Introduction
In the realm of mechanical engineering, cylindrical gears are pivotal components in parallel-axis transmissions, widely utilized for their efficiency and reliability in transmitting motion and power. Among these, involute gears are the most prevalent due to their smooth transmission, center distance variability, and consistent normal force direction. However, they are highly sensitive to installation errors, particularly cross-axis and offset-axis misalignments, which can lead to discontinuous linear transmission errors, increased vibration, noise, and edge contact, thereby reducing the lifespan of the gear transmission. Additionally, the relative sliding between tooth surfaces causes wear and heat generation, which cannot be overlooked.
Curvilinear cylindrical gears, with their unique tooth trace geometry, have been in existence for over a century and have found applications in steel mills, aluminum plants, and cement equipment factories. These gears offer advantages such as no axial force, excellent lubrication, strong self-aligning capability, and localized load-bearing contact. However, their manufacturing primarily relies on the generation method, which limits the geometric shape of the tooth surfaces to slight adjustments through tool geometry parameters. Improper parameter selection can lead to undercutting and tooth tip sharpening.
This paper introduces a novel design of pure rolling cylindrical gears with a circular arc tooth trace, which is not constrained by the generation method. This design offers greater freedom in tooth surface design and leverages additive manufacturing for rough machining, allowing the use of various materials such as polymers, ceramics, metals, and nanocomposites. The study focuses on the mathematical model and meshing characteristics of these gears, comparing their performance with traditional helical gears and pure rolling cylindrical gears with a parabolic tooth trace.
Mathematical Model and Design Parameters
The design of pure rolling cylindrical gears with a circular arc tooth trace involves a mathematical model that controls the curvilinear shape of the teeth along the face width of the gears. The tooth surfaces are formed by sweeping combined transverse tooth profiles controlled by four control points along the pure rolling contact curves. The transverse tooth profiles are a combination of a circular arc, an involute, and a Hermite curve, smoothly connected at the control points.
Key Design Parameters:
- Number of Teeth (Z): The number of teeth on the pinion and gear.
- Gear Ratio (i12): The ratio of the number of teeth on the gear to the pinion.
- Normal Module (mn): The module of the gear in the normal plane.
- Normal Pressure Angle (αn): The pressure angle in the normal plane.
- Helix Angle (β): The angle of the helix on the gear.
- Addendum Coefficient (h*an): The coefficient for the addendum height.
- Dedendum Coefficient (c*n): The coefficient for the dedendum height.
- Face Width (b): The width of the gear tooth along the axis.
Mathematical Formulations:
The basic geometric dimensions of the gear sets are derived from the following equations:mt=mncosβαt=arctan(tanαncosβ)Ri=ZimtZ2=i12Z1a=R1+R2rbi=Ricosαthai=han∗mnhfi=(han∗+cn∗)mnhd=kdmnmtαtRiZ2arbihaihfihd=cosβmn=arctan(cosβtanαn)=Zimt=i12Z1=R1+R2=Ricosαt=han∗mn=(han∗+cn∗)mn=kdmn
Where:
- mtmt: Transverse module
- αtαt: Transverse pressure angle
- RiRi: Pitch radius
- rbirbi: Base circle radius
- haihai: Addendum height
- hfihfi: Dedendum height
- hdhd: Control point height
Tooth Profile Design:
The transverse tooth profile is a combination of a circular arc, an involute, and a Hermite curve, smoothly connected at four control points Pa,Pb,Pd,Pa,Pb,Pd, and PePe. The equations for these curves are as follows:
- Circular Arc:
x(ΣCir)=ρpisinξpiy(ΣCir)=ρpicosξpi−ρpiz(ΣCir)=0x(ΣCir)y(ΣCir)z(ΣCir)=ρpisinξpi=ρpicosξpi−ρpi=0
Where ρpiρpi is the radius of the circular arc and ξpiξpi is the angle parameter.
- Involute Curve:
x(ΣInv)=rbisinui−uirbicosuiy(ΣInv)=rbicosui+uirbisinuiz(ΣInv)=0x(ΣInv)y(ΣInv)z(ΣInv)=rbisinui−uirbicosui=rbicosui+uirbisinui=0
Where rbirbi is the base circle radius and uiui is the involute parameter.
