This article focuses on the variable hyperbolic circular arc tooth trace (VHCATT) cylindrical gear. Aiming to enhance its load – bearing capacity and optimize dynamic characteristics, a tooth surface modification design method is proposed. The research details the entire process from theoretical model establishment to result analysis, including the derivation of modified tooth surface equations, construction of geometric and load – bearing contact models, and analysis of the influence of modification parameters on tooth surface load distribution and load – bearing transmission error. The findings provide a solid theoretical foundation for the design and industrial application of VHCATT cylindrical gears.
1. Introduction
1.1 Research Background
The variable hyperbolic circular arc tooth trace (VHCATT) cylindrical gear is a novel transmission form. Its tooth – line is an arc, and the tooth profile in the middle section is a standard involute, while in other sections, it is a hyperbola. This gear combines the advantages of spur gears, helical gears, and herringbone gears, showing great potential in various fields such as automotive, aerospace, and heavy machinery .
However, the current research on VHCATT cylindrical gears mainly concentrates on meshing principles, 3D modeling, contact performance, processing methods, and applications. Research on tooth surface modification design and load – bearing contact analysis is relatively scarce or insufficient. This situation restricts the optimization of the tooth surface of VHCATT cylindrical gears and the design of high – quality gear transmission systems, making it difficult to conduct noise – reduction optimization design for transmission systems and thus hindering the wide application of these gears.
1.2 Research Significance
Tooth surface modification technology can improve the tooth surface load distribution, making the gear operation more stable and reducing meshing impact. Load – bearing contact analysis can accurately evaluate the load – bearing situation of the tooth surface under real – load conditions, which is crucial for the design of gear geometric parameters and the analysis of system dynamic behavior. By conducting in – depth research on the tooth surface modification design and load – bearing contact analysis of VHCATT cylindrical gears, this study aims to fill the research gap, improve the performance of VHCATT cylindrical gears, and promote their industrial application.
2. Tooth Surface Modification Design of VHCATT Cylindrical Gear
2.1 Tooth – line Direction Modification: Cutter Inclination Method
The machining of VHCATT cylindrical gears by the large cutter – disk milling method involves the coordinated movement of cutter – disk rotation, cutter – disk translation, blank rotation, and blank indexing. To improve the gear’s performance, a modification method by tilting the cutter – disk during milling in the tooth – line direction is proposed.
As shown in Figure 1, the inner and outer cutting edges of the tilted cutter – disk form an angle of \(\alpha\pm\gamma\) with the axis of rotation. When installing the rotating cutter – disk, it is tilted by an angle to ensure that the cutting – edge pitch line is tangent to the blank’s dividing circle during standard installation. The distances from the equal – tooth – thickness points on the pitch line to the axis of rotation \(m_{d}’n_{d}’\) are \(R_{T}\), and the radii of gyration of the inner and outer cutting edges around \(m_{d}’n_{d}’\) on the pitch line are \(R_{n}=R_{T}-\frac{\pi m}{4}\cos\gamma\) and \(R_{w}=R_{T}+\frac{\pi m}{4}\cos\gamma\) respectively. [Insert Figure 1: Tilted Milling Cutter – Disk for VHCATT Cylindrical Gear Machining]
2.2 Tooth – profile Direction Modification: Parabolic Modification Blade Method
In the tooth – profile direction, a parabolic curve \(z = ax^{2n}\) is adopted for the forming cutting edge. By changing the value of parameter \(n\) (\(n = 1,2,3,\cdots\)), quadratic, quartic, or higher – order parabolic curves can be obtained. Figure 2 shows the tooth – profile modification blade curve, where \(\Delta t\) and \(\Delta r\) represent the addendum modification amount and the dedendum modification amount respectively, \(O_{dp}\) is the vertex position of the modification curve, \(a\) is the parabolic coefficient, and \(u_{0}\) is the distance between the vertex of the modification curve and the pitch line of the unmodified straight – line cutting edge. The expression of the cutting – edge modification curve in the \(O_{df}X_{df}Y_{df}Z_{df}\) coordinate system is given by specific equations. [Insert Figure 2: Tooth – Profile Modification Blade Curve of VHCATT Cylindrical Gear]
2.3 Derivation of Modified Tooth Surface Equation
Based on the gear meshing principle and the forming principle of the modified tooth surface of VHCATT cylindrical gears, the modified tooth surface equation is derived. First, the expression of the tilted cutter – disk cutting edge in the \(O_{d}X_{d}Y_{d}Z_{d}\) coordinate system is obtained. Then, according to the condition that the vector product of the normal vector of the contact point between the modified tooth surface cutting edge and the blank and the relative velocity is zero (\(n_{x}\cdot v_{d}^{(x1)} = 0\)), an equation about \(u\) is obtained and solved. Finally, through coordinate transformation, the modified tooth surface equation in the gear coordinate system \(O_{1}X_{1}Y_{1}Z_{1}\) is obtained. The detailed process of equation derivation is omitted here to avoid complex mathematical formulas.
