In the field of mechanical engineering, the design and performance analysis of gears are crucial for the efficiency and reliability of various mechanical systems. This article focuses on the modification design and load-bearing contact analysis of a cylindrical gear with variable hyperbolic circular arc tooth trace (VHCATT), providing in-depth insights into its characteristics and applications.
The VHCATT cylindrical gear is a novel type of transmission with promising applications in automotive, aerospace, and heavy machinery industries. Its tooth trace is a circular arc, the mid-section tooth profile is a standard involute, and other cross-section tooth profiles are hyperbolic, combining the advantages of spur, helical, and herringbone gears.
Previous research on VHCATT cylindrical gears has mainly focused on meshing principles, 3D modeling, contact performance, processing methods, and applications. For example, studies have been conducted to derive the tooth surface equation based on the meshing principle and establish 3D models. Additionally, research has been done on the processing methods of the gear, and its contact performance has been analyzed using finite element methods. However, the research on tooth surface modification design and load-bearing contact analysis of this gear is relatively limited or not in-depth, which restricts its application and promotion.
To improve the load-bearing capacity and dynamic characteristics of the VHCATT cylindrical gear, a tooth surface modification design method is proposed, including the cutter inclination method in the tooth line direction and the parabola modification blade method in the tooth profile direction.
The mathematical model for the modified tooth surface is derived based on the gear meshing principle and the modified tooth surface forming principle. The equation for the inclined cutter blade in the OdXdYdZd coordinate system is given by:
| Equation | Expression |
|---|---|
| x_d | {[±(u + u0) sin α ± (πm / 4) – au^(2n) cos α – RT] cos γ + (u + u0) cos α ± au^(2n) sin α} sin γ} cos θ |
| y_d | – {[±(u + u0) sin α ± (πm / 4) – au^(2n) cos α – RT] cos γ + (u + u0) cos α ± au^(2n) sin α} sin γ} sin θ |
| z_d | {[±(u + u0) sin α ± (πm / 4) – au^(2n) cos α + RT] sin γ + (u + u0) cos α ± au^(2n) sin α} cos γ |
The normal vector nx and the relative velocity v(x1)d at the contact point between the modified tooth surface blade and the blank should satisfy nx · v(x1)d = 0. By solving this equation, the expression for u can be obtained as:
| Equation | Expression |
|---|---|
| An * u^(4n – 1) + Rn * u^(2n – 1) + Pn * u + Qn | 0 |
where An, Rn, Pn, and Qn are defined as follows:
| Parameter | Expression |
|---|---|
| An | 2n * a^2 * cos θ |
| Rn | ± 2n * a * (πm / 4) |
| Pn | cos θ |
| Qn | u0 * cos θ + (πm / 4) * cos θ + 2n * a * RT * cos θ * cos(γ ± α) – 2n * a * (RT + R1 * φ1 * cos γ) * cos(γ ± α) – 2n * a * R1 * φ1 * sin γ * cos θ * sin(γ ± α) + 4 * cos θ * sin α – R1 * φ1 * sin γ * cos θ * cos(γ ± α) + (RT + R1 * φ1 * cos γ) * sin(γ ± α) – RT * cos θ * sin(γ ± α) |
The modified tooth surface equation is then obtained by transforming the equation of the tool rotary surface in the OdXdYdZd coordinate system to the gear coordinate system O1X1Y1Z1. The modified tooth surface is reconstructed using software such as MATLAB and UG.
A gear system geometric contact model is established to analyze the contact between the driving and driven gears. Considering the design efficiency and processing economy, only the tooth surface of the driven gear is modified. In the fixed coordinate system OgXgYgZg, the concave surface of the driving gear and the convex surface of the driven gear are analyzed. When the driving and driven gear tooth surfaces contact at point M, they have the same position vector and unit normal vector in the same coordinate system, but there are only 5 independent scalar equations.
