This article presents a detailed study on the modification design and load tooth contact analysis of a cylindrical gear with variable hyperbolic circular arc tooth trace (VHCATT). The VHCATT cylindrical gear is a novel transmission form with an arc tooth line in the tooth direction, a standard involute curve in the middle section tooth profile, and hyperbolic curves in other sections, combining the advantages of spur gears, helical gears, and herringbone gears. It has good application prospects in the fields of automobiles, aerospace, and heavy machinery.
1. Introduction
The research on VHCATT cylindrical gears currently mainly focuses on meshing principles, 3D modeling and contact performance, processing methods, and applications. However, studies on tooth surface modification design and load contact analysis of VHCATT cylindrical gears are relatively rare or not in-depth, which restricts the optimization of tooth surfaces and the design of high-quality gear transmission systems, as well as the noise reduction optimization design of the transmission system. To address this issue, this article proposes a tooth surface modification design method in the tooth profile and tooth line directions, derives the modified tooth surface equation, establishes a geometric contact model of the gear transmission, calculates the tooth surface clearance, calculates the contact point flexibility matrix using the finite element method, and further establishes a load contact analysis model to analyze the influence of cutter inclination angle, parabolic coefficient, and parabolic vertex position on the load distribution and load transmission error of the modified tooth surface. The research results provide a theoretical basis for the dynamic design and industrial application of VHCATT cylindrical gears.
2. Tooth Surface Modification Mathematical Model
The processing of VHCATT cylindrical gears by large cutter milling involves the close cooperation of cutter rotation, cutter translation, blank rotation, and blank indexing. To improve the load-carrying capacity and dynamic characteristics, a modification method of cutter inclination milling in the tooth line direction is proposed. The inner and outer blades of the inclined cutter are at an angle of α ± γ with the rotation axis, and the rotating cutter is inclined at an angle to ensure that the cutter pitch line is tangent to the blank indexing circle when installed in the standard way.
In the tooth profile direction, a parabolic curve z = ax^2n is used for the forming cutter. By changing the value of the parameter n (n = 1, 2, 3,…), parabolic curves of the second, fourth, or higher orders can be obtained. The expression of the cutter modification curve in the coordinate system OdXdfYdfZdf is given.
Based on the above equations and the coordinate system shown in Figure 3, the expression of the inclined cutter blade in the coordinate system OdXdYdZd can be obtained. According to the gear meshing principle, the vector product of the normal vector of the contact point between the modified tooth surface cutter and the blank and the relative velocity is zero. By calculating and simplifying, the equation about u can be obtained, and the solution of u can be found by solving this equation.
Further, the modified tooth surface equation is obtained by transforming the tool rotary surface equation in the coordinate system OdXdYdZd to the gear coordinate system O1X1Y1Z1. The transformation matrix Aid from OdXdYdZd to O1X1Y1Z1 is determined. Finally, the modified tooth surface mathematical model is established.
3. Gear System Geometric Contact Model
A gear pair meshing transmission coordinate system is shown in Figure 5. Considering design efficiency and processing economy, only the tooth surface of the driven gear is modified. The tooth surface geometric contact analysis requires expressing the tooth surfaces and normal vectors of the driving and driven gears in a certain fixed coordinate system. Here, the OgXgYgZg coordinate system is taken as the fixed coordinate system, and the concave surface of the driving gear and the convex surface of the driven gear are analyzed.
According to the meshing principle, when the tooth surfaces of the driving and driven gears contact at point M, the position vectors and unit normal vectors of point M on the tooth surfaces of the driving and driven gears in the same coordinate system are the same, but there are only 5 independent scalar equations. By taking the meshing angle of the driving gear ψ1 as the input quantity and solving the geometric contact model, the values of θ1(ψ1), φ1(ψ1), θ2(ψ1), φ2(ψ1), and ψ2(ψ2) can be obtained.
4. Load Contact Analysis Model
4.1 Gear Load Contact Model Establishment
The gear load contact deformation model is shown in Figure 6, including single-tooth pair contact deformation and double-tooth pair contact deformation. The deformation coordination equation at point j on the discrete point of the tooth surface is given. When there are n discrete contact points on the tooth surface and multi-tooth pair contact is considered, the total deformation coordination equation can be written in matrix form. The tooth surface flexibility matrix is one of the key technologies in gear load contact analysis. In this article, a finite element model is established based on the software ABAQUS and Python language for secondary development to calculate the flexibility coefficient after tooth surface loading, obtain the tooth surface node flexibility matrix, and then calculate the flexibility matrix of the contact discrete points in the direction of the major axis of the instantaneous contact ellipse using binary interpolation.
