In modern mechanical transmission systems, spur gear pairs serve as fundamental components for power and motion transfer across numerous industrial applications. With the continuous advancement of gear transmission technology, the demand for higher load capacity, improved efficiency, and enhanced reliability has driven researchers to explore innovative gear designs beyond conventional parameters. Among these innovations, the multi-modulus involute spur gear pair, where the driving and driven gears possess different module values and pressure angles, has attracted increasing attention due to its unique geometric characteristics and potential advantages in specific transmission scenarios.
Scuffing or gluing failure represents one of the most severe failure modes in spur gear transmission systems. This phenomenon occurs when the lubricating oil film between meshing tooth surfaces becomes insufficiently thick, leading to direct metallic contact, localized welding, and subsequent tearing of surface material. The consequences of gluing failure include increased vibration, excessive noise, reduced transmission efficiency, and catastrophic gear failure. Understanding the mechanisms that govern gluing characteristics and identifying effective measures to enhance gluing load capacity are therefore critical aspects of modern spur gear design and lubrication engineering.
This research investigates the gluing characteristics of multi-modulus involute spur gear pairs through a comprehensive approach that combines elastohydrodynamic lubrication (EHL) theory with finite element thermal field simulation. The study systematically analyzes how various transmission parameters affect the minimum oil film thickness, film thickness ratio, gear body temperature distribution, and gluing safety coefficient. The findings provide theoretical guidance for optimizing multi-modulus spur gear design to achieve superior lubrication performance and enhanced resistance to gluing failure.
1. Introduction and Research Background
Spur gear transmission has become one of the most widely applied mechanical transmission methods in manufacturing industries due to its high transmission efficiency, stable operation, and compact structure. The applications span high-speed railways, aerospace engineering, construction machinery, industrial robotics, and numerous other fields requiring reliable power transmission. Despite the maturity of conventional spur gear design methodologies, innovations in gear geometry and meshing theory continue to emerge, driven by the ever-increasing demands for higher power density and improved operational reliability.
The traditional design of spur gear pairs assumes that the driving and driven gears share identical module values and pressure angles. However, the fundamental condition for correct gear meshing is that the normal pitches of the two mating gears must be equal. This condition can be mathematically expressed as \( m_1 \cos \alpha_1 = m_2 \cos \alpha_2 \), where \( m_1 \) and \( m_2 \) represent the modules of the driving and driven gears respectively, and \( \alpha_1 \) and \( \alpha_2 \) denote their respective pressure angles. From this relationship, it becomes evident that gears with different modules can indeed mesh correctly if their pressure angles are appropriately adjusted to satisfy the normal pitch equality condition. Such gear pairs, where the modules of the driving and driven gears differ, are referred to as multi-modulus spur gear pairs.
The research on multi-modulus spur gear pairs represents a significant extension of conventional gear design theory. When compared with conventional equal-modulus spur gear pairs, multi-modulus configurations offer additional design flexibility that can potentially improve gear meshing performance, load distribution characteristics, and lubricating conditions. However, the gluing characteristics of multi-modulus spur gear pairs remain insufficiently understood, necessitating systematic investigation to support their practical engineering applications.
Gluing failure in spur gear transmissions is intimately related to two critical parameters: the lubricating oil film thickness between meshing tooth surfaces and the gear body temperature. If the oil film becomes excessively thin, direct metallic contact occurs between the tooth surfaces, resulting in adhesion and subsequent tearing of surface material. Conversely, elevated gear body temperatures can degrade lubricant properties, accelerate oil film breakdown, and promote gluing failure initiation. Consequently, accurate prediction of oil film thickness and temperature distribution is essential for evaluating and enhancing the gluing resistance of spur gear pairs.
The elastohydrodynamic lubrication theory, first established through the pioneering work of Reynolds and subsequently developed by scholars such as Dowson, Higginson, and Hamrock, provides a robust framework for analyzing the lubricating conditions in non-conformal contacts such as those encountered in spur gear meshing. The minimum oil film thickness, derived from EHL theory, serves as a key indicator of lubrication quality and gluing resistance. Meanwhile, finite element thermal analysis enables the determination of gear body temperature distribution, which complements the EHL analysis by providing insights into the thermal aspects of gluing failure.
This study focuses on the gluing characteristics analysis of multi-modulus involute spur gear pairs based on elastohydrodynamic lubrication theory. The research establishes the mathematical foundation for multi-modulus spur gear meshing parameters, develops computational models for minimum oil film thickness along the line of action, and performs finite element thermal field simulations to assess gear body temperature characteristics. By systematically varying the module ratio, displacement coefficients, and input torque, the influence of these parameters on spur gear lubrication performance and gluing load capacity is thoroughly investigated.
