In this thesis, I focus on the gluing characteristics of multi-modulus involute spur gears. Unlike conventional spur gears where the module and pressure angle of the two mating gears are equal, multi-modulus spur gears allow different modules and pressure angles on the pinion and gear, provided that the base pitches are equal. This condition enables a correct meshing even with unequal tooth proportions. My work establishes the no-backlash meshing equation, derives the calculation formulas for meshing angle, actual center distance, contact ratio, and then applies elastohydrodynamic lubrication (EHL) theory to compute the minimum oil film thickness along the line of action. I also perform finite element steady-state thermal analysis to evaluate the bulk temperature and gluing safety factor. The influence of modulus ratio, displacement coefficients, and torque on the gluing characteristics is systematically investigated. The results from thermal simulation are compared with the analytical film thickness ratios to validate the rationality of the EHL analysis.
1 Introduction
Gear transmissions are widely used in mechanical manufacturing due to their high efficiency and stable transmission. Among them, spur gears are the most fundamental type. In high-speed and heavy-load applications, tooth surface gluing (scuffing) is a common failure mode. Gluing occurs when the lubricating oil film between meshing tooth surfaces breaks down, leading to direct metal-to-metal contact and subsequent adhesion and tearing. This phenomenon is often accompanied by vibration and noise, and it severely affects the service life of spur gears. Therefore, it is crucial to study the gluing characteristics of spur gears, especially for advanced gear designs.
Most existing studies on gear gluing are based on gear pairs with equal module and pressure angle. However, a gear pair can also mesh correctly even if the modules and pressure angles are different, as long as the normal pitches are equal. Such a gear pair is called a multi-modulus spur gear pair. Research on gluing characteristics of multi-modulus spur gears is still limited. My work aims to fill this gap by combining analytical and numerical approaches.
A typical multi-modulus spur gear pair is shown in the following figure.
In this research, I use the minimum oil film thickness and the film thickness ratio as indicators of the gluing load capacity. The bulk temperature and gluing safety factor obtained from thermal simulation are used to verify the theoretical trends. The objective is to identify effective measures that can improve the lubrication performance and gluing resistance of multi-modulus spur gears.
2 Basic Parameters of Multi-modulus Spur Gears
For a multi-modulus spur gear pair, the pinion has module $m_1$ and pressure angle $\alpha_1$, while the gear has module $m_2$ and pressure angle $\alpha_2$. The correct meshing condition requires equal base pitch:
$$ m_1 \cos \alpha_1 = m_2 \cos \alpha_2 \tag{1} $$
I define the modulus ratio as $\delta_m = m_1/m_2$. This ratio is the key parameter that distinguishes multi-modulus spur gears from standard spur gears.
To ensure no backlash, I derive the meshing equation. The actual meshing angle $\alpha’$ can be obtained from the following equation:
$$ \mathrm{inv}\alpha’ = \frac{2(x_1 \tan\alpha_1 + x_2 \tan\alpha_2) + z_1 \mathrm{inv}\alpha_1 + z_2 \mathrm{inv}\alpha_2}{z_1 + z_2} \tag{2} $$
where $x_1$ and $x_2$ are the displacement coefficients of the pinion and gear, $z_1$ and $z_2$ are the numbers of teeth, and $\mathrm{inv}\alpha = \tan\alpha – \alpha$.
The actual center distance is:
$$ a’ = \frac{m_1 z_1 \cos\alpha_1}{2\cos\alpha’} + \frac{m_2 z_2 \cos\alpha_2}{2\cos\alpha’} \tag{3} $$
The contact ratio of the multi-modulus spur gear pair is calculated by:
$$ \varepsilon_\alpha = \frac{m_1 z_1 \cos\alpha_1 (\tan\alpha_{a1} – \tan\alpha’) + m_2 z_2 \cos\alpha_2 (\tan\alpha_{a2} – \tan\alpha’)}{2\pi m_1 \cos\alpha_1} \tag{4} $$
where $\alpha_{a1}$ and $\alpha_{a2}$ are the pressure angles at the addendum circles of the pinion and gear, respectively.
