Abstract
Based on the complex tooth surface design and actual meshing conditions of spiral bevel gears, this paper proposes a tooth profile modeling method utilizing tooth profile forming tools, which can effectively achieve precise modeling of spiral bevel gear tooth profiles. Taking spiral bevel gears as the main modeling object, an automated modeling and assembly method is proposed, and a parametric design system software is developed. This software enables automated and rapid modeling and assembly of spiral bevel gears, shortening modeling time and improving design quality and efficiency. It contributes to driving the digital transformation of the industry and the development of high-end equipment manufacturing.
1. Introduction
Spiral bevel gears are widely used in fields such as new energy vehicles and aerial vehicles due to their excellent load-bearing capacity, smooth transmission, and superior meshing effects. Industrial software is the foundation for the digitalization and intelligent development of the manufacturing industry. However, for a long time, the lagged development of industrial software has been a pain point and difficulty in the development and progress of the domestic manufacturing industry. In particular, gear design software for core components has been highly monopolized by European and American developed countries for a long time. The monopoly of industrial giants on gear industrial software has made its price exorbitant, posing a significant economic burden on domestic small and medium-sized enterprises.
To maintain the vitality of the domestic manufacturing industry, there is an urgent need to develop low-cost and high-performance domestic gear industrial software. Table 1 summarizes the research progress in gear industrial software development by domestic scholars in recent years.
| Research Area | Researchers | Key Achievements |
|---|---|---|
| Gear cutting tool fixture management | Sudan | Developed a smart selection and management system for hob fixtures, providing theoretical and technical support for smart selection and matching of hob fixtures |
| Digital management of gear processing tool fixtures | Bai Xue et al. | Designed and developed a tool fixture management system for gear processing digital workshops, enabling full-life-cycle digital management of gear processing tool fixtures |
| High-performance gear transmission database software | Liu Huaiju et al. | Designed and developed high-performance gear basic data management, data analysis and processing, transmission system optimization, and other modular functions |
2. Spiral Bevel Gear Tooth Surface Modeling
The tooth surface modeling process of spiral bevel gears is shown in Figure 1. During the tooth surface design stage of spiral bevel gears, the meshing performance requirements are linked to the processing adjustment parameters, so that the designed spatial numerical surface has practical significance. Therefore, it is necessary to calculate the machine tool processing parameters, analyze the relative motion relationship between the tool disc and the workpiece during processing, complete the coordinate transformation from the tool coordinate system established by actual processing to the workpiece coordinate system, and express the relationship between them in a functional form.
2.1 Tool Mathematical Model
The tooth surface processing process of spiral bevel gears is actually a process in which a hypothetical generating gear meshes with the gear being processed without clearance. At any instant of tooth surface generation, the points on the cutting cone correspond one-to-one with the points on the tooth surface being processed, which is the principle of plane generating gear machining, as shown in Figure 2(a) and (b).
When processing large gears, double-sided cutting edges are used. The straight part of the cutting edge processes the working tooth surface, and the arcuate part processes the transition surface, as shown in Figure 3(a) and (b).
For the straight part of the cutting edge, the expression for the position vector (ππ) and unit normal vector (ππ) of the cutting cone surface in the coordinate system ππ(π₯π, π¦π, π§π) is given by:
ππ(π π,ππ)=(π πΒ±π πsinπΌπ)cosππ(π πΒ±π πsinπΌπ)sinππβπ πcosπΌπ
ππ(ππ)=cosπΌπcosππcosπΌπsinππβsinπΌπ
For the arcuate part of the cutting edge, the expression for the position vector (ππ) and unit normal vector (ππ) of the cutting cone surface in the coordinate system ππ(π₯π, π¦π, π§π) is given by:
ππ(π,ππ)=(πΆπ₯Β±ππsinπ)cosππ(πΆπ₯Β±ππsinπ)sinππβππ(1βcosπ)
ππ(π,ππ)=sinπcosππsinπsinππΒ±cosπ
Where “Β±” corresponds to the concave and convex surfaces of the cutting cone surface, respectively.
2.2 Tooth Cutting Coordinate Transformation and Tooth Surface Equation
According to the actual tooth surface generation process and the machine tool processing parameters of the gear pair, a large wheel tooth cutting coordinate system as shown in Figure 4 is established to describe the relative motion relationship between the milling cutter head and the workpiece during processing, and to determine the coordinate transformation matrix from the tool coordinate system ππ to the workpiece coordinate system π2(π₯2, π¦2, π§2).
The position vector (ππ) and normal vector (ππ) of the cutting cone surface are transformed into the coordinate system π2 through coordinate transformation, and the large wheel tooth surface equation vector (π2) and normal vector (π2) can be obtained as:
π2=π2π2β ππ2π2β ππ2π2β ππ2π2β ππ2πβ ππ
π2=πΏ2π2β πΏπ2π2β πΏπ2π2β πΏπ2π2β πΏπ2πβ ππ
Where ππ2π, ππ2π2, ππ2π2, ππ2π2, π2π2 are the transformation matrices between the coordinate systems.
2.3 Tooth Surface Generation
The tooth surface of spiral bevel gears is generated through a complex process involving the coordinate transformation and application of the tooth surface equation derived earlier. The generation of the tooth surface commences with the division of the tooth surface into a grid of data points. These points are calculated based on the tooth surface equation and are utilized to form curves and surfaces in a three-dimensional space.
The process involves slicing and stitching together these surfaces to produce the final tooth profile. This is a crucial step as it ensures the accuracy of the tooth surface, which directly impacts the gear’s meshing performance and efficiency.
Furthermore, to achieve a smooth and continuous tooth surface, advanced interpolation techniques may be employed. These techniques help in filling in the gaps between the calculated data points, resulting in a more refined and accurate representation of the tooth surface.
3 Parametric Design and Automation in Gear Modeling
The development of a parametric design system for spiral bevel gears aims to automate the modeling process, reducing the time and complexity involved. This system leverages the principles of parametric design, where the dimensions and features of the gear are defined by parameters that can be easily modified.
By incorporating an application programming interface (API) within computer-aided design (CAD) software, the system enables users to input specific parameters for the gear, such as tooth size, spiral angle, and pressure angle, and automatically generates the corresponding tooth surface model.
Additionally, the system incorporates automation in the assembly process, allowing for the rapid and accurate assembly of multiple gears within a transmission system. This automation not only improves efficiency but also ensures consistency in the design and performance of the gears.
4 Significance and Applications
The development of this parametric design system for spiral bevel gears holds significant implications for the manufacturing industry. It addresses the long-standing issue of high costs associated with imported gear design software, providing a cost-effective and high-performance alternative.
This system is particularly beneficial for small and medium-sized enterprises (SMEs) in China, enabling them to compete more effectively in the global market. By adopting this system, SMEs can significantly reduce their design and production costs while maintaining high standards of quality and efficiency.
Moreover, the system’s capabilities in automation and parametric design align well with the trends towards digitalization and smart manufacturing. It facilitates the integration of spiral bevel gears into more complex and sophisticated machinery, contributing to advancements in various industries such as aerospace, automotive, and robotics.
In conclusion, the development of a parametric design system for spiral bevel gears represents a significant step forward in the field of gear design and manufacturing. It combines advanced modeling techniques, automation, and parametric design principles to create a powerful tool that can revolutionize the way gears are designed and produced.