. A sudden change occurs at the boundary between the single and double tooth pair contact zones because of the change in contact pressure.
| Property | Value |
|---|---|
| Elastic modulus (MPa) | $2.06 \times 10^5$ |
| Poisson’s ratio | 0.3 |
| Density (kg/m³) | 7850 |
| Specific heat (J/(kg·K)) | 452 |
| Thermal conductivity (W/(m·K)) | 48 |
| Thermal expansion coefficient (1/K) | $1.1 \times 10^{-5}$ |
3.5 Convective Heat Transfer Coefficients
The gear body exchanges heat with the surrounding lubricant or air through convection. The convective heat transfer coefficient depends on the flow regime, which is characterized by the local Reynolds number. For the gear face, the flow can be modeled as a rotating disk. The Reynolds number at a radius $r$ is:
$$ Re = \frac{\omega r^2}{\nu_f} $$
where $\omega$ is the angular velocity and $\nu_f$ is the kinematic viscosity of the lubricant. For the laminar flow regime, the convective heat transfer coefficient on the gear face is:
$$ h_s = 0.308 \left( m + 2 \right)^{0.5} \frac{\lambda}{r} Re^{0.5} Pr^{0.5} $$
where $m=2$ for a disk and $Pr$ is the Prandtl number. For the tooth face, the heat transfer is more complex because the lubricant is forced out of the tooth gaps. I use an empirical relation based on the total cooling and the local tooth height:
$$ h_s = \lambda \rho_f^{0.25} C_f^{0.75} \left( \frac{\omega}{H_c} \right)^{0.25} $$
where $C_f$ is the specific heat of the lubricant and $H_c$ is the local tooth height. For the tooth tip, the flow is approximated as flow over a flat plate:
$$ h_s = 0.664 \frac{\lambda}{L_t} Re^{0.5} Pr^{1/3} $$
where $L_t$ is the characteristic length. The calculated heat transfer coefficients increase with the rotational speed. At the rated speed, the tooth face heat transfer coefficient is higher for the pinion than for the gear, which is consistent with the higher surface velocity of the pinion.
| Property | Value |
|---|---|
| Kinematic viscosity (mm²/s) | 220 |
| Density (kg/m³) | 863.7 |
| Specific heat (J/(kg·K)) | 2129.87 |
| Temperature (K) | 338 |
| Thermal conductivity (W/(m·K)) | 0.14 |
4. Temperature Field Simulation
With the thermal boundary conditions calculated in the previous section, I perform a steady-state thermal analysis of the herringbone gear body. The heat conduction equation for a three-dimensional solid is:
$$ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} = 0 $$
For the steady-state condition, the temperature is time-independent. The boundary conditions on the tooth face combine a prescribed heat flux (frictional heat) and convective cooling. On the gear face and tooth tip, only convective cooling is present. I apply the heat flux on the meshing side of each tooth, and the convective heat transfer coefficients on all exposed surfaces.
In my finite element model, I use a segment of the herringbone gear with several teeth to reduce the computational cost while preserving the thermal interaction between adjacent teeth. The mesh consists of hexahedral elements with refined layers near the tooth surfaces to capture the high temperature gradients. The temperature field is solved using Abaqus heat transfer analysis. The results show that the maximum temperature on the pinion tooth surface reaches 134.5 °C, while the maximum temperature on the gear tooth surface is 98.9 °C. The temperature distribution along the tooth profile exhibits a double-peak pattern: the highest values occur near the tooth root and tooth tip, with a lower valley near the pitch point. This behavior directly reflects the distribution of the frictional heat flux. The temperature at the tooth core is lower than at the surface, and the thermal gradient is mainly concentrated in the near-surface layer.
To verify the correctness of the numerical results, I compare the average body temperature from the simulation with the value calculated using the ISO 6336 standard. The ISO-based body temperature is obtained from the integral temperature method, which uses empirical factors and the calculated flash temperature. The comparison shows an acceptable difference of less than 7 °C, which validates the thermal simulation approach. Compared with the standard formula, the finite element method provides more detailed temperature information, which is essential for determining the thermal deformation of the tooth profile.
| Item | ISO body temperature (°C) | Simulated maximum (°C) | Simulated average (°C) |
|---|---|---|---|
| Pinion | 109.6 | 134.5 | 104.2 |
| Gear | 94.7 | 98.9 | 87.6 |
5. Thermal-Mechanical Coupled Analysis
After obtaining the steady-state temperature field, I proceed with the thermal-mechanical coupled analysis. I use the sequential coupling method, in which the temperature field is applied as a predefined field to the structural model. The gear materials are modeled as linear elastic with temperature-independent properties. The finite element model includes contact between the tooth surfaces of the two gears. Contact constraints are defined using the surface-to-surface formulation with a friction coefficient of 0.06. The inner bore of each gear is coupled to a reference point, where the boundary conditions are applied. For the pinion, a torque of 4000 N·m is applied, while the gear has a rotational constraint. The thermal deformation is calculated by applying the temperature distribution from the thermal analysis.
