Abstract
This article presents a comprehensive analysis of the tooth contact behavior of helical gears with longitudinal modification that exhibit twist errors. Based on the generation mechanism of twist errors in modified tooth flanks, mathematical equations for the tooth flanks of these helical gears are derived. A tooth contact analysis (TCA) model is established for drum-shaped driving pinions with twist errors interacting with unmodified driven gears. The study focuses on analyzing the influence of two main design parameters, helix angle and the amount of drum-shaped modification, on transmission error and contact ellipse. The results indicate that increasing the amount of modification and helix angle exacerbates the twist phenomenon and leads to a higher transmission error amplitude, affecting transmission smoothness. While the modification amount has a minor effect on the contact ellipse area, an increase in helix angle reduces the area, thereby enhancing contact stress. Finite element analysis (FEA) is used to validate the influence of these parameters on tooth contact stress. Adjusting the modification amount and helix angle can effectively control twist, transmission error, and contact ellipse area, providing valuable insights for optimizing the design of helical gears with longitudinal modification.
Keywords
lead modification, tooth flank twist, design parameters, tooth contact analysis, transmission error, helical gear
Introduction
Gears are widely utilized in various transmission systems across industries such as automotive, machine tools, and aviation due to their ability to effectively transmit power and motion . However, misalignments and elastic deformations can deteriorate gear operation, resulting in vibrations, noise, and even premature gear failure due to uneven loading . Longitudinal modification, also known as tooth profile modification, can mitigate gear meshing impacts, improve load distribution, and subsequently reduce vibrations and noise . Nevertheless, twist errors frequently occur during the traditional grinding process of longitudinally modified helical gears, leading to distorted tooth surfaces .
Twist errors can significantly deteriorate gear meshing performance, increasing vibration and noise levels. Therefore, analyzing the twist error generation mechanism and its impact on contact mechanics is crucial for enhancing gear performance. Prior research has explored contact analysis of longitudinally modified gears, yet few studies have focused specifically on gears with twist errors . This article addresses this gap by deriving mathematical equations for twist-distorted tooth surfaces, establishing a TCA model, and analyzing the effects of design parameters on twist, transmission error, and contact ellipse.
1. Mathematical Modeling of Twist-Distorted Tooth Flanks
1.1 Standard Modified Tooth Flank Equation
When longitudinal modification is applied to helical gears, the modification curve is superimposed on the helix line at the pitch circle, as shown in Figure 1(a). The modification amount δ(y) varies along the tooth width (y-axis) . Based on the formation principle of the modified tooth flank, an additional rotation angle is introduced to formulate the tooth flank equation .
The additional rotation angle σ for a given modification amount δn at a tooth width position y can be expressed as:
sigma=rδn
where r is the radius of the pitch circle.
1.2 Twist-Distorted Tooth Flank Equation
During the continuous generating grinding process, a series of parallel contact trails are formed on the tooth surface, as shown in Figure 2(a). Due to the inclination angle βb between the contact trails and the gear end plane, different contact trails contribute to each end plane section, resulting in a twist-distorted tooth surface .
The amount of twist distortion can be calculated by integrating the modification amounts along the tooth height, considering the geometry of the grinding process:
textTwistAmount=8bsin(βb)δBB1B2
where B1B2 is the meshing line length during grinding, βb is the base helix angle, δ is the total longitudinal modification amount, and b is the tooth width.
The twist-distorted tooth flank equation is derived by incorporating the twist angle λ into the standard modified tooth flank equation:
mathbfr(1)=Mθ1+λMpr0(μ,θ1)
where r0(μ,θ1) represents the involute curve in the local coordinate system, Mp is the helical motion transformation matrix, and Mθ1+λ accounts for the additional twist rotation.
2. Tooth Contact Analysis Model
2.1 Geometry and Coordinate Systems
The TCA model considers a pinion with a twist-distorted tooth flank meshing with an unmodified gear, Multiple coordinate systems are defined to describe the gear geometry and motion:
- S0: Fixed global coordinate system
- S1 and S2: Local coordinate systems rigidly attached to the pinion and gear, respectively
- Sf and Sh: Fixed coordinate systems for the frame of the pinion and gear
- Sm: Auxiliary coordinate system for mesh analysis
2.2 Contact Conditions
During meshing, the pinion and gear tooth flanks must satisfy the conditions of common position vector and normal vector at the contact point in the fixed frame Sf:
mathbfr(1)=r(2),n(1)=n(2)
where r(1) and r(2) are the position vectors, and n(1) and n(2) are the normal vectors for the pinion and gear, respectively.
2.3 Transmission Error
Transmission error (TE) quantifies the deviation of the actual gear rotation from the ideal motion. For a helical gear pair, TE is defined as:
textTE=2πz2ϕ1−z1ϕ2
where z1 and z2 are the number of teeth on the pinion and gear, respectively, and ϕ1 and ϕ2 are their respective rotation angles.
2.4 Contact Ellipse
At each contact point, the contact area is modeled as an ellipse due to elastic deformations. The ellipse dimensions are determined by the principal curvatures and directions at the contact point, which are calculated using the first and second fundamental forms of the tooth surfaces .
3. Influence of Design Parameters
3.1 Effect on Twist Amount
Using MATLAB, a twist-distorted tooth flank model is established based on the gear parameters in Table 1 and the grinding wheel parameters in Table 2. the twist-distorted tooth surface compared to the standard modified tooth surface.
Illustrates the relationship between twist amount and modification amount (δ) as well as helix angle (β). As both parameters increase, the twist amount also increases.
3.2 Effect on Transmission Error
Transmission error curves for standard modified and twist-distorted pinions are shown in Figure 6. The twist-distorted pinion exhibits more significant fluctuations, particularly at meshing entry and exit, indicating increased vibration and noise.<figure> <img src=”figure6.png” alt=”Figure 6: Transmission Error Curves” style=”width:80%;”> <figcaption>Figure 6: Transmission Error Curves for Standard and Twist-Distorted Pinions.</figcaption> </figure>
3.3 Effect on Contact Ellipse
Compares the contact ellipse dimensions (long and short axes) and areas for standard and twist-distorted tooth surfaces. While the modification amount has a minor influence, increasing the helix angle notably reduces the contact ellipse area, leading to higher contact stresses.
4. Finite Element Analysis
To validate the analytical results, a finite element model of the helical gear pair is created using Hypermesh, as shown in Figure 8. Material properties for 17CrNiMo6 gear steel are applied, and contact simulations are performed in ABAQUS.<figure> <img src=”figure8.png” alt=”Figure 8: Finite Element Mesh Model” style=”width:80%;”> <figcaption>Figure 8: Finite Element Mesh Model of the Helical Gear Pair.</figcaption> </figure>
Contact stress distributions for various helix angles and modification amounts are presented, respectively. The maximum contact stresses increase with both helix angle and modification amount, aligning with the analytical findings.
5. Conclusion
This study presents a comprehensive analysis of twist errors in helical gears with longitudinal modification. Mathematical equations for twist-distorted tooth flanks are derived, and a TCA model is established to analyze the influence of helix angle and modification amount on transmission error and contact ellipse. Key findings include:
- Increasing the helix angle and modification amount exacerbates twist distortion and transmission error, reducing transmission smoothness.
- While the modification amount has a minor impact on contact ellipse area, the helix angle significantly influences it, with higher angles leading to smaller contact areas and increased contact stresses.
- Finite element analysis validates the analytical results, demonstrating the significance of the helix angle in controlling contact stresses.
The research provides valuable insights into optimizing the design of helical gears with longitudinal modification, particularly for minimizing twist errors and improving overall gear performance.