This paper focuses on the modification design and load tooth contact analysis of cylindrical gears with variable hyperbolic circular arc tooth trace (VHCATT). It aims to improve the load-bearing capacity and dynamic characteristics of such gears. The research involves deriving the modified tooth surface equation, establishing relevant models through geometric contact analysis and finite element method, and analyzing the influence of key parameters on tooth surface load distribution and load transmission error. The results provide valuable theoretical support for the design and application of VHCATT cylindrical gears.
1. Introduction
Cylindrical gears with variable hyperbolic circular arc tooth trace are a novel transmission form. They possess the advantages of straight gears, helical gears, and herringbone gears. Their tooth trace is an arc line in the middle section, and the tooth profile curves in other sections are hyperbolas. This unique structure gives them potential applications in various fields such as automobiles, aerospace, and heavy machinery. However, previous research on these gears mainly focused on meshing principles, 3D modeling, and contact performance, while research on tooth surface modification design and load contact analysis was relatively scarce. This paper fills this gap by proposing a tooth surface modification design method and conducting in-depth research on its load-bearing contact characteristics.
2. Gear Structure and Characteristics
2.1 Gear Geometry
The VHCATT cylindrical gear has a specific geometric structure. The tooth trace in the middle section is an arc, which is different from the traditional straight or helical tooth trace. The tooth profile in the middle section is a standard involute, while in other sections, it is a hyperbola. This combination of geometries endows the gear with unique meshing and load-bearing properties.
2.2 Advantages and Application Prospects
Compared with traditional gears, VHCATT gears have better load distribution characteristics and can withstand greater loads. They also have better dynamic performance, which can reduce vibration and noise during operation. In the automotive industry, they can improve the transmission efficiency and reliability of the gearbox. In aerospace and heavy machinery, they can meet the requirements of high load and high precision transmission.
3. Tooth Surface Modification Design Method
3.1 Cutter Inclination Method in Tooth Line Direction
In the tooth line direction, a cutter inclination milling method is proposed. The cutter is inclined at an angle during the machining process. As shown in Figure 1, the inner and outer cutting edges of the cutter form an angle of with the rotation axis. By tilting the cutter, the tooth surface curvature in the tooth line direction can be adjusted. This helps to improve the contact characteristics between the gears and reduce the load concentration on the tooth surface.
| Cutter Inclination Angle () | Effect on Tooth Surface |
|---|---|
| (Unmodified) | Standard tooth surface curvature in the tooth line direction |
| Slightly increased contact area, reduced load concentration | |
| Further improved contact characteristics, more uniform load distribution | |
| Significantly increased contact area, obvious reduction in load on the tooth surface |
3.2 Parabola Modification Blade Method in Tooth Profile Direction
In the tooth profile direction, a parabola modification blade is used. The equation of the parabola is . By changing the parameters , , and , the shape of the tooth profile can be adjusted. Different values of can result in different orders of parabola curves, such as quadratic, quartic, or higher-order curves. The parameter controls the curvature of the parabola, and determines the position of the parabola vertex. This modification method can effectively change the tooth surface structure at the tooth tip and root, improving the meshing performance and load-bearing capacity of the gear.
| Parameter | Effect on Tooth Profile |
|---|---|
| Determines the order of the parabola curve, affecting the smoothness of the tooth profile transition | |
| Controls the curvature of the parabola, influencing the amount of tooth profile modification | |
| Determines the position of the parabola vertex, changing the distribution of the tooth profile modification |
4. Gear System Modeling
4.1 Geometric Contact Model
The gear system geometric contact model is established based on the meshing principle. Considering the design efficiency and processing economy, only the tooth surface of the driven gear is modified. In a fixed coordinate system, the position vectors and unit normal vectors of the contact points on the driving and driven gears are related. By solving a set of equations, the contact conditions between the gears can be determined. This model provides a basis for calculating the tooth surface clearance and analyzing the meshing characteristics of the gears.
4.2 Loading Contact Analysis Model
4.2.1 Establishment of Gear Loading Contact Model
The gear loading contact model takes into account the deformation of the tooth surface under load. Based on the deformation coordination equation, the relationship between the load and the deformation of the contact points is established. The model also considers the flexibility matrix of the contact points, which is calculated by using the finite element method. By solving the model, the load distribution and deformation of the tooth surface can be obtained.
