1. Introduction
Helical gears play a crucial role in mechanical transmission systems. Standard symmetric involute helical gears have been widely used in various applications. However, with the development of industries, especially in high-speed and heavy-load scenarios such as electric vehicles and aerospace, the limitations of standard symmetric gears become more apparent. Asymmetric involute gears, with different pressure angles on both sides, offer potential advantages in enhancing load-carrying capacity and addressing issues like tooth tip thinning.
1.1 Research Background
Previous research on asymmetric involute gears has mainly focused on spur gears. For example, Chen et al. proposed a design method for asymmetric parabolic tooth-shaped milling cutters for straight and helical cylindrical gear pairs. Karpat et al. developed an asymmetric tooth profile generation program using Matlab based on the involute tooth profile equation and established a 3D model in SolidWorks. Bian Jingyang et al. combined the asymmetric spur gear involute equation with SolidWorks to create a 3D model and conducted dynamic analysis using Adams software. Mo et al. established a solid model of an asymmetric involute helical gears and discussed its superior performance. Dhakshain et al. analyzed asymmetric gears made of different materials and found that asymmetric gears outperformed symmetric gears in some cases. In practical applications, Zhang Xiang et al. considered asymmetric spur gears in the planetary gear transmission system of a small-diameter drill and obtained the contact stress distribution of the gear pair through Hertz theory and finite element analysis. Li Deling applied asymmetric gear transmission to a gear pump and found that a larger working pressure angle improved the trapped oil performance, enhanced the root bending strength, and reduced the contact stress.
1.2 Research Objectives and Significance
Despite these studies, there is a lack of in-depth research on the parametric modeling of asymmetric modified helical gears. The objective of this study is to establish an accurate mathematical model and a parametric design method for asymmetric modified helical gears. By analyzing the contact stress of these gears, we aim to provide a theoretical basis for their design and application. This research is significant as it can contribute to the improvement of gear transmission performance in high-speed and heavy-load applications.
2. Mathematical Model of Asymmetric Modified Helical Gears
2.1 Meshing Tooth Profile Mathematical Model
The tooth profile of an asymmetric helical gears have different pressure angles on both sides compared to a standard helical gears. Therefore, the rack cutter for an asymmetric helical gears are designed with double pressure angles.
2.1.1 Coordinate System and Parameters
We consider a normal rack cutter with a coordinate system , rigidly attached to the rack cutter and as an auxiliary coordinate system. The normal pressure angles of the two tooth profiles are and , respectively. Other parameters include the pitch , tooth thickness , tooth space width , half-tooth space width , normal module , root height , tip height , center points of the tip fillet , , fillet radius , clearance , and helix angle .
| Parameter | Symbol | Description |
|---|---|---|
| Pitch | The distance between corresponding points on adjacent teeth | |
| Tooth thickness | The width of the tooth along the pitch line | |
| Tooth space width | The width of the space between adjacent teeth | |
| Half – tooth space width | Half of the tooth space width | |
| Normal module | A standard parameter determining the size of the gear teeth | |
| Root height | The height from the root of the tooth to the pitch line | |
| Tip height | The height from the pitch line to the tip of the tooth | |
| Center points of tip fillet | , | The centers of the fillets at the tip of the tooth |
| Fillet radius | The radius of the fillet at the tip of the tooth | |
| Clearance | The distance between the tip of a tooth and the bottom of the mating tooth space | |
| Helix angle | The angle between the tooth helix and the axis of the gear |
2.1.2 Position Vectors and Normal Vectors of Rack Cutter Tooth Profiles
The position vectors of the two tooth profiles , of the rack cutter in the coordinate systems and are given by:
where , are independent parameters of the rack cutter, is an axial parameter, and , are transformation matrices from coordinate systems , to .
The unit normal vector of the rack cutter tooth profile in the coordinate system is:
2.1.3 Tooth Profile of Asymmetric Modified Helical Gears
The meshing tooth profile of an asymmetric modified helical gears are enveloped by the rack cutter. The tooth surface position vector and unit normal vector in the coordinate system (rigidly attached to the gear) are:
where is the meshing equation, , are coordinate components of the normal vector , , are coordinate components of the position vector , and is a coordinate transformation matrix from to .
2.2 Transition Tooth Profile Mathematical Model
When the cutter machines the gear, the arc segments , machine the transition part of the gear.
2.2.1 Position Vectors and Normal Vectors of Transition Tooth Profiles
The position vectors of the two transition tooth profiles , of the rack cutter in the coordinate systems and are:
where are the coordinates of the fillet centers in the coordinate system, and is a transformation matrix from to via .
The unit normal vectors of the transition tooth profiles in the coordinate systems and are:
where is the base vector transformation matrix of .
2.2.2 Tooth Profile of Asymmetric Modified Helical Gears in Transition Part
In the coordinate system , the tooth surface position vector and unit normal vector of the transition part of the asymmetric modified helical gears are:
where is the meshing equation and is a coordinate transformation matrix from to .
Combining the above derivations, we can calculate the point cloud data of the meshing and transition tooth profiles of the asymmetric modified helical gears in Matlab and fit the point cloud data to obtain the complete tooth profile surface.