A loading contact analysis model is developed to study the load-bearing characteristics of the gear. The deformation coordination equation at a discrete point j on the tooth surface is given by (Σ_(k = 1)^n f_jk + Σ_(k = 1)^n f_j’k) * F_k + w_j = s_z + d_j, where fjk and fj’k are the flexibility coefficients of the driving and driven gears, respectively. When there are n discrete contact points on the tooth surface and multiple tooth pairs are in contact, the total deformation coordination equation is written in matrix form as [S]_m * [F]_m + [w]_m = s_z * [e] + [d]_m, where [S] is the flexibility matrix of the tooth surface contact points, [F] is the load matrix of the tooth surface contact points, [w] is the initial gap matrix of the tooth surface before deformation, and [d] is the residual gap matrix at the contact points of the tooth surface after deformation.
The tooth surface flexibility matrix is a key technology in gear load-bearing contact analysis. In this study, the software ABAQUS is used for secondary development using the Python language to establish a finite element model, calculate the flexibility coefficient after loading the tooth surface, obtain the node flexibility matrix of the tooth surface, and then use binary interpolation to calculate the flexibility matrix of the contact discrete points in the direction of the major axis of the instantaneous contact ellipse.
The initial contact gap is calculated based on the geometric transmission error. The tooth surface normal gap bM at point M is given by bM = √((x2^M – x1^M)^2 + (y2^M – y1^M)^2 + (z2^M – z1^M)^2), where the coordinates of points M1 and M2 are parameter equations related to θ1 and φ1, and θ2 and φ2, respectively. The geometric transmission error Δφ2 is given by Δφ2 = φ2(φ1) – φ2(φ10) – (z1 / z2) * (φ1 – φ10), and the tooth surface gap before loading is w = b + δ.
A nonlinear programming model is established to describe the equilibrium state of the tooth surface contact under load, with the objective function of minimizing the deformation energy of the transmission system. The unknown parameters to be solved are P, sz, and d, while the known parameters are S, w, T, d, and n, among others. By solving this model and converting the normal displacement sz of the large gear under the external load to the angular displacement Δe of the driven gear, the load transmission error Δφ of the transmission system can be calculated as Δφ = Δe + Δφ2.
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 cutter inclination has a significant impact on the load distribution. As the cutter inclination increases, the width of the contact area gradually increases, and the load on the modified tooth surface decreases. However, it has no effect on the load mutation in the alternating area between single-tooth and double-tooth meshing. The reason for this is that as the cutter inclination increases, the curvature radius of the tooth surface in the tooth line direction increases, the gap between the tooth surfaces decreases, the width of the contact area after loading increases, and the tooth surface load decreases.
The parabolic coefficient also affects the load distribution. After modification, the tooth surface load in the double-tooth meshing area gradually increases at the beginning of the meshing of a pair of gears and gradually decreases in the double-tooth meshing area at the end of the meshing. The load mutation in the alternating area between single-tooth and double-tooth meshing is improved. However, as the parabolic coefficient increases, the tooth surface load gradually decreases at the beginning or end of meshing, and even the tooth surface may not be in actual contact, i.e., the tooth surface load is 0. This is because as the parabolic coefficient increases, the tooth top and root modification amounts increase, resulting in a larger gap between the tooth top and root, and there is still a certain gap between the tooth surfaces after loading, i.e., no contact.
The parabolic vertex position has a significant influence on the load distribution. When the parabolic vertex position changes from 3 mm to -3 mm, the tooth surface load gradually decreases. Especially when the parabolic vertex position is -3 mm, full-width contact is achieved near the pitch circle of the gear, and the maximum node load on the tooth surface is 114.4647 N, which is a reduction of about 77.33% compared to the maximum value of 504.9278 N when the parabolic vertex position is 3 mm. Moreover, the mutation characteristics of the tooth surface load are effectively improved. However, due to the smaller stiffness at the end of the tooth width, the end load is larger during full-width contact.