4.2.2 Calculation of Initial Contact Clearance
The initial contact clearance between the gears is calculated by analyzing the geometric relationship between the tooth surfaces. The clearance is related to the geometric transmission error. By converting the geometric transmission error into the displacement in the normal direction of the tooth surface, the initial contact clearance can be determined. This parameter is an important factor affecting the meshing performance and load-bearing capacity of the gears.
4.2.3 Nonlinear Programming Model for Tooth Loading Contact
A nonlinear programming model is established to describe the equilibrium state of the tooth surface contact under load. The objective function of the model is to minimize the deformation energy of the transmission system. By considering the conditions of deformation coordination, force balance, and non-penetration, the model can predict the load distribution and load transmission error of the gears. Solving this model can provide valuable information for optimizing the design of the gears.
5. Analysis of the Influence of Modification Parameters on Gear Performance
5.1 Influence of Cutter Inclination Angle on Load Distribution
As shown in Figure 2, with the increase of the cutter inclination angle , the contact area width of the modified tooth surface gradually increases, and the load on the tooth surface decreases. However, the cutter inclination angle has no significant effect on the load mutation in the alternating area of single-tooth and double-tooth meshing. This is because the increase of the cutter inclination angle leads to an increase in the radius of curvature of the tooth surface in the tooth line direction, a decrease in the tooth surface clearance, and an increase in the contact area width after loading, resulting in a decrease in the tooth surface load.
| Cutter Inclination Angle () | Contact Area Width (mm) | Maximum Tooth Surface Load (N) |
|---|---|---|
| (Unmodified) | [Initial width value] | [Initial load value] |
| [Increased width value] | [Reduced load value] | |
| [Further increased width value] | [Further reduced load value] | |
| [Significantly increased width value] | [Significantly reduced load value] |
5.2 Influence of Parabola Coefficient on Load Distribution
When the parabola coefficient is changed, the load distribution of the modified tooth surface also changes. As shown in Figure 3, when takes different values, the load on the tooth surface in the double-tooth meshing area at the beginning and end of meshing shows different trends. With the increase of , the load mutation in the alternating area of single-tooth and double-tooth meshing is improved. However, when is too large, the tooth surface may not be in contact at the beginning or end of meshing. This is because the increase of leads to an increase in the modification amount of the tooth tip and root, resulting in an excessive clearance between the teeth, and there is still a certain clearance between the tooth surfaces after deformation under load.
| Parabola Coefficient () | Load in Double-Tooth Meshing Area at the Beginning of Meshing (N) | Load in Double-Tooth Meshing Area at the End of Meshing (N) |
|---|---|---|
| (Unmodified) | [Initial load value] | [Initial load value] |
| [Increased load value] | [Decreased load value] | |
| [Further increased load value] | [Further decreased load value] | |
| [May cause no contact, load value approaches 0] | [May cause no contact, load value approaches 0] |
5.3 Influence of Parabola Vertex Position on Load Distribution
The position of the parabola vertex has a significant impact on the load distribution of the modified tooth surface. As shown in Figure 4, when changes from mm to mm, the tooth surface load gradually decreases. Especially when mm, full-width contact near the pitch circle of the gear can be achieved, and the maximum node load on the tooth surface is significantly reduced. However, when full-width contact occurs, the load on the end face of the tooth width may increase due to the relatively small stiffness of the end face.
| Parabola Vertex Position () (mm) | Maximum Tooth Surface Node Load (N) | Contact Status |
|---|---|---|
| [Initial maximum load value] | Partial contact | |
| [Reduced load value] | Partial contact | |
| [Further reduced load value] | Partial contact | |
| [Significantly reduced load value] | Partial contact | |
| Full-width contact near the pitch circle |
5.4 Influence of Cutter Inclination Angle on Load Transmission Error
The cutter inclination angle also affects the load transmission error of the gear. As shown in Figure 5, in the single-tooth meshing area, the load transmission error is relatively large, while in the double-tooth meshing area, it is relatively small. The amplitude and variation range of the load transmission error decrease with the increase of the cutter inclination angle. This is because the increase of the cutter inclination angle leads to an increase in the contact area and the ability to resist deformation, resulting in a decrease in the load contact deformation of the system and a reduction in the load transmission error.
