1. Introduction
1.1 Research Background and Significance
Spiral bevel gears play a crucial role in mechanical transmission systems. Their unique spiral shape endows them with excellent transmission performance, such as high load-carrying capacity, stable transmission process, and low vibration. However, due to their complex tooth shape design, it is often difficult to accurately predict whether root cutting will occur during the machining process solely based on theoretical calculations. Therefore, trial cutting operations on gears are necessary on machine tools. The trial cutting process is complex and time-consuming, increasing production costs and wasting resources, which significantly reduces manufacturing efficiency.
With the development of virtual simulation technology, it has been applied to the field of gear machining to replace the traditional trial cutting method. By establishing models of machine tools, cutter heads, and gear blanks in a virtual three-dimensional environment and controlling the movement of the machine tool to drive the cutter head to cut the gear blank, the actual machining process can be simulated. This not only verifies the correctness of relevant parameters but also improves the production efficiency of spiral bevel gears and reduces production costs.
1.2 Current Research Status at Home and Abroad
1.2.1 Machining Technology of Spiral Bevel Gears
The machining technology of spiral bevel gears has a close relationship with machining machine tools and has experienced three stages: from traditional mechanical to partially numerical control and then to modern full numerical control. Foreign countries have made significant progress in this field, with companies like Gleason continuously introducing advanced machine tools and manufacturing expert systems. In China, research on spiral bevel gears started relatively late in the 1970s. Although starting later compared to foreign countries, remarkable research achievements have been made under the research of many domestic experts and scholars. Domestic research has covered aspects such as gear design theory, machining principles, and the development of machining equipment.
1.2.2 Optimization of Milling Process Parameters
Both domestic and foreign scholars have conducted in-depth research on the optimization of milling process parameters. In terms of cutting force research, various cutting force models have been established for different gear machining processes. Based on the research of cutting force, scholars have also studied the optimization of process parameters, aiming to improve the stability and efficiency of the machining process and ensure the quality of the processed parts.
1.2.3 Machining Simulation Technology of Spiral Bevel Gears
Researchers at home and abroad have made certain achievements in the machining simulation technology of spiral bevel gears. They have used various methods and software to achieve the simulation of the gear machining process. However, existing simulation methods still have problems such as slow simulation speed and low accuracy of the obtained models.
1.3 Research Contents and Ideas of This Topic
This topic focuses on spiral bevel gears and aims to address the problems of long time and low efficiency in forming the gear tooth surface during machining simulation. The main research contents include: analyzing the cutting principle and the complex structure of machining machine tools, establishing a cutting coordinate system, deriving the calculation expression of the cutter head’s motion trajectory during numerical control milling, establishing STL models of the cutter head and gear blank, proposing a topology reconstruction method to handle data redundancy, using a VTK visualization function library to display models, proposing a machining simulation method based on Boolean operation principles and CGAL geometric algorithm libraries, analyzing the actual finishing process to complete the finishing simulation process of tool setting, establishing a cutting force calculation model based on the oblique cutting principle, using an AABB tree fast slicing algorithm to obtain the cutting area of the inner and outer cutter heads, analyzing the simulation results to verify the accuracy of the simulation method, and optimizing the feed rate based on the relationship between cutting force and feed rate to improve milling efficiency.
2. Construction of Machining Simulation Model
2.1 Calculation of Cutter Location Points in Numerical Control Milling
2.1.1 Cutting Coordinate System of Traditional Mechanical Milling Machine
When the spiral bevel gear blank is machined on a traditional mechanical milling machine, the actual machining process can be regarded as a meshing process between an imaginary generating gear and the gear blank at a certain transmission ratio. The generating gear is simulated by the machine tool’s cradle mechanism, and the motion trajectory of the cutter teeth during the machining process is equivalent to a tooth of the generating gear. The relative position and relative motion between the generating gear and the gear blank determine the shape of the gear tooth surface. Based on the structure of the traditional mechanical milling machine, a cutting coordinate system is established, including the machine tool coordinate system, the dynamic coordinate system during cutting, and the coordinate system of the cutter head (generating surface).
2.1.2 Conversion of Milling Coordinates
Compared with the traditional mechanical milling machine, the numerical control milling machine has a more concise structure design and introduces a multi-axis numerical control motion system, which provides highly accurate control for milling machining. Taking a six-axis numerical control milling machine as an example, the six axes of the machine tool are defined, and the motion calculation formulas of each axis during the cutting process are derived through the conversion of coordinates from the traditional mechanical milling machine to the numerical control milling machine.
