Effect of Geometric Parameters of High-Speed Helical Gears on Friction Flash Temperature and Scuffing Load Capacity in Electric Vehicles

Abstract

High-speed reducers in electric vehicles, characterized by high rotation speeds, heavy loads, large helix angles, and high contact ratios, are prone to tooth surface scuffing due to high sliding speeds. This scuffing is caused by adhesion wear from excessive instantaneous friction flash temperatures. The prevailing approach to gear scuffing analysis relies on the standard formula method, which is a relatively rudimentary technique. This method lacks the precision required to accurately assess the intricate distribution of tooth surface flash temperature (TSFT), limiting its efficacy in targeted tooth optimization. This study introduces an enhanced semi-analytical method to calculate TSFT and analyzes its variation under different conditions: increased tooth number and reduced module, altered pressure angle, and varied helix angle. The aim is to understand how these geometric parameters affect TSFT and the scuffing load capacity of high-speed reducer gears. This study calculates load distribution and TSFT under peak operating conditions and shows that increasing the tooth number, pressure angle, and helix angle can reduce maximum TSFT by more than 30%, improving scuffing safety and load capacity. However, these improvements must consider the gear’s allowable bending safety factor and bearing service life. The research concludes that optimizing these geometric parameters can significantly enhance the scuffing load capacity of gearsets.

1. Introduction

The development of electric vehicles (EVs) is accelerating rapidly. In comparison to internal combustion engine vehicles, the operating characteristics of electric vehicle drive motors are completely different. The motor is capable of outputting its maximum torque at startup, and its highest speed can reach over 20,000 rpm, with a trend towards even higher speeds. It is therefore evident that electric vehicle motors require a high-speed reducer with a large gear ratio [1,2]. In such extreme conditions, high-speed reducer gears are prone to tooth surface wear and scuffing failure. The reliability of the transmission system is a determining factor in the safety, comfort, and reliability of the electric powertrain and the vehicle as a whole [3,4].

The reduction gears of electric vehicles are evolving towards high speed and heavy load directions. However, this technological evolution has also given rise to a number of challenges, including gear scuffing, tooth profile fracture, and gear whine [5]. Surface scuffing represents a significant failure mode of gear transmission systems; this phenomenon is caused by the instantaneous friction flash temperature of the surface being too high, which results in the contact temperature exceeding the scuffing temperature of the gears and lubricating oil system [6].

The power loss during gear meshing is also caused by frictional loss on the tooth surface. When designing a gear system in order to prevent scuffing failure and achieve high meshing efficiency, it is essential to accurately calculate the flash temperature distribution on the tooth surface. Standard methods such as ISO 6336-20 [7] and AGMA 925-A03 [8], both developed from Blok’s flash temperature theory [9,10,11], are the current common calculation method for TSFT. The surface load specified in the equation represents the average unit load, and the detailed distribution of the tooth load is not considered. Despite the introduction of a tooth surface load uneven distribution factor K_Hβ, the precise location of the maximum flash temperature remains uncertain. Consequently, the flash temperature cannot be reduced by micro-geometry modifying directly. The TSFT is closely associated with operational parameters, including torque and boundary conditions. These factors influence the position of the maximum load on the tooth surface, which in turn effects the location of the maximum flash temperature. Therefore, to obtain an accurate distribution of flash temperature on the tooth surface, it is first necessary to analyze the working conditions and boundary conditions of the transmission system. In addition, the friction coefficient used in the calculation of TSFT is usually an average friction coefficient, which is a constant value. Due to the different pressures, radii of curvature, and sliding speeds at different contact positions on the tooth surface, the oil film thickness and lubrication state will vary, making the friction coefficient a variable. Zhou et al. [12] put forth a novel approach to predicting gear friction coefficients, employing a computational inverse technique that encompasses both rolling and sliding friction on the tooth surface. Xu [13] put forward an equation for the instantaneous friction coefficient during gear meshing based on EHL theory and corrected the accuracy of the friction coefficient equation with a number of experiments. The friction coefficients for Zhou and Xu are similar. However, Zhou’s method is more complicated and requires double verification by finite element calculation and experimental tests, which is time-consuming and costly. Xu’s model is based on EHL and a large number of tests to determine the correction coefficient. This makes it quick and accurate, and it is more suitable for gear design and calibration.

High flash temperatures on the tooth flank during gear meshing can adversely affect the meshing properties. Gou et al. [14] established the TSFT equations based on Blok’s theory and calculated the tooth surface deformation caused by contact temperature; Pan et al. [15] found that the TSFT changes the time-varying mesh stiffness, which leads to the change of cyclic kinematic characteristics of the gear rotor system. Liu et al. [16] found that the thermal deformation of the tooth in a planetary gearset due to the equilibrium temperature of the system also changes the time-varying mesh stiffness and increases the vibration of the system. Bai et al. [17] demonstrated that tooth modification can improve the surface thermal distribution and reduce the temperature and friction loss during gear rotation. Yu et al. [18] derived calculation methods for instantaneous TSFT based on the Hertz contact theory and thermal shock model. They also analyzed the effect of the variable factor and tooth modification on flash temperature and demonstrated that a reasonable choice of variable coefficients and modification values can improve the gearset scuffing load capacity. Wei et al. [19] investigated the mechanisms by which surface topography, rolling-slip contact stress, subsurface shear stress, and depth of spur gears under EHL conditions influence the formation of tooth pitting. Chen et al. [20] proposed a new method based on the pressure–velocity–temperature (PVT) limit to assess the anti-scuffing of gears. Experimental results showed a maximum deviation of 6.6%, proving that the PVT limit evaluates gear anti-scuffing capacity effectively under different conditions. Miltenović et al. [21] employed the finite element method to simulate the effects of friction in the contact zone during the worm-gear meshing on heat transfer on the gear tooth. The tooth modification of gears can improve certain problems encountered during meshing to reduce failure rates and increase equipment reliability. Chen et al. [22] investigated the solution for meshing stiffness, taking into account tooth modifications, misalignment, and machining errors, as well as the tooth surface optimization method based on that stiffness model. Using a statistical model, Ganti et al. [23] studied the effects of tooth modification on the distribution of contact stress and flash temperature in order to predict the risk of scuffing in vehicle transmissions and discussed the sensitivity of the tooth modification to scuffing temperature.