2. Fundamental Parameters and Meshing Theory of Multi-Modulus Spur Gear Pairs
2.1 Correct Meshing Condition
Based on the gear meshing principle, the correct meshing condition for a pair of involute spur gears requires equality of normal pitches. For multi-modulus spur gear pairs, this condition is expressed as:
$$m_1 \cos \alpha_1 = m_2 \cos \alpha_2$$
where \( m_1 \) and \( \alpha_1 \) are the module and pressure angle of the driving gear, while \( m_2 \) and \( \alpha_2 \) correspond to the driven gear. The module ratio \( \delta_m \) is defined as \( \delta_m = m_1 / m_2 \). This fundamental relationship enables the coordinated selection of modules and pressure angles for the two mating gears to ensure correct meshing despite their geometric differences.
2.2 No-Backlash Meshing Equation
In practical spur gear transmissions, although a certain amount of backlash is necessary to accommodate manufacturing tolerances and thermal expansion, the calculation of nominal gear dimensions and center distance conventionally assumes zero backlash. For multi-modulus spur gear pairs to achieve no-backlash meshing, the tooth thickness of one gear on its pitch circle must equal the tooth space width of the other gear on its pitch circle. Through geometric analysis of the meshing geometry, the following no-backlash meshing equation for multi-modulus spur gear pairs is established:
$$z_1 (\text{inv} \, \alpha’ – \text{inv} \, \alpha_1) + z_2 (\text{inv} \, \alpha’ – \text{inv} \, \alpha_2) = 2(x_1 + x_2) \tan \alpha_n$$
where \( z_1 \) and \( z_2 \) are the tooth numbers of the driving and driven gears, \( \alpha’ \) is the operating pressure angle under the actual mounting condition, \( x_1 \) and \( x_2 \) are the displacement coefficients of the driving and driven gears respectively, and \( \alpha_n \) represents the equivalent normal pressure angle.
Rearranging this equation yields the explicit expression for the involute function of the operating pressure angle:
$$\text{inv} \, \alpha’ = \frac{2(x_1 \tan \alpha_1 + x_2 \tan \alpha_2) + z_1 \text{inv} \alpha_1 + z_2 \text{inv} \alpha_2}{z_1 + z_2}$$
This equation demonstrates that the operating pressure angle of a multi-modulus spur gear pair depends not only on the displacement coefficients but also significantly on the individual pressure angles of the two gears, which differs fundamentally from conventional equal-modulus gear pairs.
2.3 Actual Center Distance
For multi-modulus spur gear pairs, because the modules and pressure angles of the two gears differ, the actual center distance differs from the standard center distance regardless of whether modification is applied. The actual center distance, defined as the sum of the pitch circle radii, is calculated as:
$$a’ = r_1′ + r_2′ = \frac{m_1 z_1 \cos \alpha_1}{2 \cos \alpha’} + \frac{m_2 z_2 \cos \alpha_2}{2 \cos \alpha’} = \frac{\cos \alpha_1}{2 \cos \alpha’} (m_1 z_1 + m_2 z_2)$$
This expression highlights the coupling between the module ratio, pressure angle difference, and operating pressure angle in determining the mounting center distance.
2.4 Contact Ratio
The contact ratio, which quantifies the average number of tooth pairs in contact during meshing, is a critical parameter for transmission continuity and smoothness. For multi-modulus spur gear pairs, the contact ratio is derived based on the geometric relationships along the line of action:
$$\varepsilon_\alpha = \frac{r_{b1} (\tan \alpha_{a1} – \tan \alpha’) + r_{b2} (\tan \alpha_{a2} – \tan \alpha’)}{p_b}$$
where \( r_{b1} \) and \( r_{b2} \) are the base circle radii, \( \alpha_{a1} \) and \( \alpha_{a2} \) are the tip circle pressure angles, and \( p_b \) is the base pitch. Alternatively, the contact ratio can be expressed using the module and pressure angle parameters as:
$$\varepsilon_\alpha = \frac{m_1 z_1 \cos \alpha_1 (\tan \alpha_{a1} – \tan \alpha’) + m_2 z_2 \cos \alpha_2 (\tan \alpha_{a2} – \tan \alpha’)}{2 \pi m_1 \cos \alpha_1}$$
The contact ratio must exceed unity to ensure continuous transmission, with a value of at least 1.2 generally recommended for practical engineering applications.
2.5 Constraints on Displacement Coefficients
When investigating the gluing characteristics of multi-modulus spur gears, the selection of displacement coefficients must satisfy several essential constraints to ensure proper meshing without interference or operational issues:
(1) Tooth Thickness Constraint: The tooth tip thickness must not become excessively thin during gear modification. The constraint condition ensures that the tip thickness remains greater than 0.25 times the module:
$$\frac{x_1 \pi}{2} + \frac{2x_1 \tan \alpha_1}{z_1} + (\text{inv} \, \alpha_1 – \text{inv} \, \alpha_{a1}) \geq 0.25$$
(2) Non-Interference Condition: To prevent interference between the tooth profiles during meshing, the following conditions must be satisfied:
$$\tan \alpha’ – \frac{4(h_a^* – x_1)}{z_1 \sin 2\alpha’} \geq \tan \alpha_2 – \frac{z_2}{z_1} (\tan \alpha_{a2} – \tan \alpha’)$$
(3) Contact Ratio Requirement: The contact ratio of the gear pair must be maintained above 1.2 to ensure transmission smoothness and continuity.
