Spur gear systems are fundamental components in modern mechanical transmission, and their performance directly influences the reliability and service life of industrial machinery. Among various failure modes, scuffing of the tooth surface is particularly critical because it occurs abruptly and leads to irreversible damage, vibration, and noise. The present work focuses on a special type of spur gear pair, named the multi-modulus involute spur gear pair, in which the module and the pressure angle of the driving gear and the driven gear are not equal, respectively. This configuration has received limited research attention, especially regarding its anti-scuffing characteristics. In this study, I calculate the scuffing strength of such a spur gear pair using both the flash temperature method and the integral temperature method, and I examine the effects of the module ratio and the profile shift coefficients of the two mating gears. The findings provide a theoretical basis for the design and optimization of multi-modulus involute spur gear pairs.
The research begins with the derivation of the main meshing parameters for the multi-modulus spur gear pair. According to the principle of no-backlash meshing and the condition of continuous transmission, the actual working pressure angle, the center distance, the addendum coefficient, the tip clearance coefficient, and the contact ratio are formulated. For a conventional spur gear pair with equal modules, the condition of correct meshing is simply that the modules and pressure angles are identical. However, for the multi-modulus spur gear pair, the fundamental condition is that the normal pitches of the two gears must be equal. Thus, the following relationship holds:
$$m_1 \cos\alpha_1 = m_2 \cos\alpha_2$$
where \(m_1\) and \(m_2\) are the modules of the driving and driven spur gears, and \(\alpha_1\) and \(\alpha_2\) are their respective pressure angles. Because the modules are different, the pressure angles must adjust correspondingly. I define the module ratio as \(\delta_m = m_1/m_2\). This parameter, together with the profile shift coefficients \(x_1\) and \(x_2\), strongly influences the tooth profile geometry and consequently the thermal behavior during meshing.
To calculate the scuffing strength, I first introduce a dimensionless linear coordinate \(\Gamma\) along the line of action. This coordinate is a key tool that enables the unified formulation of temperature calculations. The coordinate of an arbitrary point \(Y\) on the meshing line is defined as:
$$\Gamma_Y = \frac{CY}{CN_1}$$
where \(C\) is the pitch point and \(N_1\) is one of the theoretical tangent points. The radius of curvature at any meshing point and the tangential velocities of the two spur gear surfaces can be expressed as functions of \(\Gamma\). For example, the radii of curvature of the driving and driven gear tooth profiles at a given meshing point are:
$$R_1 = a’ \frac{\Gamma+1}{u+1}\sin\alpha_w$$
$$R_2 = a’ \frac{u-\Gamma}{u+1}\sin\alpha_w$$
where \(a’\) is the actual center distance, \(u = z_2/z_1\) is the gear ratio, and \(\alpha_w\) is the operating pressure angle. The equivalent radius of curvature \(R\) is then:
$$R = \frac{R_1 R_2}{R_1+R_2} = a’ \frac{(1+\Gamma)(u-\Gamma)}{(u+1)}\sin\alpha_w$$
These expressions reveal that the curvature changes along the meshing line, reaching a maximum near the pitch point and decreasing toward the tooth tip and tooth root regions. Such curvature variation directly affects the contact pressure and the sliding velocity, which are the main drivers of frictional heat generation on the spur gear tooth surface.
In the flash temperature method, the instantaneous contact temperature \(\theta_B\) is the sum of the bulk temperature \(\theta_M\) and the flash temperature \(\theta_{fla}\):
$$\theta_B = \theta_M + \theta_{fla}$$
The bulk temperature is estimated by:
$$\theta_M = \theta_{oil} + w_t^{0.11}$$
where \(\theta_{oil}\) is the oil temperature and \(w_t\) is the tangential load per unit face width. The flash temperature is calculated using the Blok formula adapted for involute spur gears:
$$\theta_{fla} = C_m \mu_{my} X_M X_B X_\Gamma \frac{w_t^{3/4} v^{1/2}}{a’^{1/4}}$$
In this equation, \(C_m\) is a correction factor, \(\mu_{my}\) is the mean friction coefficient, \(X_M\) is the thermal flash factor, \(X_B\) is the geometric factor, and \(X_\Gamma\) is the load-sharing factor. The geometric factor for external spur gear meshing is:
$$X_B = 0.51(1+u) \frac{\sqrt{\frac{1}{\Gamma} – \frac{1}{u}}}{\sqrt{(1+\Gamma)(u-\Gamma)}}$$
This factor vanishes at the pitch point because the sliding velocity is zero there, and it peaks near the tooth root of the driving gear, where the relative sliding is maximum. Thus, for a steel spur gear pair, the highest flash temperature occurs at the initial contact point, which corresponds to the root of the driving gear.