In conclusion, a substantial body of research has been conducted on the flash temperature of gear surfaces in transmission systems. These studies are typically conducted in accordance with the calculation method of scuffing load-carrying capacity and/or through the scuffing test method. The calculation method employs equations from international standards, and the gear surface load calculation generally uses average line load without considering factors such as system deformation that may lead to edge contact and uneven load distribution. This results in an inaccurate calculation of the TSFT distribution on the gear surface. The aim of this study is to investigate the influence of geometric parameters on the flash temperature and scuffing load-carrying capacity of gears. This paper puts forward a dense teeth design method and evaluates its potential to reduce the flash temperature and enhance the load-carrying capacity. The effects of changing the pressure angle and helix angle on the flash temperature are investigated. The gears studied in this paper take into account the system misalignment and have made a micro-modification on the tooth surface to improve the distribution of tooth surface loads. Consequently, the paper assesses the influence of three geometric parameters of gears on the contact ratio, specific sliding, surface load distribution, flash temperature distribution, and scuffing safety factor.

2. Semi-Analytical Calculation Model for Gear Surface Flash Temperature

2.1. Local Geometry and Velocity Parameters of Gear Tooth Surface

During the gearset operation, there is both rolling and sliding between the meshing tooth surfaces, but pure rolling only occurs at pitch point C, as shown in Figure 1. When the pinion rotates clockwise with an angular velocity ω₁ to drive the gear, the line segment T₁T₂ is their line of action. When the current contact point is at an arbitrary point Y, a pair of conjugate tooth surfaces of the pinion and the gear are tangent at point Y. At this time, the circumferential velocity of the pinion at this point is v_t1, and the circumferential velocity of the gear is v_t2. They have the same normal velocity v_n at point Y, but the tangential velocities v_g1 and v_g2 of the pinion and the gear at point Y are not equal. Their difference is the relative sliding velocity v_s at point Y.

Based on the properties of involute and gear meshing relationships, the sliding velocity at any meshing point on the tooth surface is calculated using the following equation, with the unit m/s.

$$v_s(i)=\mathrm{abs}(v_{g1}(i)-v_{g2}(i))=\mathrm{abs}\left({\omega }_1\cdot {\rho }_{y1}-{\omega }_2\cdot {\rho }_{y2}\right)/1000$$

(1)

The local radii of curvature at the contact point for the pinion and the gear are given by the following equations, respectively:

$${\rho }_{y1}={\displaystyle \frac{\sqrt{d_{y1}^2-d_{b1}^2}}{2}}$$

(2)

$${\rho }_{y2}={\displaystyle \frac{\sqrt{d_{y2}^2-d_{b2}^2}}{2}}$$

(3)

According to the definition of the tooth surface-specific sliding [24], the specific sliding for the pinion and the gear at any contact point are shown below:

$${\zeta }_1(i)={\displaystyle \frac{v_s(i)}{v_{g1}(i)}}=\mathrm{abs}(1-{\displaystyle \frac{{\rho }_{y2}}{u{\rho }_{y1}}})$$

(4)

$${\zeta }_2(i)={\displaystyle \frac{v_s(i)}{v_{g2}(i)}}=\mathrm{abs}(1-{\displaystyle \frac{u{\rho }_{y1}}{{\rho }_{y2}}})$$

(5)

where u = |z₂/z₁| represents the gear ratio.

It can be seen that gears with different geometric parameters have different local radii of curvature on their tooth surfaces, which results in different tangential velocities and relative sliding velocities during rotation, thus leading to different tooth surface-specific sliding.

As shown in Figure 2, during the operation of a pair of meshing gears, the tooth of the pinion enters meshing from the root to the tip and then exits meshing, with the tooth surface sliding velocity v_s changing from large to small and then back to large. At the pitch point, the tangential velocity components of the tooth surfaces are equal, indicating pure rolling, while at other positions there is relative sliding velocity, which increases further away from the pitch point, therefore the maximum value of v_s occurring near the tooth tip and root. The same pattern applies to a single instantaneous contact line on the tooth surface, where the sliding velocity is zero along the pitch line and then gradually increases towards the tooth tip and root as it moves along the height of the tooth.

