Research Background: Spiral bevel gears are widely used in aerospace, automotive and other industries due to their high meshing ratio, large load capacity, low vibration, low noise and high strength. However, problems such as large transmission error, high contact temperature and high contact Hertzian stress still exist.
Previous Research Review:
Parametric Modeling: Existing research mainly focuses on using spherical involute theory and spatial geometry to establish 3D models, but these methods have challenges in data processing and integration with simulation software.
Gear Optimization: Current research mainly concentrates on single-objective optimization design and cannot fully consider multiple conflicting design requirements.
Research Objectives and Contributions:
To solve the problems of spiral bevel gears by combining parametric modeling and multi-objective optimization.
Contributions include accurate parametric modeling using Kiss soft and effective multi-objective optimization with NSGA-II algorithm.
2. Parametric Modeling of Spiral Bevel Gear
Reliability of Kiss soft: Kiss soft is widely recognized in the industry for its professionalism and high efficiency in spiral bevel gear parametric modeling. It follows the gear design theory, reduces human errors, and provides accurate calculation and comprehensive evaluation tools.
Modeling Process and Results:
Parameter Calculation: Calculate gear parameters using formulas such as R=d/sinδ1, d=m✖z.
Modeling Steps: Open Kiss soft, create a project, select gear type, input parameters and generate the model. The main and driven gear 3D models and assembly model.
Project Name
Driving Wheel
Driven Wheel
Number of Teeth
8
37
Tooth Width (mm)
70
70
Module (mm)
8.98
8.98
Pressure Angle ()
20
20
Helix Angle ()
30
30
Axis Angle ()
90
90
Pitch Cone Angle ()
12.20
77.80
Pitch Diameter (mm)
71.87
332.39
Addendum Height (mm)
10.86
4.07
Dedendum Height (mm)
5.79
12.58
Top Cone Angle (mm)
4.85
1.82
Root Cone Angle (mm)
1.82
4.85
Outer Cone Distance (mm)
170.04
170.04
Rotation Direction
Left-handed
Right-handed
3. Contact Analysis of Spiral Bevel Gear
Analysis Steps:
Build the gear model in Kiss soft.
Define working conditions including speed, service life, power and torque.
Conduct contact analysis and evaluate the results.
Result Analysis: Before optimization, the maximum fluctuation of transmission error is 32.94 μm with an average value of 6.61 μm), the maximum contact temperature is 221.79 C, and the maximum contact Hertzian stress is 2676.08 MPa.
4. Multi-objective Optimization Design
Optimization Strategy: NSGA-II algorithm is used to optimize spiral bevel gears considering multiple conflicting objectives. It can search and maintain a set of diverse non-dominated solutions for better trade-offs.
NSGA-II Algorithm Introduction:
Algorithm Principle: Combine the offspring population Qn with the parent population Pn, perform non-dominated sorting and calculate crowding degree to select the new parent population Pn+1. Then use genetic operators to generate the new offspring population Qn+1.
Algorithm Flow:
Mathematical Model Establishment:
Optimization Variables: Helix angle β, pressure angle α and tooth width b.
Objective Functions: Minimize transmission error and maximize longitudinal contact ratio εT.
Constraints: 20°≤β≤35°, 15°≤α≤25°, 70mm≤b≤75mm.
Algorithm Parameter Settings: Population size is 150, mutation probability is 0.20 and crossover probability is 0.80.
Optimization Results and Analysis:
Parameter Values: After optimization, β=35°, α=22°, b=75mm .
Objective Function Results: Transmission error is reduced by 46.90% and longitudinal contact ratio is increased by 21.92%. The optimized transmission error curve, contact temperature curve and contact Hertzian stress curve .
Parameter
Before Optimization
After Optimization
Optimization Percentage (%)
Transmission Error (m)
6.61
3.51
46.90
Longitudinal Contact Ratio
1.46
1.78
21.92
5. Conclusion
The proposed method based on Kiss soft and NSGA-II effectively improves the performance of spiral bevel gears. The optimized transmission error, contact temperature and contact Hertzian stress are significantly reduced, while the longitudinal contact ratio is increased. This study provides important guidance and reference for gear system design.
