In modern automotive electric power steering (EPS) systems, the demand for efficient, quiet, and cost-effective transmission mechanisms has led to the widespread adoption of helical gears paired with steel worms. These helical gears, often made from nylon, offer significant advantages in terms of reduced noise, smooth operation, and lower production costs compared to traditional worm gear systems. However, the unique material properties of nylon introduce challenges that render conventional worm gear design criteria inadequate. This study focuses on optimizing the design of helical gears in worm-type transmissions by analyzing structural parameters, safety factors, and modification coefficients. Using Kisssoft software for initial design and analysis, and Workbench for simulation validation, I explore how specific adjustments, such as profile shift coefficients, enhance performance while ensuring durability and efficiency.
The fundamental principle behind helical gears in worm drives lies in their ability to transmit motion between non-parallel shafts with high efficiency and minimal backlash. Helical gears engage with worm shafts in a manner similar to crossed helical gear pairs, where the worm acts as a multi-start helical gear with a large helix angle. This configuration allows for continuous tooth engagement, which is critical for reducing vibration and noise in EPS applications. The material combination of nylon for the helical gear and steel for the worm necessitates a tailored approach to design, as nylon’s lower modulus of elasticity and strength require careful consideration of load distribution and tooth geometry to prevent premature failure.
To address these challenges, I begin with a theoretical analysis of the bending stress in helical gears. The root bending stress formula for helical gears is derived from standard gear theory and accounts for factors such as load distribution, tooth geometry, and material properties. The expression for bending stress $\sigma_F$ is given by:
$$\sigma_F = \frac{K F_t Y_{Fa} Y_{Sa} Y_{\beta}}{b m_n \epsilon_{\alpha}}$$
where $K$ is the load factor, $F_t$ represents the tangential force calculated as $F_t = 2T / d$ (with $T$ being torque and $d$ the pitch diameter), $Y_{Fa}$ is the form factor for helical gears, $Y_{Sa}$ is the stress correction factor, $Y_{\beta}$ is the helix angle factor, $b$ denotes the face width, $m_n$ is the normal module, and $\epsilon_{\alpha}$ is the transverse contact ratio. This equation highlights the sensitivity of helical gears to geometric parameters, emphasizing the need for precise design to minimize stress concentrations.
In standard helical gear and worm pairs, the tooth thickness is initially uniform. For a standard helical gear, the tooth thickness $s_a$ is expressed as:
$$s_a = \frac{\pi m_n}{2}$$
However, when applying profile shifts to optimize performance, the tooth thickness changes. For a profile-shifted helical gear, the tooth thickness $s_{la}$ becomes:
$$s_{la} = \frac{\pi m_n}{2} + 2x m_n \tan \alpha$$
where $x$ is the profile shift coefficient and $\alpha$ is the pressure angle. Correspondingly, for the worm, the tooth thickness $s_{ga}$ after profile shift is:
$$s_{ga} = \frac{\pi m_n}{2} – 2x m_n \tan \alpha$$
To prevent tooth tip sharpening in the steel worm, which could lead to reduced strength and increased wear, I adopt a criterion where the worm’s tip thickness $s_{a}^*$ is set to a minimum of $0.4m_n$ (typically between 0.6 mm and 1 mm for practical applications). This gives:
$$s_{a}^* = \frac{\pi m_n}{2} – 2m_n \tan \alpha – 2x m_n \tan \alpha$$
Setting $s_{a}^* = 0.4m_n$ and solving for the profile shift coefficient $x^*$ yields:
$$x^* = \frac{\pi – 0.8}{4 \tan \alpha} – 1$$
This formulation ensures that the worm maintains sufficient tip thickness while allowing for an optimized tooth profile in the helical gear. In contrast, some literature suggests a tooth thickness ratio of 7:3 for polymer helical gears and metal worms, which corresponds to a profile shift coefficient of:
$$x = \frac{\pi}{10 \tan \alpha}$$
Comparing these two approaches, I find that for pressure angles below 15°, the profile shift coefficient derived from the tip thickness criterion ($x^*$) is slightly larger, resulting in improved strength for the helical gear without compromising the worm’s integrity. This insight forms the basis for my optimization strategy in designing helical gears for EPS systems.