| Cutter Inclination Angle () | Maximum Load Transmission Error (“) | Minimum Load Transmission Error (“) | Variation Range (“) |
|---|---|---|---|
| [Reduced value] | [Reduced value] | [Reduced value] | |
| [Further reduced value] | [Further reduced value] | [Further reduced value] | |
5.5 Influence of Parabola Coefficient on Load Transmission Error
The parabola coefficient has an important influence on the load transmission error of the gear. As shown in Figure 6, within a certain range, increasing the parabola coefficient can reduce the mutation amplitude of the load transmission error at the alternating position of single-tooth and double-tooth meshing and the variation amplitude of the load transmission error. However, when the parabola coefficient is too large, the system load contact error will deteriorate rapidly. This is because the change of the parabola coefficient affects the modification amount of the tooth profile, which in turn affects the tooth surface clearance and the geometric transmission error of the system, ultimately affecting the load transmission error.
| Parabola Coefficient () | Mutation Amplitude of Load Transmission Error at Alternating Position (“) | Variation Amplitude of Load Transmission Error (“) |
|---|---|---|
| (Large value causing deterioration) | [Significantly increased error value] | [Significantly increased error value] |
5.6 Influence of Parabola Vertex Position on Load Transmission Error
The position of the parabola vertex also affects the load transmission error of the gear. When , increasing has little effect on the load transmission error in the double-tooth meshing area at the beginning of meshing, but the load transmission error in the double-tooth meshing area at the end of meshing increases. Overall, the amplitude variation of the load transmission error and the mutation amount at the alternating position of single-tooth and double-tooth meshing increase with the increase of . When , increasing reduces the load transmission error in the double-tooth meshing area at the beginning and end of meshing and the mutation amount at the alternating position of single-tooth and double-tooth meshing, but the amplitude variation of the load transmission error slightly increases.
| Parabola Vertex Position () (mm) | Amplitude Variation of Load Transmission Error (“) | Mutation Amount of Load Transmission Error at Alternating Position (“) |
|---|---|---|
| [Initial amplitude variation value] | [Initial mutation amount value] | |
| [Increased value] | [Increased value] | |
| [Further increased value] | [Further increased value] | |
| [Reduced value] | [Reduced value] | |
| [Slightly increased value] | [Reduced value] |
5.7 Influence of Load on Gear Load Transmission Error
The load has a direct impact on the gear load transmission error. As shown in Figure 7, with the increase of the load, the system load transmission error and its fluctuation range increase. This is because the increase of the load leads to an increase in the comprehensive elastic deformation of the gear system, resulting in an increase in the load transmission error.
| Load (N·m) | Maximum Load Transmission Error (“) | Minimum Load Transmission Error (“) | Fluctuation Range (“) |
|---|---|---|---|
| [Initial maximum error value] | [Initial minimum error value] | [Initial fluctuation range value] | |
| [Increased value] | [Increased value] | [Increased value] | |
| [Further increased value] | [Further increased value] | [Further increased value] | |
| [Significantly increased value] | [Significantly increased value] | [Significantly increased value] | |
| [Even more increased value] | [Even more increased value] | [Even more increased value] |
6. Conclusion
In this paper, a tooth surface modification design method for VHCATT cylindrical gears is proposed. Through theoretical analysis and model establishment, the influence of key modification parameters on the gear’s tooth surface load distribution and load transmission error is studied. The results show that reasonable selection of the cutter inclination angle and parabola vertex position can effectively reduce the tooth surface load and improve the system load transmission error characteristics. The appropriate parabola coefficient can improve the load mutation situation during the alternation of single-tooth and double-tooth meshing. These research results provide a theoretical basis for the dynamic design and industrial application of VHCATT cylindrical gears, which is of great significance for improving the performance and reliability of gear transmission systems. Future research can further explore the optimization of modification parameters under different working conditions and expand the application scope of VHCATT cylindrical gears.
