1. Introduction
Variable hyperbolic circular arc tooth trace (VHCATT) cylindrical gears, a novel type of transmission component, have attracted significant attention in recent years. Their unique tooth profile and trace characteristics endow them with a series of advantages, such as the combination of the merits of spur, helical, and herringbone gears. These gears hold great potential in various fields, including automotive, aerospace, and heavy machinery industries.
However, like other gears, VHCATT cylindrical gears face challenges in practical applications. One of the key issues is how to improve their load – bearing capacity and dynamic performance. Tooth surface modification design and load – tooth contact analysis play crucial roles in addressing these problems. By optimizing the tooth surface shape, it is possible to enhance the load – bearing capacity, reduce vibration and noise during operation, and improve the overall reliability and service life of the gear system.
This paper aims to comprehensively introduce the research on VHCATT cylindrical gears, including their modification design methods, load – tooth contact analysis models, and the impact of modification parameters on gear performance. The research results are expected to provide a theoretical basis for the further design and industrial application of VHCATT cylindrical gears.
2. Modification Design of VHCATT Cylindrical Gears
2.1 Tooth Line Direction Modification – Cutter Inclination Method
The cutter inclination method is proposed to improve the load – bearing capacity and dynamic characteristics of VHCATT cylindrical gears in the tooth line direction. As shown in Figure 1, the large cutter for milling the VHCATT cylindrical gear is tilted during the machining process. The inner and outer cutting edges of the cutter form an angle α±γ with the axis of rotation. By tilting the cutter, the curvature of the tooth line can be adjusted, which affects the contact area and load distribution of the gear teeth.
| Parameter | Symbol | Explanation |
|---|---|---|
| Inner – edge radius | Rn | Rn=RT−4πmcosγ, related to the position of the inner cutting edge of the cutter |
| Outer – edge radius | Rw | Rw=RT+4πmcosγ, related to the position of the outer cutting edge of the cutter |
| Cutter inclination angle | γ | The angle by which the cutter is tilted, a key parameter affecting tooth line modification |
| Distance from the pitch – line point of equal tooth thickness to the axis md′nd′ | RT | Determines the basic position of the cutter during machining |
| Module | m | A fundamental parameter of the gear, related to the size and shape of the gear teeth |
Figure 1: Inclined Milling Cutter for VHCATT Cylindrical Gear Machining
[Insert an image here showing an inclined milling cutter for machining VHCATT cylindrical gears, with clear labels for the cutter, the gear blank, and relevant angles and distances]
When the cutter is tilted, the tooth line of the gear becomes curved. A larger cutter inclination angle γ will increase the curvature radius of the tooth line in the modified tooth surface. This change leads to a decrease in the clearance between the tooth surfaces. As a result, when the gear is under load, the contact area between the teeth increases, and the tooth surface load decreases. This can be clearly seen from Figure 2, which shows the comparison of tooth line profiles before and after modification with different cutter inclination angles.
Figure 2: Comparison of Tooth Line Profiles Before and After Modification with Different Cutter Inclination Angles
[Insert an image here with multiple sub – images showing the tooth line profiles of VHCATT cylindrical gears before and after modification with cutter inclination angles of 0°, 3°, 5°, and 7°. Each sub – image should have clear markings to distinguish the tooth lines]
2.2 Tooth Profile Direction Modification – Parabola Modification Blade Method
In the tooth profile direction, a parabola – shaped blade is used for modification. The equation of the parabola is z=ax2n, where n can take values such as 1, 2, 3, etc., to obtain different – order parabola curves. As shown in Figure 3, Δt and Δr represent the tooth tip modification amount and tooth root modification amount respectively, Odp is the position of the vertex of the modification curve, a is the parabola coefficient, and u0 is the distance between the vertex of the modification curve and the pitch line of the unmodified straight – blade cutter.