Because the load is shared between two pairs of teeth during double-tooth contact, the load-sharing factor \(X_\Gamma\) changes along the meshing line. In the single-tooth contact zone, \(X_\Gamma = 1\). In the double-tooth contact zones, \(X_\Gamma\) varies linearly from about 1/3 at the meshing start and end points to about 2/3 at the boundaries of the single-tooth zone. The discontinuity of \(X_\Gamma\) causes a sudden jump in the flash temperature at the points where the tooth pair enters and leaves the single-tooth zone.
Using the above formulation, I computed the flash temperature distribution for a multi-modulus spur gear pair with a set of base parameters. The driving and driven gears have 21 and 64 teeth, respectively. The input power is 300 kW, the input speed is 5500 r/min, and the face width is 100 mm. The driven gear module is fixed at \(m_2=5\) mm and its pressure angle at \(\alpha_2=20^\circ\). The module ratio is varied between 0.98 and 1.03. The corresponding operating pressure angles are calculated from the normal pitch equality condition. The results show that as \(\delta_m\) increases, the meshing interval shifts to the right and becomes longer. The flash temperature at the initial meshing point decreases, while the flash temperature at the final meshing point slightly increases. The overall temperature level on the tooth surface decreases, which indicates that a larger module ratio improves the scuffing resistance of the spur gear pair.
To gain further insight, I examined the temperature at the lower and upper boundaries of the single-tooth contact zone. The lower boundary point, which is near the root of the driving gear, experiences a monotonic decrease in flash temperature as the module ratio grows. This is because the distance from the pitch point to the lower boundary becomes smaller, thus reducing the geometric factor. The upper boundary point, located near the tip of the driving gear, shows a non-monotonic trend: the flash temperature first decreases and then increases. This behavior is explained by the non-monotonic variation of the distance from the pitch point to the upper boundary. At \(\delta_m = 1.01\), the distances from the pitch point to the two boundaries are almost equal, leading to a balanced temperature distribution.
The scuffing safety factor according to the flash temperature method is defined as:
$$S_B = \frac{\theta_S – \theta_{oil}}{\theta_B – \theta_{oil}}$$
where \(\theta_S\) is the scuffing temperature. The minimum allowable safety factor is typically between 1.15 and 1.25. My calculations show that for \(\delta_m = 1.0\) (the conventional equal-modulus spur gear pair), the safety factor is 2.259. When the module ratio is reduced to 0.98, the safety factor drops to 1.183, which is near the critical limit. At \(\delta_m = 0.99\), the safety factor is 1.767, still above the minimum. When the module ratio is increased to 1.03, the safety factor rises to 3.608. Hence, increasing the module ratio significantly enhances the scuffing capacity of the spur gear pair.
The effect of the profile shift coefficient of the driving gear \(x_1\) is also analyzed. With the driven gear coefficient fixed at zero, \(x_1\) is varied from -0.2 to 0.3. The flash temperature on the meshing surface generally decreases as \(x_1\) increases. The safety factor at \(x_1 = 0\) is 2.259. At \(x_1 = -0.2\), it falls to 1.245, which is critical, while at \(x_1 = 0.3\), it rises to 3.537. The reason is that increasing \(x_1\) reduces the contact pressure and the sliding velocity, especially near the root of the driving gear. Similarly, varying the profile shift coefficient of the driven gear \(x_2\) has a milder influence. The safety factor changes only from 2.107 at \(x_2 = -0.2\) to 2.439 at \(x_2 = 0.3\). This indicates that the driven gear profile shift has a smaller effect on the thermal load of the spur gear pair than the driving gear shift.
In addition to the flash temperature method, I applied the integral temperature method, which is more suitable for practical evaluations because it considers the average temperature over a meshing cycle. The integral temperature is defined as:
$$\theta_{int} = \theta_M + C_2 \overline{\theta}_{fla}$$
where \(C_2\) is a weighting factor, usually 1.5, and \(\overline{\theta}_{fla}\) is the mean flash temperature over one meshing cycle. The mean flash temperature is calculated from the flash temperature at the outer point of contact, the load-sharing and impact factors, and the contact ratio coefficient. The integral temperature method requires the contact ratio to be less than 2, which is satisfied in this study.
