In modern all-wheel-drive vehicles, the inter-axle limited slip differential plays a critical role in distributing torque between the front and rear axles, ensuring optimal traction and stability under varying road conditions. Traditional differentials, such as those using bevel gears or viscous couplings, often face limitations in locking capability, structural complexity, and manufacturing costs. This article proposes a novel design centered on face gear technology, which offers superior load-bearing capacity, reduced noise, and enhanced locking performance. By leveraging the unique axial force characteristics of face gears, this design improves the friction torque capacity in the clutch packs, leading to higher locking coefficients and better torque distribution. The following sections detail the working principles, mathematical modeling, simulation setup, and performance analysis of the face gear inter-axle limited slip differential, with a focus on the role of the gear shaft in torque transmission and limiting slip.
The face gear inter-axle limited slip differential consists of key components including the input shaft, an integrated planetary carrier, planetary gears, front and rear face gears, internal and external friction plates, and output shafts for the front and rear axles. A schematic of the structure is shown below, illustrating the arrangement of these elements around the central gear shaft.
In this configuration, the planetary gears are straight spur gears that mesh with the front and rear face gears. The differential is designed with an asymmetric torque distribution ratio of 40:60 between the front and rear axles, achieved by setting the tooth count ratio of the front to rear face gears as 4:6. This bias ensures that the rear axle receives more torque, which is beneficial for handling and acceleration. The friction plate assemblies include four working surfaces on the front side and five on the rear side, enhancing the locking capability during slip conditions. The gear shaft, which connects the input to the planetary carrier, is crucial for transmitting torque and enabling differential action.
Kinematic and Torque Characteristics
The kinematic behavior of the differential can be analyzed using the principles of planetary gear systems. Let $\omega_0$, $\omega_1$, and $\omega_2$ represent the angular velocities of the input shaft (planetary carrier), front face gear, and rear face gear, respectively. The relationship between these velocities is derived from the gear ratios and the geometry of the gear shaft assembly. For a system with $n_3$ planetary gears, the angular velocity equations are:
$$ \omega_0 = i_{201} \omega_1 + i_{102} \omega_2 $$
$$ \omega_1 = \omega_0 + \omega_3 \frac{z_3}{z_1} $$
$$ \omega_2 = \omega_0 – \omega_3 \frac{z_3}{z_2} $$
where $z_1$, $z_2$, and $z_3$ are the tooth numbers of the front face gear, rear face gear, and planetary gear, respectively, and $\omega_3$ is the angular velocity of the planetary gear. The transmission ratios $i_{201}$ and $i_{102}$ are calculated as:
$$ i_{201} = \frac{1}{1 – i_{012}} \quad \text{and} \quad i_{102} = \frac{1}{1 – i_{021}} $$
with $i_{012} = -\frac{z_2}{z_1}$ and $i_{021} = -\frac{z_1}{z_2}$. Substituting the tooth ratio $z_1/z_2 = 4/6$ yields:
$$ \omega_0 = \frac{2}{5} \omega_1 + \frac{3}{5} \omega_2 $$
This equation defines the speed relationship under ideal conditions, where the planetary gears do not rotate relative to the carrier. The torque characteristics are governed by the equilibrium of forces acting on the gear shaft and face gears. The input torque $T_0$ is distributed between the front and rear output torques $T_1$ and $T_2$:
$$ T_0 = T_1 + T_2 $$
During steady-state operation on a uniform road surface, the friction plates are inactive, and the torque distribution follows the geometric ratio of the face gears:
$$ T_1 = \frac{r_1}{r_1 + r_2} T_0 \quad \text{and} \quad T_2 = \frac{r_2}{r_1 + r_2} T_0 $$
where $r_1$ and $r_2$ are the effective radii at the mesh points of the front and rear face gears, respectively. When one axle experiences slip, the friction plates engage, modifying the torque distribution. For instance, if the front axle slips ($\omega_1 > \omega_2$), the friction torque $T_{f1}$ acts as a resistance on the front side, while $T_{f2}$ assists the rear side:
$$ T_1 = \frac{r_1}{r_1 + r_2} T_0 – T_{f1} \quad \text{and} \quad T_2 = \frac{r_2}{r_1 + r_2} T_0 + T_{f2} $$
Conversely, if the rear axle slips, the roles of $T_{f1}$ and $T_{f2}$ are reversed. This dynamic adjustment ensures that torque is redirected to the axle with better traction, enhancing vehicle stability.