The displacement coefficients must satisfy several constraints to ensure proper tooth geometry and meshing. These include the tooth tip thickness condition, the non-interference condition, and the contact ratio condition. For example, the tooth tip thickness should not be less than $0.25m$ to avoid sharp tips. The non-interference condition prevents any overlap of the tooth profiles during meshing, while the contact ratio condition ensures that the contact ratio remains greater than or equal to 1.2 for smooth transmission. These constraints are used to select a reasonable range of displacement coefficients in my analysis.
Table 1 lists the basic parameters of the multi-modulus spur gear pair used in my study.
| Parameter | Value |
|---|---|
| Number of teeth $z_1/z_2$ | 23/30 |
| Face width $b$ (mm) | 20 |
| Input speed $n_1$ (r/min) | 2000 |
| Input torque $T_1$ (N·mm) | 400,000 |
| Elastic modulus $E$ (MPa) | 206,000 |
| Poisson’s ratio $\nu$ | 0.3 |
| Density $\rho$ (kg/m³) | 7850 |
| Specific heat $c$ (J/(kg·℃)) | 465 |
| Thermal conductivity $\lambda$ (W/(m·℃)) | 46 |
In the present work, the module and pressure angle of the gear are fixed at $m_2 = 4$ mm and $\alpha_2 = 20^\circ$, respectively. By varying the modulus ratio $\delta_m$, I determine the corresponding module and pressure angle of the pinion according to Equation (1). To avoid unrealistic tooth profiles, the modulus ratio is restricted to the range $0.98 \leq \delta_m \leq 1.03$. Table 2 presents the six combinations used in the parametric study.
| No. | $m_1$ (mm) | $\alpha_1$ (°) | $m_2$ (mm) | $\alpha_2$ (°) | $\delta_m$ |
|---|---|---|---|---|---|
| 1 | 3.92 | 16.49 | 4 | 20 | 0.98 |
| 2 | 3.96 | 18.34 | 4 | 20 | 0.99 |
| 3 | 4.00 | 20.00 | 4 | 20 | 1.00 |
| 4 | 4.04 | 21.50 | 4 | 20 | 1.01 |
| 5 | 4.08 | 22.89 | 4 | 20 | 1.02 |
| 6 | 4.12 | 24.17 | 4 | 20 | 1.03 |
3 Elastohydrodynamic Lubrication and Minimum Oil Film Thickness
In the EHL analysis of spur gears, the minimum oil film thickness emerges at each meshing point along the line of action. I use the line-contact EHL formula proposed by Yang and Wen, which is suitable for engineering applications. The formula is based on the Reynolds equation, the film thickness equation, the elastic deformation equation, and the viscosity-pressure relationship. For multi-modulus spur gears, the geometry and kinematics vary along the meshing line, so I must evaluate the relevant parameters at every point of contact.
First, I establish a dimensionless linear coordinate $\Gamma$ along the meshing line. The origin is at the pitch point, and the positive direction points toward the gear tip. For any meshing point, the coordinate is defined as:
$$ \Gamma = \frac{N_1 C – PN_1}{PN_1} = \frac{\tan\alpha_c}{\tan\alpha’} – 1 \tag{5} $$
where $\alpha_c$ is the pressure angle at the meshing point on the pinion. This coordinate completely describes the position along the meshing line, from the start of engagement to the end of engagement.
The comprehensive curvature radius $\rho_\Sigma$ at any meshing point is given by:
$$ \rho_\Sigma = \frac{\rho_1 \rho_2}{\rho_1 + \rho_2} \tag{6} $$
where $\rho_1 = r_1′ (1+\Gamma)\sin\alpha’$ and $\rho_2 = r_1′ (u – \Gamma)\sin\alpha’$ are the radii of curvature of the pinion and gear tooth surfaces at the meshing point, with $u = z_2/z_1$.