2.2 Establishment and Display of STL Models of Gear Blank and Cutter Head
2.2.1 STL File Data Analysis
The STL file is a standard format for three-dimensional model files. It represents the model surface as a polygon composed of many triangular patches. The precision of the STL model can be controlled by adjusting parameters such as chord height and step length. There are two types of STL file formats: binary and ASCII code. This paper adopts the ASCII code format to express the three-dimensional model.
2.2.2 Establishment of STL Model of Gear Blank
The actual gear blank can be regarded as a revolving body formed by rotating its axial section around its own axis. According to the gear blank parameters, the gear blank section is constructed, and then the three-dimensional model of the gear blank is obtained by rotating the section around the x-axis. The coordinates of key points on the gear blank are calculated, and the STL model of the gear blank is established using relevant software libraries.
2.2.3 Establishment of STL Model of Cutter Head
There are two common types of cutter heads for machining spiral bevel gears: double-sided cutter heads and single-sided cutter heads. In the machining simulation process, the bottom of the cutter head can be removed to simplify the model and reduce the calculation burden. The STL models of the outer cutter, inner cutter, and the whole cutter head are established respectively according to the geometric parameters of the cutter head.
2.2.4 Model Visualization
Using the established STL models, the VTK function library is combined with the VS environment under Windows to read and render the STL models of the cutter head and gear blank and display them in the simulation environment. The visualization process of VTK includes steps such as data source, filter, mapper, actor, renderer, rendering window, and rendering window interactors. Through these steps, the cutter head and gear blank can be displayed, and their relative positions can be controlled according to the axis position information to achieve dynamic simulation.
2.3 Reconstruction of STL Model Topology Structure
The redundant data of the STL model not only burdens the file storage but also brings inconvenience to subsequent data processing. To address this problem, a topology reconstruction method is proposed. The basic principle of topology reconstruction is to sort the vertices on each triangular patch of the STL model in a coordinate system, delete duplicate vertex coordinates, and establish a regular triangular grid while recording the coordinate information of adjacent triangles to complete the reconstruction of the triangular grid model. In the process of topology reconstruction, two main linked lists (VertexList and FacetList) and two sub-linked lists (VerList and FaceList) are created to store vertex and patch information. By comparing the coordinates of vertices, redundant vertices are removed, and an STL model without redundant vertex data is established. This reconstruction method not only clears the redundant vertex data but also establishes the structural information of the triangular grid, providing a more convenient basis for subsequent data processing, visualization, and analysis.
3. Machining Simulation Method of Spiral Bevel Gear Machining Process
3.1 Tooth Surface Forming Simulation Based on CGAL
3.1.1 Introduction to CGAL
The CGAL library is a powerful and highly flexible C++ geometric algorithm library. It covers various aspects from basic geometric object operations to advanced geometric calculations and provides accurate and efficient algorithm support for geometric operations such as point, line, and surface operations, as well as complex triangulation, mesh generation, and polygon Boolean operations. It is widely used in the field of computational geometry and provides important tools for related practical applications.
3.1.2 Principle of Boolean Operations
The spatial Boolean operation between three-dimensional data models involves in-depth geometric analysis of two or more three-dimensional spatial bodies. By calculating their intersections, unions, and differences, new three-dimensional forms are obtained. The principle of Boolean operations is to rearrange and combine the triangular meshes of two models, involving the in-depth reconstruction of the topological structure. Different Boolean operations generate different models, demonstrating the diversity and flexibility of Boolean operations. The Boolean operation relations between two STL models are expressed as equations, and the key to realizing three-dimensional Boolean operations lies in the accurate calculation of the boundaries and intersections of geometric bodies.
3.1.3 AABB Bounding Box Intersection Detection
To improve the simulation efficiency, an intersection detection is carried out on two models during the simulation process. The AABB bounding box algorithm is used as an effective spatial optimization technique. The AABB bounding box tightly wraps an object through an axis-aligned cuboid, enabling rapid positioning and intersection detection of the object. The intersection test of the AABB bounding box determines the relative position relationship between two bounding boxes in three-dimensional space. If the bounding boxes do not intersect, it can be quickly determined that the two model entities do not intersect, avoiding subsequent complex Boolean operation processes. When the bounding boxes intersect, the intersection detection is further refined. In this paper, the AABB (axis-aligned bounding box) intersection detection principle in the hierarchical bounding box method is used, and the AABB tree in the CGAL library is utilized to effectively solve the intersection problem between models, improving the accuracy of collision detection and the calculation efficiency.