To validate this theoretical framework, I proceed with a case study using Kisssoft software. The design requirements for the helical gear and worm transmission in an automotive EPS application are summarized in the table below. These parameters include module, number of starts, transmission ratio, pressure angle, center distance, input power, and speed, which are critical for ensuring the helical gears meet operational demands.
| Parameter | Helical Gear | Worm |
|---|---|---|
| Module (mm) | 2 | 2 |
| Number of Starts | — | 2 or 3 |
| Transmission Ratio | 20.5 | — |
| Pressure Angle (°) | 13.5 | 13.5 |
| Center Distance (mm) | 50 | 50 |
| Input Power (kW) | — | 0.3 |
| Speed (r/min) | — | 1000 |
Using Kisssoft’s crossed helical gears module, I input the normal module of 2 mm, pressure angle of 13.5°, center distance of 50 mm, helical gear tooth count of 41, and worm starts of 2. The software calculates the helix angle as 74.3795°, which ensures proper meshing conditions. Additional settings include material selection: PA66 for the helical gear and 40Cr for the worm, with their properties detailed in the following table. Lubrication is specified as grease-based using Grafloscon C-SG 2000 ULTRA, and tooth profile parameters are set to a standard of 1.25/0.25/1.00 for root height coefficient, clearance coefficient, and addendum coefficient, respectively.
| Material | Elastic Modulus (GPa) | Poisson’s Ratio | Density (kg/m³) | Yield Strength (MPa) |
|---|---|---|---|---|
| PA66 (Helical Gear) | 1.4 | 0.38 | 1140 | 83 |
| 40Cr (Worm) | 21.17 | 0.3 | 7850 | 785 |
After initial calculations, the results indicate a contact ratio of 2.184 and safety factors for tooth root strength. For the helical gear, the root safety factor is 1.6002, while the worm exhibits a higher value of 10.701, reflecting the material disparity. To enhance the helical gear’s strength, I apply profile shifts based on the derived coefficients. By selecting “Allow large profile shift” and maintaining tip and root circles in Kisssoft, I set the worm’s profile shift coefficient to -1.4383 (from $x^*$) and -1.3086 (from $x$), which effectively increases the helical gear’s tooth thickness without altering external dimensions.
The recalculated safety factors demonstrate significant improvements. For instance, with $x = 1.4383$, the helical gear’s root safety factor rises to 2.5138, a 57% increase compared to the standard design. Similarly, with $x = 1.3086$, it reaches 2.4309, a 52% improvement. This confirms that the profile shift coefficient derived from the tip thickness criterion provides superior reinforcement for helical gears, aligning with the goal of achieving equal strength and preventing failure in EPS applications.
| Component | Profile Shift Coefficient | Root Safety Factor |
|---|---|---|
| Helical Gear | 1.3086 | 2.4309 |
| Worm | 1.3086 | 5.3453 |
| Helical Gear | 1.4383 | 2.5138 |
| Worm | 1.4383 | 4.8974 |
For a detailed strength assessment, I employ Ansys Workbench to perform static structural analysis. Three-dimensional models of the helical gear and worm are created in Catia based on the Kisssoft dimensions, including both standard and profile-shifted versions. The models are imported into Workbench in STEP format, and material properties are assigned as per Table 2. Contact pairs are configured with a bonded type to simulate the meshing of two tooth pairs, consistent with the calculated contact ratio of 2.184. Meshing is automated globally to ensure accuracy.
Boundary conditions are set to replicate real-world loading: the worm is fixed at one end, and a torque of 60 N·m is applied to the helical gear’s output shaft. This setup mirrors experimental validation tests, allowing for a direct comparison of stress distributions. The results from Workbench show von Mises stress contours for the helical gear under different profile shift conditions. In the standard design ($x = 0$), the maximum equivalent stress at the tooth root is 62.24 MPa. For profile-shifted helical gears, the stress reduces to 41.122 MPa with $x = 1.3086$ and 38.177 MPa with $x = 1.4383$. Given the yield strength of PA66 is 83 MPa, the safety factors for the profile-shifted designs are 2.018 and 2.174, respectively, indicating a robust design that prevents yielding under operational loads.
The reduction in stress concentrations with profile shifts underscores the effectiveness of this optimization approach for helical gears. It is important to note that the Workbench analysis assumes two-tooth engagement, whereas actual operation involves alternating between two and three teeth in contact, leading to slightly lower safety factors in practice compared to Kisssoft predictions. Nonetheless, the trend confirms that increasing the profile shift coefficient enhances the durability of helical gears without adversely affecting the worm’s performance.
In conclusion, this study establishes a practical methodology for designing helical gears in automotive EPS worm transmissions. By deriving the profile shift coefficient from the worm tip thickness criterion, I achieve a balance between ease of manufacturing, failure prevention, and strength equality. The iterative use of Kisssoft for parameter optimization and Workbench for simulation validation ensures that the helical gears meet stringent performance requirements. Future work could explore dynamic loading conditions and thermal effects on nylon helical gears to further refine the design process. Overall, the optimized helical gear and worm transmission demonstrates superior comprehensive performance, making it well-suited for modern EPS systems where reliability and efficiency are paramount.