2.4 Tooth Surface Reconstruction
Taking a VHCATT cylindrical gear with specific parameters (number of teeth \(z = 29\), tooth width \(b = 80\) mm, pressure angle \(\alpha=20^{\circ}\), modulus \(m = 8\) mm, and tooth – line radius \(R_{T}=200\) mm) as an example, software such as MATLAB and UG is used to realize tooth surface reconstruction. Figure 3 shows the comparison of the tooth surface before and after modification in the tooth – line direction and the tooth – profile direction. It can be seen that the cutter – disk inclination angle mainly affects the arc – bending degree of the tooth, and the tooth – profile modification mainly changes the tooth surface structure characteristics in the addendum and dedendum regions.
3. Geometric Contact Model of Gear System
3.1 Establishment of Meshing Transmission Coordinate System
As shown in Figure 4, a gear pair meshing transmission coordinate system is established. Considering design efficiency and processing economy, only the tooth surface of the driven gear is modified. In the figure, \(\sum ^{I}\) represents the tooth surface of the driving gear, \(\sum ^{II}\) represents the tooth surface of the driven gear, \(M\) is the tooth surface contact point, \(n_{c}\) is the common normal at the contact point, \(\psi_{1}\) is the meshing rotation angle of the driving gear, and \(\psi_{2}\) is the meshing rotation angle of the driven gear. The \(O_{g}X_{g}Y_{g}Z_{g}\) coordinate system is selected as the fixed coordinate system, and the concave surface of the driving gear and the convex surface of the driven gear are analyzed. [Insert Figure 4: Meshing Transmission Coordinate System of Gear Pair]
3.2 Geometric Contact Analysis
According to the meshing principle, when the tooth surfaces of the driving and driven gears are in contact at point \(M\), in the same coordinate system, the two tooth surfaces have the same position vector and unit normal vector at point \(M\), resulting in five independent scalar equations. There are six unknown variables in the contact model (\(\theta_{1}\), \(\varphi_{1}\), \(\psi_{1}\), \(\theta_{2}\), \(\varphi_{2}\), \(\psi_{2}\)). By taking the rotation angle \(\psi_{1}\) of the driving gear as the input quantity and solving the geometric contact model, the values of \(\theta_{1}(\psi_{1})\), \(\varphi_{1}(\psi_{1})\), \(\theta_{2}(\psi_{1})\), \(\varphi_{2}(\psi_{1})\), \(\psi_{2}(\psi_{2})\) can be obtained.