| Parameter | Symbol | Explanation |
|---|---|---|
| Tooth tip modification amount | Δt | The amount of modification at the tooth tip, affects the contact condition at the tooth tip |
| Tooth root modification amount | Δr | The amount of modification at the tooth root, affects the contact condition at the tooth root |
| Parabola coefficient | a | Determines the shape of the parabola – shaped modification curve, a key parameter for tooth profile modification |
| Vertex position of the modification curve | Odp | The position of the vertex of the parabola – shaped modification curve, affects the distribution of modification amount along the tooth profile |
| Distance between the vertex of the modification curve and the pitch line of the unmodified straight – blade cutter | u0 | Influences the overall shape and position of the modified tooth profile |
Figure 3: Parabola – Shaped Modification Blade Curve for Tooth Profile
[Insert an image here showing the parabola – shaped modification blade curve for the tooth profile, with clear labels for Δt, Δr, Odp, a, and u0]
The parabola – shaped modification of the tooth profile can change the structure characteristics of the tooth surface in the tooth tip and tooth root regions. When the parabola coefficient a changes, the modification amount of the tooth tip and tooth root also changes. A larger a value will lead to a greater modification amount at the tooth tip and tooth root. However, if a is too large, it may cause the tooth tip and tooth root to have excessive clearance, resulting in non – contact between the tooth surfaces at the start or end of meshing.
3. Mathematical Model of Modified Tooth Surfaces
3.1 Establishment of Coordinate Systems
To accurately describe the modified tooth surface, multiple coordinate systems are established, as shown in Figure 4. O1X1Y1Z1 is the moving coordinate system of the gear blank, Od0Xd0Yd0Zd0 is the moving coordinate system of the cutter, OdXdYdZd is the static coordinate system of the cutter, OfXfYfZf is the static coordinate system of the gear blank, and OXYZ is the auxiliary coordinate system.
| Coordinate System | Symbol | Explanation |
|---|---|---|
| Moving coordinate system of the gear blank | O1X1Y1Z1 | Used to describe the position and orientation of the gear blank during machining |
| Moving coordinate system of the cutter | Od0Xd0Yd0Zd0 | Used to describe the position and orientation of the cutter during machining |
| Static coordinate system of the cutter | OdXdYdZd | A fixed – reference coordinate system for the cutter |
| Static coordinate system of the gear blank | OfXfYfZf | A fixed – reference coordinate system for the gear blank |
| Auxiliary coordinate system | OXYZ | Helps in the transformation between different coordinate systems |
| Module | m | A fundamental parameter of the gear, related to the size and shape of the gear teeth |
| Pitch – circle radius | R1 | Determines the basic size of the gear |
| Tooth width | b | The width of the gear teeth along the axial direction |
| Cutter rotation speed | ω | The speed at which the cutter rotates during machining |
| Gear blank rotation speed | ω1 | The speed at which the gear blank rotates during machining |
| Gear blank rotation angle | φ1 | Represents the angular position of the gear blank during rotation |
| Cutter rotation angle | θd | Represents the angular position of the cutter during rotation |
Figure 4: Coordinate Systems for Tooth Surface Modification Design
[Insert an image here showing all the coordinate systems for tooth surface modification design, with clear labels for each coordinate system and relevant geometric parameters]
3.2 Derivation of the Modified Tooth Surface Equation
Based on the gear meshing principle and the forming principle of the modified tooth surface of VHCATT cylindrical gears, the equation of the modified tooth surface can be derived. First, the expression of the inclined cutter edge in the cutter static coordinate system OdXdYdZd is obtained. Then, according to the condition that the normal vector of the contact point between the modified tooth surface edge and the gear blank and the relative velocity vector are perpendicular (nx⋅vd(x1)=0), an equation about u is obtained and solved.
Finally, through coordinate transformation, the equation of the modified tooth surface in the gear coordinate system O1X1Y1Z1 is obtained:
{x1ny1n=[(u+u0)sin(γ∓α)−au2ncos(γ∓α)−RT∓4πmcosγ]cosθcos(γ+φ1)−[(u+u0)cos(γ∓α)+au2nsin(γ∓α)±4πmsinγ]sin(γ+φ1)+(RTcosγ+R1φ1)cosφ1−RTsinφ1sinγ−R1sinφ1=[(u+u0)sin(γ∓α)−au2ncos(γ∓α)−RT∓4πmcosγ]cosθsin(γ+φ1)+[(u+u0)cos(γ∓α)+au2nsin(γ∓α)±4πmsinγ]cos(γ+φ1)+(RTcosγ+R1φ1)sinφ1+RTcosφ1sinγ+R1cosφ1
where n can take values such as 1, 2, 3, etc., corresponding to different – order parabola – modified tooth surface equations.
