1. Introduction
Hypoid gears with low crossed shaft angle have unique advantages in various fields. However, the current research on them is still in the initial stage, lacking systematic design and analysis methods. This article aims to address these issues.
| Researcher | Contribution |
|---|---|
| DOONER et al. | Proposed the application idea of hypoid gears in low crossed shaft angle transmission |
| STADTFELD | Conducted preliminary derivation of geometric parameters |
| YANG et al. | Explored the calculation method of machining parameters |
| LIU et al. | Discussed the value range of basic geometric parameters and drew the Closed Circle |
| DOONER | Proposed a design method suitable for 0° – 90° working conditions based on evolvents |
2. Geometric Design of Hypoid Gears with Low Crossed Shaft Angle
The geometric design of hypoid gears with low crossed shaft angle is based on the instantaneous axis of the uniparted hyperboloid and the meshing mechanism of hypoid gears.
2.1 Spatial Relationship of Uniparted Hyperboloids
The relative motion of spatial crossed shaft transmission is a spiral motion. The contact area of two uniparted hyperboloids is defined as the instantaneous contact axis. The geometric relationships are as follows:
| Parameter | Definition | Formula |
|---|---|---|
| Ε | Distance between the rotation axes of two uniparted hyperboloids | – |
| ∑ | Crossed shaft angle | – |
| m21 | Rotational speed ratio of two uniparted hyperboloids | – |
| β | Angle between the instantaneous axis and the -axis in the global coordinate system Sf | |
| Coordinates of any point on the instantaneous axis in Sf | (x1,y1,z1) |
2.2 Conversion to Pitch Cone Model
The pitch cone model is established based on the uniparted hyperboloid model. The geometric relationships of the pitch cone are as follows:
| Parameter | Definition | Formula |
|---|---|---|
| γm1 | Pitch cone angle of the pinion | – |
| γm2 | Pitch cone angle of the gear | – |
| βm1 | Helix angle of the pinion at the reference point | – |
| βm2 | Helix angle of the gear at the reference point | – |
| βm12 | Difference between the helix angles of the pinion and the gear | |
| i12 | Transmission ratio of the gear pair | |
| ηm | Offset angle of the pinion | |
| εm | Offset angle of the gear | |
| E(derived from pitch cone geometric relationship) | Offset distance |
3. Optimization of Local Synthesis Method for Face-milled Hypoid Gears with Low Crossed Shaft Angle
The local synthesis method for hypoid gears with low crossed shaft angle is optimized to improve the meshing characteristics.
3.1 Gear Tooth Surface Generation Mechanism
The gear tooth surface is generated by the milling machine. The machining parameters of the pinion and the gear are as follows:
| Gear | Machining Parameter | Symbol |
|---|---|---|
| Pinion | Tool tilt angle | Tt1,i |
| Tool rotation angle | Wr1i | |
| Radial tool position | Sx1i | |
| Angular tool position | Or1i | |
| Vertical wheel position | Erii | |
| Horizontal wheel position | Arii | |
| Bed position | Br1i | |
| Machine root cone angle | Mrl | |
| Roll ratio | Vrl | |
| Gear | Radial tool position | Sr2 |
| Angular tool position | Q:2 | |
| Horizontal wheel position | Ar2 | |
| Bed position | Br2 | |
| Machine root cone angle | M12 |
3.2 Modification of Machine Root Cone Angle
The machine root cone angle is modified to eliminate the step in the gear root transition area. The correction method is as follows:
| Parameter | Definition | Formula |
|---|---|---|
| γmc | Angle between the projection line of the concave tooth root and the rotation axis | |
| γmsv | Angle between the projection line of the convex tooth root and the rotation axis |
4. Study on Meshing Characteristics of Tooth Surface Based on Tooth Contact Analysis
The meshing characteristics of the hypoid gear pair are studied by establishing a meshing model and analyzing the influence of external loads.
4.1 Gear Pair Meshing Model
The basic geometric parameters and preset meshing characteristics parameters of the hypoid gear pair are as follows:
| Parameter | Pinion (i – 1) | Gear (i – 2) |
|---|---|---|
| Number of teeth Ni | 29 | 37 |
| Module at the large end (mm) | 2.4324 | – |
| Shaft angle (°) | 15.0000 | – |
| Offset distance E (mm) | 25.0000 | – |
| Pitch cone angle γ (°) | 7.3421 | 6.8907 |
| Face cone angle γ (°) | 8.0087 | 7.0967 |
| Root cone angle γ (°) | 7.1362 | 6.2238 |
| Helix angle at the reference point (°) | 24.7482 | 20.0000 |
| Tooth width bi (mm) | 25.5721 | 25.0000 |
| Axial installation distance A (mm) | 297.8371 | 296.8270 |
| Cutter radius rca (mm) | 95.2500 | – |
| Contact ellipse major axis length Lc (mm) | 8 | 8 |
| Angle between contact trace and the first principal direction θ (°) | 80 | 100 |
| Peak-to-peak value of transmission error 2 (“) | 12 | -12 |
4.2 Influence of External Load on Gear Meshing Characteristics
The influence of external load on gear meshing characteristics is as follows:
| Load (N·m) | Contact Ellipse Major Axis Length | Contact Region Percentage | Transmission Error Peak-to-Peak Value |
|---|---|---|---|
| 50 | – | – | – |
| 100 | – | – | – |
| 150 | – | – | – |
| 200 | – | – | – |
| 250 | – | – | – |
| 300 | – | – | – |
4.3 Influence of External Load on Tooth Surface Contact Stress and Tooth Root Bending Stress
The influence of external load on tooth surface contact stress and tooth root bending stress is as follows:
| Load (N·m) | Tooth Surface Contact Stress | Tooth Root Bending Stress |
|---|---|---|
| 50 | – | – |
| 100 | – | – |
| 150 | – | – |
| 200 | – | – |
| 250 | – | – |
| 300 | – | – |
5. Prototype of Hypoid Gear with Low Crossed Shaft Angle
The prototype of the hypoid gear with low crossed shaft angle is designed and manufactured to verify the correctness of the design method.
| Component | Specification |
|---|---|
| Hypoid gear pair | Manufactured by 3D printing with a precision of 0.05mm |
| Transmission system | Input shaft is the small wheel rotation axis of the hypoid gear pair, output shaft is the small wheel of the thickening gear pair, and the two shafts are parallel |
| Indexing plate | Controls the angle between the two gear pairs with a precision of 0.02° |
| Thickening gear pair | Offset distance is 60mm, number of teeth of the small wheel and the large wheel are 25 and 39 respectively |
6. Conclusion
- The geometric relationships and design process of hypoid gears with low crossed shaft angle are derived.
- The machining parameter calculation method considering the meshing characteristics is proposed.
- The influence of external load on the meshing characteristics and stress of the gear is studied.
- The prototype verifies the correctness of the design and analysis methods.