3.3.4 Implementation of Cutting Simulation Process
In the cutting simulation of the spiral bevel gear tooth surface forming, the spatial Boolean operations between the cutter head and gear blank models are mainly solved by using the spatial computing functions of the CGAL library. The data structure of the three-dimensional model needs to be converted to be applicable to the CGAL operation. The process includes converting the data structure of the three-dimensional model to CGAL’s Polyhedral_3, then using CGAL’s functions to convert it to Nef_Polyhedral_3 for spatial operations, and finally converting it back to Polyhedral_3 as needed. According to the coordinate information in the numerical control file, the relative positions of the gear blank and cutter head models are accurately adjusted. Through Boolean difference and intersection operations, the new model of the gear blank after machining and the material model removed during the cutting process are calculated, improving the accuracy of the cutting simulation and providing data support for subsequent analysis and optimization.
3.2 Tool Setting for Tooth Surface Finish Cutting Simulation
3.2.1 Extraction of Tooth Surface Data Points
In the finishing process of the small gear, problems such as finish cutting allowance and tool setting for finishing need to be addressed. Based on the rough machining simulation model of the gear blank, the process of tool setting for finishing simulation is as follows: first, construct the tooth surface according to the concave and convex tooth surface points obtained after rough machining simulation; then, construct the single-sided inner and outer cutter cone surfaces for finish cutting; next, return the concave and convex tooth surfaces to the initial position and determine the relative positions between the tooth surface and the inner and outer cutter cone surfaces according to the finishing tool positions to judge whether they intersect; if not, rotate the tooth surface by a small angle until it intersects with the inner or outer cutter cone surface; finally, use an iterative method to calculate the rotation angle of the tooth surface corresponding to each tool position after rotation when it intersects with the inner or outer cutter cone surface, and take the maximum value to ensure the complete machining of the tooth surface. The extraction process of the concave and convex tooth surface points of the gear blank after rough cutting includes steps such as rotating the model, separating the tooth surface points according to geometric parameters, rotating the points back to the three-dimensional space, and using functions to separate the concave and convex tooth surface points.
3.2.2 Reconstruction of Concave and Convex Tooth Surfaces
To accurately calculate the rotation angle when the tooth surface intersects with the cutter head cone surface, the concave and convex tooth surface points calculated in the previous section are reconstructed to form a three-dimensional curved surface. The implicit surface method is used for reconstruction. The VTKSurfaceReconstructionFilter class in the VTK library is used to construct the implicit surface model of the tooth surface according to the tooth surface point cloud data through mesh division. Parameters such as the number of neighborhood points and mesh spacing are set during the reconstruction process. Then, the VTKContourFilter class is used to extract the zero isosurface grid of the corresponding tooth surface.
3.2.3 Inner and Outer Cutter Head Cone Surfaces
During the cutting process, the cutting edges of the cutter teeth are calibrated with the generatrix of the conical surface. The rotation axis of the conical surface is the same as that of the cutter head. The outer cutter teeth cutting edges rotate to form the outer cutter cone surface, and the inner cutter teeth cutting edges rotate to form the inner cutter cone surface. The mathematical models of the inner and outer cutter cone surfaces are established, and according to these models, the cone point clouds are generated and reconstructed into the solid curved surface models of the inner and outer cutter cone surfaces. Then, the finishing simulation process is carried out, which includes steps such as adjusting the positions of the cutter head and gear blank models, judging the intersection using Newton’s iteration method, and performing Boolean difference operations.
3.2.4 Interference Judgment in the Cutting Process
In the machining process of spiral bevel gears, the interference detection between the cutter head and gear blank is crucial. In the actual manufacturing process, interference between the cutter head and gear blank can lead to workpiece scrapping and damage to the machining equipment, affecting the stable operation of the production line. To avoid interference in the actual finishing process and waste of materials, virtual simulation means are used for interference detection. In the finishing process of the small gear, the AABB bounding box intersection judgment method is used to judge whether the non-working surface of the single-sided cutter head for finish cutting interferes with the non-finishing tooth surface of the gear blank. Through this process, potential problems can be detected in time, and errors in the numerical control program can be modified to ensure the correctness of the numerical control program.
3.3 Analysis of Cutting Force in the Machining Process
3.3.1 Prediction Model of Cutting Force
The cutting force mainly comes from two aspects: the resistance generated by the elastic and plastic deformation of the cutting layer material, chips, and the surface layer material of the workpiece, and the frictional resistance between the tool and chips and the workpiece surface. It can be decomposed into tangential force (main cutting force), radial force (thrust), and axial force (feed force). Based on the oblique cutting model, the cutting force in the spiral bevel gear cutting process is analyzed. The oblique cutting model is applicable to the milling process of spiral bevel gears because the cutting process of spiral bevel gears using the generating method is an intermittent cutting process, and the side edges and top edges participate in the cutting process together. The expressions for calculating the tangential force, radial force, and axial force are derived according to the analysis of the oblique cutting process. The size of the cutting force is mainly determined by factors such as shear stress, cutting area, and relevant cutting angles. The shear stress is calculated using the Johoson – Cook constitutive equation, and the cutting area is calculated based on the model of the chip and the AABB bounding tree fast slicing method.