The average tangential velocity of the two tooth surfaces is:
$$ V_m = \frac{V_1 + V_2}{2} \tag{7} $$
where $V_1 = \omega_1 r_1′ (1+\Gamma)\sin\alpha’$ and $V_2 = \omega_1 r_1′ (u – \Gamma)\sin\alpha’ / u$. At the pitch point ($\Gamma = 0$), the two velocities are equal, and the teeth perform pure rolling, which generates almost no frictional heat.
The normal load per unit tooth width $W$ is determined by the load-sharing factor $X_\Gamma$:
$$ W = \frac{T_1}{r_1′ b \cos\alpha’} X_\Gamma \tag{8} $$
The load-sharing factor $X_\Gamma$ equals 1 in the single-tooth-pair contact zone and varies linearly from 1/3 to 1 in the double-tooth-pair contact zones. The transition causes a sudden change in the load, which directly affects the film thickness.
Finally, the minimum oil film thickness is calculated using the formula:
$$ h_{\min} = 6.76 \frac{(\alpha \eta_0 V_m)^{0.53} \rho_\Sigma^{0.41}}{E^{0.06} W^{0.16}} \tag{9} $$
where $\alpha$ is the pressure-viscosity coefficient and $\eta_0$ is the dynamic viscosity of the lubricant. In my analysis, I use $\alpha = 0.021 \ \mathrm{MPa}^{-1}$ and $\eta_0 = 2.88 \times 10^{-7} \ \mathrm{MPa \cdot s}$.
Using MATLAB, I computed the variation of these parameters along the meshing line. The comprehensive curvature radius increases from the start of engagement to a maximum near the pitch point and then decreases. The average tangential velocity increases linearly along the meshing line. The load fluctuates due to the transition between single and double tooth pair contact. The resulting minimum film thickness along the meshing line is minimal at the double-single transition where the load is high, and maximal near the end of engagement. This behavior is characteristic of spur gears with a contact ratio between 1 and 2, which is the case for the multi-modulus gear pairs considered here.
Table 3 lists the lubricant parameters used in the EHL and thermal analyses.
| Parameter | Value |
|---|---|
| Lubricant type | SCH632 |
| Kinematic viscosity $v_f$ (m²/s) | $92.5 \times 10^{-6}$ |
| Density $\rho_f$ (kg/m³) | 870 |
| Specific heat $c_f$ (J/(kg·℃)) | 2000 |
| Thermal conductivity $\lambda_f$ (W/(m·℃)) | 0.14 |
| Dynamic viscosity $\eta_0$ (MPa·s) | $2.88 \times 10^{-7}$ |
| Pressure-viscosity coefficient $\alpha$ (MPa⁻¹) | 0.021 |
4 Film Thickness Ratio and Gluing Characteristics
To link the oil film thickness to gluing load capacity, I introduce the film thickness ratio $\lambda$ at the pitch point, where the gear teeth perform pure rolling and the load is fully carried by one pair of teeth. It is defined as:
$$ \lambda = \frac{h_{\min-p}}{\sqrt{R_{a1}^2 + R_{a2}^2}} \tag{10} $$
where $h_{\min-p}$ is the minimum film thickness at the pitch point, and $R_{a1}$, $R_{a2}$ are the arithmetic mean surface roughness values of the pinion and gear tooth surfaces. In my calculations, I set $R_{a1}=R_{a2}=0.8\ \mu\mathrm{m}$.
The gluing risk is assessed as follows:
- If $\lambda > 3$, the contact is in full elastohydrodynamic lubrication, and gluing is unlikely.
- If $1 < \lambda < 3$, the contact is in mixed lubrication, with possible mild wear.
- If $\lambda < 1$, boundary lubrication occurs, and gluing is highly possible.
For a well-designed spur gear pair, it is recommended to keep $\lambda \geq 1.5$. In the following, I present the results of the parametric study on the pitch-point film thickness and film thickness ratio for multi-modulus spur gears.