3. Minimum Oil Film Thickness Calculation for Multi-Modulus Spur Gear Pairs
3.1 Elastohydrodynamic Lubrication Theory
The elastohydrodynamic lubrication theory for line contact problems has been established through decades of research evolution. The fundamental governing equations include the Reynolds equation, film thickness equation, elastic deformation equation, viscosity-pressure relationship, density-pressure relationship, and load balance equation. For spur gear applications, the minimum oil film thickness formula developed by Yang and Wen based on numerical solutions with Roelands viscosity-pressure relationship is employed:
$$h_{\min} = \frac{6.76 (\alpha \eta_0)^{0.53} V_m^{0.75} \rho_\Sigma^{0.41}}{E^{0.06} W^{0.16}}$$
where \( \alpha \) is the pressure-viscosity coefficient of the lubricant, \( \eta_0 \) is the dynamic viscosity at atmospheric pressure, \( V_m \) is the average tangential velocity of the tooth surfaces at the meshing point, \( \rho_\Sigma \) is the comprehensive curvature radius, \( E \) is the equivalent elastic modulus, and \( W \) is the normal load per unit tooth width.
3.2 Linear Coordinate Along the Line of Action
To analyze the variation of oil film thickness along the meshing cycle, a dimensionless linear coordinate parameter \( \Gamma \) is established on the line of action. The coordinate is defined such that the meshing node P corresponds to \( \Gamma = 0 \), the theoretical meshing limit point \( N_1 \) corresponds to \( \Gamma = -1 \), and the opposite limit point \( N_2 \) corresponds to \( \Gamma = u \), where \( u = z_2/z_1 \) is the gear ratio. For any arbitrary meshing point c, the coordinate is given by:
$$\Gamma_c = \frac{N_1 C}{N_1 P} = 1 – \frac{\tan \alpha_c}{\tan \alpha’}$$
The special points along the line of action include the beginning of engagement \( B_2 \), the end of engagement \( B_1 \), and the single-tooth engagement boundaries C and D. Their coordinate values are determined from the gear geometry as:
$$\Gamma_{B_2} = \frac{r_{a2}^2 – r_{b2}^2}{N_1 P} – 1, \quad \Gamma_{B_1} = \frac{r_{a1}^2 – r_{b1}^2}{N_1 P} – 1$$
$$\Gamma_C = \Gamma_{B_2} + \frac{p_b}{N_1 P}, \quad \Gamma_D = \Gamma_{B_1} – \frac{p_b}{N_1 P}$$
3.3 Comprehensive Curvature Radius
Based on the gear meshing geometry, the curvature radii of the driving and driven gear tooth surfaces at any meshing point, along with the comprehensive curvature radius of the gear pair, are calculated as:
$$\rho_1 = r_1′ (1 + \Gamma) \sin \alpha’, \quad \rho_2 = r_2′ (u – \Gamma) \sin \alpha’$$
$$\rho_\Sigma = \frac{\rho_1 \rho_2}{\rho_1 + \rho_2}$$
The comprehensive curvature radius exhibits a non-monotonic distribution along the line of action. It initially increases, reaches its maximum value at the point where the individual curvature radii of the two gears become equal, and subsequently decreases. This characteristic distribution significantly influences the oil film thickness variation during the meshing cycle.
3.4 Average Tangential Velocity
The tangential velocities of the tooth surfaces at any meshing point are determined from the rotational speeds and the distance from the instantaneous center of rotation:
$$V_1 = \omega_1 \rho_1 = \frac{\pi n_1}{30} r_1′ (1 + \Gamma) \sin \alpha’$$
$$V_2 = \omega_2 \rho_2 = \frac{\pi n_1}{30 u} r_2′ (u – \Gamma) \sin \alpha’$$
The average tangential velocity, which represents the entraining velocity responsible for generating the hydrodynamic pressure, is calculated as:
$$V_m = \frac{V_1 + V_2}{2}$$
The average tangential velocity increases linearly along the line of action from the beginning of engagement to the end of engagement. At the pitch point, where \( \Gamma = 0 \), the two tangential velocities become equal, and the gear surfaces undergo pure rolling with zero sliding velocity.