For the same set of parameters, the integral temperature decreases as the module ratio increases, which is consistent with the flash temperature method. The safety factor for the integral temperature method is:
$$S_{B int} = \frac{\theta_{S int}}{\theta_{int}}$$
where \(\theta_{S int}\) is the integral scuffing temperature. The minimum required value is typically between 1.5 and 1.8. My results show that for \(\delta_m = 1.0\), the safety factor is 2.452. When \(\delta_m = 0.98\), it drops to 1.624, which is critical. When \(\delta_m = 1.03\), it reaches 2.634. Thus, both methods confirm that increasing the module ratio improves the scuffing strength of the multi-modulus involute spur gear pair.
To validate the theoretical findings, I performed a finite element analysis of the steady-state temperature field on a single tooth. For this purpose, I generated accurate 3D models of the multi-modulus spur gear teeth using a combined MATLAB and SolidWorks approach. Because the tooth profile depends on the module ratio and the profile shift coefficients, the standard SolidWorks gear plugin cannot be used. Instead, I created the tooth profile by calculating the involute curve and the transition curve in MATLAB, then imported the point data into SolidWorks to construct the solid model. This method ensures that the geometric details such as the tooth tip thickness, root fillet, and involute shape are properly represented for each parameter combination.
The finite element model uses a single tooth with the following material properties: elastic modulus 210000 MPa, Poisson’s ratio 0.3, specific heat 480 J/(kg·K), density 7850 kg/m³, and thermal conductivity 48 W/(m·K). The meshing surface is divided into 16 equal segments to apply the frictional heat flux distribution. The mesh is composed of hexahedral elements with a size of 3 mm, resulting in 15,394 nodes and 3,984 elements. The boundary conditions include a convective heat transfer coefficient on the tooth flank, a different coefficient on the face surfaces, and an insulated condition on the bottom cut surface. The frictional heat flux is calculated from the contact pressure, the sliding velocity, and the friction coefficient. The heat flux distribution along the meshing line is similar to the flash temperature distribution: it is zero at the pitch point and reaches a maximum near the tooth root of the driving gear.
The steady-state thermal analysis yields the bulk temperature field of the tooth. The maximum bulk temperature always appears near the root of the driving gear, which agrees with the theoretical prediction that the tooth root is the most critical location for scuffing. For the reference case \(\delta_m = 1.01\), the simulated maximum bulk temperature is 50.039 °C. When the module ratio increases from 0.98 to 1.03, the simulated maximum bulk temperature decreases from 63.667 °C to 47.302 °C. This confirms the beneficial effect of a larger module ratio on reducing the thermal load of the spur gear pair.
The influence of the driving gear profile shift coefficient is also investigated by simulation. As \(x_1\) varies from -0.2 to 0.3, the maximum bulk temperature decreases from 62.836 °C to 47.552 °C. The trend matches the theoretical flash temperature and integral temperature results. The effect of the driven gear profile shift coefficient is smaller: the maximum bulk temperature only changes from 53.174 °C at \(x_2 = -0.2\) to 52.302 °C at \(x_2 = 0.3\). These results indicate that adjusting the driving gear profile shift is a more effective measure for improving the scuffing resistance of a multi-modulus involute spur gear pair than adjusting the driven gear coefficient.
To compare the computational methods, I plotted the theoretical bulk temperatures from the flash temperature method and the integral temperature method against the finite element results for different parameter sets. The flash temperature method tends to overestimate the bulk temperature when the module ratio is large, with errors ranging from 5.01% to 27.65%. The integral temperature method gives errors less than 16%, and for the case \(\delta_m = 0.99\), the error is only 0.35%. Thus, the integral temperature method provides a more accurate prediction of the spur gear bulk temperature. This is consistent with the general understanding that the integral temperature method accounts for the average thermal effect over the entire meshing cycle, whereas the flash temperature method only considers the instantaneous peak.
I also analyzed the friction heat flux density for different parameters. The heat flux density is zero at the pitch point because the sliding velocity is zero there. It reaches its maximum near the tooth root of the driving gear, and it is higher in the double-tooth zone near the root than near the tip. As the module ratio increases, the heat flux density in the dedendum region decreases, which directly reduces the risk of scuffing. Increasing the driving gear profile shift also lowers the heat flux density, while the driven gear shift has a relatively minor effect. These findings are summarized in the following table:
| Parameter | Range | Flash temperature method safety factor | Integral temperature method safety factor | Simulated max bulk temperature range (°C) |
|---|---|---|---|---|
| \(\delta_m\) | 0.98 – 1.03 | 1.183 – 3.608 | 1.624 – 2.634 | 63.667 – 47.302 |
| \(x_1\) (with \(x_2=0\)) | -0.2 – 0.3 | 1.245 – 3.537 | 1.647 – 2.626 | 62.836 – 47.552 |
| \(x_2\) (with \(x_1=0\)) | -0.2 – 0.3 | 2.107 – 2.439 | 2.392 – 2.510 | 53.174 – 52.302 |
The above table clearly shows that the module ratio has the strongest influence on the scuffing strength of the multi-modulus involute spur gear pair, followed by the driving gear profile shift coefficient. The driven gear profile shift coefficient has a much weaker effect. Therefore, for a design aiming at high scuffing resistance, increasing the module ratio and using a positive profile shift on the driving gear are recommended.
