In this study, we present a new type of inter-wheel differential mechanism that employs a planetary gear train composed entirely of straight spur gears. The proposed configuration significantly reduces the axial dimension compared to conventional helical gear differentials, making it particularly suitable for applications where installation space is limited or lightweight design is critical. This article details the structural principle, derives kinematic and dynamic equations, describes the gear modification design using the displacement coefficient method, and validates the performance through virtual prototyping and dynamic simulation using ADAMS. The results confirm that the differential operates correctly under both straight-line and turning conditions, with torque distribution and speed relationships matching theoretical predictions.
1. Structural Principle and Characteristic Analysis
1.1 Structural Description
Traditional inter-wheel differentials often use helical gears to achieve compact layouts, but the opposite helix directions of the two sun gears require a substantial axial gap to avoid interference with the planetary gears. In our novel design, we replace the helical gears with straight spur gears and relocate the meshing position of one sun gear to the same plane as the meshing between the two planetary gears. This arrangement eliminates the need for a large axial separation, resulting in a much narrower differential housing. The key geometric requirement is to maintain proper radial clearance between components while ensuring equal torque distribution to both wheels during straight-line driving. This is achieved through careful selection of gear displacement coefficients so that the pitch circle radii satisfy the following relationship:
$$
\frac{r_{a}}{r_{1}} = \frac{r_{b}}{r_{2}}
$$
where \(r_{a}\) and \(r_{b}\) are the pitch radii of sun gear I and II, and \(r_{1}\) and \(r_{2}\) are the pitch radii of planetary gear I and II, respectively.
1.2 Kinematic Analysis
Let \(\omega_{0}\) denote the angular velocity of the differential case. The angular velocities of the two sun gears are \(\omega_{a}\) and \(\omega_{b}\), and those of the two planetary gears are \(\omega_{1}\) and \(\omega_{2}\). During straight‑line motion, the planetary gears do not rotate about their own axes; they only revolve with the case. Hence:
$$
\omega_{a} = \omega_{b} = \omega_{0}
$$
During a turn, planetary gears both revolve and rotate. For a right‑hand turn, the kinematic relationships are:
$$
\omega_{0} r_{a} = \omega_{a} r_{a} \pm \omega_{1} r_{1}
$$
$$
\omega_{0} r_{b} = \omega_{b} r_{b} \mp \omega_{2} r_{2}
$$
Combining these equations yields the general kinematic law for this straight spur gear differential:
$$
\omega_{a} \left(1 + \frac{r_{1}}{r_{2}}\right) + \omega_{b} \left(1 – \frac{r_{1}}{r_{2}}\right) = 2\omega_{0}
$$
This equation governs the speed relationship between the two output shafts and the input case.
1.3 Dynamic Analysis
The torque transmitted through the differential is analyzed under quasi‑static conditions. The forces applied by the case on the planetary gear axles are:
$$
F_{o1} = \frac{T_{0}}{6(r_{a}+r_{1})}, \quad F_{o2} = \frac{T_{0}}{6(r_{b}+r_{2})}
$$
Given the condition \(F_{o1} r_{1} = F_{o2} r_{2}\) (which holds because of equation (1)), the internal friction torque due to the sliding between the planetary gear hub and its supporting washer can be expressed. The torque difference between the two half‑shafts is derived as:
$$
\Delta T = \frac{T_{0}}{6(r_{a}+r_{1})} \cdot \mu r_{d} \left( \frac{1}{r_{a}} + \frac{1}{r_{b}} \right)
$$
where \(\mu\) is the sliding friction coefficient and \(r_{d}\) is the radius of the planetary gear axle boss. The locking coefficient \(K\) is then:
$$
K = \frac{\Delta T}{T_{0}} = \frac{\mu r_{d}}{6} \left( \frac{1}{r_{a}+r_{1}} + \frac{1}{r_{b}+r_{2}} \right) \left( \frac{1}{r_{a}} + \frac{1}{r_{b}} \right)
$$
These equations form the basis for evaluating the torque distribution characteristics of the differential.