3.3.2 Calculation of Cutting Area
In cutting machining, the cutting area is a crucial parameter that directly reflects the contact area between the tool and the workpiece and affects the size of the cutting force. In this paper, the AABB bounding tree fast slicing method is used to calculate the cutting area. The basic principle of slicing is to cut the three-dimensional model with a specific slicing plane to obtain the section profile. The process includes determining the slicing plane, calculating the intersection lines with the triangular patches, and obtaining the slice profile. The chip model is sliced layer by layer, and the cutting area is calculated. The cutter head and gear blank models are used to calculate the cutting area at each tool position through Boolean intersection operations. The three-dimensional contour points of the removed material slices of the inner and outer cutter heads at each tool position are obtained, and these points are projected onto the xOz plane to calculate the area.
4. Geometric Analysis of Simulation Results
4.1 Tooth Surface Error Analysis
4.1.1 Calculation of Theoretical Tooth Surface Points
According to the analysis in Chapter 2, the motion modes of the workpiece and cutter head in a numerical control milling machine change significantly compared to those in a traditional mechanical milling machine. Based on the geometric characteristics of the generating surface section and the coordinate system transformation, the geometric feature vectors of the generating surface section are defined. The relative motion speed between the cutter head and gear blank at the meshing point is calculated, and according to the meshing principle, the distance from the instantaneous contact point along the generatrix to the cutter tip point is obtained. By changing the parameters related to the cutter head and gear blank, a series of tooth surface points are obtained, and these points are converted into the gear coordinate system to form the tooth surface. In this paper, the tooth surface is finely meshed, and the parameters corresponding to the mesh points are calculated based on the actual machining parameters of the gear blank, machine tool, and cutter head. The contact point parameters corresponding to these mesh points are calculated using the binary Newton two-point secant method for iterative calculation.
4.1.2 Error Calculation
The tooth surface error is defined as the normal distance between the actual tooth surface and the theoretical tooth surface at the same mesh point. It reflects the deviation or error that may exist in the tooth surface machining process and is an important evaluation index for assessing the machining accuracy and tooth surface quality. Because the theoretical tooth surface points and the actual tooth surface are not in the same coordinate system, the actual tooth surface data points need to be rotated to the same coordinate system as the theoretical tooth surface points according to the coordinate transformation. The tooth surface error is calculated as the dot product of the difference between the two discrete point vectors and the inner normal vector.
4.2 Tooth Height Error Analysis
The tooth height is an important parameter in gear design, and its machining accuracy has a significant impact on the working performance of the gear. In the calculation of the tooth height error, the vertex data points of the tooth top of the concave and convex tooth surfaces need to be obtained. Based on the rough machining simulation model, the steps to extract the vertex data points of the tooth top are as follows:
- Rotate the simulated model around the Y-axis by a root cone angle -Γ to make the center axis of the gear blank coincide with the x-axis.
- Determine the data points of the tooth top of the concave and convex tooth surfaces according to the face cone angle.
- Separate the vertices of the concave and convex tooth surfaces.
The specific process of extracting the tooth top data points is as follows:
- According to the steps of extracting tooth surface data points in Chapter 3, rotate the gear blank to make its rotation center coincide with the x-axis, and then traverse all the vertices of the model and access their coordinates.
- Rotate the points to the xOz plane, and according to the face cone angle δa, the distance from the face cone vertex to the axis intersection point Ga and other parameters, separate the vertices of the concave and convex tooth surfaces according to the geometric relationship. Select the points whose angle is lower than the face cone angle and higher than 5% of the face cone angle as the tooth top data points.
- After the above two steps, the vertices of the concave and convex tooth surfaces are obtained, but they are not separated yet. According to the separation process of tooth surface data points in Section 3.2, separate the vertex data.
- Select 12 tooth top data points at equal distances along the tooth width direction.
The tooth height error Δδh is defined as the error between the vertical distance from the actually machined tooth top vertex to the root cone and the designed theoretical full tooth height parameter. For the convenience of calculation, the points are rotated to the xOz plane and the distance from the vertex to the root cone is calculated. The tooth height error Δδh is calculated as the difference between the theoretical full tooth height he and the calculated distance d.