Spur gears are widely used in various industrial applications due to their compact structure and high transmission efficiency. However, most rolling bearings, gear transmissions, and cam mechanisms operate under mixed lubrication conditions, making it essential to study the lubrication performance of spur gears to ensure reliability and longevity. This article explores the transient thermal mixed lubrication properties of involute spur gears under line contact conditions, considering factors such as non-Newtonian fluids, transient effects, thermal effects, and surface roughness. The analysis is based on statistical models and asperity contact models, incorporating the Patir and Cheng average flow model and the ZMC asperity contact model to account for elastic, elastoplastic, and fully plastic deformation stages. The influence of integrated surface roughness, roughness texture parameters, gear geometric parameters, and operating conditions on lubrication performance is systematically investigated.
The geometric parameters of involute spur gears are crucial for understanding their lubrication behavior. In a Cartesian coordinate system, a static coordinate system XPY is used to determine the position of points on the gear teeth, while a dynamic coordinate system xKy ensures that the origin coincides with the meshing point at any time. The equivalent curvature radius R at the meshing point is given by the formula: $$ R = \frac{R_1 R_2}{R_1 + R_2} $$ where R1 and R2 are the radii of curvature at the meshing point for the two gears. The entrainment velocity U and sliding velocity Us are defined as: $$ U = \frac{U_1 + U_2}{2} $$ and $$ U_s = |U_1 – U_2| $$ respectively. These parameters vary along the line of action, influencing the lubrication state of spur gears.
The mixed lubrication model for spur gears involves solving the average Reynolds equation, which accounts for transient and thermal effects. For a Ree-Eyring non-Newtonian fluid, the average Reynolds equation is expressed as: $$ \frac{\partial}{\partial x} \left[ \phi_x \frac{\rho}{\eta_e} h^3 \frac{\partial p_h}{\partial x} \right] = 12\phi_c \frac{\partial}{\partial x} (\rho^* U h) + 12\sigma \frac{\partial}{\partial x} \left[ \phi_s \rho^* \frac{U_1 – U_2}{2} \right] + 12\phi_c \frac{\partial}{\partial t} (\rho_e h) $$ where ph is the film pressure, h is the film thickness, ρ is density, η is viscosity, and φx, φs, and φc are pressure flow factor, shear flow factor, and contact factor, respectively. The equivalent parameters are defined as: $$ \left( \frac{\rho}{\eta} \right)_e = 12 \left( \frac{\eta_e \rho’_e}{\eta’_e} – \rho”_e \right) $$ $$ \rho^* = \frac{\rho’_e \eta_e (U_b – U_a) + \rho_e U_a}{U_e} $$ with additional integrals for density and viscosity variations. The pressure flow factor φx is given by: $$ \phi_x = \begin{cases} 1 – C_1 e^{-r\lambda}, & \gamma \leq 1 \\ 1 + C_1 \lambda^{-r}, & \gamma > 1 \end{cases} $$ where λ = h/σ is the film thickness ratio, and γ is the surface texture parameter. The shear flow factor φs is: $$ \phi_s = \begin{cases} A_1 \lambda^{\alpha_1} e^{-\alpha_2 \lambda + \alpha_3 \lambda^2}, & \lambda \leq 5 \\ A_2 e^{-0.25\lambda}, & \lambda > 5 \end{cases} $$ and the contact factor φc is: $$ \phi_c = \begin{cases} e^{-0.6912 + 0.782\lambda – 0.304\lambda^2 + 0.040\lambda^3}, & 0 \leq \lambda < 3 \\ 1, & \lambda \geq 3 \end{cases} $$ These factors play a critical role in modeling the mixed lubrication of spur gears.
The film thickness equation for spur gears, considering transient effects, is: $$ h(x,t) = h_0(t) + \frac{x^2}{2R(t)} – \frac{2}{\pi E’} \int_{x_{in}}^{x_{out}} p(x’,t) \ln(x – x’)^2 dx’ $$ where h0 is the rigid central film thickness, R is the equivalent curvature radius, and E’ is the composite elastic modulus. The viscosity and density of the lubricant are pressure- and temperature-dependent, given by: $$ \eta = \eta_0 \exp \left[ (\ln \eta_0 + 9.67) \left( (1 + 5.1 \times 10^{-9} p_h)^{Z_0} \left( \frac{T – 138}{T_0 – 138} \right)^{-S_0} – 1 \right) \right] $$ and $$ \rho = \rho_0 \left[ 1 + \frac{0.6 \times 10^{-9} p_h}{1 + 1.7 \times 10^{-9} p_h} – 0.00065 (T – T_0) \right] $$ where η0 and ρ0 are ambient viscosity and density, T0 is ambient temperature, Z0 is the viscosity-pressure coefficient, and S0 is the viscosity-temperature coefficient. For Ree-Eyring fluids, the equivalent viscosity is η* = η (τ/τ0) / sinh(τ/τ0), where τ0 is the characteristic shear stress.
