Precision 5-Axis Machining Solutions for Toroidal Gear Profiles

Planar secondary enveloping toroidal worms are typically produced using various machine tools; however, there are challenges such as difficulty in improving processing accuracy, a complex manufacturing process, and high production costs. To address these issues, we propose a method for fine milling planar secondary enveloping toroidal worms using a stick milling cutter on a five-axis machine tool.

This approach is based on precise solid modeling of the toroidal worm. The tool path planning algorithm is implemented using UG/Open GRIP, and the post-processing algorithm is derived based on the motion relationship between the axes of the five-axis machine tool. A post-processing program is developed in C++, and five-axis machining simulations are conducted with software to verify the accuracy of the algorithm and evaluate performance.

This method provides a platform for the digital manufacturing of planar secondary enveloping toroidal worms.

 

PART 1 Introduction

The planar secondary enveloping toroidal worm transmission offers several advantages, including multi-tooth meshing, instantaneous double-line contact, and a large overall curvature. As a result, it is increasingly being utilized in the field of mechanical transmission.

Currently, the production of toroidal worms primarily involves rough machining through turning or milling, followed by fine machining with a grinding wheel. Li Haitao and colleagues have proposed methods for the rough machining of the spiral surface of toroidal worms using both turning and milling techniques. Meanwhile, Feng Xingxin and his team suggested a grinding approach that incorporates a structure design with a built-in grinding head and an inclined support for the grinding wheel.

 

However, these methods generally require rough and fine machining to be performed on different machine tools. This can lead to repeated clamping, which in turn reduces machining accuracy. Additionally, the complexity of the machining process and the higher costs associated with it do not contribute to improved efficiency.

To address these challenges, a novel machining method is proposed, which involves fine milling a planar secondary enveloping toroidal worm using a stick milling cutter on a five-axis machine tool. This approach includes tool path planning based on precise solid modeling of the toroidal worm. A post-processing algorithm has been developed to reflect the motion relationship between the axes of the five-axis machine. Furthermore, a post-processing program has been created, and five-axis machining simulations have been conducted using specialized software to verify the accuracy of the algorithm and assess the precision of the tooth surface.

 

PART 2 Pre-processing technology for NC machining of toroidal worm

The process of planning tool paths and creating tool location files based on the positional relationship between the tool and the part, as well as the machining parameters, prior to NC (Numerical Control) machining is known as pre-processing. Tool path planning plays a crucial role in the functionality of the NC system and the quality of the generated NC program. An effective tool path generation method should exhibit characteristics such as reasonable step length distribution, uniform cutting span, smooth tool paths, and high machining efficiency.

 

2.1 Basic principle of machining toroidal worm tooth surface with rod milling cutter
Research has shown that when machining a plane secondary enveloping toroidal worm using traditional methods, the instantaneous contact of the grinding wheel with the tooth surface is represented by a straight line, as illustrated in Figure 1a. To reduce machining costs and enhance both efficiency and precision, it is suggested to employ rod milling cutters for CNC machining the tooth surface of the plane secondary enveloping toroidal worm. In this approach, the generatrix of the rod milling cutter aligns with the tooth surface envelope during the machining process, as depicted in Figure 1b.

During the machining process, it is essential to prevent interference caused by the rod milling cutter colliding with adjacent tooth surfaces. To achieve this, the diameter of the rod milling cutter must be smaller than the minimum spacing of the toroidal worm tooth roots. The specific parameters of the toroidal worm are presented in Table 1, and the three-dimensional model of the plane secondary enveloping toroidal worm is illustrated in Figure 2.

2.2 Tool path planning algorithm for CNC machining of toroidal worm
The machining of plane secondary enveloping toroidal worms involves three main stages: rough machining, semi-finishing, and finishing. This paper focuses specifically on the finishing process of the worm tooth surface.

During the finishing of the worm tooth surface, the tool is positioned so that it is tangent to the tooth surface along the feed path. The process begins at the top of the right tooth of the worm and involves cutting continuously along the spiral line of the worm until reaching the top of the left tooth. The tool then feeds along the tooth surface and cuts from left to right. This process is repeated until the entire worm tooth surface is processed.

This machining strategy helps to shorten the tool path, reduce idle strokes, and ultimately increases machining efficiency.

 

The position of the tool during machining is primarily determined by the tool center point and the direction of the tool axis. The tool center point defines the tool’s position in space, while the tool axis direction specifies the angle and orientation of the tool at that center point. Consequently, planning the tool trajectory requires calculating both the coordinates of the tool center point and the vector for the tool axis direction.