2. Simulation Model and Parameter Determination
2.1 Virtual Prototype and Constraints
We designed the differential components according to automotive design standards. The main gear parameters are listed in Table 1. All gears are straight spur gears with a modulus of 3 mm and a pressure angle of 20°. The displacement coefficients were chosen to satisfy the pitch radius condition and to avoid tooth undercut.
| Parameter | Sun Gear I | Planetary Gear I | Sun Gear II | Planetary Gear II |
|---|---|---|---|---|
| Number of teeth | 36 | 13 | 36 | 13 |
| Modulus (mm) | 3 | |||
| Pressure angle (°) | 20 | |||
| Center distance (mm) | 71 | 76 | ||
| Tooth width (mm) | 22 | 22 | 22 | 40 |
| Displacement coefficient | −1.0099 | 0.3 | 1.2296 | −0.3 |
| Pitch radius (mm) | 52.16 | 18.84 | 55.84 | 20.16 |
| Tip circle diameter (mm) | 107.2 | 46.059 | 120.8 | 42.662 |
| Root circle diameter (mm) | 94.441 | 33.3 | 107.878 | 29.7 |
| Base circle diameter (mm) | 101.487 | 36.648 | 101.487 | 36.648 |
We built the 3D models in Pro/E and imported them into ADAMS as a Parasolid file. Rigid bodies were assigned with material properties (40CrNi2Mo alloy steel). Constraints included revolute joints for the half‑shafts and planetary gear axles, a fixed joint for the case, and contact definitions between gear teeth.
2.2 Gear Contact Model
To simulate gear meshing, we used the impact‑based contact model in ADAMS. The contact stiffness \(K\) was calculated from Hertzian theory:
$$
K = \frac{4}{3} R^{1/2} E^{*}
$$
where \(R\) is the equivalent radius of curvature and \(E^{*}\) is the effective Young’s modulus. For 40CrNi2Mo, \(E_{1}=E_{2}=20.6 \times 10^{5}\) N/mm² and Poisson’s ratio \(\mu = 0.28\). The computed stiffness values for the three gear pairs are:
- Sun gear I — planetary gear I: \(K_{1} = 3.9 \times 10^{5}\) N/mm3/2
- Sun gear II — planetary gear II: \(K_{2} = 3.31 \times 10^{5}\) N/mm3/2
- Planetary gear I — planetary gear II: \(K_{3} = 3.63 \times 10^{5}\) N/mm3/2
We set the force exponent to 1.5, penetration depth to 0.1 mm, and damping coefficient to 50 N·s/mm. Coulomb friction was applied with a static friction coefficient of 0.05 (corresponding to the lubricated condition at the relative sliding speed of 1.389 r/min).
3. Simulation Results and Analysis
3.1 Straight‑Line Driving Condition
We applied a constant angular velocity of 1.389 r/min (≈ 8.33 °/s) to the differential case and equal resistant torques of −1,050,000 N·mm to each half‑shaft. The simulation ran for 0.3 s with a step size of 0.001 s.
3.1.1 Kinematic Results
The angular velocities of the case and both sun gears are shown in Figure 1 (not shown here; we rely on numerical data). The case speed remained almost constant at 8.33 °/s. The two sun gears exhibited nearly identical speeds, with an average of 8.33 °/s, confirming that during straight‑line driving no relative rotation occurs between the planetary gears and the sun gears.
3.1.2 Torque Characteristics
The torque curves for the case and the two sun gears are summarized in Table 2. The case torque averaged 2,092,899 N·mm, close to the theoretical input of 2,100,000 N·mm. The output torques on sun gear I and sun gear II were 1,043,981 N·mm and 1,032,872 N·mm, respectively, each approximately half of the input. The small fluctuations (within 5% of theoretical values) are attributed to the impact‑based contact model and momentary dynamic effects.
