1. Introduction
1.1 Research Background and Significance
Full-section hard rock tunnel boring machines (shield machines) are critical equipment in underground engineering. Their main bearing drive spur gear frequently endure extreme loads, such as impacts and overloads, leading to tooth fracture, pitting, and operational interruptions. Existing research predominantly focuses on high-speed, light-load conditions, leaving low-speed, heavy-load spur gear understudied due to their complex loading states. This study aims to enhance the meshing stability and load-bearing capacity of internal spur gear transmissions by analyzing tooth profile and axial modifications for shield machine main bearing gears.
1.2 Current Research Status
1.2.1 Tooth Profile Modification
- International Studies: Walker (1938) pioneered gear modification concepts. Subsequent researchers like Bruyère et al. developed simplified models to estimate modification quantities, while Wang et al. optimized dynamic performance using parabolic modifications.
- Domestic Studies: Chinese scholars, such as Xue Jianhua and Chen Siyu, explored how profile modifications affect transmission error and load distribution.
1.2.2 Tooth Direction Modification
- International Studies: Walker’s foundational work was expanded by Tesfahunegn et al., who linked helical line modifications to stress reduction.
- Domestic Studies: Li Guolong and Hou Qing optimized load distribution through drum-shaped modifications for heavy-duty applications.
1.3 Research Objectives
This study investigates spur gear modification under low-speed, heavy-load conditions. A 3m-class shield machine main bearing spur gear is selected as the research object. Key objectives include:
- Establishing modification principles for tooth profile and axial adjustments.
- Conducting quasi-static contact analysis to evaluate fatigue strength and load distribution.
- Proposing optimal modification schemes through multi-objective optimization.
- Validating results via scaled-down models and strain tests.
2. Theoretical Analysis of Spur Gear Modification
2.1 Tooth Profile Modification
2.1.1 Principles
Spur gear under heavy loads experience abrupt load transitions during single-to-double tooth meshing phases, causing dynamic impacts. Profile modifications mitigate these effects by adjusting the involute shape.
Key Parameters:
- Maximum modification amount (ΔmaxΔmax)
- Modification length (LL)
- Modification curve (linear, parabolic, or arc-shaped)
Empirical Formulas:
- ISO Standard:Δmax=KAFtbCγϵαΔmax=bCγϵαKAFtwhere KAKA= application factor, FtFt= tangential force, bb= face width, CγCγ= mesh stiffness, ϵαϵα= transverse contact ratio.
- Walker’s Formula:Δmax=0.53FtbΔmax=0.53bFt
2.1.2 Modification Methods
- Linear Modification: Simplifies machining but may not fully eliminate edge contact.
- Arc Modification: Smooths load transitions but requires precise manufacturing.
Table 1: Comparison of Profile Modification Methods
| Method | Advantages | Limitations |
|---|---|---|
| Linear | Low cost, easy implementation | Limited impact reduction |
| Arc | Optimal stress distribution | Complex machining |
| Parabolic | Balanced performance | Requires iterative optimization |
2.2 Tooth Direction Modification
2.2.1 Principles
Misalignment due to shaft deflection or installation errors causes uneven load distribution. Axial modifications, such as drum or helical adjustments, compensate for these deviations.
Key Parameters:
- Drum modification amount (CcCc)
- Helical modification (ChCh)
Empirical Formulas:
- ISO 6336:Cc=0.5FβyCc=0.5Fβywhere FβyFβy= meshing skewness.
- Song Lemin’s Formula:Cc=bcalbFpv=2FmFpvCγbCc=bbcalFpv=Cγb2FmFpv
2.2.2 Modification Methods
- Drum Modification: Centralizes load distribution.
- Helical Modification: Addresses misalignment-induced skewness.