4. Load – Bearing Contact Analysis Model
4.1 Establishment of Gear Load – Bearing Contact Deformation Model
The gear load – bearing contact deformation model is shown, including single – tooth – pair contact deformation and double – tooth – pair contact deformation. Assuming that the small gear is fixed and the large gear has a normal displacement \(s_{2}\) under the action of an external load, the deformation – coordination equation at the discrete point \(j\) is established based on the research of Fang Zongde. When considering multiple discrete contact points on the tooth surface and multi – tooth – pair contact, the overall deformation – coordination equation is written in matrix form. [Insert Figure 5: Load – Bearing Contact Deformation Model of Gear]
4.2 Calculation of Tooth Surface Flexibility Matrix
The tooth surface flexibility matrix is a key technology in gear load – bearing contact analysis. In this study, based on the software ABAQUS and using Python language for secondary development, a finite – element model is established to calculate the flexibility coefficient after tooth surface loading, obtain the tooth surface node flexibility matrix, and then use binary interpolation to calculate the flexibility matrix of the contact discrete points in the long – axis direction of the instantaneous contact ellipse. Figure 6 shows the node loading and tooth surface deformation.
4.3 Calculation of Initial Contact Clearance
The calculation of the initial contact clearance is shown in Figure 7. The normal clearance \(b_{M}\) at point \(M\) on the tooth surface is calculated by the distance formula between two points on the main and driven tooth surfaces. The tooth – to – tooth clearance depends on the geometric transmission error. The geometric transmission error is converted into the displacement in the normal direction of the tooth surface according to the geometric relationship of the gear pair, which is the tooth – to – tooth clearance \(\delta\). The tooth surface clearance before gear loading deformation is \(w = b+\delta\). [Insert Figure 7: Schematic Diagram of Calculating Tooth Surface Normal Clearance]
4.4 Non – linear Programming Model of Tooth Loading Contact
Based on the conditions of deformation coordination, force balance, and non – penetration, a non – linear programming model is established to describe the equilibrium state of tooth surface contact under load. The objective function of this non – linear programming is to minimize the deformation energy of the transmission system. The known parameters include \(S\), \(w\), \(T\), \(d\), and \(n\), etc., and the unknown parameters to be solved include \(P\), \(s_{2}\), and \(d\). The normal displacement \(s_{x}\) of the large gear under the external load is converted into the angular displacement \(\Delta e\) of the driven gear, and the load – bearing transmission error \(\Delta\varphi\) of the transmission system is calculated as \(\Delta\varphi=\Delta e+\Delta\varphi_{2}\).
5. Analysis of Load Distribution on Modified Tooth Surface
5.1 Influence of Cutter Inclination Angle on Load Distribution
Figure 8 shows the influence of the cutter – disk inclination angle on the load distribution of the modified tooth surface. When the cutter – disk inclination angle \(\gamma\) takes values of \(0^{\circ}\) (unmodified), \(3^{\circ}\), \(5^{\circ}\), and \(7^{\circ}\) respectively, and the tooth – profile direction is unmodified with a load of \(1000\) N·m, it can be seen that as the cutter – disk inclination angle increases, the width of the contact area gradually increases, and the load on the modified tooth surface gradually decreases. The cutter – disk inclination angle has no effect on the load mutation in the single – tooth and double – tooth meshing alternating area. The reason for the change in the load distribution of the modified tooth surface is that as the cutter – disk inclination angle increases, the radius of curvature of the tooth surface in the tooth – line direction increases, the clearance between the tooth surfaces decreases, and the width of the contact area increases after loading, resulting in a decrease in the tooth surface load.