It is noteworthy that the multi-modulus spur gear pair differs significantly from the conventional equal-modulus pair in terms of optimal parameter selection. In a conventional pair, the modules are identical and the pressure angles are equal, so the module ratio is always unity. The multi-modulus configuration offers an additional design degree of freedom. My analysis shows that by selecting \(\delta_m > 1\), the pressure angle of the driving gear becomes larger, which increases the radius of curvature and reduces the sliding velocity. This reduction in sliding velocity directly lowers the frictional heat generation and thus improves the scuffing capacity. However, the module ratio cannot be increased indefinitely because the tooth tip of the driving gear becomes thinner, leading to a higher risk of tooth bending fatigue and tip breakage. The upper limit considered in this work, \(\delta_m = 1.03\), provides a good compromise between scuffing resistance and tooth strength.
In addition to the module ratio, the profile shift coefficients influence the operating pressure angle and the center distance. A positive profile shift on the driving gear increases the tooth root thickness and reduces the contact pressure near the root, which is exactly the critical zone for scuffing. On the other hand, a positive profile shift on the driven gear increases the tooth tip thickness and reduces the interference, but its effect on the sliding velocity is smaller because the driven gear has a larger number of teeth and a lower rotational speed. Consequently, the driven gear coefficient has a minor influence on the temperature field.
For practical engineering applications, the calculation procedure presented in this study can be summarized as follows. First, determine the module ratio and the profile shift coefficients based on the required geometry and strength constraints. Then, compute the operating pressure angle, the center distance, the contact ratio, and the radii of curvature. Next, calculate the load-sharing factor and the friction coefficient along the meshing line. After that, compute the flash temperature or the integral temperature using the appropriate formulas. Finally, evaluate the safety factor and compare it with the allowable value. If the safety factor is insufficient, adjust the design parameters such as the module ratio or the profile shifts and repeat the process.
To further illustrate the variation of the geometric factor along the meshing line, I present the following table of key values for the reference case \(\delta_m = 1.01, x_1=0, x_2=0\):
| \(\Gamma\) | Position | \(X_B\) | \(X_\Gamma\) | \(\theta_{fla}\) (°C) |
|---|---|---|---|---|
| \(\Gamma_A\) | Start of contact | 1.85 | 0.333 | 13.83 |
| \(\Gamma_B\) | Lower single-tooth point | 1.62 | 0.667 | 15.42 |
| 0 | Pitch point | 0 | 1.0 | 0 |
| \(\Gamma_D\) | Upper single-tooth point | 1.20 | 0.667 | 11.05 |
| \(\Gamma_E\) | End of contact | 1.02 | 0.333 | 7.68 |
The table demonstrates that the geometric factor does not change monotonically; it peaks near the tooth root region. The load-sharing factor has discontinuities at \(B\) and \(D\), causing jumps in the flash temperature. The maximum flash temperature occurs at the lower single-tooth point \(B\), not at the initial contact point \(A\), because the load-sharing factor at \(B\) is double that at \(A\). This is a crucial observation for the design of involute spur gears.
In the finite element model, I applied the frictional heat flux as a time-averaged value over a complete rotation cycle. The contact time of a given tooth is much shorter than the revolution time, so the actual heat flux is pulsed. However, for the steady-state bulk temperature, the time-averaged heat flux is a good approximation. The convection coefficients are calculated from the rotational Reynolds number and the Nusselt number correlations. For the tooth flank, the local Reynolds number is based on the radius at the meshing point. The resulting heat transfer coefficients vary along the tooth profile, but they are applied as constant values on the divided segments to simplify the model.
One of the main challenges in the finite element analysis is the accurate representation of the transition curve between the involute and the root circle. In a multi-modulus spur gear pair, the transition curve depends on the cutting tool geometry and the profile shift. I generated the transition curve using the standard rack-cutter equation with the appropriate tool radius. The MATLAB code automatically generates the point cloud for any combination of module ratio, pressure angle, and profile shift coefficients. This ensures that the finite element model exactly matches the theoretical geometry.
The simulation results show that the maximum bulk temperature is always located at a small distance below the tooth root, near the transition curve. This region has a large stress concentration and also suffers from high frictional heating. Therefore, in practical spur gear design, special attention should be paid to the root fillet area. Increasing the tip radius of the cutter or using a positive profile shift can reduce the stress and the temperature simultaneously.