Table 2: Axial Modification Parameters
| Parameter | Formula | Application |
|---|---|---|
| CcCc | 0.7×10−3Fmb0.7×10−3bFm | Heavy-load spur gear |
| ChCh | Fpv−FmCγbFpv−CγbFm | High skewness conditions |
3. Quasi-Static Contact Analysis
3.1 Gear Pair Modeling
A parametric model of a 3m-class shield machine spur gear was developed using KISSsoft. Key parameters include:
Table 3: Gear Pair Parameters
| Parameter | Pinion | Ring Gear |
|---|---|---|
| Module (mnmn) | 22 mm | 22 mm |
| Face Width (bb) | 250 mm | 240 mm |
| Teeth (zz) | 15 | 105 |
| Pressure Angle | 20° | 20° |
3.2 Analysis of Unmodified Gears
Under the worst-case load spectrum (173.5 kN·m torque), unmodified spur gear exhibited:
- Contact Stress: 2439 N/mm² (peak)
- Transmission Error: 122.65 μm (fluctuation)
- Root Stress: 662.28 N/mm²
Table 4: Safety Factors (Unmodified)
| Parameter | Value | Allowable | Status |
|---|---|---|---|
| Contact Safety (SHSH) | 1.032 | ≥1.2 | Unsafe |
| Bending Safety (SFSF) | 0.901 | ≥1.1 | Unsafe |
3.3 Optimization of Modification Parameters
3.3.1 Profile Modification
Four schemes were evaluated:
- Linear tip modification.
- Arc tip modification.
- Linear tip + arc root modification.
- Arc tip + arc root modification.
Table 5: Profile Modification Results
| Scheme | ΔmaxΔmax (μm) | Transmission Error (μm) | SHSH | SFSF |
|---|---|---|---|---|
| 1 | 285.6 | 136.3 | 1.15 | 1.08 |
| 2 | 310.1 | 139.4 | 1.22 | 1.12 |
| 3 | 285.6 + 95 | 63.8 | 1.31 | 1.25 |
| 4 | 310.1 + 164.5 | 105.8 | 1.28 | 1.20 |
Optimal Scheme: Linear tip (285.6 μm) + arc root (95 μm).
3.3.2 Axial Modification
Drum (Cc=86.6 μmCc=86.6μm) and helical (Ch=−275.5 μmCh=−275.5μm) modifications achieved:
- Contact Stress Reduction: 30.8% (1688.36 N/mm²)
- Load Distribution: Centralized, eliminating edge effects.
4. Transient Dynamic Analysis
4.1 Finite Element Model
A simplified spur gear model was analyzed in ANSYS Workbench using implicit transient dynamics. Key settings:
- Mesh: 91,156 elements (5 mm refined at teeth).
- Load: 173.5 kN·m torque, 1.8 rpm speed.
4.2 Results
Table 6: Dynamic Performance Comparison
| Parameter | Unmodified | Modified | Improvement |
|---|---|---|---|
| Peak Meshing Stress | 2387 N/mm² | 1571 N/mm² | 34.2% |
| Vibration Acceleration | 45 m/s² | 20 m/s² | 55.6% |
| Transmission Error | 122.6 μm | 63.8 μm | 47.9% |
5. Scaled Model Validation
5.1 Similarity Criteria
Using dimensional analysis, scaling laws for root bending stress were derived:
A 1:2.75 scaled model was tested under .
5.2 Strain Test Results
Table 7: Strain Comparison
| Condition | Max Strain (με) | Stress (N/mm²) |
|---|---|---|
| Unmodified | 3002 | 618.4 |
| Modified | 2765 | 569.6 |
6. Conclusions and Outlook
6.1 Conclusions
- Profile modification (linear tip + arc root) reduced transmission error by 47.9% and improved SHSH by 17.3%.
- Axial modification (drum + helical) centralized load distribution, lowering contact stress by 30.8%.
- Scaled model tests validated the effectiveness of modifications, showing a 9.8% reduction in root strain.
6.2 Future Work
- Incorporate manufacturing tolerances in misalignment simulations.
- Conduct full-scale vibration and noise tests.
- Explore AI-driven optimization for modification parameters.