1. Introduction
Cylindrical gears play a crucial role in mechanical transmission systems. Among them, the Variable hyperbolic circular-arc-tooth-trace (VH-CATT) cylindrical gear has unique characteristics and advantages. This article focuses on the study of the contact fatigue crack propagation characteristics of VH-CATT cylindrical gears.
1.1 Gear Failure and Contact Fatigue Cracks
Gear failure is a significant concern in mechanical engineering. Contact fatigue cracks are one of the main causes of gear failure. The process of gear contact fatigue failure generally includes crack initiation, propagation, and fracture. Different crack propagation directions and extents can lead to various failure forms such as pitting, spalling, and tooth breakage.
1.2 VH-CATT Cylindrical Gear
The VH-CATT cylindrical gear has a unique tooth profile. Its tooth line is a circular arc in a certain direction, and the tooth profile in the middle section is an involute, while in other sections, it is an envelope of a family of hyperbolas. Compared with traditional involute spur gears and helical gears, it has better meshing performance, larger contact ratio, stronger load-carrying capacity, higher transmission efficiency, no additional axial force, and lower requirements for installation accuracy.
1.3 Research Significance
Although there have been many studies on VH-CATT cylindrical gears in terms of curvature characteristics, contact characteristics, friction and wear, and machining methods, research on the contact fatigue crack propagation characteristics based on gear contact analysis is scarce. Understanding these characteristics is essential for optimizing gear design, preventing fatigue failure, and extending the service life of gears. It also has important guiding significance for the popularization and application of VH-CATT cylindrical gears in engineering fields.
2. Theoretical Analysis of VH-CATT Cylindrical Gear
2.1 VH-CATT Cylindrical Gear Forming Principle
The machining method of VH-CATT cylindrical gear is similar to that of hypoid bevel gears. This article focuses on the analysis of the double-edged milling method. As shown in Figure 2, the cutter rotates around its axis, and the gear blank rotates around its own axis and moves horizontally at the same time. The inner and outer cutting edges of the tapered milling cutter form the concave and convex tooth surfaces of the VH-CATT cylindrical gear through a generating motion.
2.2 VH-CATT Cylindrical Gear Tooth Surface Equation
To obtain the tooth surface mathematical model of the VH-CATT cylindrical gear, it is necessary to establish the coordinate systems of the gear workpiece and the cutter and use the transformation relationship between them to derive the tooth surface mathematical expression. The equations of the working tooth surface and the transition tooth surface are derived, and different symbols are used to represent the concave and convex tooth surfaces of the main and driven wheels.
2.3 VH-CATT Cylindrical Gear Contact Ellipse Determination
Based on the curvature characteristics of the VH-CATT cylindrical gear in the tooth profile direction and the tooth line direction, the comprehensive curvature radii in the two directions are calculated. Then, according to the elliptical contact numerical calculation equation, the long and short axis radii of the contact ellipse and the stress distribution on the tooth surface contact area at different contact positions of the VH-CATT cylindrical gear pair can be obtained. This provides a theoretical basis for accurately locating the high-stress concentration area on the tooth surface and the dangerous position of tooth surface fatigue crack initiation.
3. Gear Contact Fatigue Crack Propagation Model
3.1 Extended Finite Element Theory (XFEM)
XFEM is an important method for analyzing crack propagation problems. It is based on the finite element method and has the advantage of not needing to update the mesh on the crack surface in real-time during the simulation of crack propagation. It can use the level set method to detect the real-time position and propagation angle of the crack and accurately predict the next step of crack propagation through interpolation functions.
3.2 Gear Contact Fatigue Dangerous Position
By analyzing the tooth surface contact stress obtained from the simulation, it is found that the maximum contact stress in the single-tooth meshing state is relatively close and shows an irregular change trend. By substituting the maximum contact stress in the single-tooth meshing state into the VH-CATT cylindrical gear contact ellipse numerical calculation model, it is found that the contact ellipse is the smallest at the initial position of single-tooth meshing, and this position is more likely to be the dangerous position of fatigue crack initiation.
3.3 Gear Contact Fatigue Crack Model
According to the size definition of crack initiation and propagation, a semicircular crack perpendicular to the tooth surface tangent direction is preset at the contact fatigue dangerous position of the VH-CATT cylindrical gear. The crack size is set to 0.2 mm. The XFEM module of the simulation software is used to numerically simulate the contact fatigue crack propagation of the VH-CATT cylindrical gear.