3.5 Normal Load Per Unit Tooth Width
During the spur gear meshing cycle, the load is shared between one or two pairs of teeth depending on the engagement zone. The load distribution coefficient \( X_\Gamma \) is introduced to account for this variation:
$$X_\Gamma = \begin{cases} \frac{1}{3} + \frac{1}{3} \frac{\Gamma – \Gamma_{B_2}}{\Gamma_C – \Gamma_{B_2}}, & \Gamma_{B_2} \leq \Gamma \leq \Gamma_C \\ 1, & \Gamma_C \leq \Gamma \leq \Gamma_D \\ \frac{1}{3} + \frac{1}{3} \frac{\Gamma_{B_1} – \Gamma}{\Gamma_{B_1} – \Gamma_D}, & \Gamma_D \leq \Gamma \leq \Gamma_{B_1} \end{cases}$$
The normal load per unit tooth width is then expressed as:
$$W = \frac{T_1 X_\Gamma}{r_1′ b \cos \alpha’}$$
where \( T_1 \) is the input torque and \( b \) is the tooth width. The load distribution demonstrates abrupt changes at the transitions between single and double tooth engagement zones, which directly causes corresponding discontinuities in the oil film thickness distribution.
| Parameter | Symbol | Value |
|---|---|---|
| Number of teeth (driving/driven) | \( z_1 / z_2 \) | 23 / 30 |
| Tooth width (mm) | \( b \) | 20 |
| Input speed (r/min) | \( n_1 \) | 2000 |
| Input torque (N·mm) | \( T_1 \) | 400,000 |
| Elastic modulus (MPa) | \( E \) | 206,000 |
| Poisson’s ratio | \( \nu \) | 0.3 |
| Density (kg/m³) | \( \rho \) | 7850 |
| Specific heat capacity [J/(kg·℃)] | \( c \) | 465 |
| Thermal conductivity [W/(m·℃)] | \( \lambda \) | 46 |
| Lubricant dynamic viscosity (MPa·s) | \( \eta_0 \) | 2.88×10⁻⁷ |
| Pressure-viscosity coefficient (MPa⁻¹) | \( \alpha \) | 0.021 |
4. Oil Film Thickness Variation Characteristics Analysis
4.1 Film Thickness Ratio and Gluing Criterion
To establish a direct correlation between the elastohydrodynamic oil film thickness and the gluing characteristics of spur gear pairs, the film thickness ratio \( \lambda \) is adopted as a quantitative indicator:
$$\lambda = \frac{h_{\min – p}}{\sqrt{R_{a1}^2 + R_{a2}^2}}$$
where \( h_{\min-p} \) represents the minimum oil film thickness at the pitch point, and \( R_{a1} \), \( R_{a2} \) are the surface roughness values of the driving and driven gear tooth surfaces. The pitch point is selected as the representative position because it lies within the single-tooth engagement zone and experiences the full load. The lubricating condition and gluing risk are categorized according to the following criteria:
- When \( \lambda > 3 \): Full EHL regime, minimal gluing risk
- When \( 1 < \lambda < 3 \): Mixed lubrication regime, moderate wear possible
- When \( \lambda < 1 \): Boundary lubrication regime, high gluing risk
For general spur gear design, a film thickness ratio of at least 1.5 is typically recommended to ensure adequate protection against gluing failure.
4.2 Parameter Combinations for Multi-Modulus Spur Gear Cases
In this analysis, the driven gear parameters are maintained constant with \( m_2 = 4 \) mm, \( \alpha_2 = 20° \), while the driving gear module and pressure angle vary according to the module ratio. The module ratio range is constrained between 0.98 and 1.03 to maintain geometric integrity and satisfy the design constraints discussed earlier. Table 2 presents the parameter combinations employed in the study.
| Case | \( m_1 \) (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 |
4.3 Influence of Module Ratio on Minimum Oil Film Thickness
The computational analysis reveals that the minimum oil film thickness of multi-modulus spur gear pairs varies significantly along the line of action, with the peak value occurring near the end of engagement where the driving gear tooth tip meshes with the driven gear tooth root. Abrupt changes in film thickness are observed at the transitions between single and double tooth engagement zones due to load discontinuities. As the module ratio increases, the contact ratio decreases, resulting in a shorter total meshing length and more pronounced distribution of film thickness variation.
For the case of \( \delta_m = 0.98 \), the pitch point film thickness is calculated as \( h_{\min-p} = 1.893 \, \mu m \). Using a surface roughness of \( R_{a1} = R_{a2} = 0.8 \, \mu m \), the film thickness ratio is determined to be \( \lambda = 1.67 \). Table 3 summarizes the results for different module ratios.
| \( \delta_m \) | \( h_{\min-p} \) (μm) | \( \lambda \) | Film Thickness Increase (%) | \( \lambda \) Increase (%) |
|---|---|---|---|---|
| 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 |
The results clearly demonstrate that increasing the module ratio from 0.98 to 1.03 enhances both the minimum oil film thickness and the film thickness ratio by approximately 23.4% and 24.0% respectively. This improvement is attributed to the increased comprehensive curvature radius and average tangential velocity that accompany larger module ratios. Therefore, increasing the module ratio constitutes an effective measure for improving the lubrication performance and gluing load capacity of multi-modulus spur gear pairs.