The load balance equation for spur gears ensures that the total load is shared between the film and asperity contacts: $$ \int_{x_{in}}^{x_{out}} p_h(x,t) dx + \int_{x_{in}}^{x_{out}} p_a(x,t) dx = w(t) $$ where w(t) is the load per unit width at time t. The energy equation for the lubricant film is: $$ c \left( \rho \frac{\partial T}{\partial t} + \rho u \frac{\partial T}{\partial x} + \rho w \frac{\partial T}{\partial z} \right) – k \frac{\partial^2 T}{\partial t^2} = – \frac{T}{\rho} \frac{\partial \rho}{\partial T} \left( \frac{\partial p}{\partial t} + u \frac{\partial p_h}{\partial x} \right) + \tau \frac{\partial u}{\partial z} + Q_c $$ where Qc = Us fa pa / h is the heat generated by asperity contacts, and fa is the friction coefficient. The heat conduction equations for the gear solids are: $$ c_1 \rho_1 \left( \frac{\partial T}{\partial t} + U_1 \frac{\partial T}{\partial x} \right) = k_1 \frac{\partial^2 T}{\partial z_1^2} $$ and $$ c_2 \rho_2 \left( \frac{\partial T}{\partial t} + U_2 \frac{\partial T}{\partial x} \right) = k_2 \frac{\partial^2 T}{\partial z_2^2} $$ with boundary conditions ensuring continuity of heat flux at the interfaces.
The asperity contact pressure pa is modeled using the ZMC model, which accounts for elastic, elastoplastic, and plastic deformations: $$ p_a = \frac{2}{3} E’ n \beta^{0.5} \sigma^{1.5} \left( \frac{\sigma}{\sigma_s} \right) \frac{1}{\sqrt{2\pi}} \int_{h^* – y_s^* + w_1^*}^{h^* – y_s^*} w^{*1.5} e^{-0.5 (\sigma / \sigma_s z^*)^2} dz^* + 2\pi \cdot HV \cdot n \beta \sigma \frac{1}{\sqrt{2\pi}} \left( \frac{\sigma}{\sigma_s} \right) \int_{h^* – y_s^* + w_2^*}^{\infty} w^* e^{-0.5 (\sigma / \sigma_s z^*)^2} dz^* + \pi \cdot HV \cdot n \beta \sigma \frac{1}{\sqrt{2\pi}} \left( \frac{\sigma}{\sigma_s} \right) \int_{h^* – y_s^* + w_1^*}^{h^* – y_s^* + w_2^*} w^* e^{-0.5 (\sigma / \sigma_s z^*)^2} \left[ 1 – 0.6 \frac{\ln w_2^* – \ln w^*}{\ln w_2^* – \ln w_1^*} \right] \left[ 1 – 2 \left( \frac{w^* – w_1^*}{w_2^* – w_1^*} \right)^3 + 3 \left( \frac{w^* – w_1^*}{w_2^* – w_1^*} \right)^2 \right] dz^* $$ where HV is the Vickers hardness, nβσ is typically 0.05, and σs and ys are given by: $$ \sigma_s = \left[ 1 – 3.7169 \times 10^{-4} (n \beta \sigma)^2 \right] \sigma $$ $$ y_s = \frac{0.0459}{n \beta \sigma} \sigma $$ This comprehensive model allows for accurate prediction of contact pressures in spur gears under mixed lubrication.
Numerical methods are employed to solve the governing equations for spur gears. The multigrid method, multigrid integral method, and row-by-row scanning technique are used to compute film pressure, film thickness, and temperature fields. The computational domain is set as Xin = -4.6 and Xout = 1.4, with a W-cycle and six grid layers. The number of nodes in the x-direction from coarse to fine grids are 31, 61, 121, 241, 481, and 961, while the z-direction has 23 nodes (9 in the film, 6 in each gear, and 1 at each interface). The meshing process of spur gears is divided into 180 instants, with convergence criteria of relative errors less than 1×10⁻³ for pressure and load, and 1×10⁻⁴ for temperature. A simplified load spectrum is used, representing key points along the line of action, such as the start of meshing, pitch point, and end of meshing.
The validation of the model is performed using the M-K formula for minimum film thickness: $$ H_{\text{min}} = \frac{h_{\text{min}}}{R} = 1.652 W^{-0.077} U^{0.716} G^{0.695} \times \left( 1 + 0.026 \sigma^{1.120} V^{0.185} W^{-0.312} U^{-0.809} G^{-0.977} \right) $$ where W, U, G, V, and σ are dimensionless parameters. The numerical results show good agreement with the M-K empirical solutions, with relative errors below 20%, confirming the model’s accuracy for spur gears.