4.1 Effect of Modulus Ratio
I first investigate the influence of the modulus ratio $\delta_m$ on the gluing characteristics. The pinion and gear both have zero displacement coefficients, and the input torque is set to $4.0 \times 10^5$ N·mm. Table 4 presents the pitch-point film thickness and film thickness ratio for different values of $\delta_m$.
| $\delta_m$ | $h_{\min-p}$ (μm) | $\lambda$ | Increase in $h$ (%) | Increase in $\lambda$ (%) |
|---|---|---|---|---|
| 0.98 | 1.893 | 1.67 | 0 | 0 |
| 0.99 | 1.979 | 1.75 | 4.54 | 4.79 |
| 1.00 | 2.067 | 1.83 | 9.19 | 9.58 |
| 1.01 | 2.156 | 1.91 | 13.89 | 14.37 |
| 1.02 | 2.247 | 1.99 | 18.70 | 19.16 |
| 1.03 | 2.336 | 2.07 | 23.40 | 23.95 |
It is clear that increasing the modulus ratio increases both the minimum film thickness and the film thickness ratio. This improvement is mainly due to the larger comprehensive curvature radius and the higher average tangential velocity that accompany a larger $\delta_m$. Therefore, adjusting the modulus ratio is an effective way to enhance the gluing capacity of multi-modulus spur gears.
4.2 Effect of Pinion Displacement Coefficient
Next, I study the influence of the pinion displacement coefficient $x_1$ while keeping $\delta_m = 1.01$, $x_2 = 0$, and $T_1 = 4.0 \times 10^5$ N·mm. Table 5 lists the results.
| $x_1$ | $h_{\min-p}$ (μm) | $\lambda$ | Increase in $h$ (%) | Increase in $\lambda$ (%) |
|---|---|---|---|---|
| -0.3 | 1.898 | 1.68 | 0 | 0 |
| -0.2 | 1.989 | 1.76 | 4.79 | 4.76 |
| -0.1 | 2.075 | 1.83 | 9.33 | 8.93 |
| 0.0 | 2.156 | 1.91 | 13.59 | 13.69 |
| 0.1 | 2.233 | 1.97 | 17.65 | 17.26 |
| 0.2 | 2.307 | 2.04 | 21.55 | 21.43 |
Increasing $x_1$ increases both $h_{\min-p}$ and $\lambda$. A positive profile shift on the pinion enlarges the equivalent curvature radius and improves the entraining velocity, which benefits the oil film formation. Thus, a positive displacement on the pinion can enhance the gluing resistance of multi-modulus spur gears.
4.3 Effect of Gear Displacement Coefficient
Similarly, I vary the gear displacement coefficient $x_2$ while keeping $x_1 = 0$ and $\delta_m = 1.01$. Table 6 shows the results.
| $x_2$ | $h_{\min-p}$ (μm) | $\lambda$ | Increase in $h$ (%) | Increase in $\lambda$ (%) |
|---|---|---|---|---|
| -0.3 | 1.919 | 1.70 | 0 | 0 |
| -0.2 | 2.003 | 1.77 | 4.38 | 4.12 |
| -0.1 | 2.081 | 1.84 | 8.44 | 8.24 |
| 0.0 | 2.156 | 1.91 | 12.35 | 12.35 |
| 0.1 | 2.227 | 1.97 | 16.05 | 15.88 |
| 0.2 | 2.296 | 2.03 | 19.65 | 19.41 |
The trend is the same as for the pinion displacement: increasing $x_2$ improves the film thickness and the film thickness ratio. Therefore, both pinion and gear profile shifts can be used to optimize the lubrication condition of multi-modulus spur gears.
4.4 Effect of Input Torque
Finally, I examine the influence of the input torque $T_1$ on the gluing capacity. For this parameter, I set $\delta_m = 1.01$, $x_1 = x_2 = 0$, and vary the torque from $6.5 \times 10^5$ N·mm to $4.0 \times 10^5$ N·mm. Table 7 displays the results.