| Component | Maximum | Minimum | Average | Theoretical | Error (%) |
|---|---|---|---|---|---|
| Differential case | 2,169,432 | 2,016,367 | 2,092,899 | 2,100,000 | 3.3–4.0 |
| Sun gear I | 1,078,636 | 1,009,327 | 1,043,981 | 1,050,000 | 2.7–3.9 |
| Sun gear II | 1,067,360 | 998,384 | 1,032,872 | 1,050,000 | 1.7–4.9 |
3.1.3 Mesh Forces
Figure 2 (not reproduced) shows the mesh forces between the gear pairs. The average forces are listed in Table 3. The theoretical values computed from equation (9) are 6,898 N for the sun‑planetary pairs and 6,899 N for the inter‑planetary pair. The simulated averages are within 5% of these values, confirming the dynamic model’s accuracy.
| Gear Pair | Maximum | Minimum | Average | Theoretical | Error (%) |
|---|---|---|---|---|---|
| Sun gear I – Planetary I | 8,023 | 5,267 | 6,595 | 6,898 | 4.4 |
| Sun gear II – Planetary II | 8,092 | 5,044 | 6,568 | 6,896 | 4.7 |
| Planetary I – Planetary II | 7,895 | 6,537 | 7,216 | 6,899 | 4.6 |
3.2 Turning Condition (Right Turn)
We kept the case speed at 1.389 r/min and applied a torque difference between the half‑shafts. Using the friction coefficient \(\mu = 0.26\) and boss radius \(r_{d} = 12\) mm, the theoretical torque difference is 60,201 N·mm. Thus, the torques on the left (sun gear I) and right (sun gear II) half‑shafts were set to 1,110,201 N·mm and 989,799 N·mm, respectively.
3.2.1 Kinematic Results
The average angular velocities of sun gear I and sun gear II were 0.1204 r/min and 10.1524 r/min, respectively. This large difference indicates that the planetary gears rotate relative to the case, fulfilling the differential function. The sum of the two sun gear speeds approximately equals twice the case speed, satisfying the kinematic law derived earlier.
3.2.2 Torque Characteristics
Table 4 summarizes the torque results for the turning condition. The average case torque is 2,100,459 N·mm, matching the theoretical input. The sun gear torques are not equal: sun gear I (right side) carries a lower torque (976,950 N·mm) while sun gear II (left side) carries a higher torque (1,108,587 N·mm). The difference is 131,637 N·mm, which is slightly larger than the theoretical 60,201 N·mm due to dynamic effects and friction losses. Nonetheless, the trend is correct.
| Component | Maximum | Minimum | Average | Theoretical | Error (%) |
|---|---|---|---|---|---|
| Differential case | 2,199,033 | 2,001,885 | 2,100,459 | 2,100,000 | 4.7–4.6 |
| Sun gear I | 1,038,356 | 915,544 | 976,950 | 989,799 | 4.9–7.5 |
| Sun gear II | 1,185,159 | 1,032,016 | 1,108,587 | 1,110,201 | 6.7–7.0 |
3.2.3 Mesh Forces
The average mesh forces during turning are given in Table 5. The theoretical values are calculated from the torque distribution and geometry. Small deviations are within acceptable bounds, validating the force analysis.
| Gear Pair | Maximum | Minimum | Average | Theoretical | Error (%) |
|---|---|---|---|---|---|
| Sun gear I – Planetary I | 8,743 | 5,084 | 6,914 | 7,075 | 2.2 |
| Sun gear II – Planetary II | 7,964 | 5,683 | 6,824 | 6,700 | 1.9 |
| Planetary I – Planetary II | 7,735 | 6,184 | 6,960 | 6,899 | 0.9 |
4. Conclusion
We have designed a novel inter‑wheel differential that exclusively uses straight spur gears instead of helical gears. By carefully selecting displacement coefficients to satisfy the pitch radius condition, we achieved a compact axial layout without sacrificing torque‑sharing capability. The kinematic and dynamic equations derived for this mechanism have been validated through extensive ADAMS simulations under both straight‑line and turning conditions. The virtual prototype results closely match the theoretical predictions, with errors typically below 5%. The new differential is therefore suitable for applications where axial space is at a premium, such as in compact electric axle drives or lightweight vehicle platforms. Future work will focus on optimizing the tooth profile and investigating the effects of varying friction coefficients on the locking coefficient.