The total deformation of the herringbone gear under the combined thermal and mechanical loads consists of the thermal expansion and the elastic deformation caused by the contact force. The maximum total deformation of the pinion is 28 µm, and that of the gear is 25 µm. The deformation pattern is dominated by the tangential component, which increases with the radius. The axial deformation also exists due to the helical tooth shape, but its magnitude is smaller. The presence of the thermal expansion makes the gear teeth slightly thicker, which can lead to a reduction of the backlash and even cause interference at certain mesh positions. This is particularly important at the entry and exit of meshing, where the teeth experience the largest sliding and load fluctuations.
The von Mises stress distribution in the herringbone gear during the mesh-in and mesh-out positions is shown below. When the gear enters mesh, the maximum von Mises stress is located at the point of contact near the tooth tip of the driven gear. When the pinion exits mesh, the maximum von Mises stress occurs near the tooth tip of the driving gear. The results demonstrate that the thermal load contributes significantly to the total stress. Compared with a purely mechanical analysis without thermal effects, the maximum von Mises stress under thermal-mechanical coupling is higher, and the stress concentration at the tooth tip is more pronounced. This suggests that ignoring thermal effects would underestimate the risk of tooth damage.
The contact pressure distribution on the tooth surfaces is also extracted at critical mesh positions. The maximum contact pressure occurs at the moment of entry into mesh for the gear and at the moment of exit for the pinion. The pinion has a slightly lower maximum contact pressure than the gear, which is consistent with the larger number of teeth on the gear and the corresponding smaller load per contact line. In addition, due to the thermal deformation, some tooth pairs may experience contact on both flanks simultaneously, which is a phenomenon that cannot be captured by a purely mechanical analysis. This finding emphasizes the need to include thermal effects when designing tooth modifications for herringbone gears.
6. Tooth Profile Modification Based on Thermal-Mechanical Coupled Analysis
The results of the thermal-mechanical coupled analysis provide a realistic basis for determining the optimum tooth profile modification. The main idea is to choose the modification parameters such that the combined elastic and thermal deformation is compensated, thereby reducing the meshing impact and avoiding premature contact at the tooth tip.
6.1 Determination of the Maximum Modification Amount
For a gear pair, the maximum modification amount on the pinion should be equal to the total deformation of the pinion and gear at the moment when the pinion exits mesh. Similarly, the maximum modification amount on the gear should be equal to the total deformation of the two gears at the moment when the gear enters mesh. From the finite element results, I extract the tangential deformations of the driving and driven teeth at these two critical positions. The maximum modification amounts are obtained as:
$$ \Delta_{\max,1} = \delta_{1,exit} + \delta_{2,exit} = 6.56 \ \mu m + 18.24 \ \mu m = 24.80 \ \mu m $$
$$ \Delta_{\max,2} = \delta_{1,entry} + \delta_{2,entry} = 4.72 \ \mu m + 17.15 \ \mu m = 21.87 \ \mu m $$
Thus, I set the maximum modification amount on the pinion to 24.80 µm and that on the gear to 21.87 µm.
6.2 Modification Length
For high-speed wind turbine herringbone gears, long profile modification is preferred. The modification starts at the lowest point of single-tooth-pair contact and extends to the tooth tip. This length corresponds to the distance along the line of action from the alternating contact point to the outer point of meshing. In the studied gear pair, the modification length on both gears is about 4.5 mm. This value is in agreement with the empirical formula $L = 0.6 m_n$ for a normal module of 4.5 mm, which gives 2.7 mm, but the longer modification length is chosen to better smooth the load transition under high speed.