3.4 VH-CATT Cylindrical Gear Contact Fatigue Crack Propagation Law
The crack propagation trajectory at the contact fatigue dangerous position of the VH-CATT cylindrical gear is a symmetrical circular arc, which is opposite to the direction of the variable hyperbolic circular-arc tooth line. The crack first slowly propagates towards the tooth core, then mainly propagates towards the two end faces of the gear, and finally propagates back towards the tooth core until the tooth breaks.
3.5 Crack Propagation Rate under Different Tooth Line Radii
The influence of different tooth line radii on the crack propagation rate of VH-CATT cylindrical gears is studied. It is found that as the tooth line radius increases, the crack propagation rate in the tooth width direction decreases, while the crack propagation rate in the tooth core direction increases.
3.6 Crack Propagation Rate under Different Moduli
The influence of different moduli on the crack propagation rate of VH-CATT cylindrical gears is analyzed. The results show that as the gear modulus increases, the crack propagation rate in the tooth width direction and the tooth core direction both increase.
3.7 Crack Propagation Rate under Different Torques
The influence of different torques on the crack propagation rate of VH-CATT cylindrical gears is investigated. It is found that as the torque increases, the crack propagation rate in the tooth width direction and the tooth core direction both increase, and the length of the crack tending to be stable in the tooth core direction also increases.
4. Stress Intensity Factor of Crack Propagation
4.1 Stress Intensity Factor Theory
Stress intensity factor is a core concept in fracture mechanics. It is used to predict the stress state near the crack or notch tip caused by changes in service conditions or gear parameters and evaluate the anti-fatigue crack propagation ability of VH-CATT cylindrical gears. The stress intensity factor can be divided into opening mode stress intensity factor , sliding mode stress intensity factor , and tearing mode stress intensity factor .
4.2 Influence of Modulus on Crack Front Stress Intensity Factor
The influence of different moduli on the type I stress intensity factor of the crack front of VH-CATT cylindrical gears is analyzed. The results show that in the long crack propagation stage, the larger the gear modulus, the larger the stress intensity factor of the crack in the tooth width and tooth core directions.
4.3 Influence of Tooth Line Radius on Crack Front Stress Intensity Factor
The influence of different tooth line radii on the type I stress intensity factor of the crack front of VH-CATT cylindrical gears is studied. The results show that when the crack propagates from a short crack to a long crack, the smaller the tooth line radius, the larger the stress intensity factor. In the long crack propagation stage, a larger tooth line radius can effectively reduce the stress intensity factor of the crack front.
4.4 Influence of Crack Preset Angle on Stress Intensity Factor
The influence of different crack preset angles on the type I stress intensity factor of the crack front of VH-CATT cylindrical gears is investigated. The results show that when the crack preset angle is 90° and 135°, in the initial stage of crack propagation, the stress intensity factor in the tooth width and tooth core directions of the 90° preset angle crack is greater than that of the 135° preset angle crack. In the long crack propagation stage, the stress intensity factor in the tooth width and tooth core directions of the 135° preset angle crack is larger.
5. Conclusion
5.1 VH-CATT Cylindrical Gear Contact Ellipse Model
The VH-CATT cylindrical gear contact ellipse numerical calculation model is established based on the tooth surface equation and the contact ellipse numerical calculation method. The single-tooth meshing position of the VH-CATT cylindrical gear is determined as the dangerous position for contact fatigue crack initiation.
5.2 Influence of Parameters on Crack Propagation Rate
The XFEM method is used to establish a numerical simulation model of contact fatigue crack propagation of VH-CATT cylindrical gears. The influence of tooth line radius, modulus, and torque on the crack propagation rate is analyzed. The results show that the crack propagation rate in the tooth width direction is affected by these parameters as described above, and the crack propagation rate in the tooth core direction is also affected accordingly.
5.3 Influence of Parameters on Crack Front Stress Intensity Factor
The VH-CATT cylindrical gear contact fatigue crack front stress intensity factor analysis model is established. The influence of modulus, tooth line radius, and crack preset angle on the type I stress intensity factor is analyzed. The results show that different moduli, tooth line radii, and crack preset angles have different effects on the stress intensity factor of the crack front in different stages of crack propagation.
In summary, this study provides a theoretical basis and reference for the design and optimization of VH-CATT cylindrical gears, which is beneficial to reducing the crack propagation speed and improving the service life and reliability of gears.