In mechanical transmission systems, spur gears are widely used due to their simplicity and efficiency. However, their performance is significantly influenced by lubrication conditions, particularly under mixed lubrication where both fluid film and asperity contacts occur. This study focuses on analyzing the friction characteristics and transmission efficiency of spur gears operating in mixed lubrication regimes. I develop a comprehensive model that incorporates contact geometry, load distribution, surface roughness, and lubricant properties to calculate the friction coefficient and power losses. The findings provide insights into optimizing gear design for improved efficiency and durability.
Spur gears are characterized by their straight teeth parallel to the gear axis, making them susceptible to high sliding friction in mixed lubrication. The transition between boundary, mixed, and elastohydrodynamic lubrication (EHL) regimes is critical for understanding gear performance. In this work, I derive a friction coefficient model based on load-sharing theory, where the total normal load is supported by both the lubricant film and asperity contacts. The model accounts for non-Newtonian lubricant behavior and real surface topography measured from gear specimens.
The contact analysis for spur gears involves determining the instantaneous contact ellipse, curvature, and sliding velocities. For a pair of spur gears, the equivalent radius of curvature $R_x$ and $R_y$ in the contact zone can be calculated using the gear geometry. The contact ellipse dimensions, semi-major axis $a$ and semi-minor axis $b$, are given by:
$$a = k_a \left[ \frac{3w}{2E(A+B)} \right]^{1/3}$$
$$b = k_b \left[ \frac{3w}{2E(A+B)} \right]^{1/3}$$
where $w$ is the load per unit width, $E$ is the modulus of elasticity, and $A$ and $B$ are the curvature sums. The parameters $k_a$ and $k_b$ depend on the ellipticity ratio $e = \sqrt{1 – (b/a)^2}$. The sliding velocity $v_r$ and entrainment velocity $u_e$ are critical for friction calculations and vary with the gear rotation angle $\phi$.
Under mixed lubrication, the total friction force $F$ is the sum of asperity friction $F_c$ and fluid shear friction $F_\tau$:
$$F = F_c + F_\tau$$
The friction coefficient $\mu$ is then:
$$\mu = \frac{F_c}{F_n} + \frac{F_\tau}{F_n}$$
where $F_n$ is the normal load. The asperity friction is modeled using the GW statistical model, considering the surface roughness parameters such as asperity height $\delta_s$ and density $n$. The fluid shear friction incorporates the Ree-Eyring model for non-Newtonian lubricants:
$$\tau = \tau_0 \arcsinh\left( \frac{\eta u_s}{h \tau_0} \right)$$
where $\tau_0$ is the characteristic shear stress, $\eta$ is the viscosity, $u_s$ is the sliding velocity, and $h$ is the film thickness.
The load-sharing between asperities and fluid film is governed by the Reynolds equation for mixed lubrication:
$$\frac{\partial}{\partial x} \left( \frac{\rho h^3}{12 \eta} \frac{\partial p}{\partial x} \right) + \frac{\partial}{\partial y} \left( \frac{\rho h^3}{12 \eta} \frac{\partial p}{\partial y} \right) = u_e \frac{\partial (\rho h)}{\partial x} + \frac{\partial (\rho h)}{\partial t}$$
The film thickness equation includes surface roughness effects:
$$h(x,y) = h_0(t) + \frac{x^2}{2R_x} + \frac{y^2}{2R_y} + v(x,y) + \delta_1 + \delta_2$$
where $v(x,y)$ is the elastic deformation, and $\delta_1$ and $\delta_2$ are roughness amplitudes.
To analyze the transmission efficiency, I calculate the power loss due to sliding friction. The instantaneous friction power loss $P_f(\phi_i)$ at rotation angle $\phi_i$ is:
$$P_f(\phi_i) = \sum_{j=1}^{n(i)} F_f(\phi_i) \cdot u_s(M_j)(\phi_i)$$
where $F_f(\phi_i)$ is the sliding friction force, and $u_s(M_j)(\phi_i)$ is the sliding velocity at discrete points. The transmission efficiency $\eta$ is:
$$\eta(\phi_i) = 1 – \frac{P_f(\phi_i)}{P_{in}}$$
where $P_{in}$ is the input power.
In the results, I examine the friction coefficient variation with rotation angle, speed, lubricant viscosity, and load. For spur gears, the friction coefficient decreases with increasing speed due to enhanced hydrodynamic effects. The Stribeck curve illustrates the transition between lubrication regimes, influenced by surface roughness and lubricant properties.
| Parameter | Value |
|---|---|
| Number of teeth | 20 |
| Module (mm) | 4 |
| Pressure angle (°) | 20 |
| Face width (mm) | 30 |
| Young’s modulus (GPa) | 210 |
| Poisson’s ratio | 0.3 |
| Lubricant viscosity (Pa·s) | 0.05–0.1 |
| Surface roughness $\delta_s$ (μm) | 0.02–0.1 |
The friction coefficient for spur gears exhibits a U-shaped trend over the mesh cycle, with minimum values near the pitch point due to pure rolling. As speed increases, the friction coefficient decreases, as shown in the following equation derived from the model:
$$\mu = \frac{f_c p_a + \tau_0 \arcsinh\left( \frac{\eta u_s}{h \tau_0} \right) A_E}{F_n}$$
where $f_c$ is the boundary friction coefficient, $p_a$ is the asperity pressure, and $A_E$ is the fluid film area.
The Stribeck curve for spur gears demonstrates how the friction coefficient varies with the Sommerfeld number $S = \frac{\eta u_e}{F_n}$. At low $S$, boundary lubrication dominates with high friction; at high $S$, EHL prevails with lower friction. Surface roughness parameters $\delta_s$ and $n$ shift the curve, expanding the mixed lubrication region.
| Parameter | Effect on Friction Coefficient | Effect on Stribeck Curve |
|---|---|---|
| Increased speed | Decreases | Shifts to EHL region |
| Increased viscosity | Increases in EHL, decreases in mixed | Leftward shift |
| Increased load | Increases near pitch point | Expands mixed region |
| Increased roughness | Increases in mixed region | Rightward shift |
Transmission efficiency improves with speed and stabilizes at higher velocities. For spur gears, efficiency is lowest at the mesh inlet and outlet due to high sliding friction. The efficiency model shows that lubricant viscosity and surface roughness have significant impacts at low speeds, but diminish at high speeds where fluid film dominates.
To quantify the efficiency, I use the following formula based on power loss calculations:
$$\eta = 1 – \frac{\sum P_f}{P_{in}}$$
where $\sum P_f$ is the total friction power loss over the mesh cycle. For spur gears, the average efficiency can exceed 95% under optimal lubrication conditions.
In conclusion, this study provides a detailed analysis of friction and efficiency in spur gears under mixed lubrication. The models developed here can guide the design of spur gears for applications requiring high efficiency and reliability. Future work could explore the effects of tooth modifications and advanced lubricants on spur gear performance.
The analysis underscores the importance of considering mixed lubrication in spur gear design, as it directly impacts energy consumption and operational lifespan. By optimizing surface topography and lubricant selection, manufacturers can enhance the performance of spur gears in various mechanical systems.