The cutter inclination also has an impact on the load transmission error. In the single-tooth meshing area, the load transmission error is larger, while in the double-tooth meshing area, it is smaller. The variation law of the transmission error is rectangular in shape. Both the amplitude and variation range of the load transmission error decrease with the increase of the cutter inclination. For example, when the cutter inclination is 7°, the maximum load transmission error is -14.53″, and the minimum is -8.90″. When the cutter inclination is 0°, the maximum load transmission error is 23.69″, and the minimum is -14.35″, with variation ranges of 5.63″ and 9.34″, respectively. The main reason for this is that as the cutter inclination increases, the contact area increases, the ability to resist deformation increases, the load-bearing contact deformation of the system decreases, and therefore the load transmission error decreases.
The parabolic coefficient affects the load transmission error as well. Within a certain range, increasing the parabolic coefficient reduces the mutation amplitude of the load transmission error at the alternating point between single-tooth and double-tooth meshing and also reduces the variation range of the load transmission error. However, the amplitude of the load transmission error at the entry and exit of meshing increases. For example, when the parabolic coefficients are 0, -0.00005, and -0.00010, the corresponding mutation quantities of the load transmission error are 5.95″, 4.85″, and 3.83″, and the variation ranges of the load transmission error are 7.37″, 5.29″, and 7.74″, respectively. The load transmission errors at the entry or exit moments are -12.99″, -18.245″, and -22.96″, respectively. However, when the parabolic coefficient is too large, the load-bearing contact error of the system deteriorates sharply. When the parabolic coefficient is 0.0005 in the figure, the maximum load-bearing transmission error of the gear is -57.21″. According to the influence of the parabolic coefficient on the tooth surface load, different parabolic coefficients result in different modification amounts of the tooth profile curve. Since the load-bearing transmission error of the gear is the sum of the geometric transmission error and the transmission error caused by the system deformation after loading, numerical calculations show that the increase of the parabolic coefficient increases the tooth surface gap, thereby increasing the geometric transmission error of the system and affecting the load-bearing transmission error of the system.
The parabolic vertex position also influences the load transmission error. When the parabolic vertex position is greater than 0, an increase in the parabolic vertex position has little effect on the load transmission error in the double-tooth meshing area during entry but increases the load transmission error in the double-tooth meshing area during exit. In general, the variation range of the load transmission error amplitude and the mutation quantity of the load transmission error at the alternating moment between single and double teeth increase with the increase of the parabolic vertex position. When the parabolic vertex position is less than 0, the load transmission error in the double-tooth meshing area during entry and exit and the mutation quantity of the load transmission error at the alternating moment between single and double teeth decrease with the increase of the parabolic vertex position, but the variation range of the load transmission error amplitude slightly increases. Changing the parabolic vertex position changes the geometric transmission error of the system.
The load also has an effect on the load-bearing transmission error. The larger the load, the greater the load-bearing transmission error of the system, and the larger the fluctuation range. This is because as the load increases, the comprehensive elastic deformation of the gear system increases.
In conclusion, the tooth surface load gradually decreases with the increase of the cutter inclination, and the cutter inclination has no effect on the load mutation at the alternating point between single and double teeth. The increase of the parabolic coefficient improves the load mutation at the alternating point between single and double teeth, but when the parabolic coefficient is too large, the tooth surface may not be in actual contact during entry or exit of meshing. When the parabolic vertex position changes from 3 mm to -3 mm, the tooth surface load gradually decreases.
The amplitude and variation range of the load transmission error both decrease with the increase of the cutter inclination. The increase of the parabolic coefficient reduces the transmission error and the error mutation at the alternating point between single and double teeth, but when the parabolic coefficient is too large, the transmission error deteriorates sharply. In general, the smaller the parabolic vertex position u0, the smaller the transmission error amplitude. The larger the load, the greater the transmission error and the larger the fluctuation range.
These research results provide a theoretical basis for the further design and industrial application of VHCATT cylindrical gears, and also contribute to the optimization of gear transmission systems and the reduction of noise in mechanical systems. Future research can focus on further improving the accuracy and reliability of the tooth surface modification design and load-bearing contact analysis methods, as well as exploring the application of new materials and manufacturing technologies to enhance the performance and durability of the gears.