4.4 Influence of Driving Gear Displacement Coefficient
When the driving gear displacement coefficient \( x_1 \) varies while other parameters remain constant (with \( \delta_m = 1.01 \) and \( x_2 = 0 \)), the minimum oil film thickness demonstrates consistent distribution trends along the line of action. As \( x_1 \) increases from -0.3 to 0.2, the contact ratio decreases, the single-tooth engagement zone expands while shifting toward the driving gear tooth tip, and the oil film thickness at every meshing position increases due to the enhanced comprehensive curvature radius and average tangential velocity.
| \( x_1 \) | \( h_{\min-p} \) (μm) | \( \lambda \) | Film Thickness Increase (%) | \( \lambda \) Increase (%) |
|---|---|---|---|---|
| -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 | 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 |
The analytical results confirm that increasing the driving gear displacement coefficient from -0.3 to 0.2 improves the minimum oil film thickness by 21.55% and the film thickness ratio by 21.43%, thereby enhancing the gluing load capacity of the multi-modulus spur gear pair.
4.5 Influence of Driven Gear Displacement Coefficient
Similar parametric analysis is conducted for the driven gear displacement coefficient \( x_2 \) with \( \delta_m = 1.01 \) and \( x_1 = 0 \). The results are presented in Table 5.
| \( x_2 \) | \( h_{\min-p} \) (μm) | \( \lambda \) | Film Thickness Increase (%) | \( \lambda \) Increase (%) |
|---|---|---|---|---|
| -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 | 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 |
Increasing the driven gear displacement coefficient from -0.3 to 0.2 results in a 19.65% improvement in minimum oil film thickness and a 19.41% improvement in film thickness ratio, confirming that both driving and driven gear modifications positively influence the lubrication performance of multi-modulus spur gear pairs.
4.6 Influence of Input Torque
Since the input torque affects the normal load per unit tooth width while having negligible influence on curvature radius and tangential velocity, its impact on film thickness operates through the load term in the EHL formula. The results for varying torques are summarized in Table 6.
| \( T_1 \) (×10⁵ N·mm) | \( h_{\min-p} \) (μm) | \( \lambda \) | Film Thickness Increase (%) | \( \lambda \) Increase (%) |
|---|---|---|---|---|
| 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 |
Reducing the input torque from 6.5×10⁵ N·mm to 4.0×10⁵ N·mm leads to an 8.07% increase in minimum oil film thickness and an 8.52% increase in film thickness ratio. These findings suggest that operating multi-modulus spur gears at reduced torque levels, when permitted by the application requirements, can enhance lubrication performance and gluing resistance.
5. Finite Element Thermal Field Analysis of Multi-Modulus Spur Gear Pairs
5.1 Theoretical Foundation of Gear Temperature Field
Heat transfer in spur gear transmission systems occurs through three fundamental mechanisms: conduction, convection, and radiation. In the context of gear meshing, conduction and convection are the dominant modes. The frictional heat generated at the contact interface is conducted into the gear body while simultaneously being convected away by the lubricating oil and surrounding air.
For a spur gear pair operating under steady-state conditions, the thermal energy balance achieves equilibrium when the rate of frictional heat generation equals the rate of heat dissipation through convection and conduction. The steady-state heat balance equation for gears can be simplified to the Laplace equation:
$$\frac{\partial^2 T_B}{\partial x^2} + \frac{\partial^2 T_B}{\partial y^2} + \frac{\partial^2 T_B}{\partial z^2} = 0$$
where \( T_B \) represents the gear body temperature at any spatial location within the gear body.
5.2 Boundary Conditions for Gear Temperature Field Analysis
Due to the symmetry of gear structures, the temperature field analysis can be performed on a single tooth. The tooth surfaces are divided into distinct regions based on their thermal boundary characteristics:
- Meshing surface (m-region): Subject to both friction heat flux and convective heat transfer, with boundary condition:
$$-\lambda \frac{\partial T_B}{\partial n} = h_m (T_B – T_m) – q_m$$
- Gear face surfaces (e-region): Subject to convective heat transfer only:
$$-\lambda \frac{\partial T_B}{\partial n} = h_e (T_B – T_e)$$
- Tooth tip, root, and non-meshing surfaces (d-region): Subject to convective heat transfer:
$$-\lambda \frac{\partial T_B}{\partial n} = h_d (T_B – T_d)$$
- Tooth contact surfaces and bottom surface (f-region): Considered adiabatic:
$$\frac{\partial T_B}{\partial n} = 0$$
5.3 Convective Heat Transfer Coefficient Calculation
The convective heat transfer coefficients for various gear surfaces are determined using established empirical correlations. For the tooth meshing surface, considering the flow of lubricant over the gear surface:
$$h_m = 0.228 \, Re_f^{0.731} \, Pr_f^{0.333} \frac{\lambda_f}{d’}$$
where \( Re_f \) is the Reynolds number, \( Pr_f \) is the Prandtl number, \( \lambda_f \) is the thermal conductivity of the lubricant, and \( d’ \) is the pitch circle diameter.