- Hermite Curve:
The Hermite curve is defined by control points PdPd and PePe, with the curve parameters determined by the Hermite curve tangent vector weights THpTHp and THgTHg.
Meshing Performance Analysis
The meshing performance of the pure rolling cylindrical gears with a circular arc tooth trace was analyzed through tooth contact analysis (TCA) and stress analysis. The results were compared with those of pure rolling cylindrical gears with a parabolic tooth trace and traditional helical gears with modified tooth surfaces.
Tooth Contact Analysis (TCA):
The TCA was performed to determine the contact patterns and transmission error curves. The analysis showed that the contact patterns of the pure rolling cylindrical gears with a circular arc tooth trace were similar to those of the gears with a parabolic tooth trace. Both exhibited symmetric contact patterns about the mid-face width, which helps eliminate axial forces. The contact ellipses were located on the pitch radius, with the major axis length increasing from the ends to the middle of the tooth face.
Stress Analysis:
The stress analysis was conducted using the ABAQUS software. The finite element models included five pairs of teeth for the curvilinear gears and seven pairs for the helical gears to account for the higher contact ratio of the latter. The analysis focused on the von Mises stress on the tooth surfaces and the maximum bending stress at the tooth root.
Results:
- Contact Stress:
- The maximum von Mises stress on the pinion tooth surfaces was similar for the pure rolling cylindrical gears with a circular arc and parabolic tooth trace.
- The contact stress distribution was consistent with the TCA contact patterns.
- The maximum von Mises stress was slightly higher for the pure rolling cylindrical gears compared to the traditional helical gears.
- Bending Stress:
- The maximum bending stress at the tooth root was lower for the pure rolling cylindrical gears compared to the traditional helical gears.
- The bending stress curves for the pinion and gear were similar for both types of pure rolling cylindrical gears.
- The bending stress was higher for the pinion than the gear in traditional helical gears, whereas it was lower for the pinion in pure rolling cylindrical gears.
- Loaded Transmission Error:
- The loaded transmission error curves for the pure rolling cylindrical gears showed a parabolic shape due to the parabolic modification along the face width.
- The amplitude of the loaded transmission error was higher for the pure rolling cylindrical gears compared to the traditional helical gears.
Conclusion
The study demonstrated that pure rolling cylindrical gears with a circular arc tooth trace exhibit similar meshing characteristics to those with a parabolic tooth trace, including contact patterns, maximum bending and contact stresses, and loaded transmission error functions. Compared to traditional helical gears, these gears offer lower maximum bending stress, similar maximum contact stress, and higher loaded transmission error amplitude. These characteristics make them a promising alternative for applications requiring high performance and reliability in parallel-axis transmissions.
Tables:
Table 1: Basic Design Parameters of Curvilinear and Helical Gear Sets
| Parameter | Value |
|---|---|
| Number of Teeth (Z1) | 30 |
| Gear Ratio (i12) | 2.0 |
| Normal Module (mn) | 2.0 mm |
| Normal Pressure Angle (αn) | 20° |
| Helix Angle (β) | 22.1474° |
| Addendum Coefficient (h*an) | 1.0 |
| Dedendum Coefficient (c*n) | 0.25 |
| Face Width (b) | 50 mm |
Table 2: Tooth Profile Parameters for Transverse Tooth Profiles of Curvilinear Gear Sets
| Parameter | Value |
|---|---|
| Motion Coefficient (kφ) | π |
| Maximum Parameter (tmax) | 0.1 |
| Control Point Coefficient (kd) | 0.75 |
| Rotation Angle Coefficient (kχa1) | 0.11 |
| Rotation Angle Coefficient (kχa2) | 0.04 |
| Hermite Curve Weight (THp) | 0.5 |
| Hermite Curve Weight (THg) | 0.7 |
| Coefficient for η (kη) | 0.02 |
| Coefficient for λ (kλ) | 0.02 |
Table 3: Micro-geometry Modifications Applied for the Four Cases of Design Gear Sets
| Case | Modification Type | Modification Amount (μm) |
|---|---|---|
| 1 | Parabolic | 2 |
| 2 | Parabolic | 2 |
| 3 | Circular Arc | 10 |
| 4 | Parabolic | 60 |