2.2. Calculation Model for Tooth Surface Load and Flash Temperature

Due to the elastic deformation that occurs after the mechanical transmission system is loaded, a pair of conjugate involute helicoid surfaces will not mesh at the theoretical position after being loaded but will exhibit uneven load distribution, stress concentration, or edge contact, as shown in Figure 3a. This can lead to early tooth surface pitting, scuffing, wear, or gear tooth flank fracture, and can also cause whine noise and meshing impact problems in the gear system.

Gear surfaces without micro-modification often experience edge contact, particularly near the tooth tips, roots, or at the ends of the tooth face. This occurs due to elastic deformation, leading to high contact stress and, consequently, larger friction heat. This consumption of input energy can generate high flash temperatures, which can lead to scuffing failures. The advent of gear modification technology addresses these issues by making minor adjustments at the micron level to the gear surface geometry. This results in a deviation from the theoretical gear surface, as illustrated in Figure 3c. This modified surface no longer conforms to the theoretical involute helix. Gear micro-modification enhances transmission accuracy, improves the stability of gear operation, mitigates phenomena like load bias and stress concentration, and thereby extends the service life and enhances the load-bearing capacity of gears. The flash temperature distribution across the gear surface correlates positively with local load and relative sliding velocity. This is tied to the gear’s rotation speed and macro-geometry and thus cannot be altered through minor modifications. However, micro-modifications can reduce contact loads in areas with high relative sliding velocities, thereby minimizing flash temperatures on the gear surface. For electric vehicle high-speed reducers, gear tooth modification can lower the flash temperature of gear meshing surfaces, effectively enhancing their anti-scuffing capacity. Additionally, it can decrease power loss during meshing, improve drive efficiency, and extend the vehicle’s driving range.

During the meshing process of gears, flash temperatures can be caused by friction on the tooth surfaces. When the contact temperature exceeds the allowable scuffing temperature, it can lead to scuffing on the tooth surfaces. Scuffing is different from fatigue; it can occur almost instantaneously. There are typically two methods of calculation TSFT: the Block flash temperature and the integral temperature method. The integral temperature is an average value of TSFT. It is based on the flash temperature at the pinion tip and incorporates the load-sharing coefficient into influencing factors. The integral temperature value closely approximates the maximum flash temperature, making the risk analysis for scuffing similar to that obtained from the flash temperature method. In instances where local temperature peaks are present, such as in gears with low contact ratios, edge contact at the tip and root, or contact at geometrically sensitive areas, the integral temperature exhibits reduced sensitivity and is unable to reflect the distribution of flash temperatures on the tooth surfaces. To investigate the state of TSFT distribution in gear pairs following elastic deformation and under conditions of uneven loads, an enhanced flash temperature method is more suitable for analysis.

The equation for calculating the TSFT at the instantaneous contact position of the gearset is as follows [7]:

$${\Theta }_{fl}=1.11\cdot {\displaystyle \frac{\mathsf{\mu }\cdot X_{\Gamma }\cdot X_J\cdot w_n}{\sqrt{2\cdot b_H}}}\cdot {\displaystyle \frac{abs\left(v_{g1}-v_{g2}\right)}{B_{M1}\cdot \sqrt{v_{g1}}+B_{M2}\cdot \sqrt{v_{g2}}}}$$

(6)

The tangential velocity components v_g1 and v_g2 of the tooth surfaces of involute cylindrical gears are related to the local geometric parameters and rotational speed. Therefore, the aforementioned equation can be replaced equivalently with the following equation:

$${\Theta }_{fl}=2.52\cdot \mathsf{\mu }\cdot {\displaystyle \frac{X_M}{50}}\cdot X_J\cdot \sqrt[4]{{(X_{\Gamma }\cdot w_n)}^3}\cdot \sqrt{n_1/60}\cdot {\displaystyle \frac{abs\left(\sqrt{{\rho }_{y1}}-\sqrt{{\rho }_{y2}/u}\right)}{\sqrt[4]{{\rho }_{rely}}}}$$

(7)

where X_J represents the engage-in coefficient, X_M denotes the thermoelastic coefficient, and X_Γ denotes the load-sharing factor. The specific calculation methods for these three coefficients are detailed in Ref. [7]. The local comprehensive curvature radius is as follows:

$${\rho }_y={\displaystyle \frac{{\rho }_{y1}\cdot {\rho }_{y2}}{{\rho }_{y1}+{\rho }_{y2}}}$$

(8)

In this paper, the friction coefficient used for calculating the TSFT does not adopt the calculation equation recommended by Ref. [7], but instead uses a more accurate empirical equation derived from the EHL theory through extensive experimental regression analysis [13] as follows:

$$\left\{\begin{array}{l}\mathsf{\mu }=e^{f(\xi ,P_h,{\nu }_0,R_q)}{Ph}_{c_2}{\left|\xi \right|}^{c_3}{V}_e^{c_6}{\nu }_0^{c_7}{\rho }_y^{c_8}\hfill \\ f(\xi ,P_h,{\nu }_0,R_q)=c_1+c_4\left|\xi \right|P_h{\log }_{10}(v_0)+c_5{e}^{-\left|\xi \right|P_h{\log }_{10}(v_0)}+c_9{e}^{R_q}\hfill \end{array}\right.$$

(9)

where ξ denotes the slip-roll ratio; P_h represents the maximum Hertzian stress in GPa; ν₀ represents the dynamic viscosity of the lubricating oil in cP; R_q denotes the comprehensive roughness on the tooth surface, in µm; V_e denotes the entrainment velocity, in m/s. Coefficients b_i are the regression coefficients, the values of which are shown in the following

Table 1. Coefficients for the Equation (9).