| Cutter Inclination Angle (\(\gamma\)) | Contact Area Width | Tooth Surface Load | Influence on Load Mutation in Meshing Alternating Area |
|---|---|---|---|
| \(0^{\circ}\) (unmodified) | Smallest | Largest | None |
| \(3^{\circ}\) | Slightly increased compared to \(0^{\circ}\) | Slightly decreased compared to \(0^{\circ}\) | None |
| \(5^{\circ}\) | Further increased | Further decreased | None |
| \(7^{\circ}\) | Largest among the values | Smallest among the values | None |
5.2 Influence of Parabolic Coefficient on Load Distribution
Figure 9 shows the influence of the parabolic coefficient on the load distribution of the modified tooth surface. When the parabolic coefficient takes values of \(0\) (unmodified), \(- 0.00005\), \(-0.00010\), and \(-0.00050\) respectively, the vertex position of the parabola \(u_{0}\) is \(0\), the cutter – disk inclination angle \(\gamma\) is \(5^{\circ}\), and the load is \(1000\) N·m. After modification, the tooth surface load of a pair of teeth gradually increases in the double – tooth meshing area when starting to mesh and gradually decreases in the double – tooth meshing area when exiting meshing. The load mutation in the single – tooth and double – tooth meshing alternating area is improved. However, as the parabolic coefficient increases, the tooth surface load gradually decreases at the start or end of meshing, and even the phenomenon of non – contact between the tooth surfaces may occur. The reason is that as the parabolic coefficient increases, the modification amounts of the addendum and dedendum increase, resulting in an excessive clearance between the teeth at the addendum and dedendum. Even after the tooth surface is deformed under load, there is still a certain clearance between the tooth surfaces, that is, non – contact.
| Parabolic Coefficient (\(a\)) | Load Change in Double – Tooth Meshing Area at Start of Meshing | Load Change in Double – Tooth Meshing Area at End of Meshing | Load Mutation in Meshing Alternating Area | Possibility of Tooth Surface Non – Contact |
|---|---|---|---|---|
| \(0\) (unmodified) | No obvious change | No obvious change | Large | No |
| \(-0.00005\) | Slightly increased | Slightly decreased | Reduced | No |
| \(-0.00010\) | More obvious increase | More obvious decrease | Further reduced | Slightly possible |
| \(-0.00050\) | Significant increase | Significant decrease | Significantly reduced | More likely |
5.3 Influence of Parabolic Vertex Position on Load Distribution
Figure 10 shows the influence of the vertex position of the parabolic cutting edge on the load distribution of the modified tooth surface. When the vertex position of the parabola takes values of \(-3.0\) mm, \(-1.5\) mm, \(0\), \(1.5\) mm, and \(3.0\) mm respectively, the parabolic coefficient \(a\) is \(-0.00005\), the cutter – disk inclination angle \(\gamma\) is \(5^{\circ}\), and the load is \(1000\) N·m. The vertex position of the parabola has a significant impact on the magnitude of the tooth surface load. When the vertex position of the parabola changes from \(3\) mm to \(- 3\) mm, the tooth surface load gradually decreases. Especially when the vertex position of the parabola is \(-3\) mm, full – tooth – width contact is achieved near the pitch circle of the gear, and the maximum value of the tooth surface node load is \(114.4647\) N, which is a decrease of about \(77.33\%\) compared with the maximum value of \(504.9278\) N when the vertex position of the parabola is \(3\) mm. The mutation characteristics of the tooth surface load are effectively improved. However, due to the low stiffness of the tooth – width end face, the end – face load is relatively large during full – tooth – width contact.