For the gear face surfaces, the rotating disk model is employed with consideration of laminar, transition, and turbulent flow regimes:
$$h_e = \begin{cases} 0.308 (m + 2)^{0.5} Pr_f^{0.5} \lambda_f \sqrt{\frac{\omega}{\nu_f}}, & 2 \times 10^5 \\ 10^{-4} \lambda_f \frac{Re_f}{r’}, & 2 \times 10^5 \leq Re_f \leq 2.5 \times 10^5 \\ 0.0197 (2.6m + 2.6)^{0.2} Pr_f^{0.6} \lambda_f Re_f^{0.8} \frac{1}{r’}, & Re_f \geq 2.5 \times 10^5 \end{cases}$$
For the tooth tip, root, and non-meshing surfaces, a simplified relation is applied:
$$h_d = 0.664 \, Pr_f^{0.333} \lambda_f \sqrt{\frac{\omega}{\nu_f}}$$
| Parameter | Value |
|---|---|
| Lubricant type | SCH632 |
| Kinematic viscosity (m²/s) | 92.5×10⁻⁶ |
| Density (kg/m³) | 870 |
| Specific heat capacity [J/(kg·℃)] | 2000 |
| Thermal conductivity [W/(m·℃)] | 0.14 |
5.4 Friction Heat Generation Analysis
The frictional heat generated during spur gear meshing is directly related to the contact stress and relative sliding velocity at each meshing point. Based on Hertzian contact theory, the average contact stress at any meshing position is calculated as:
$$\sigma_h = \sqrt{\frac{\omega_e E’}{4\pi \rho_\Sigma}}$$
where \( \omega_e \) is the line load per unit length, \( E’ \) is the equivalent elastic modulus, and \( \rho_\Sigma \) is the comprehensive curvature radius at the meshing point.
The relative tangential velocity between the two tooth surfaces at any meshing point is:
$$V_c = |V_1 – V_2| = \left| \frac{\pi n_1}{30} r_1′ (1 + \Gamma) \sin \alpha’ – \frac{\pi n_1}{30u} r_2′ (u – \Gamma) \sin \alpha’ \right|$$
The instantaneous frictional heat flux at any meshing point is then expressed as:
$$Q = \delta f \sigma_h V_c$$
where \( \delta \) is the heat conversion coefficient (typically 0.95) and \( f \) is the friction coefficient.
Since the driving and driven gears experience different numbers of meshing cycles per revolution, a heat partition coefficient is introduced to distribute the frictional heat between the two gears:
$$\beta_1 = \frac{\lambda_1 \rho_1′ c_{v1}}{\lambda_1 \rho_1′ c_{v1} + \lambda_2 \rho_2′ c_{v2}}, \quad \beta_2 = 1 – \beta_1$$
The average frictional heat flux over one complete meshing cycle is calculated by integrating the instantaneous heat flux and dividing by the cycle period:
$$q_1 = \frac{\omega_1}{2\pi} \int \frac{Q_1}{V_1} \, ds, \quad q_2 = \frac{\omega_1}{2\pi} \int \frac{Q_2}{V_2 u} \, ds$$
5.5 Average Friction Heat Flux Characteristics
Computational analysis reveals that the average friction heat flux exhibits distinct distribution patterns along the line of action. For multi-modulus spur gear pairs, the active wheel demonstrates higher average friction heat flux compared with the driven wheel due to fewer teeth and consequently more frequent meshing cycles per revolution. The heat flux peaks near the tooth root and tip regions, while attaining minimum values near the pitch point where sliding velocity approaches zero.
Parametric analysis demonstrates that increasing the module ratio, increasing the displacement coefficients, or reducing the input torque all result in decreased average friction heat flux along the meshing line. These reductions are most pronounced near the tooth root and tip regions, which are critical zones for gluing failure initiation.
5.6 Three-Dimensional Modeling of Multi-Modulus Spur Gears
The finite element thermal analysis requires accurate three-dimensional models of the multi-modulus spur gears. Due to the complex tooth profiles resulting from different modules, pressure angles, and displacement coefficients, a hybrid parameterized modeling approach combining MATLAB and SolidWorks is employed.
In this approach, MATLAB generates the involute tooth profile curve data points based on the gear geometric equations. The curve data are then imported into SolidWorks through the “Curve through XYZ points” command. The involute curve, along with transition curves, is mirrored to create complete tooth profile outlines. The three-dimensional gear models are generated through the extrusion of tooth profiles and circular pattern operations. The assembled multi-modulus spur gear pair model is verified to ensure proper meshing without interference.
5.7 Steady-State Temperature Field Simulation
Taking advantage of the geometric symmetry of spur gears, the thermal field analysis is performed on a single tooth model. The meshing surface is divided into multiple rectangular strips to accommodate the varying heat flux distribution along the line of action. The computed convective heat transfer coefficients and friction heat fluxes are applied to the respective surface regions in the ANSYS Workbench steady-state thermal analysis module.