Coefficients	c1	c2	c3	c4	c5	c6	c7	c8	c9
Values	–8.916	1.033	1.036	–0.354	2.812	–0.1	0.753	–0.391	0.62

.

$$\xi =V_s/V_e$$

(10)

$$R_q=\sqrt{{\displaystyle \frac{R_{q1}^2+R_{q2}^2}{2}}}$$

(11)

where R_q1 and R_q2 represent the RMS roughness of the tooth surfaces for pinion and gear, respectively, and w_n represents the normal unit load. The normal unit load, as defined in ISO 6336-20:2022, is the ratio of the normal force at the current meshing position to the tooth width. This value represents a mean value and is therefore unable to reflect the distribution of load along the instantaneous contact line. As a consequence of meshing misalignment and tooth surface modification, the load distribution along the instantaneous contact line is necessarily uneven. In order to calculate the actual tooth surface load distribution, this paper employs Loaded Tooth Contact Analysis (LTCA) based on the FEM to study the distribution of w_n, as illustrated in Figure 4.

If a meshing cycle is divided into N instantaneous positions and the tooth width direction is divided into M grid nodes, then after achieving load balance at the i-th instantaneous meshing position, the normal load vector on the instantaneous contact line can be obtained according to Equation (12).

$$w_i^n=(w_{i1}^n,w_{i2}^n,w_{i3}^n,\cdot \cdot \cdot \cdot \cdot \cdot ,w_{iM}^n)$$

(12)

After solving for the N meshing positions, they are assembled into the form of a load matrix as shown in Equation (13).

$$w^n=\left[\begin{array}{ccccc}{w}_{11}^n& w_{12}^n& w_{13}^n& \cdot \cdot \cdot & w_{1M}^n\\ w_{21}^n& w_{22}^n& w_{23}^n& \cdot \cdot \cdot & w_{24}^n\\ w_{31}^n& w_{32}^n& w_{33}^n& \cdot \cdot \cdot & w_{3M}^n\\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ w_{N1}^n& w_{N2}^n& w_{N3}^n& \cdot \cdot \cdot & w_{NM}^n\end{array}\right]$$

(13)

Then, the transverse unit load w_t can be calculated according to Equation (14).

$$w_t=w_n\cdot \cos {\alpha }_{\mathrm{wn}}\cdot \cos {\beta }_{\mathrm{w}}$$

(14)

In summary, the specific calculation process for the flash temperature distribution on the tooth surface involves the following steps: Firstly, the gear LTCA, which is based on the FEM model, is employed for the calculation of the load and stress distribution on the micro-modified tooth surface. Secondly, the load distribution state is calculated iteratively, and the optimality of the modification parameters is evaluated. Thirdly, the instantaneous contact position line load obtained from the aforementioned calculation is substituted into Equations (7) to (14), thereby determining the TSFT at the current meshing position. Subsequently, following the calculation of the N meshing positions within a single meshing cycle, the maximum flash temperature value should be stored at each node position. Ultimately, the integration of the maximum flash temperatures of the nodes onto a rotational projection diagram of the tooth surface enables the acquisition of the flash temperature distribution across the entire tooth surface.

3. Results and Discussion

3.1. The Effects of Tooth Number and Module on Scuffing Load Capacity

3.1.1. Gearsets Geometry Parameters Planning

When designing gear transmissions, we initially determine the center distance of the gear pair based on the load and spatial size boundary conditions. Subsequently, the transmission ratio and module were established. Once the transmission ratio is set, the gear module is inversely proportional to the number of teeth. Following this, we explore the relationship between the flash temperature on the tooth surface and the module. As detailed in

Table 2. Gearsets parameters of variable modules.

Gearset	Symbol	G2263		G2469		G2675		G2881		G3087
Gear		Z22	Z63	Z24	Z69	Z26	Z75	Z28	Z81	Z30	Z87
Tooth number	z	22	63	24	69	26	75	28	81	30	87
Normal module	mn/mm	1.925		1.759		1.62		1.501		1.398
Gear ratio	u	2.864		2.875		2.885		2.893		2.9
Pressure angle	αn	19.5°
Helical angle	β	20°
Face width	b	40 mm
Center distance	a	87 mm
Profile shift	x	0.208	–0.245	0.208	–0.245	0.208	–0.245	0.208	–0.245	0.208	–0.245
Tip diameter	da/mm	50.6	132.7	50	132.8	49.6	132.7	49.1	132.5	48.7	132.4
Root diameter	df/mm	39.678	122.282	40	122.933	40.287	123.498	40.522	124.01	40.717	124.427

, we have planned parameters for five sets of gear pairs.

Each gear set was designed with a center distance of 87 mm, a pressure angle of 19.5°, and a right-hand helix angle of 20°. The approximate gear ratio is around 2.88. The normal module decreases from 1.925 to 1.398 as the number of teeth increases. In the design process for the gear profile shift coefficient, addendum coefficient, and dedendum coefficient, the fundamental principle is to balance the specific sliding between the pinion and the gear.

For the analysis of extreme conditions in the electric drive, only the maximum speed and maximum torque conditions were selected to study the trends of change. The load conditions and material parameters are shown in

Table 3. The parameters of gear material and load conditions.