| Parabolic Vertex Position (\(u_{0}\)) | Tooth Surface Load Magnitude | Maximum Tooth Surface Node Load | Improvement of Load Mutation Characteristics | End – Face Load during Full – Tooth – Width Contact |
|---|---|---|---|---|
| \(3.0\) mm | Largest | \(504.9278\) N | Poor | Small |
| \(1.5\) mm | Larger | – | General | Slightly larger |
| \(0\) | Medium | – | Moderate | Moderate |
| \(-1.5\) mm | Smaller | – | Good | Larger |
| \(-3.0\) mm | Smallest | \(114.4647\) N | Best | Largest |
6. Load – Bearing Transmission Error of Modified Gears
6.1 Influence of Cutter Inclination Angle on Load – Bearing Transmission
Figure 11 shows the influence of the cutter – disk inclination angle on the load – bearing contact transmission error of the modified gear. The selected parameters are the same as those in the analysis of the influence of the cutter – disk inclination angle on the tooth surface load distribution. In the single – tooth meshing area, the load – bearing transmission error is relatively large, and in the double – tooth meshing area, it is relatively small. The change rule of the transmission error is in a rectangular shape. The amplitude and variation range of the load – bearing transmission error both decrease as the cutter – disk inclination angle increases. For example, when \(\gamma = 7^{\circ}\), the maximum value of the load – bearing transmission error is \(14.53”\), and the minimum value is \(-8.90”\). When \(\gamma = 0^{\circ}\), the maximum value of the load – bearing transmission error is \(-23.69”\), and the minimum value is \(-14.35”\). The variation ranges are \(5.63”\) and \(9.34”\) respectively. The main reason for the
| Cutter Inclination Angle (\(\gamma\)) | Maximum Load – Bearing Transmission Error | Minimum Load – Bearing Transmission Error | Variation Range of Load – Bearing Transmission Error |
|---|---|---|---|
| \(0^{\circ}\) | \(-23.69”\) | \(-14.35”\) | \(9.34”\) |
| \(3^{\circ}\) | – | – | Slightly reduced compared to \(0^{\circ}\) |
| \(5^{\circ}\) | – | – | Further reduced |
| \(7^{\circ}\) | \(14.53”\) | \(-8.90”\) | \(5.63”\) |
6.2 Influence of Parabolic Coefficient on Load – Bearing Transmission
Figure 12 shows the influence of the parabolic coefficient on the load – bearing contact transmission error. The selected parameters are the same as those in the analysis of the influence of the parabolic coefficient on the tooth surface load distribution. Within a certain range, increasing the parabolic coefficient can reduce the mutation amplitude of the load – bearing transmission error at the alternation of single – tooth and double – tooth meshing and also reduce the amplitude variation of the load – bearing transmission error. However, at the entry and exit of meshing, the amplitude of the load – bearing transmission error increases. For example, when \(a\) takes values of \(0\), \(-0.00005\), and \(-0.00010\), the mutation amounts of the load – bearing transmission error are \(5.95”\), \(4.85”\), and \(3.83”\) respectively; the amplitude variations of the load – bearing transmission error are \(7.37”\), \(5.29”\), and \(7.74”\) respectively; the load – bearing transmission errors at the entry or exit moment are \(-12.99”\), \(-18.245”\), and \(-22.96”\) respectively. But when the parabolic coefficient is relatively large, the load – bearing contact error of the system deteriorates rapidly. In Figure 12, when the parabolic coefficient is \(0.0005\), the maximum value of the gear load – bearing transmission error 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 contact is the sum of the geometric transmission error and the transmission error caused by the deformation of the system under load, numerical calculations show that the modification parameter, the parabolic coefficient, increases the tooth surface clearance, thus increasing the geometric transmission error of the system and affecting the load – bearing transmission error of the system. Figure 13 shows the influence of the parabolic coefficient on the geometric transmission error.
| Parabolic Coefficient (\(a\)) | Mutation Amount of Load – Bearing Transmission Error at Meshing Alternation | Amplitude Variation of Load – Bearing Transmission Error | Load – Bearing Transmission Error at Entry/Exit of Meshing | Influence on System Load – Bearing Contact Error |
|---|---|---|---|---|
| \(0\) | \(5.95”\) | \(7.37”\) | \(-12.99”\) | Normal |
| \(-0.00005\) | \(4.85”\) | \(5.29”\) | \(-18.245”\) | Slightly improved |
| \(-0.00010\) | \(3.83”\) | \(7.74”\) | \(-22.96”\) | Improved in some aspects, but increased error at entry/exit |
| \(0.0005\) | – | – | \(-57.21”\) | Deteriorated rapidly |
6.3 Influence of Parabolic Vertex Position on Load – Bearing Transmission
Figure 14 shows the influence of the parabolic vertex position on the load – bearing contact transmission error. The selected parameters are the same as those in the analysis of the influence of the parabolic vertex position on the tooth surface load distribution. When the value of the parabolic vertex position is greater than zero, increasing the value of the parabolic vertex position has little effect on the load – bearing transmission error in the double – tooth meshing area at the entry of meshing, but the load – bearing transmission error in the double – tooth meshing area at the exit of meshing increases. Overall, the amplitude variation of the load – bearing transmission error and the mutation amount of the load – bearing transmission error at the single – double – tooth alternation increase as the parabolic vertex position increases. When the value of the parabolic vertex position is less than zero, the load – bearing transmission errors in the double – tooth meshing areas at the entry and exit of meshing and the mutation amount of the load – bearing transmission error at the single – double – tooth alternation decrease as the parabolic vertex position increases, but the amplitude variation of the load – bearing transmission error increases slightly. Figure 15 shows the change rule of the geometric transmission error corresponding to each parabolic vertex position. The change of the parabolic vertex position will change the geometric transmission error of the system.