4.2 Initial Contact Gap Calculation
As shown in Figure 8, the normal gap bM at point M on the tooth surface is calculated by the coordinates of the intersection points M1 and M2 of the straight line L passing through point M parallel to the normal vector ng(M0) and the driving and driven tooth surfaces. The tooth gap depends on the geometric transmission error. The geometric transmission error of the gear pair is shown in Figure 9, and the tooth gap δ is obtained by converting the geometric transmission error into the displacement in the normal direction of the tooth surface according to the geometric relationship of the gear pair. Then, the tooth surface gap before gear loading deformation is w = b + δ.
4.3 Nonlinear Programming Model of Tooth Load Contact
A nonlinear programming model is established to describe the equilibrium state of tooth surface contact under load with conditions such as deformation coordination, force balance, and non-embedding. The objective function of this nonlinear programming is to minimize the deformation energy of the transmission system. Known parameters include S, w, T, d, and n, and the parameters to be solved are P, s2, and d. By solving this model, the normal displacement sz of the large gear under the external load is converted into the angular displacement Δe of the driven gear, and then the load transmission error Δφ of the transmission system can be calculated by Δφ = Δe + Δφ2.
5. Analysis of Load Distribution on Modified Tooth Surface
5.1 Influence of Cutter Inclination on Load Distribution
Figure 10 shows the influence of the cutter inclination on the load distribution of the modified tooth surface. As the cutter inclination γ increases, the width of the contact area gradually increases, and the load on the modified tooth surface gradually decreases. However, the cutter inclination has no effect on the load mutation in the area where single-tooth meshing and double-tooth meshing alternate. The reason for the change in the load distribution of the modified tooth surface is that as the cutter inclination increases, the curvature radius of the tooth line direction of the modified tooth surface increases, the gap between the tooth surfaces decreases, the width of the contact area increases after loading, and the tooth surface load decreases.
5.2 Influence of Parabolic Coefficient on Load Distribution
Figure 11 shows the influence of the parabolic coefficient on the load distribution of the modified tooth surface. After modification, the load on the tooth surface of a pair of gears gradually increases in the double-tooth meshing area at the beginning of meshing, and the load on the tooth surface gradually decreases in the double-tooth meshing area at the stage of exiting meshing, improving the load mutation in the area where single-tooth and double-tooth meshing alternate. However, as the parabolic coefficient increases, the load on the tooth surface gradually decreases at the beginning or exiting meshing moment, and even the phenomenon of actual non-contact on the tooth surface occurs, that is, the tooth surface load is 0 at this time. The reason is that as the parabolic coefficient increases, the tooth top and root modification amounts increase when starting and exiting meshing, resulting in excessive tooth gap between the tooth top and root, and there is still a certain gap between the tooth surfaces after loading and deformation, that is, non-contact.
5.3 Influence of Parabolic Vertex Position on Load Distribution
Figure 12 shows the influence of the parabolic vertex position on the load distribution of the modified tooth surface. It can be seen that the parabolic vertex position has a significant impact on the tooth surface load. When the parabolic vertex position changes from -3.0 mm to 3.0 mm, the load on the tooth surface first increases and then decreases. When the parabolic vertex position is -1.5 mm or 1.5 mm, the load distribution on the tooth surface is relatively uniform.
6. Conclusion
This article proposes a tooth surface modification design method for VHCATT cylindrical gears, including the cutter inclination method in the tooth line direction and the parabola modification blade method in the tooth profile direction. The modified tooth surface equation is derived, the gear transmission geometric contact model is established, the tooth surface clearance is calculated, the contact point flexibility matrix is calculated using the finite element method, and the load contact analysis model is established. The influence of the cutter inclination, parabolic coefficient, and parabolic vertex position on the load distribution and load transmission error of the modified tooth surface is analyzed. The research results show that a reasonable cutter inclination angle and parabolic vertex position can effectively reduce the tooth surface load and improve the characteristics of the system load transmission error, while a reasonable parabolic coefficient can improve the load mutation when single-tooth and double-tooth meshing alternate. These results provide a theoretical basis for the noise reduction design of VHCATT cylindrical gears and have important guiding significance for their further design and industrial application.