	Density	Elastic Modulus	Poisson Ratio	Thermal Conductivity	Surface Roughness
Gear material	7890 kg/m3	206 GPa	0.3	49 W/m/K	Ra 0.8 µm
Peak speed load case	Input torque: 99 Nm		Input speed: 25,000 rpm
Peak torque load case	Input torque: 450 Nm		Input speed: 5000 rpm

.

High-speed gears for electric drives utilize spray oil for lubrication, with oil parameters detailed in

Table 4. The parameters of gear lubrication oil.

Kinematic Viscosity @40 °C	Kinematic Viscosity @100 °C	Density @15 °C	FZG Scuffing Test Grade
18 mm2/s	4.6 mm2/s	850 kg/m3	12

.

3.1.2. Comparison of Contact Ratio, Specific Sliding, and Load-Carrying Capacity

First, the contact ratio and tooth root bending safety factor for the five aforementioned gear sets were calculated. Figure 5a illustrates that the total contact ratio increases linearly with decreasing module and increasing number of teeth (Denser Teeth Design), ranging from 3.884 to 4.814, which is a 23.94% increase. The most notable enhancement is observed in the overlap ratio ε_β, while the transverse contact ratio remains largely stable. This indicates that the design strategy of Denser Teeth Design is effective for enhancing the gear contact ratio. Consequently, this can positively influence the smoothness of gear motion and other NVH (Noise, Vibration, and Harshness) performance.

Following the design of a gear pair with denser teeth, the tooth root bending stress and safety factor become the critical constraints, especially when the module is reduced and the external load remains constant. The bending fatigue safety factor at the tooth root is generally required to be no less than 1.2 to ensure reliability. Figure 5b demonstrates that, with the speed ratio kept essentially constant, the bending safety factor under peak load conditions decreases from 1.71 to 1.24 as the number of teeth on the pinion increases from 22 to 30. This trend clearly indicates that the combination of tooth numbers 30 and 87 represents an extreme value for this specific center distance, highlighting the need for careful consideration in gear design to maintain safety margins.

The curves in Figure 6 illustrate the variation in sliding velocity and specific sliding with respect to the module. It is observable that below the pitch circle, during the entry phase, the sliding velocity decreases with an increase in the number of teeth. Conversely, above the pitch circle, during the exit phase, the sliding velocity increases. The specific sliding is the same trend as the sliding velocity. However, there is a distinction: the sliding velocity at the instantaneous contact point increases linearly with the diameter distance from the point to the pitch circle. In contrast, the sliding rate at the instantaneous contact point shows a nonlinear increase with the diameter distance from the point to the pitch circle. Furthermore, the sliding rate’s rate of change is more significant below the pitch circle and less pronounced above it.

Tooth surface load distribution is a critical indicator for assessing the load-carrying capacity of gears. Following the design modifications that increase tooth number and reduce module, it is essential to examine the resulting changes in tooth surface load. This examination also explores the impact of various parameters on the load-carrying capacity of gears, thereby providing a foundation for the subsequent analysis of TSFT.

Table 5. Tooth surface load distribution of variable module gearsets.

Gearset	Tooth Surface Load Distribution Peak Speed Load Case: 99 Nm, 25,000 rpm	Tooth Surface Load Distribution Peak Torque Load Case: 450 Nm, 5000 rpm
G2263
G2469
G2675
G2881
G3087

presents the tooth surface pressure distribution for five gear sets, with the left column indicating the highest speed condition and the right column indicating the peak torque condition. In the highest speed condition, the tooth surface pressure remains largely stable despite the reduction in the module, with a minimum of 877.9 MPa and a maximum of 894 MPa and fluctuations not exceeding 1.8%. In contrast, under the peak torque condition, there is a reduction in tooth surface pressure with module reduction, from a maximum of 2419 MPa to a minimum of 2194.5 MPa, representing a reduction rate of 9.3%. It is noteworthy that the position and shape of the load distribution on the tooth surface have remained largely unchanged. The optimization strategy of increasing tooth number and reducing module not only improves the total contact ratio but also reduces contact pressure under peak load conditions, thereby enhancing the load-carrying capacity and fatigue life of the gear set.

3.1.3. Tooth Flash Temperature and Scuffing Safety Assessment

The distribution of flash temperature on the tooth surface is a critical indicator for the assessment of gear scuffing load capacity.

Table 6. Flash temperature distribution on tooth surface of variable module gearsets.

Gearset	Load Case: Peak Speed 99 Nm @ 25,000 rpm	Load Case: Peak Torque 450 Nm @ 5000 rpm
G2263
G2469
G2675
G2881
G3087

illustrates the distribution of flash temperature on the tooth surfaces of five sets of control gears, which were designed with an increased number of teeth and a reduced module. The aim of this study is to investigate the influence of different parameters on the TSFT.

The left column of Table 6 enumerates the conditions at the maximum rotational speed, while the right column enumerates the conditions at the maximum torque. At the maximum rotational speed, the maximum flash temperature on the tooth surface is observed to decrease with a decreasing module, with a reduction rate of 33.7%. The maximum flash temperature observed at this condition was 54.3 °C, while the minimum was 36 °C. Similarly, under the maximum torque condition, the maximum flash temperature on the tooth surface decreases as the module decreases, with values ranging from a maximum of 217.7 °C to a minimum of 153 °C, indicating a reduction rate of 29.7%. Moreover, the position and shape of the maximum flash temperature on the tooth surface remain largely unaltered. It is evident that the strategy of Denser Teeth Design significantly reduces the frictional flash temperature on the tooth surface under extreme conditions, thereby offering a substantial advantage for gear performance.