The effects of integrated surface roughness σ on lubrication performance are significant for spur gears. As σ increases, the asperity contact pressure rises, and the load-bearing area slightly expands. However, the film thickness and minimum film thickness increase due to the “pump effect,” while the film thickness ratio decreases. This indicates that while larger roughness may enhance film thickness, it reduces the ratio, potentially leading to increased wear. Therefore, selecting an appropriate σ is crucial for optimizing the lubrication of spur gears.
| σ (μm) | Asperity Contact Pressure (Pa) | Minimum Film Thickness (μm) | Film Thickness Ratio |
|---|---|---|---|
| 0.2 | 1.2e8 | 0.75 | 3.75 |
| 0.4 | 1.5e8 | 0.83 | 2.08 |
| 0.6 | 1.8e8 | 0.90 | 1.50 |
The surface texture parameter γ also plays a vital role in the mixed lubrication of spur gears. As γ increases, the second pressure peak of the film pressure grows and shifts toward the contact center. The asperity contact pressure increases, while the minimum film thickness and film thickness ratio decrease. The maximum temperature rise ratio of the oil film first decreases and then increases with γ, reaching a minimum at γ = 1/3. This suggests that transverse roughness textures (γ < 1) outperform longitudinal ones (γ > 1) in terms of lubrication performance, with γ = 1/3 being optimal for spur gears.
| γ | Second Pressure Peak (Pa) | Minimum Film Thickness (μm) | Maximum Temperature Rise Ratio |
|---|---|---|---|
| 1/6 | 2.5e8 | 0.85 | 1.15 |
| 1/3 | 2.8e8 | 0.80 | 1.10 |
| 1 | 3.2e8 | 0.75 | 1.20 |
| 3 | 3.5e8 | 0.70 | 1.25 |
Geometric parameters of spur gears, such as module and pressure angle, significantly affect lubrication. Increasing the module m reduces the load, leading to lower central pressure, higher central and minimum film thickness, and a lower maximum temperature rise ratio. Similarly, a larger pressure angle θ decreases central pressure and increases film thickness, improving lubrication. However, excessive pressure angles may increase gear stiffness and noise, so moderation is key for spur gears.
| Parameter | Value | Central Pressure (Pa) | Minimum Film Thickness (μm) | Max Temperature Rise Ratio |
|---|---|---|---|---|
| Module m (mm) | 3 | 1.0e9 | 0.70 | 1.30 |
| 4 | 8.5e8 | 0.83 | 1.15 | |
| 5 | 7.0e8 | 0.95 | 1.05 | |
| Pressure Angle θ (°) | 20 | 8.5e8 | 0.83 | 1.15 |
| 25 | 7.8e8 | 0.88 | 1.10 | |
| 30 | 7.2e8 | 0.92 | 1.05 |
Operating conditions, including rotational speed and input power, also influence the lubrication of spur gears. Higher rotational speeds increase entrainment velocity, resulting in greater film thickness and reduced central pressure. However, the maximum temperature rise ratio may slightly increase, particularly in the meshing-in and meshing-out regions. Conversely, higher input power raises the load, leading to increased central pressure, decreased film thickness, and higher temperatures. Thus, operating spur gears at moderate speeds and lower power levels is beneficial for lubrication.
| Parameter | Value | Central Pressure (Pa) | Minimum Film Thickness (μm) | Max Temperature Rise Ratio |
|---|---|---|---|---|
| Speed n1 (rpm) | 500 | 9.0e8 | 0.78 | 1.12 |
| 620 | 8.5e8 | 0.83 | 1.15 | |
| 750 | 8.0e8 | 0.87 | 1.18 | |
| Input Power P (kW) | 30 | 7.5e8 | 0.90 | 1.08 |
| 40 | 8.5e8 | 0.83 | 1.15 | |
| 50 | 9.5e8 | 0.75 | 1.22 |
In conclusion, the transient thermal mixed lubrication analysis of spur gears reveals that surface roughness and texture parameters are critical for performance. The integrated surface roughness σ should be carefully selected to balance film thickness and wear. The surface texture parameter γ optimally at 1/3 for transverse roughness minimizes temperature rise and enhances lubrication. Geometric parameters like module and pressure angle, when increased, improve lubrication, while operating conditions such as higher speed and lower power are favorable. These findings provide valuable insights for designing and operating spur gears in mixed lubrication regimes, ensuring durability and efficiency in various applications.
The comprehensive model developed for spur gears, incorporating non-Newtonian fluids, transient effects, and thermal aspects, offers a robust framework for predicting lubrication behavior. Future work could explore the effects of lubricant additives or surface coatings on the mixed lubrication of spur gears to further enhance performance. Overall, this study underscores the importance of considering multiple factors in the lubrication analysis of spur gears for industrial applications.