Comparing the two theoretical methods, the flash temperature method gives a conservative estimate of the scuffing risk because it only looks at the instantaneous maximum temperature. The integral temperature method smooths out the peaks and provides a more realistic average temperature. For this reason, many international standards, such as ISO 13989, recommend the integral temperature method for scuffing load capacity calculations. The present study confirms that for multi-modulus involute spur gear pairs, the integral temperature method yields results that are closer to the finite element simulations than the flash temperature method.
The sensitivity analysis reveals that a 1% increase in the module ratio can reduce the maximum bulk temperature by about 10%. This is a significant improvement. However, the module ratio is limited by the tooth tip thickness. The minimum allowable tooth tip thickness is usually \(0.2m\). For the case \(\delta_m = 1.03\), the tooth tip thickness of the driving gear is still above this limit, but for \(\delta_m = 1.04\), it would be too thin. Therefore, the range \(0.98 \le \delta_m \le 1.03\) is safe in terms of both scuffing and tooth strength.
The influence of the profile shift coefficient on the driving gear is also significant. A shift of \(+0.3\) can improve the safety factor from 2.259 to 3.537 in the flash temperature method. The reason is that a positive shift moves the tooth contact closer to the pitch point, reducing the sliding distance and the relative velocity. Additionally, the positive shift increases the radius of curvature at the root, which lowers the contact pressure. Thus, a positive profile shift on the pinion is a classic and effective measure to improve scuffing resistance.
On the other hand, the driven gear profile shift has a limited effect because the driven gear has more teeth, and the relative sliding near its root is smaller. The safety factor changes by only about 8% over the whole range of \(x_2\). Therefore, in the design of a multi-modulus spur gear pair, it is more efficient to adjust the module ratio and the pinion shift than the gear shift.
Finally, I compared the theoretical bulk temperatures with the finite element results for all parameter combinations. The error of the integral temperature method is below 16%, while the error of the flash temperature method can exceed 27% at high module ratios. The larger error for the flash temperature method is expected because the bulk temperature formula \(\theta_M = \theta_{oil} + w_t^{0.11}\) is a simple empirical relation that does not account for the actual cooling effect of the lubricant and the gear body geometry. In contrast, the integral temperature method includes a more detailed model of the average thermal cycle. Nevertheless, both methods correctly predict the trend: the bulk temperature decreases with increasing module ratio or increasing profile shift coefficients.
In summary, this research provides a comprehensive analysis of the scuffing strength of multi-modulus involute spur gear pairs based on flash temperature and integral temperature criteria. The key conclusions are:
- The module ratio has a dominant effect on the tooth surface temperature of the spur gear pair. Increasing \(\delta_m\) from 0.98 to 1.03 reduces the maximum flash temperature and the bulk temperature, thus improving the scuffing resistance.
- Increasing the driving gear profile shift coefficient \(x_1\) also significantly reduces the thermal load, while the driven gear coefficient \(x_2\) has a smaller influence.
- The highest temperature occurs at the root of the driving gear, which is therefore the critical zone for scuffing initiation.
- The integral temperature method provides a more accurate prediction of the bulk temperature than the flash temperature method when compared with finite element simulations.
- The finite element steady-state thermal analysis confirms the theoretical results, validating the proposed calculation procedure for multi-modulus involute spur gear pairs.
The findings of this study can be directly applied to the design of high-speed and heavy-duty spur gear transmissions where scuffing is a major concern. By selecting a suitable module ratio and an optimal positive profile shift on the driving gear, it is possible to significantly increase the scuffing load capacity without changing the center distance or the gear ratio. This offers an additional design flexibility compared with conventional equal-modulus spur gear pairs.
For future work, I plan to extend the analysis to multi-modulus helical gear pairs and to include the effect of tooth profile modifications such as tip relief and root relief. Experimental validation using a gear test rig with thermocouple measurement is also necessary to confirm the theoretical predictions. In addition, the interaction between scuffing and pitting or bending fatigue in multi-modulus spur gears should be investigated, as the optimal parameters for one failure mode may be detrimental to another. A multi-objective optimization framework could then be developed to obtain the best compromise among different performance criteria.
I believe that the present contribution fills a gap in the existing literature and offers a useful engineering tool for the analysis and design of multi-modulus involute spur gear pairs. The formulas and the numerical procedure presented here are general and can be easily adapted to other gear geometries and operating conditions. The use of the dimensionless coordinate \(\Gamma\) and the systematic derivation of the temperature and safety factor formulas make the analysis transparent and reproducible.