Figure 7 illustrates that both the flash temperature value and the scuffing safety factor demonstrate a tendency to enhance the gear scuffing load capacity with the increase of tooth number following the dense tooth design. The minimum scuffing safety factor for the pinion with 22 teeth is 1.071. An increase in the number of teeth on the pinion to 30 would result in a safety factor of 1.399, representing a 30.6% improvement. It can thus be concluded that, provided that the tooth root bending load capacity and the speed ratio remain unaltered, an increase in the number of teeth will result in a reduction in the likelihood of gear scuffing failure. Furthermore, it can be observed that the flash temperature under the peak rotational speed condition is significantly lower than that under the peak torque condition, with the corresponding scuffing safety factor approximately twice as large. This suggests that the peak torque condition represents a critical point with regard to scuffing in the high-speed reducer of electric vehicles.

3.2. The Effects of Pressure Angle on Scuffing Load Capacity

3.2.1. Gearsets Geometry Parameters Planning

The subsequent study examines the effects of varying pressure angles on a number of key variables, including tooth surface sliding velocity, specific sliding, load distribution, and flash temperature. These variables are considered under a constant center distance. The specific parameters for the five sets of gear pairs are outlined in

Table 7. Gearset parameters of variable pressure angle.

Gearset	Symbol	A16		A17.5		A19.5		A21.5		A23
Gear		Z26	Z75	Z26	Z75	Z26	Z75	Z26	Z75	Z26	Z75
Tooth number	z	26	75	26	75	26	75	26	75	26	75
Normal module	mn/mm	1.62
Gear ratio	u	2.885
Pressure angle	αn	16°		17.5°		19.5°		21.5°		23°
Helical angle	β	20°
Face width	b	40 mm
Center distance	a	87 mm
Profile shift coefficient	x	0.408	–0.445	0.308	–0.345	0.208	–0.245	0.208	–0.245	0.208	–0.245
Tip diameter	da/mm	50.26	132.08	49.9	132	49.6	132.7	49.5	132.8	49.3	132.6
Root diameter	df/mm	40.935	122.785	40.611	123.142	40.287	123.498	40.287	123.563	40.287	123.563

. Each gear pair has a center distance of 87 mm, with the number of teeth consistently set at 26 and 75, a helix angle of 20° to the right, and a normal module of 1.62 mm. The pressure angles are systematically varied as 16°, 17.5°, 19.5°, 21.5°, and 23° across the five sets. In the design process, the guiding principle for determining the variation coefficient, tooth tip height, and tooth root height coefficients is to achieve a balance in the sliding rates between the large and small wheels.

3.2.2. Comparison of Contact Ratio, Specific Sliding, and Load-Carrying Capacity

Figure 8 illustrates the contact ratio and bending safety factor for the five gear sets described. It can be observed that the total contact ratio ε_γ decreases in a linear fashion with an increasing pressure angle, falling from 4.579 to 4.184, which represents an 8.6% reduction. The transverse contact ratio ε_α exhibits the most pronounced decline, whereas the overlap ratio ε_β remains unaltered. It is evident that an increase in pressure angle has a detrimental effect on the enhancement of gear contact ratios, which subsequently impacts the smoothness of gear motion and other NVH performance indicators.

Figure 8b depicts the correlation between the pressure angle and the tooth root bending safety factor for the G2675 gearset, with all other variables maintained at a constant value except for the pressure angle. As the pressure angle increases from 16° to 23°, the bending safety factor under peak load conditions decreases from 1.43 to 1.32. It is noteworthy that the safety factor for gearsets with a pressure angle of 21.5° or less remains relatively stable, whereas a significant decline is observed for the 23° gearset. In order to ensure that the tooth root bending safety factor is at least 1.2, it would be advisable to limit the pressure angle to 23°. Furthermore, the safety factor would fall below the permissible value. Moreover, an elevated pressure angle gives rise to augmented radial forces on the gear, which, in turn, intensifies the load on the bearings and may potentially diminish their fatigue life.

Figure 9 illustrates the correlation between sliding velocity and specific sliding in relation to pressure angle. Irrespective of whether the point in question is situated below or above the pinion’s pitch circle during the tooth exit phase, the sliding velocity is observed to decrease in accordance with the increase in pressure angle. This trend is reflected in the corresponding specific sliding. A significant distinction is that the sliding velocity at the instantaneous contact point demonstrates a linear increase in the diameter distance from this point to the pitch circle. In contrast, the specific sliding at the instantaneous contact point displays a nonlinear increase with increasing diameter distance. Furthermore, the rate of change in the specific sliding is more pronounced below the pitch circle and more gradual above it.

Table 8. Tooth surface load distribution of variable pressure angle gearsets.