| $T_1$ ($10^5$ N·mm) | $h_{\min-p}$ (μm) | $\lambda$ | Increase in $h$ (%) | Increase in $\lambda$ (%) |
|---|---|---|---|---|
| 6.5 | 1.995 | 1.76 | 0 | 0 |
| 6.0 | 2.021 | 1.79 | 1.30 | 1.71 |
| 5.5 | 2.049 | 1.81 | 2.71 | 2.84 |
| 5.0 | 2.081 | 1.84 | 4.31 | 4.55 |
| 4.5 | 2.116 | 1.87 | 6.07 | 6.25 |
| 4.0 | 2.156 | 1.91 | 8.07 | 8.52 |
Lowering the torque increases the film thickness because the normal load per unit width is reduced. This reduction in load leads to a thicker oil film and a higher film thickness ratio, which is beneficial for avoiding gluing. In practice, reducing the transmitted torque may not always be possible, but the result indicates that overloaded spur gears are more prone to scuffing failure.
5 Finite Element Thermal Analysis
To verify the analytical results, I perform a steady-state thermal analysis of the multi-modulus spur gear body. The gear bulk temperature is a critical factor that influences the lubricant viscosity and the stability of the oil film. If the bulk temperature becomes too high, the oil film may rupture, leading to gluing. Therefore, I use the finite element method to solve the steady-state heat conduction equation for the gear body:
$$ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} = 0 \tag{11} $$
Boundary conditions are defined for the tooth surfaces. The meshing surface receives a heat flux generated by friction, while the other surfaces exchange heat with the ambient environment through convection. The convection coefficients for the tooth flank, end face, and tooth gap are determined from empirical formulas. For the tooth flank, the convection coefficient is expressed as:
$$ h_m = \frac{\lambda_f}{d’} \cdot C \cdot Re_f^{0.731} Pr_f^{0.333} \tag{12} $$
where $d’$ is the pitch diameter, $Re_f$ is the Reynolds number based on the pitch line velocity, and $Pr_f$ is the Prandtl number of the lubricant. The end face convection is approximated by the rotating disk model, and the tooth gap convection uses a simple laminar-flow expression.
The average frictional heat flux on the tooth flank is calculated from the contact stress and the relative sliding velocity:
$$ Q = \beta \cdot f \cdot \sigma_h \cdot V_c \tag{13} $$
where $f$ is the friction coefficient (taken as 0.06 in this work), $\sigma_h$ is the average contact stress according to Hertzian theory, $V_c$ is the relative sliding velocity, and $\beta$ is the heat partition factor between the pinion and gear. The heat partition factor depends on the thermal properties of the two gears. Since the pinion has fewer teeth, it meshes more often and thus carries a higher average heat flux. This is why the pinion generally experiences higher temperatures than the gear in a spur gear pair.
The finite element model is built from a single tooth due to the cyclic symmetry of spur gears. The meshing surface is subdivided into small rectangular regions, and the mean heat flux for each region is applied as a boundary condition. Figure 3 in the original thesis (not shown here) illustrates the meshed single-tooth model. In this paper, I use ANSYS Workbench to perform the steady-state thermal analysis. The gear material is steel with a thermal conductivity of 46 W/(m·℃) and a specific heat of 465 J/(kg·℃). The ambient temperature is set to the oil temperature (60℃).
From the thermal simulation, I obtain the maximum bulk temperature of the pinion for various parameters. The gluing safety factor $S_{\mathrm{intS}}$ is defined according to ISO standards:
$$ S_{\mathrm{intS}} = \frac{\theta_{\mathrm{intS}}}{\theta_{\mathrm{int}}} \tag{14} $$
where $\theta_{\mathrm{intS}}$ is the integral temperature limit and $\theta_{\mathrm{int}}$ is the gear integral temperature calculated from the bulk temperature and the mean flash temperature. A value of $S_{\mathrm{intS}} > 1$ indicates a low risk of gluing.