3. Solid Modeling of Asymmetric Helical Gears
3.1 SolidWorks API and Secondary Development
SolidWorks provides API functions that can be called by programming languages such as C#. This allows for the development of custom programs to improve the design efficiency and accuracy.
3.2 Parametric Modeling Process
The parametric solid modeling process of asymmetric helical gears in C# and SolidWorks.API is as follows:
- Calculate the diameter of the tip circle based on geometric parameters and stretch the sketch with the gear width to obtain the gear blank.
- The key to constructing the 3D model of the asymmetric helical gears are to create an accurate tooth profile. The data of the two end faces can define the accurate spline curve of the gear tooth space.
- Cut the tooth space part of the gear blank at the root using a guide line.
- Create an array feature and array the tooth spaces to obtain an accurate 3D model of the asymmetric involute helical gears.
However, during the modeling process, to avoid incomplete tooth profile formation when cutting the tooth space, interpolation may be required to extend the tooth profile of the tooth space. For example, Lagrange interpolation polynomial can be used to interpolate the meshing tooth profile points to obtain interpolation points, and a construction point is located on the midline between the interpolation points. The sketches of the end faces then include the spline curves created by the interpolation points and the construction point.
4. Contact Stress Analysis of Asymmetric Helical Gears Tooth Surface
4.1 Finite Element Method
The finite element method is used to calculate the contact stress of the gear tooth surface. The gear pair model is imported into Workbench software, and the materials, elastic moduli, and Poisson’s ratios of the gears are defined. The model is then meshed. The large gear is rotated around the Z-axis by a certain angle, and a certain load torque is set for the small gear as boundary conditions. Through simulation, the stress cloud diagram near the pitch circle is obtained. To ensure the accuracy of stress calculation, tetrahedral elements are often chosen for meshing as they provide more accurate results compared to hexahedral elements when analyzing gear contact stress. In this study, the grid of the gear pair is divided such that the contact surfaces of the large and small gears are refined, while the non-contact surfaces are less refined and diverge from the contact surfaces.
4.2 Formula Method
The formula method is also used to calculate the contact stress of gears. For standard symmetric involute gears, the Hertz stress is a key indicator of the tooth surface contact stress. The Hertz stress can be calculated using the Hertz formula:
where is the node area coefficient, is the elastic coefficient, is the coincidence degree coefficient, is the helix angle coefficient, is the tangential force on the pitch circle, is the transmission ratio, is the pitch circle diameter, and is the meshing tooth width.
4.3 Comparison and Analysis
A comparison is made between the contact stress calculated by the finite element method and the formula method for asymmetric modified helical gears. Geometric and material parameters of a certain asymmetric involute modified helical gear pair are considered, and the working side pressure angle is varied.
| Parameter | Value |
|---|---|
| Number of teeth of small gear | 30 |
| Number of teeth of large gear | 48 |
| Normal module (mm) | 2 |
| Working side normal pressure angle () | 15 – 25 |
| Non – working side normal pressure angle () | 20 |
| Helix angle () | 8 |
| Tip height coefficient | 1 |
| Clearance coefficient | 0.25 |
| Small gear modification coefficient | 0.1 |
| Large gear modification coefficient | – 0.1 |
| Gear width (mm) | 30 |
| Poisson’s ratio | 0.3 |
| Elastic modulus (MPa) | |
| Rotational speed of small gear (r/min) | 2500 |
| Rated power (kW) | 25 |
The results show that when the pressure angle on one side of the tooth surface is the same, the tooth surface contact stress of the asymmetric involute modified helical gears are very close to that of the symmetric modified helical gears, and the increase and decrease trends of contact stress with changes in the pressure angle are basically the same. The difference between the two methods is less than 10%. This indicates that the traditional formula method used to calculate the tooth surface contact stress of symmetrical modified helical gears can be used to approximate the tooth surface contact stress of asymmetric involute modified helical gears.
5. Conclusions
5.1 Summary of Research Results
- A mathematical model of the asymmetric involute modified helical gears have been established, including the meshing tooth profile and transition tooth profile models. This provides a theoretical basis for further analysis using analytical and finite element methods.
- An accurate 3D solid model of the asymmetric involute modified helical gears have been created through secondary development in C# and SolidWorks.API, which is beneficial for analyzing the mechanical properties of the gear.
- The contact stress of the asymmetric involute modified helical gears have been analyzed using the finite element method and compared with the results of the formula method for symmetric modified helical gears. The two methods show similar trends, and the traditional formula method can be used for approximate calculation of the contact stress of asymmetric involute modified helical gears.
5.2 Future Research Directions
Future research could focus on further optimizing the design of asymmetric modified helical gears, considering factors such as wear resistance and lubrication. Additionally, more in-depth studies on the dynamic characteristics of these gears under different operating conditions could be conducted to improve their performance and reliability in practical applications.
In conclusion, this study has made significant contributions to the understanding and design of asymmetric modified helical gears, providing valuable insights for their application in various mechanical transmission systems.