Abstract
To explore the propagation mechanism of contact fatigue cracks in the Cylindrical Gear with Variable Hyperbolic Circular-arc-tooth-trace (VH-CATT), a numerical simulation and analysis approach is presented. Based on the gear’s curvature characteristics, a numerical model for calculating the contact ellipse is established. Using the Finite Element Method (FEM) and numerical calculation, the elliptical contact trajectory of the tooth surface is analyzed to determine the dangerous positions where cracks are prone to initiate. The Extended Finite Element Method (XFEM) is employed to study the propagation law of contact fatigue cracks in VH-CATT cylindrical gears. Furthermore, an analysis model for the stress intensity factor of VH-CATT gears is developed to investigate the influence of modulus, tooth line radius, and crack preset angle on the Mode I stress intensity factor. Results indicate that in the long crack propagation stage, the larger the gear modulus, the higher the stress intensity factor in both the tooth width and tooth core directions. A larger tooth line radius effectively reduces the stress intensity factor at the crack front during the long crack propagation stage. Meanwhile, a larger crack preset angle increases the stress intensity factor in both the tooth width and tooth core directions.
1. Introduction
Cylindrical gears play a crucial role in mechanical transmission systems, and their performance and reliability are significantly affected by contact fatigue cracks. These cracks directly impact the overall performance and lifespan of the gears. The formation and propagation of these cracks follow a well-defined process: initiation, propagation, and eventual fracture. Contact fatigue failure in gears typically originates from the initiation of microscopic cracks, which then propagate and can lead to surface pitting, spalling, or even tooth fracture, depending on factors such as crack direction, gear material hardness, and loading conditions [1].
1.1 Background and Motivation
Gears undergo complex loading conditions during operation, including cyclic normal and tangential loads, surface friction, and impact loads. These factors, combined with the geometric characteristics of the tooth surface, make the process of contact fatigue crack initiation and propagation highly complex. VH-CATT cylindrical gears, with their unique hyperbolic circular-arc tooth trace, offer several advantages over traditional gears, including improved meshing performance, higher load-carrying capacity, and increased contact ratio. However, the specific behavior of contact fatigue cracks in VH-CATT gears has not been extensively studied.
1.2 Research Objectives
The primary objectives of this study are:
- To establish a numerical model for calculating the contact ellipse of VH-CATT cylindrical gears based on their curvature characteristics.
- To identify the dangerous positions on the gear tooth surface where cracks are most likely to initiate using FEM and numerical calculations.
- To investigate the propagation law of contact fatigue cracks in VH-CATT gears using XFEM.
- To analyze the influence of gear parameters (modulus, tooth line radius, and crack preset angle) on the stress intensity factor using the established analysis model.
2. Theoretical Analysis of VH-CATT Cylindrical Gears
2.1 VH-CATT Gear Formation Principle
The manufacturing process of VH-CATT cylindrical gears is similar to that of hypoid gears, involving a rotary cutting tool with double-edged blades. The gear blank rotates about its axis while the tool moves along its own axis, forming the gear teeth through a generating motion.
2.2 Tooth Surface Equation
To establish the mathematical model of the VH-CATT gear tooth surface, a coordinate system is defined for both the tool and the gear blank, The tooth surface equations are derived based on the relative positions and orientations of these coordinate systems.
The equations for the working and transitional tooth surfaces are derived as follows:
Working Surface:zA=Rm(1−cos(2θi))
Transitional Surface:zB=Rm2−r2sin2αiRmhmrB
where Rm, θi, hmr, B, r, and αi are the gear parameters defined in the coordinate system.
2.3 Contact Ellipse Calculation
The contact ellipse between mating VH-CATT gears is determined using the Hertzian contact theory. The combined curvature radii in the tooth trace and profile directions are calculated, and the contact ellipse dimensions (long and short axes) are obtained using numerical methods.
3. Crack Propagation Modeling using XFEM
3.1 XFEM Theory
XFEM extends the capabilities of traditional FEM by allowing discontinuities and singularities within the element mesh, without the need for remeshing during crack propagation. The displacement field is enriched with additional functions to represent crack surfaces and crack tips.
3.2 Gear Model and Boundary Conditions
A simplified five-tooth model of the VH-CATT gear pair is imported into a FEM software for analysis. A torque of 140 N·m is applied to the driven gear, while the driving gear is rotated at 1.968 rad/s. The tooth surfaces are defined as frictionless contact pairs.
3.3 Crack Initiation and Propagation
An initial semi-circular crack of 0.2 mm radius is pre-inserted at the dangerous position identified through contact stress analysis. The crack is allowed to propagate using the XFEM module, and the crack path is recorded. The crack propagation trajectories for various cases.
4. Crack Propagation Rate Analysis
4.1 Effect of Tooth Line Radius
The crack propagation rates in the tooth width and tooth core directions are analyzed for different tooth line radii (100 mm, 200 mm, and 300 mm). The results, indicate that the crack propagation rate in the tooth width direction decreases with increasing tooth line radius, while the rate in the tooth core direction increases.
4.2 Effect of Modulus
The effect of gear modulus on crack propagation is investigated for values ranging from 1.5 mm to 5 mm. both the tooth width and tooth core propagation rates increase with modulus.
4.3 Effect of Torque
The analysis is extended to investigate the effect of applied torque on crack propagation. Torques of 140 N·m, 250 N·m, and 360 N·m are considered. higher torques lead to faster crack propagation rates in both directions.
5. Stress Intensity Factor Analysis
5.1 Stress Intensity Factor Theory
The stress intensity factor (SIF) is a critical parameter in fracture mechanics, used to quantify the stress state near the crack tip. The SIFs for Modes I, II, and III are calculated using the M-integral method.
5.2 Analysis Model
An analysis model is developed using ABAQUS and FRANC3D to calculate the SIFs during crack propagation. The initial crack is inserted at the determined dangerous position, and the crack is allowed to propagate in increments of 0.03 mm.
5.3 Results and Discussion
The SIFs in the tooth width and tooth core directions are analyzed for different crack sizes and gear parameters. the variation of Mode I SIF with crack size for different moduli and tooth line radii, respectively.
The results indicate that the Mode I SIF is the dominant factor in crack propagation and increases with crack size and gear modulus. A larger tooth line radius effectively reduces the SIF during long crack propagation.
6. Conclusion
This study investigated the contact fatigue crack propagation characteristics of VH-CATT cylindrical gears using numerical simulation and XFEM analysis. The key findings are summarized below:
- A numerical model for calculating the contact ellipse of VH-CATT gears was established based on their curvature characteristics.
- The dangerous positions for crack initiation were identified using FEM and numerical calculations.
- The crack propagation law in VH-CATT gears was studied using XFEM, revealing crack trajectories and propagation rates under various conditions.
- The effects of gear modulus, tooth line radius, and applied torque on crack propagation rates were analyzed.
- The stress intensity factors during crack propagation were calculated using the M-integral method, indicating that Mode I SIF is the dominant factor.
The results provide valuable insights into the design and optimization of VH-CATT cylindrical gears to mitigate contact fatigue cracks and enhance their performance and reliability.