5.1 Effect of Modulus Ratio on Temperature Field
Table 8 compares the thermal simulation results with the ISO theoretical values for different modulus ratios. The displacements are zero and the torque is $4.0 \times 10^5$ N·mm.
| $\delta_m$ | ISO bulk temp (°C) | Sim. max temp (°C) | Sim. mean temp (°C) | $S_{\mathrm{intS}}$ | $\lambda$ |
|---|---|---|---|---|---|
| 0.98 | 75.42 | 78.58 | 70.05 | 1.73 | 1.67 |
| 0.99 | 74.47 | 76.48 | 68.95 | 1.79 | 1.75 |
| 1.00 | 73.75 | 74.82 | 68.06 | 1.84 | 1.83 |
| 1.01 | 73.25 | 73.10 | 67.14 | 1.89 | 1.91 |
| 1.02 | 72.95 | 72.83 | 66.96 | 1.91 | 1.99 |
| 1.03 | 72.83 | 72.74 | 66.87 | 1.92 | 2.07 |
The maximum temperature decreases as the modulus ratio increases. The gluing safety factor increases, which is consistent with the increase of the film thickness ratio from the EHL analysis. The agreement between the simulated and ISO temperatures is satisfactory, with a maximum deviation of about 4.2% for the maximum temperature and 8.5% for the mean temperature.
5.2 Effect of Pinion Displacement Coefficient on Temperature Field
Table 9 shows the results for different pinion displacement coefficients $x_1$ with $\delta_m = 1.01$, $x_2 = 0$, and $T_1 = 4.0 \times 10^5$ N·mm.
| $x_1$ | ISO bulk temp (°C) | Sim. max temp (°C) | Sim. mean temp (°C) | $S_{\mathrm{intS}}$ | $\lambda$ |
|---|---|---|---|---|---|
| -0.3 | 75.25 | 77.61 | 69.54 | 1.75 | 1.68 |
| -0.2 | 74.34 | 75.61 | 68.47 | 1.81 | 1.76 |
| -0.1 | 73.69 | 74.23 | 67.74 | 1.85 | 1.83 |
| 0.0 | 73.25 | 73.10 | 67.14 | 1.89 | 1.91 |
| 0.1 | 72.99 | 72.84 | 66.97 | 1.90 | 1.97 |
| 0.2 | 72.89 | 72.76 | 66.90 | 1.91 | 2.04 |
Again, the safety factor grows with increasing $x_1$, matching the film thickness ratio trend. A positive pinion displacement not only improves the lubrication but also reduces the gear body temperature, which further lowers the gluing risk.
5.3 Effect of Gear Displacement Coefficient on Temperature Field
Table 10 shows the results for different gear displacement coefficients $x_2$ with $\delta_m = 1.01$, $x_1 = 0$, and $T_1 = 4.0 \times 10^5$ N·mm.
| $x_2$ | ISO bulk temp (°C) | Sim. max temp (°C) | Sim. mean temp (°C) | $S_{\mathrm{intS}}$ | $\lambda$ |
|---|---|---|---|---|---|
| -0.3 | 76.20 | 74.61 | 67.92 | 1.73 | 1.70 |
| -0.2 | 74.95 | 73.01 | 67.61 | 1.79 | 1.77 |
| -0.1 | 73.99 | 73.44 | 67.31 | 1.84 | 1.84 |
| 0.0 | 73.25 | 73.10 | 67.14 | 1.89 | 1.91 |
| 0.1 | 72.69 | 72.97 | 67.07 | 1.92 | 1.97 |
| 0.2 | 72.28 | 72.90 | 66.02 | 1.94 | 2.03 |
Increasing $x_2$ also reduces the bulk temperature and increases the safety factor. The effect is slightly weaker than that of $x_1$, but still evident.
5.4 Effect of Torque on Temperature Field
Table 11 shows the thermal results for different input torques with $\delta_m = 1.01$ and no profile shift.
| $T_1$ ($10^5$ N·mm) | ISO bulk temp (°C) | Sim. max temp (°C) | Sim. mean temp (°C) | $S_{\mathrm{intS}}$ | $\lambda$ |
|---|---|---|---|---|---|
| 6.5 | 79.07 | 81.28 | 71.60 | 1.56 | 1.76 |
| 6.0 | 77.96 | 79.64 | 70.71 | 1.61 | 1.79 |
| 5.5 | 76.83 | 78.00 | 69.81 | 1.67 | 1.81 |
| 5.0 | 75.67 | 76.37 | 68.92 | 1.74 | 1.84 |
| 4.5 | 74.48 | 74.73 | 68.03 | 1.81 | 1.87 |
| 4.0 | 73.25 | 73.10 | 67.14 | 1.89 | 1.91 |
The safety factor increases with decreasing torque, which aligns with the film thickness ratio behavior. The thermal simulation confirms that reducing the torque is an effective way to avoid scuffing, although this might not always be practically feasible.