Mathematical Modeling of the Differential
To analyze the performance, a comprehensive mathematical model is developed, including the mechanical balance, friction plate behavior, and locking coefficient. The mechanical model considers the inertia of components connected to the gear shaft. The equations of motion are:
$$ J_0 \dot{\omega}_0 = T_0 – T_L – n_3 r_1 F_1 – n_3 r_2 F_2 $$
$$ J_1 \dot{\omega}_1 = n_3 r_1 F_1 – T_1 – T_{f1} $$
$$ J_2 \dot{\omega}_2 = n_3 r_2 F_2 – T_2 + T_{f2} $$
$$ J_3 \dot{\omega}_3 = -r_3 F_1 – r_3 F_2 $$
where $J_0$, $J_1$, $J_2$, and $J_3$ are the moments of inertia of the input assembly, front output, rear output, and planetary gears, respectively; $F_1$ and $F_2$ are the forces at the gear meshes; and $T_L$ accounts for friction losses. Solving these equations under steady-state conditions ($\dot{\omega}_0 = \dot{\omega}_3 = 0$) gives the output torques:
$$ T_1 = \gamma_1 (T_0 – T_L) – T_{f1} \quad \text{and} \quad T_2 = \gamma_2 (T_0 – T_L) + T_{f2} $$
with $\gamma_1 = r_1 / (r_1 + r_2)$ and $\gamma_2 = r_2 / (r_1 + r_2)$. The friction torques $T_{f1}$ and $T_{f2}$ are derived from the clutch capacity equation:
$$ T_f = \frac{2}{3} n F_p \mu \frac{R_o^3 – R_i^3}{R_o^2 – R_i^2} $$
where $n$ is the number of friction surfaces, $\mu$ is the friction coefficient, $R_o$ and $R_i$ are the outer and inner radii of the friction plates, and $F_p$ is the axial pressure. This pressure combines the preset force $F_s$ from the thread ring and the axial component $F_z$ from the gear mesh force:
$$ F_p = F_s + F_z \quad \text{and} \quad F_z = \frac{T_c \tan \alpha}{z_3 m} $$
Here, $T_c$ is the torque on the face gear, $\alpha$ is the pressure angle, and $m$ is the module of the gears. Substituting these into the friction torque equation yields:
$$ T_f = K_f F_s + K_f K_x T_c $$
where $K_f = \frac{2}{3} n \mu \frac{R_o^3 – R_i^3}{R_o^2 – R_i^2}$ and $K_x = \frac{2 \tan \alpha}{z_3 m}$ are the clutch capacity and axial force coefficients, respectively. The locking coefficient $K$, defined as the ratio of the higher torque to the lower torque ($K \geq 1$), is a key performance metric. For front axle slip ($\omega_1 > \omega_2$):
$$ K_1 = \frac{T_2}{T_1} = \frac{T_{c2} + T_{f2}}{T_{c1} – T_{f1}} $$
and for rear axle slip ($\omega_1 < \omega_2$):
$$ K_2 = \frac{T_1}{T_2} = \frac{T_{c1} + T_{f1}}{T_{c2} – T_{f2}} $$
Using the parameters from the design, the locking coefficients are calculated and compared with conventional bevel gear differentials. The results, summarized in the table below, show that the face gear differential achieves significantly higher locking coefficients, demonstrating its superior slip-limiting capability. This is attributed to the axial forces generated by the face gears, which increase the friction plate pressure without inducing radial loads on the gear shaft.
| Differential Type | Locking Coefficient (Front) | Locking Coefficient (Rear) |
|---|---|---|
| Open Differential | 1.0 | 1.0 |
| Bevel Gear LSD | 2.0–3.0 | 2.0–3.0 |
| Face Gear LSD (Proposed) | 6.5 | 5.0 |
The mathematical model confirms that the face gear design enhances the locking performance, making it suitable for demanding applications in all-wheel-drive vehicles. The gear shaft, as the central torque-transmitting element, ensures efficient force distribution and contributes to the overall robustness of the system.
Simulation Model Development
To validate the theoretical findings, a simulation model of the face gear inter-axle limited slip differential is developed. The face gear tooth surface is generated based on the principle of gear shaping, where a shaper cutter mimics the meshing with a cylindrical gear. The position vector of the shaper or cylindrical gear tooth surface is given by:
$$ \mathbf{R}_w(u_w, \theta_w) = \begin{bmatrix}
r_{bw} [ \sin(\theta_{ow} + \theta_w) – \theta_w \cos(\theta_{ow} + \theta_w) ] \\
-r_{bw} [ \cos(\theta_{ow} + \theta_w) + \theta_w \sin(\theta_{ow} + \theta_w) ] \\
u_w \\
1
\end{bmatrix} $$
where $r_{bw}$ is the base radius, $u_w$ and $\theta_w$ are axial and angular parameters, and $\theta_{ow}$ defines the initial angle. The coordinate transformation from the shaper to the face gear is represented by the matrix $\mathbf{M}_{2s}$, which accounts for rotations and offsets. The meshing equation and the theoretical tooth surface equation are:
$$ f_1(u_s, \theta_s, \phi_s) = \mathbf{n} \cdot \mathbf{v}_{s2s} = r_{bs} (1 – m_{2s} \cos \gamma_n) – u_s m_{2s} \sin \gamma_n \cos(\phi_s + \theta_s + \theta_{os}) = 0 $$
$$ \mathbf{R}_2(u_s, \theta_s, \phi_s) = \mathbf{M}_{2s}(\phi_s) \mathbf{R}_s(u_s, \theta_s) $$
where $\mathbf{v}_{s2s}$ is the relative velocity, $m_{2s} = n_s / n_2$ is the ratio of shaper to face gear teeth, and $\gamma_n$ is the shaft angle. These equations are solved numerically to generate the tooth profile, which is then imported into CAD software to assemble the differential components, including the gear shaft, planetary gears, and face gears.