| Parabolic Vertex Position (\(u_{0}\)) | Influence on Load – Bearing Transmission Error in Double – Tooth Meshing Area at Entry of Meshing | Influence on Load – Bearing Transmission Error in Double – Tooth Meshing Area at Exit of Meshing | Amplitude Variation of Load – Bearing Transmission Error | Mutation Amount of Load – Bearing Transmission Error at Single – Double – Tooth Alternation | Influence on Geometric Transmission Error |
|---|---|---|---|---|---|
| \(>0\) | Little effect | Increases | Increases | Increases | Changes |
| \(<0\) | Decreases | Decreases | Increases slightly | Decreases | Changes |
6.4 Influence of Load on Gear Load – Bearing Transmission Error
Figure 16 shows the influence of the load on the load – bearing transmission error. The modification parameters are: parabolic coefficient \(a=-0.00005\), parabolic vertex position \(=-1.5\) mm, cutter – disk inclination angle \(=5^{\circ}\), and the loads are \(400\) N·m, \(700\) N·m, \(1000\) N·m, \(1500\) N·m, and \(1800\) N·m respectively. It can be seen that the greater the load, the greater the load – bearing transmission error of the system, and the greater the fluctuation range. The reason is that as the load increases, the comprehensive elastic deformation of the gear system increases. [Insert Figure 16: Influence of Load on Load – Bearing Transmission Error]
| Load (N·m) | Load – Bearing Transmission Error of the System | Fluctuation Range of Load – Bearing Transmission Error |
|---|---|---|
| \(400\) | Smallest | Smallest |
| \(700\) | Slightly increased compared to \(400\) N·m | Slightly increased |
| \(1000\) | Further increased | Further increased |
| \(1500\) | Significantly increased | Significantly increased |
| \(1800\) | Largest among the values | Largest among the values |
7. Conclusion
This article proposes a tooth surface modification design method for VHCATT cylindrical gears, derives the modified tooth surface equation, constructs a three – dimensional model, and obtains the load – bearing contact characteristics of the gear based on geometric contact analysis, finite element method, and load – bearing contact analysis. The influence of modification parameters on the tooth surface load and the load – bearing transmission error of the system is analyzed. The main conclusions are as follows: (1) The tooth surface load gradually decreases as the cutter – disk inclination angle increases. The cutter – disk inclination angle has no effect on the load mutation at the single – double – tooth alternation. As the parabolic coefficient increases, the load mutation at the single – double – tooth alternation is improved, but when the parabolic coefficient is too large, the tooth surfaces do not actually contact at the entry or exit of meshing. When the parabolic vertex position changes from \(3\) mm to \(-3\) mm, the tooth surface load gradually decreases. (2) The amplitude and variation range of the load – bearing transmission error both decrease as the cutter – disk inclination angle increases. As the parabolic coefficient increases, the transmission error and the mutation amplitude of the error at the single – double – tooth alternation decrease, but when the parabolic coefficient is too large, the transmission error deteriorates rapidly. Generally, the smaller the value of the parabolic vertex position \(u_{0}\), the smaller the amplitude of the transmission error. The greater the load, the greater the transmission error and the greater the fluctuation range.
