1. Introduction
With the increasing application of Electric Power Steering (EPS) systems and the growing demand for driving comfort, the steering feel issues in Column-type EPS (C-EPS) systems have become critical. Research indicates that the most significant factor affecting steering perception in C-EPS systems is the inter-tooth friction within the worm gear transmission of the reduction mechanism. Excessive friction torque leads to heavy steering, while fluctuations in friction torque cause variations in steering feel, compromising operational comfort. Therefore, studying the frictional torque of worm gear meshing in EPS systems holds substantial engineering value.
This thesis establishes a finite element model of an EPS worm gear with backlash adjustment, calculates friction torque using an exponential decay friction model, and validates the model against experimental data. Key factors such as tooth surface errors, lubrication conditions, gear deformation, preload parameters, and tooth profile modifications are systematically analyzed to optimize meshing performance.
2. Finite Element Model Development and Validation
2.1 Model Setup
The C-EPS reduction mechanism comprises a steel worm, plastic helical gear, bearings, and backlash adjustment components. Key parameters of the helical gear are summarized in Table 1.
Table 1: Basic Parameters of the Plastic Helical Gear
| Parameter | Symbol | Value |
|---|---|---|
| Number of Teeth | z | 42 |
| Normal Module | mn | 2.055 mm |
| Pressure Angle | α | 14.5° |
| Helix Angle | β | 14.995° |
| Face Width | b | 17.6 mm |
| Tip Diameter | da | 93.30 mm |
| Pitch Diameter | d | 89.352 mm |
2.2 Finite Element Modeling
- Mesh Generation: HyperMesh was used to create hexahedral meshes for critical regions (tooth surfaces) and tetrahedral meshes for non-critical areas.
- Material Properties: Elastic and rigid body definitions were assigned based on component materials (Table 2).
Table 2: Material Parameters of Components
| Component | Material | Elastic Modulus (MPa) | Poisson’s Ratio |
|---|---|---|---|
| Metal Bracket | 51CrV4 | 210,000 | 0.3 |
| Plastic Gear Teeth | PA66 | 2,200 | 0.4 |
| Bearing Bush | HC380LA | 68,144 | 0.3 |
- Contact Interactions: Frictional contact pairs were defined between the worm and helical gear using an exponential decay friction model:μ=μk+(μs−μk)e−dcγeqμ=μk+(μs−μk)e−dcγeqwhere μs=0.08μs=0.08, μk=0.038μk=0.038, and dc=0.2dc=0.2.
2.3 Model Validation
Simulated friction torque was compared with experimental measurements. Results showed a close match in mean values, though fluctuations were underestimated due to simplifications in bearing dynamics and manufacturing tolerances.
Table 3: Simulated vs. Experimental Friction Torque
| Parameter | Simulation | Experiment |
|---|---|---|
| Starting Torque | 2.707 Nm | 3.205 Nm |
| Mean Rotating Torque | 2.136 Nm | 1.801 Nm |
| Torque Fluctuation | 0.088 Nm | 1.640 Nm |
3. Impact of Tooth Surface Errors and Lubrication Conditions
3.1 Tooth Surface Measurement
Gear tooth profiles were measured using a Zeiss coordinate measuring machine. Deviations in tooth form and alignment were incorporated into the finite element model via nodal offsets.
3.2 Effect of Tooth Surface Errors
Tooth surface errors increased transmission error (TE) and friction torque fluctuations. For example, a measured gear with a 0.04 mm form error exhibited a 35% increase in torque fluctuation compared to an ideal gear (Table 4).
Table 4: Impact of Tooth Surface Errors
| Condition | Mean Friction Torque | Torque Fluctuation | TE (rad) |
|---|---|---|---|
| Ideal Tooth Surface | 2.136 Nm | 0.088 Nm | -0.010 |
| Measured Gear 1 | 2.093 Nm | 0.119 Nm | -0.012 |
| Measured Gear 2 | 2.087 Nm | 0.127 Nm | -0.013 |
3.3 Effect of Friction Coefficient
Increasing the friction coefficient (μμ) from 0.02 to 0.05 raised the mean friction torque by 25% and amplified fluctuations due to enhanced contact area and sliding friction.
4. Influence of Gear Deformation and Preload Parameters
4.1 Thermal Deformation of Plastic Gears
Thermal deformation reduced preload, increasing the number of meshing teeth from 2–3 to 3–4. This lowered mean friction torque but increased fluctuations due to uneven load distribution.
Table 5: Deformed vs. Undeformed Gear Performance
| Condition | Mean Torque | Torque Fluctuation | Meshing Stiffness (N/mm) |
|---|---|---|---|
| Undeformed Gear | 2.136 Nm | 0.088 Nm | 1.5×10⁴ |
| Deformed Gear | 0.937 Nm | 0.236 Nm | 2.8×10⁴ |
4.2 Structural Preload Analysis
A DOE study evaluated the effects of center distance, eccentricity, and bracket stiffness. Center distance error had the most significant impact on friction torque (Table 6).
Table 6: Orthogonal Experiment Results
| Center Distance (mm) | Eccentricity (mm) | Bracket Thickness (mm) | Torque Fluctuation |
|---|---|---|---|
| 52.34 | 0.55 | 2.602 | 0.463 Nm |
| 52.52 | 0.50 | 2.500 | 0.088 Nm |
| 52.70 | 0.45 | 2.602 | 0.078 Nm |
5. Tooth Profile Modification for Friction Reduction
5.1 Modification Strategies
Linear and parabolic profile modifications were applied with varying amounts (0.01–0.04 mm). Key parameters included:
- Modification Length: Short modification to avoid reducing preload.
- Modification Curve: Linear (Δ=Δmax(X/L)Δ=Δmax(X/L)) vs. parabolic (Δ=Δmax(X/L)1.5Δ=Δmax(X/L)1.5).
5.2 Results
Linear modification with 0.04 mm reduced torque fluctuation by 15% but increased meshing stiffness variability. Parabolic modification worsened transmission error.
Table 7: Performance of Modified Gears
| Modification Type | ΔmaxΔmax (mm) | Torque Fluctuation | TE (rad) |
|---|---|---|---|
| Linear | 0.04 | 0.083 Nm | -0.008 |
| Parabolic | 0.04 | 0.100 Nm | -0.014 |
6. Conclusions and Future Work
6.1 Key Findings
- The finite element model effectively predicts worm gear friction torque, validated against experimental data.
- Tooth surface errors and higher friction coefficients exacerbate torque fluctuations and transmission error.
- Gear deformation reduces preload but increases meshing stiffness variability.
- Center distance error is the dominant factor in preload-related torque fluctuations.
- Linear tooth profile modification marginally reduces torque fluctuations but increases meshing stiffness variability.
6.2 Future Directions
- Incorporate dynamic bearing effects to improve fluctuation prediction accuracy.
- Optimize tooth profile modifications using higher-order curves or machine learning.
- Explore thermal-structural coupling for plastic gear deformation analysis.