The simulation results demonstrate that the temperature distribution trends for driving and driven gears are similar in general characteristics: meshing surfaces exhibit the highest temperatures, gear hubs exhibit the lowest temperatures, and clear temperature gradients are established. High-temperature zones concentrate near the tooth root and tip regions on the meshing surface, with temperatures distributed uniformly along the tooth width direction and decreasing toward the gear face surfaces.
5.8 ISO Standard Temperature Calculation and Gluing Safety Coefficient
To validate the finite element simulation results, the ISO standard provides empirical formulas for calculating gear body integral temperature. The ISO gear body temperature can be expressed as:
$$\theta_M = \theta_{oil} + X_s \theta_{flaint}$$
where \( \theta_{oil} \) is the lubricant temperature, \( X_s \) is the lubrication coefficient, and \( \theta_{flaint} \) is the average instantaneous temperature rise along the meshing line.
The gluing safety coefficient, which provides a quantitative measure of the resistance to gluing failure, is calculated as:
$$S_{intS} = \frac{\theta_{intS}}{\theta_{int}}$$
where \( \theta_{intS} \) is the critical gluing integral temperature and \( \theta_{int} \) is the actual gear integral temperature. A safety coefficient greater than 1 indicates low gluing risk, while values below 1 signify high gluing risk.
The gear integral temperature is composed of the gear body temperature and the weighted average flash temperature rise:
$$\theta_{int} = \theta_M + C_2 \theta_{flaint}$$
where \( C_2 \) is a weighting coefficient taken as 1.5 in this analysis.
5.9 Influence of Module Ratio on Temperature Field Characteristics
The temperature field simulations are performed for various module ratios while maintaining all other parameters constant. Table 8 summarizes the ISO theoretical temperatures, simulation results, gluing safety coefficients, and corresponding film thickness ratios.
| \( \delta_m \) | ISO Temperature (°C) | Max Simulated Temperature (°C) | Average Simulated Temperature (°C) | Safety Coefficient \( S_{intS} \) | Film Thickness Ratio \( \lambda \) |
|---|---|---|---|---|---|
| 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 |
As the module ratio increases from 0.98 to 1.03, both the ISO theoretical temperature and simulation temperatures decrease. The maximum simulated temperature decreases by approximately 7.43%, the average simulated temperature by 4.54%, and the ISO theoretical temperature by 3.43%. The gluing safety coefficient increases from 1.73 to 1.92, representing an improvement of 10.98%. Meanwhile, the film thickness ratio increases by 23.95%. Both indicators consistently demonstrate that increasing the module ratio enhances the gluing resistance and lubrication quality of multi-modulus spur gear pairs.
5.10 Influence of Driving Gear Displacement Coefficient on Temperature Field
The temperature field analysis is extended to investigate the influence of the driving gear displacement coefficient, with results summarized in Table 9.
| \( x_1 \) | ISO Temperature (°C) | Max Simulated Temperature (°C) | Average Simulated Temperature (°C) | Safety Coefficient \( S_{intS} \) | Film Thickness Ratio \( \lambda \) |
|---|---|---|---|---|---|
| -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 | 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 |
As the driving gear displacement coefficient increases from -0.3 to 0.2, the gear body temperatures decrease, with the maximum simulated temperature dropping by 6.25% and the average simulated temperature by 3.80%. The gluing safety coefficient increases by 9.14%, while the film thickness ratio increases by 21.43%. These findings confirm that positive displacement modification of the driving gear effectively reduces thermal loading and enhances gluing resistance.
5.11 Influence of Driven Gear Displacement Coefficient on Temperature Field
The analysis of the driven gear displacement coefficient yields similar qualitative trends, as summarized in Table 10.
| \( x_2 \) | ISO Temperature (°C) | Max Simulated Temperature (°C) | Average Simulated Temperature (°C) | Safety Coefficient \( S_{intS} \) | Film Thickness Ratio \( \lambda \) |
|---|---|---|---|---|---|
| -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 | 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 the driven gear displacement coefficient from -0.3 to 0.2 results in the safety coefficient increasing by 12.14% and the film thickness ratio increasing by 19.41%. The results consistently support the conclusion that positive modification of either gear enhances the gluing resistance of multi-modulus spur gear pairs.
5.12 Influence of Input Torque on Temperature Field
Table 11 presents the simulation results for varying input torques with \( \delta_m = 1.01 \) and both displacement coefficients set to zero.
| \( T_1 \) (×10⁵ N·mm) | ISO Temperature (°C) | Max Simulated Temperature (°C) | Average Simulated Temperature (°C) | Safety Coefficient \( S_{intS} \) | Film Thickness Ratio \( \lambda \) |
|---|---|---|---|---|---|
| 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 |
Reducing the input torque from 6.5×10⁵ N·mm to 4.0×10⁵ N·mm decreases the maximum simulated gear body temperature by 10.06% and the average simulated temperature by 6.23%. The gluing safety coefficient increases by 21.15%, while the film thickness ratio increases by 8.52%. Both the thermal field simulations and the elastohydrodynamic lubrication calculations consistently demonstrate that operating multi-modulus spur gears at reduced torque levels improves their gluing resistance.