The Adams virtual prototype model is constructed by assigning material properties, applying constraints, and defining contacts between gears. The input shaft is connected to the planetary carrier, and the output shafts are linked to the face gears. Friction plates are modeled with preset forces to simulate their engagement during slip. The contact between gears is defined using Hertzian theory, with stiffness set to $10^5$ N/mm, damping to 50 N·s/mm, and friction coefficients of 0.08 (static) and 0.05 (dynamic). The model parameters are listed in the table below, based on the gear shaft dimensions and vehicle specifications.
| Parameter | Planetary Gear | Front Face Gear | Rear Face Gear |
|---|---|---|---|
| Number of Teeth | 18 | 48 | 72 |
| Module (mm) | 3.4 | 3.4 | 3.4 |
| Pressure Angle (°) | 20 | 20 | 20 |
| Face Width (mm) | 25 | 15 | 20 |
The vehicle parameters include a mass of 2000 kg, wheel radius of 0.3255 m, axle load distribution of 45:55, and a final drive ratio of 2.5. These are used to set the loads and speeds in the simulation, ensuring realistic conditions for evaluating the differential’s performance.
Simulation Results and Analysis
The simulation examines three scenarios: steady-state driving on a uniform road, and straight-line driving on split-μ roads with different adhesion coefficients. For each case, the input speed and output loads are applied based on the vehicle speed and axle loads. The results are analyzed in terms of angular velocities, friction torques, and torque distribution.
In the first scenario, the vehicle drives at 20 km/h on a good road. The input shaft speed is set to 2446.25 °/s, and the output loads are 878.50 N·m for the front axle and 1073.72 N·m for the rear axle. The angular velocities of the input and output shafts converge quickly after initial fluctuations, indicating that the planetary gears cease relative rotation and the system reaches steady state. The output torques stabilize at approximately 960 N·m for the front and 1350 N·m for the rear, giving a distribution ratio of 41.5:58.5, which aligns closely with the design ratio of 40:60. This confirms that the gear shaft effectively transmits torque according to the geometric ratio when no slip occurs.
For split-μ conditions, two cases are simulated: front axle on low-μ surface (ice) and rear axle on low-μ surface. In the first case, the front load is reduced to 200.96 N·m, while the rear load remains at 1073.72 N·m. The angular velocities show that the front axle initially accelerates due to slip, but the friction plates engage, causing the speed to decrease and synchronize with the input. The friction torques $T_{f1}$ and $T_{f2}$ exhibit periodic fluctuations due to gear meshing impulses but overall increase during slip and stabilize after locking. The output torques settle at 340.65 N·m (front) and 1260.75 N·m (rear), resulting in a distribution ratio of 21.3:78.7. This demonstrates the differential’s ability to redirect torque to the high-traction axle, with the gear shaft facilitating the transfer through the planetary system.
In the second split-μ case, the rear load is set to 245.62 N·m, and the front load is 878.50 N·m. Similar velocity and friction patterns are observed, with the rear axle slipping initially and then locking. The output torques stabilize at 1025 N·m (front) and 385 N·m (rear), giving a ratio of 72.7:27.3. The torque balance across the input and output shafts validates the simulation model, as the sum of output torques matches the input torque within acceptable limits. The table below summarizes the torque distribution ratios from simulation and theory, showing minor deviations due to dynamic effects and inertia.
| Operating Condition | Simulated Torque Ratio (Front:Rear) | Theoretical Torque Ratio | Deviation |
|---|---|---|---|
| Good Road (20 km/h) | 41.5:58.5 | 40:60 | 1.5 |
| Front Axle Slip | 21.3:78.7 | 16:84 | 5.3 |
| Rear Axle Slip | 72.7:27.3 | 78:22 | 5.3 |
These results highlight the effectiveness of the face gear differential in managing torque distribution under adverse conditions. The high locking coefficients enable rapid response to slip, and the gear shaft design ensures reliable torque transmission without excessive complexity. The simulation also confirms that the friction plates contribute significantly to the locking action, with their capacity enhanced by the axial forces from the face gears.
Conclusion
The face gear inter-axle limited slip differential presented in this article offers a robust solution for all-wheel-drive vehicles, addressing the limitations of conventional designs. The use of face gears in the planetary system reduces radial loads on the gear shaft and increases the friction torque capacity, resulting in locking coefficients of up to 6.5 for the front axle and 5.0 for the rear axle. These values surpass those of bevel gear differentials, providing superior slip control and traction management. The mathematical models developed for kinematics, torque distribution, and locking performance are validated through dynamic simulations, which show consistent torque ratios under various road conditions. The differential efficiently redistributes torque to axles with better adhesion, enhancing vehicle stability and off-road capability. Future work could focus on optimizing the gear shaft geometry and friction plate materials to further improve performance and durability. Overall, this design demonstrates the potential of face gear technology in advancing differential systems for high-performance automotive applications.