Gearset	Tooth Surface Load Distribution Peak Speed Load Case: 99 Nm, 25,000 rpm	Tooth Surface Load Distribution Peak Torque Load Case: 450 Nm, 5000 rpm
A16
A17.5
A19.5
A21.5
A23

presents the tooth surface pressure distribution for five gear sets with varying pressure angles. The left column indicates peak speed conditions, while the right column indicates peak torque conditions. In conditions of peak speed, the tooth surface pressure remains relatively stable with an increasing pressure angle, with a minimum of 872.2 MPa and a maximum of 883.4 MPa and with fluctuations within a 1.3% margin. In contrast, under peak torque conditions, tooth surface pressure decreases with an increase in pressure angle from a maximum of 2592.9 MPa to a minimum of 2062 MPa, representing a 20.5% reduction. Additionally, the load distribution across the tooth surface becomes increasingly uniform. In these gear sets, where the pressure angle is variable, the contact ratio is inversely correlated with the pressure angle. However, the contact pressure under peak load conditions notably decreases with the pressure angle increase, which is advantageous for improving the gearsets’ load-carrying capacity and contact fatigue life.

3.2.3. Tooth Flash Temperature and Scuffing Safety Assessment

Table 9. Flash temperature distribution on tooth surface of pressure angle gearsets.

Gearset	Tooth Surface Load Distribution Peak Speed Load Case: 99 Nm, 25,000 rpm	Tooth Surface Load Distribution Peak Torque Load Case: 450 Nm, 5000 rpm
A16
A17.5
A19.5
A21.5
A23

illustrates the calculated flash temperature distribution on the tooth surfaces of five gear sets with varying pressure angles. The left column represents the highest speed operating conditions, while the right column represents the peak torque operating conditions. In the highest speed operating conditions, the maximum flash temperature on the tooth surface decreases with an increasing pressure angle, with a range of 54.3 °C to 36 °C, representing a 33.7% reduction. Similarly, under peak torque conditions, the maximum flash temperature decreases from 236 °C to 129 °C, representing a decrease of 107 °C, or a 45.3% reduction. It is evident that an increase in pressure angle represents an effective design strategy for the reduction of tooth surface friction flash temperature under extreme operating conditions.

Figure 10 illustrates that both the flash temperature value and the scuffing safety factor are positively influenced by an increase in the pressure angle, thereby enhancing the gear’s scuffing load-carrying capacity. At a pressure angle of 16°, the scuffing safety factor reaches its minimum value of 1.0. An increase in the pressure angle to 23° results in an improvement in this factor to 1.569, representing a 56.9% increase. Therefore, provided that the minimum transverse contact ratio and bearing life are maintained, employing a larger pressure angle significantly enhances the gear’s scuffing load-carrying capacity.

3.3. The Effects of Helix Angle on Scuffing Load Capacity

3.3.1. Gearsets Geometry Parameters Planning

Previous research has investigated the influence of varying the module and pressure angle on flash temperature, load distribution, contact ratio, and specific sliding of the tooth surface. These investigations were conducted under conditions of a fixed center distance and gear ratio. The subsequent phase of the investigation will entail an examination of the impact of the helix angle on the aforementioned meshing performance. Accordingly, five sets of gear pair parameters were designed and are presented in

Table 10. Gearset parameters of variable helical angle.

Gearset	Symbol	B10		B15		B20		B25		B30
Gear		Z26	Z75	Z26	Z75	Z26	Z75	Z26	Z75	Z26	Z75
Tooth number	z	26	75	26	75	26	75	26	75	26	75
Normal module	mn/mm	1.62
Gear ratio	u	2.885
Pressure angle	αn	19.5°
Helical angle	β	10°		15°		20°		25°		30°
Face width	b	40 mm
Center distance	a	83 mm		85 mm		87 mm		90.2 mm		94.4 mm
Profile shift coefficient	x	0.239	–0.283	0.375	–0.185	0.208	–0.245	0.316	–0.357	0.196	–0.237
Tip diameter	da/mm	47.5	126.25	48.7	129.5	49.6	132.7	51.5	137.2	53.5	143.9
Root diameter	df/mm	38.61	117.65	39.576	120.484	40.287	123.498	42.25	128.03	44.025	134.47

. Each gearset is composed of 26 and 75 teeth, a pressure angle of 19.5°, and a normal module of 1.62 mm. The helix angles for the five sets were designed to be 10°, 15°, 20°, 25°, and 30°, respectively, with all pinions being right-handed. The corresponding center distances are calculated as 83 mm, 85 mm, 87 mm, 90.2 mm, and 94.4 mm, respectively. In designing the profile shift coefficient, addendum coefficient, and dedendum coefficient, the fundamental objective is to achieve balance in the specific sliding of the pinion and wheel.

3.3.2. Comparison of Contact Ratio, Specific Sliding, and Load-Carrying Capacity

Figure 11 illustrates the contact ratios and bending safety factors for the five gear sets previously discussed. It is evident that the total overlap ratio increases in a linear fashion with the helix angle, rising from 3.105 to 5.493, which corresponds to a 76.9% increase. The most notable enhancement is observed in the overlap ratio ε_β, while the transverse contact ratio exhibits a minor fluctuation of approximately 7.4%. A correlation exists between larger helix angles and higher overlap ratios, which contribute to enhanced gear motion and improved NVH performance. However, a larger helix angle also results in greater axial forces during meshing, which can negatively impact the life of the supporting bearings. Consequently, for power transmission gears, a helix angle exceeding 30° is typically not advisable.