6.3 Influence of the Modification Curve on the Temperature Field
I investigate the effect of different modification curves on the temperature field of the herringbone gears. The modification curves considered are $p=1$, $p=1.22$, $p=1.5$, and a special curve given by the equation discussed earlier. The finite element thermal analysis is repeated for each modified gear model. The results show that the maximum tooth surface temperature of the pinion decreases from 134.5 °C to about 127.5 °C for $p=1$, 120.9 °C for $p=1.22$, 117.6 °C for $p=1.5$, and 109.2 °C for the special curve. For the gear, the maximum temperature drops from 98.9 °C to 75.5 °C for the special curve. The special curve therefore provides the best reduction in tooth surface temperature among the tested curves. The reason is that this curve removes more material near the tooth tip, which is the region with the highest sliding velocity and frictional heat generation. Consequently, the contact pressure at the tip is reduced, leading to a lower heat flux and a cooler tooth surface.
| Modification curve | Pinion max. temp (°C) | Gear max. temp (°C) |
|---|---|---|
| No modification | 134.5 | 98.9 |
| $p=1$ | 127.5 | 94.8 |
| $p=1.22$ | 120.9 | 88.3 |
| $p=1.5$ | 117.6 | 84.1 |
| Special curve | 109.2 | 75.5 |
6.4 Influence of the Modification Curve on the Structural Field
The effect of the modification curve on the total deformation and stress is also evaluated. The maximum total deformation of the pinion drops from 28 µm to 20 µm for $p=1$, to 18 µm for $p=1.22$, to 16 µm for $p=1.5$, and to 13 µm for the special curve. The deformation of the gear drops correspondingly. The von Mises stress at the entry position decreases from 1004.8 MPa to 709.2 MPa with the special curve, which is a reduction of about 29.4%. At the exit position, the maximum von Mises stress decreases from 1279.6 MPa to 763.6 MPa, a reduction of about 40.3%. The tooth surface contact pressure also decreases significantly: the maximum contact pressure on the pinion drops by about 433 MPa, and that on the gear drops by about 586 MPa. These improvements demonstrate that the special modification curve is highly effective in reducing the stress concentration and improving the load carrying capacity of high-speed herringbone gears.
| Modification curve | Pinion deformation (µm) | Gear deformation (µm) | Max. von Mises at entry (MPa) | Max. von Mises at exit (MPa) |
|---|---|---|---|---|
| No modification | 28 | 25 | 1004.8 | 1279.6 |
| $p=1$ | 20 | 19 | 832.5 | 1012.3 |
| $p=1.22$ | 18 | 17 | 768.4 | 893.7 |
| $p=1.5$ | 16 | 15 | 745.2 | 812.5 |
| Special curve | 13 | 13 | 709.2 | 763.6 |
7. Conclusion and Future Work
In this research, I have systematically studied the thermal-mechanical coupled behaviour and tooth profile modification of high-speed herringbone gears used in wind turbine gearboxes. The main conclusions are summarized as follows:
First, the frictional heat flux on the tooth surfaces of herringbone gears is highest near the tooth root and tooth tip, and lowest near the pitch point, where the relative sliding velocity is zero. The heat flux distribution is consistent with the temperature distribution, which shows a double-peak pattern along the tooth profile. The average friction heat flux on the pinion is higher than that on the gear because of the higher rotational speed and more frequent meshing.
Second, the steady-state temperature field of the herringbone gear body has been successfully simulated using the finite element method. The maximum temperature of the pinion is 134.5 °C, while that of the gear is 98.9 °C. The simulated average body temperature is within 7 °C of the value predicted by the ISO 6336 method, which confirms the reliability of the thermal analysis.
Third, the thermal-mechanical coupled analysis reveals that the total deformation of the pinion is 28 µm and that of the gear is 25 µm under the rated torque. The thermal effect increases the tooth deformation and can cause additional flank contact, which would not be captured by a purely mechanical analysis. Therefore, thermal effects must be considered when designing tooth modifications for high-speed herringbone gears.
Fourth, a tooth profile modification strategy based on the thermal-mechanical coupled deformation has been proposed. The maximum modification amounts are determined from the coupled deformation at the mesh-in and mesh-out positions. The special modification curve with the equation $\Delta(x) = \Delta_{\max} [0.44(x/L) + 0.56(x/L)^2]$ provides the best performance among the tested curves. It reduces the maximum tooth surface temperature of the pinion by about 25 °C, the maximum von Mises stress at the entry by about 29.4%, and that at the exit by about 40.3%, while also reducing the maximum contact pressure significantly.
Future work should extend the analysis to the complete gearbox system, including the influence of the gearbox housing on heat dissipation and the effect of time-varying operating conditions. A fully coupled thermos-elastohydrodynamic model could provide more accurate predictions of the flash temperature and the local friction coefficient. In addition, experimental validation using thermocouple measurements or infrared thermography would be valuable to confirm the simulation results. Finally, an optimization framework could be developed to determine the optimal modification curve coefficients or even the full three-dimensional tooth surface modification for herringbone gears under varying wind loads.