3.3 Tooth Surface Reconstruction
Using software such as MATLAB and UG, the tooth surface reconstruction of VHCATT cylindrical gears can be realized. Taking a gear with specific parameters (number of teeth z=29, tooth width b=80mm, pressure angle α=20∘, module m=8mm, tooth line radius RT=200mm) as an example, Figure 5 shows the comparison of the tooth surface before and after modification.
Figure 5: Comparison of Tooth Surfaces Before and After Modification
[Insert an image here with two sub – images, one showing the tooth surface before modification and the other showing the tooth surface after modification. The differences in tooth line and tooth profile should be clearly visible]
As can be seen from Figure 5(a), the cutter inclination angle mainly affects the bending degree of the tooth arc. A larger cutter inclination angle makes the tooth arc more curved. Figure 5(b) shows that the tooth profile modification mainly changes the structural characteristics of the tooth surface in the tooth tip and tooth root regions. The modified tooth surface can better adapt to the load distribution during meshing, improving the performance of the gear.
4. Geometric Contact Model of the Gear System
4.1 Establishment of the Meshing Transmission Coordinate System
For the geometric contact analysis of the gear system, a meshing transmission coordinate system is established, as shown in Figure 6. ∑I represents the tooth surface of the driving gear, ∑II represents the tooth surface of the driven gear, M is the contact point of the tooth surfaces, nc is the common normal at the contact point, ψ1 is the meshing rotation angle of the driving gear, and ψ2 is the meshing rotation angle of the driven gear.
| Symbol | Explanation |
|---|---|
| ∑I | Tooth surface of the driving gear |
| ∑II | Tooth surface of the driven gear |
| M | Contact point of the tooth surfaces |
| nc | Common normal at the contact point |
| ψ1 | Meshing rotation angle of the driving gear |
| ψ2 | Meshing rotation angle of the driven gear |
Figure 6: Meshing Transmission Coordinate System of the Gear Pair
[Insert an image here showing the meshing transmission coordinate system of the gear pair, with clear labels for ∑I, ∑II, M, nc, ψ1, and ψ2]
4.2 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 position vectors and unit normal vectors of point M on the tooth surfaces of the two gears are the same. There are 5 independent scalar equations:
⎩⎨⎧xg1(θ1,φ1,ψ1)yg1(θ1,φ1,ψ1)zg1(θ1,φ1,ψ1)ngx1(θ1,φ1,ψ1)ngy1(θ1,φ1,ψ1)=xg2(θ2,φ2,ψ2)=yg2(θ2,φ2,ψ2)=zg2(θ2,φ2,ψ2)=ngx2(θ2,φ2,ψ2)=ngy2(θ2,φ2,ψ2)
In the contact model, there are 6 unknown variables θ1, φ1, ψ1, θ2, φ2, ψ2. By taking the rotation angle ψ1 of the driving gear as the input quantity and solving the geometric contact model, the values of θ1(ψ1), φ1(ψ1), θ2(ψ1), φ2(ψ1), ψ2(ψ2) can be obtained. This analysis is crucial for understanding the contact relationship between the tooth surfaces of the driving and driven gears during meshing.
5. Load – Contact Analysis Model
5.1 Establishment of the Gear Load – Contact Deformation Model
The gear load – contact deformation model is established to analyze the contact state of the gear teeth under load. As shown in Figure 7, Figure 7(a) represents the contact deformation of a single – tooth pair, and Figure 7(b) represents the contact deformation of a double – tooth pair.
| Symbol | Explanation |
|---|---|
| ∑1 | Tooth surface of the driving gear |
| ∑2 | Tooth surface of the driven gear |
| Solid line | Tooth surface before deformation |
| Dashed line | Tooth surface after deformation |
| M | Contact point of the tooth surfaces |
| j | Arbitrary discrete point on the instantaneous contact line |
| wj | Normal clearance of the tooth surface at point j |
| Fj | Normal load at point j |
| uj | Contact deformation at point j under load P |
| s2 | Normal displacement of the large gear under external load |