Figure 11b depicts the correlation between the helix angle and the root bending safety factor for the G2675 gear pair, with all other variables maintained at their respective constant values. The bending safety factor under peak load conditions demonstrates a nonlinear increase from 1.44 to 1.51 as the helix angle is increased from 10° to 30°. Gear pairs with helix angles of 25° or less exhibit relatively similar safety factors. While a 30° helix angle enhances the overlap ratio and the bending safety factor, it also increases the axial force on the gear, thereby increasing the load on the bearings and potentially affecting their fatigue life. Therefore, the helix angle for high-speed gears is typically maintained below 30°.

Figure 12 illustrates the variation in sliding velocity and specific sliding as a function of the helix angle. The pitch circles of the gears exhibit variation due to different helix angles, resulting in the sliding velocity and specific sliding curves for the five gear sets being distributed approximately as normal equidistant lines. While the highest sliding velocities remain essentially equal, the maximum specific sliding decreases with an increase in the helix angle. The patterns of sliding velocity and specific sliding variation at the instantaneous contact point with distance from the pitch circle follow the same trends as previously analyzed cases involving variable modules and pressure angles. Therefore, they will not be reiterated here.

Table 11. Tooth surface load distribution of variable pressure angle gearsets.

Gearset	Tooth Surface Load Distribution Peak Speed Load Case: 99 Nm, 25,000 rpm	Tooth Surface Load Distribution Peak Torque Load Case: 450 Nm, 5000 rpm
B10
B15
B20
B25
B30

presents the tooth surface pressure distribution for five gear sets with varying helix angles. The left column indicates the operating conditions corresponding to peak speed, while the right column indicates the operating conditions corresponding to peak torque. In accordance with the data presented in Table 11, it can be observed that under peak speed operating conditions, there is a notable decrease in tooth surface pressure with an increase in helix angle. This reduction ranges from a maximum of 996.84 MPa to a minimum of 739.5 MPa, which represents a 25.8% reduction. Similarly, under peak torque conditions, the pressure decreases from a maximum of 2588.7 MPa to a minimum of 1737.8 MPa, representing a 32.9% reduction. As the helix angle increases, the load distribution across the tooth surface becomes more uniform. This phenomenon can be attributed to the fact that an increased helix angle results in a greater overlap ratio, which in turn reduces contact stress. This reduction is advantageous for enhancing the gearset load-carrying capacity and contact fatigue life.

3.3.3. Tooth Flash Temperature and Scuffing Safety Assessment

Table 12. Flash temperature distribution on tooth surface of pressure angle gearsets.

Gearset	Tooth Surface Load Distribution Peak Speed Load Case: 99 Nm, 25,000 rpm	Tooth Surface Load Distribution Peak Torque Load Case: 450 Nm, 5000 rpm
B10
B15
B20
B25
B30

illustrates the calculated flash temperature distribution on the tooth surfaces of five gear sets with different helix angles. The left column represents the highest speed operating conditions, while the right column represents the peak torque operating conditions. Under the highest speed operating conditions, the maximum flash temperature on the tooth surface decreases with an increasing helix angle, from 58.8 °C to 31.5 °C, which represents a 46.4% reduction. Similarly, under peak torque conditions, the maximum flash temperature decreases from 208 °C to 122.6 °C, representing an 85.4 °C reduction and a 41% decrease. It is evident that increasing the helix angle represents an effective design strategy for reducing the tooth surface friction flash temperature under extreme operating conditions.

Figure 13 indicates that the scuffing safety factor exhibits a fluctuating pattern of improvement with an increase in the helix angle. At a helix angle of 10°, the scuffing safety factor reaches its minimum value of 1.118. Upon increasing the helix angle to 30°, the factor rises to 1.602, representing a 43.3% improvement. While an increased helix angle enhances the scuffing load-carrying capacity, as previously discussed, it also increases the axial force on the gear, which may result in a reduction in the bearing’s lifespan. Therefore, in order to ensure the service life of the bearings, it is possible to utilize a larger helix angle in order to enhance the gearsets scuffing load-carrying capacity.

4. Conclusions

In order to investigate the impact of macro-geometry parameter design of high-speed reducers for electric vehicle gearboxes on TSFT and scuffing load-carrying capacity, an improved semi-analytical TSFT calculation method was proposed and studied how gear design affects performance. Gear parameters such as tooth number and module were varied while the gear ratio was kept constant to analyze the effects on sliding velocity, load distribution, and flash temperature. The following conclusions were derived from the analysis.

More teeth and smaller modules increase the contact ratio, improving gear motion smoothness. This design also reduces tooth surface friction and flash temperature, enhancing scuffing resistance.

(1)

Increasing the pressure angle decreases the contact ratio, especially affecting the transverse contact ratio. It also reduces sliding velocity and tooth-bending safety factor, potentially impacting bearing life. However, a larger pressure angle can enhance scuffing load-carrying capacity.

(2)

A higher helix angle significantly increases the total contact ratio, improving load-carrying capacity and contact fatigue life. It also reduces TSFT and improves the scuffing safety factor. However, higher helix angles increase axial force, which could reduce bearing life.

(3)

In summary, gear design significantly impacts the performance of electric vehicle gearboxes. Increasing teeth number and using larger pressure and helix angles can improve certain performance metrics, but care must be taken to balance these improvements against potential reductions in bearing life. In the next step, we will carry out a large number of experimental tests on the specific effects of the macro-parameters of the gears on the scuffing load-carrying capacity, as well as on how to design better gear parameters, to verify the reasonableness of the results obtained in this paper from theoretical analyses and numerical simulations.