6. Discussion of Results
The comprehensive analysis of multi-modulus spur gear gluing characteristics through both elastohydrodynamic lubrication theory and finite element thermal field simulation has yielded consistent and complementary results. The film thickness ratio, calculated from EHL theory, and the gluing safety coefficient, obtained from thermal field simulations, serve as complementary indicators for evaluating the gluing resistance of multi-modulus spur gears.
The results demonstrate that increasing the module ratio \( \delta_m \) provides threefold benefits: it increases the comprehensive curvature radius and tangential velocity, thereby enhancing hydrodynamic lubrication; it reduces the average friction heat flux, thereby lowering the thermal loading; and it increases both the film thickness ratio and gluing safety coefficient, indicating superior gluing resistance. When \( \delta_m \) increases from 0.98 to 1.03, the film thickness ratio improves from 1.67 to 2.07, moving the spur gear lubrication regime from mixed lubrication toward fully elastohydrodynamic lubrication.
Positive displacement modification of either the driving or driven gear similarly improves gluing characteristics through combined geometric and thermal effects. The displacement coefficient modifies the tooth profile curvature distribution, affecting both the entraining velocity and the load sharing characteristics. The positive displacement coefficients also modify the meshing geometry such that the friction heat flux distribution becomes more favorable, reducing peak temperatures in critical zones near the tooth root and tip.
Input torque reduction demonstrates a clear monotonic improvement in gluing resistance primarily through reduced contact stress and consequently lower friction heat generation. While the film thickness improvement through torque reduction is relatively modest compared with geometric modifications, the thermal benefits are substantial, as evidenced by the significant decrease in gear body temperature.
The validation of finite element simulation results against ISO theoretical calculations provides confidence in the reliability of the thermal analysis. The temperature differences between simulation and ISO calculations remain within reasonable bounds, with typical deviations in the range of 2.5% to 4.2% for maximum temperatures and up to 8.5% for average temperatures. These deviations are expected given the simplifying assumptions inherent in both approaches.
7. Conclusions and Future Perspectives
This research presents a comprehensive investigation of the gluing characteristics of multi-modulus involute spur gear pairs through the integration of elastohydrodynamic lubrication theory and finite element thermal field analysis. The following conclusions can be drawn from this study:
(1) The no-backlash meshing equation for multi-modulus spur gears has been established based on the correct meshing condition \( m_1 \cos \alpha_1 = m_2 \cos \alpha_2 \), enabling the calculation of the operating pressure angle, actual center distance, and contact ratio. The displacement coefficient selection for such gear pairs must satisfy tooth thickness, non-interference, and contact ratio constraints.
(2) Based on the elastohydrodynamic lubrication theory, the minimum oil film thickness of multi-modulus spur gears along the line of action exhibits non-uniform distribution, with abrupt changes at the transitions between single and double tooth engagement zones. The minimum oil film thickness is maximum near the end of engagement and minimum in the double-tooth engagement zone near the beginning of engagement.
(3) Increasing the module ratio \( \delta_m \) from 0.98 to 1.03 increases the film thickness ratio by approximately 24%, significantly improving the lubrication quality and moving the spur gear toward full elastohydrodynamic lubrication conditions while reducing the gluing risk.
(4) Increasing either the driving or driven gear displacement coefficient from -0.3 to 0.2 improves the film thickness ratio by approximately 21% and 19% respectively, while simultaneously reducing the gear body temperature and improving the gluing safety coefficient. Positive gear modification is therefore confirmed as an effective measure for enhancing the gluing resistance of multi-modulus spur gears.
(5) Reducing the input torque from 6.5×10⁵ N·mm to 4.0×10⁵ N·mm improves the film thickness ratio by approximately 8.5% and the gluing safety coefficient by approximately 21%, demonstrating that operational parameter optimization can significantly enhance the gluing resistance of multi-modulus spur gear pairs.
(6) The finite element thermal field simulation results exhibit consistent trends with the elastohydrodynamic lubrication calculations. Both approaches confirm that increasing the module ratio, increasing the displacement coefficients, and reducing the input torque enhance the gluing load capacity of multi-modulus spur gears, thereby verifying the rationality and validity of both analytical methods.
Future research directions may include experimental validation of the theoretical and numerical predictions using appropriate test rigs, extension of the analysis to consider the effects of tooth profile modifications and surface treatments, and investigation of the dynamic behavior of multi-modulus spur gear systems under transient operating conditions. Additionally, the influence of lubrication parameters such as viscosity grade, additive chemistry, and oil supply temperature on the gluing characteristics of multi-modulus spur gears warrants further systematic investigation.