6 Comparison and Discussion
The finite element thermal results and the EHL analytical results show consistent trends. In all parameter studies, an increase in the film thickness ratio $\lambda$ corresponds to an increase in the gluing safety factor $S_{\mathrm{intS}}$. For example, when the modulus ratio increases from 0.98 to 1.03, $\lambda$ rises by about 24%, while $S_{\mathrm{intS}}$ rises by about 11%. Similarly, increasing the pinion displacement from -0.3 to 0.2 raises $\lambda$ by about 21% and $S_{\mathrm{intS}}$ by about 9%. The torque reduction from $6.5 \times 10^5$ to $4.0 \times 10^5$ N·mm increases $\lambda$ by about 8.5% and $S_{\mathrm{intS}}$ by about 21%. The quantitative difference between the two indicators is expected because they represent different physical quantities: $\lambda$ directly measures the oil film thickness relative to surface roughness, while $S_{\mathrm{intS}}$ combines the bulk temperature and flash temperature. Nevertheless, both indicators point to the same conclusion: enhancing the film thickness and reducing the thermal load will improve the gluing resistance of multi-modulus spur gears.
The reason for the beneficial effect of a larger modulus ratio lies in the geometry of the teeth. A larger $\delta_m$ increases the equivalent curvature radius of the tooth surfaces and the entraining velocity, both of which promote thicker oil films. A positive displacement coefficient similarly increases the curvature radius and shifts the contact point away from the unfavorable root region. On the other hand, a lower torque reduces the contact load, which directly improves the film thickness according to the EHL formula. These insights are valuable for the design of multi-modulus spur gears, especially in applications where high power density and reliability are required.
7 Conclusion
In this thesis, I have systematically analyzed the gluing characteristics of multi-modulus involute spur gears based on elastohydrodynamic lubrication theory and finite element thermal simulation. The main conclusions are as follows:
- I derived the no-backlash meshing equation and the calculation formulas for the meshing angle, actual center distance, and contact ratio of multi-modulus spur gears with unequal modules and pressure angles. These formulas are essential for the subsequent analysis.
- I established a linear coordinate system along the meshing line and calculated the minimum oil film thickness using the line-contact EHL formula. The film thickness varies along the meshing line and is strongly affected by the load-sharing transitions. The pitch-point film thickness serves as a representative value for evaluating the lubrication condition.
- Increasing the modulus ratio $\delta_m$ from 0.98 to 1.03 increases the pitch-point film thickness by about 23.5% and the film thickness ratio by about 24%, thereby improving the gluing load capacity of multi-modulus spur gears.
- Increasing the pinion displacement coefficient $x_1$ from -0.3 to 0.2 increases the film thickness ratio by about 21.4%, while increasing the gear displacement coefficient $x_2$ over the same range increases it by about 19.4%. Positive profile shifts are beneficial for improving lubrication and reducing the gluing risk.
- Reducing the input torque from $6.5 \times 10^5$ to $4.0 \times 10^5$ N·mm increases the film thickness ratio by about 8.5%. Lower loads favor film formation and reduce the bulk temperature, which are both beneficial for avoiding gluing.
- The finite element thermal simulations show that the bulk temperature of the gear body decreases with increasing modulus ratio, displacement coefficients, and decreasing torque. The gluing safety factor increases accordingly, which is consistent with the EHL predictions. The agreement between the thermal simulation and the analytical film thickness approach validates the reliability of the analytical model.
The results provide a useful reference for the design of multi-modulus spur gear pairs to prevent scuffing and improve lubrication performance. Future work may include experimental validation and consideration of dynamic effects, gear modification, and thermal elastohydrodynamic lubrication models.