(1) Determining the tool center point coordinates: The geometric relationship between the rod milling cutter and the worm tooth surface is illustrated in Figure 4. The line segment p1p2 represents an envelope line on the worm tooth surface. We define the N direction as the normal direction to the tooth surface at point p1. By offsetting the line p1p2 along this normal direction by the radius of the rod milling cutter (rt), we obtain the line segment p3p4. Here, point p3 serves as the tool center point, and the straight line p3p4 indicates the direction of the tool axis.

 

(2) Solving the cutter axis direction vector

The coordinates of the tool center point determine the specific position of the center of the rod milling cutter, while the direction vector of the cutter axis defines the cutter’s spatial orientation at the tool center point. In the case of machining the tooth surface of a toroidal worm, the cutter axis direction of the rod milling cutter is always parallel to the envelope line of the tooth surface. This means that the direction vector of the cutter axis should match or be proportional to the direction vector of the envelope line during the machining process. The envelope line of the worm tooth surface is illustrated in Figure 5.

 

Each data point on the tooth surface is calculated using this method to determine its corresponding tool center point coordinates and tool axis direction vector. All tool positions are then combined to produce the tool trajectory.

 

2.3 Implementation of CNC machining tool trajectory

According to the algorithm described, UG/Open GRIP is utilized for secondary development. Since the GRIP language can compile programs using the built-in Notepad on the computer, it is essential to change the Notepad encoding to ANSI and save the file in the .grs format. Compiling the program in this format ensures the accuracy of the file structure. The tool motion trajectory for machining a single-sided toroidal worm, obtained through this secondary development, is illustrated in Figure 6. Additionally, some relevant data from the tool position file is presented in Table 2.

 

The tool path file plays a crucial role in the quality of CNC machining products, as its accuracy and effectiveness significantly affect the final outcome. Therefore, verifying the tool path is essential. The tool path planning algorithm is deemed correct and effective when the tool path is simulated in UG (refer to Figure 7).

 

PART 3 Post-processing technology for CNC machining of toroidal worm gears

The purpose of post-processing is to convert the trajectory data of tool movement from the workpiece coordinate system to the machine tool coordinate system. This conversion is based on the spatial transformation relationship between the two coordinate systems. Given the geometric characteristics of the toroidal worm gear, the machine tool utilizes a five-axis setup with a swing head rotary table. This configuration includes three translation axes (X, Y, and Z), as well as the workpiece rotation axis (A) and the tool swing axis (B).

 

3.1 Establishment of the coordinate system of the swing head rotary table five-axis machine tool

The establishment of the five-axis machine tool coordinate system is illustrated in Figure 8. We define the following coordinate systems:

1. OwXwYwZw: The workpiece coordinate system, which is fixed to the workpiece.
2. OtXtYtZt: The tool coordinate system, with its origin located at the tool center point.
3. Om1Xm1Ym1Zm1: The coordinate system fixed to the workpiece rotation axis (A axis), with its origin, Om1, positioned on the rotation axis.
4. Om2Xm2Ym2Zm2: The coordinate system fixed to the tool swing axis (B axis), where the origin, Om2, is at the intersection of the swing axis and the tool axis.

The coordinate axis directions for each system are consistent with one another.

In the initial state, the tool axis is parallel to the Zt axis of the tool coordinate system, and the coordinates of OwXwYwZw coincide with the origin of OtXtYtZt. The position of the midpoint Om2 in the tool coordinate system is given as rm2 (0, 0, L), while its position in the workpiece coordinate system is r m1 (mx, my, mz). The machine tool translation axis is represented as rs (sx, sy, sz) relative to this initial state. The A and B axes are denoted as θA and θB, respectively, relative to the initial state, with the negative direction indicated by an arc arrow in Figure 8.

3.2 Post-processing algorithm of the five-axis machine tool with a swing head and a rotary table

In the tool coordinate system, the tool position vector is represented as \([0, 0, 0]^T\), and the tool axis direction vector is \([0, 0, 1]^T\). In the workpiece coordinate system, let the tool position vector be expressed as \(\mathbf{r}_p (p_x, p_y, p_z)\), and let the tool axis direction vector be denoted as \(\mathbf{u} (u_x, u_y, u_z)\).

These vectors can be transformed from the coordinate system \(O_tX_tY_tZ_t\) to the coordinate system \(O_{m2}X_{m2}Y_{m2}Z_{m2}\), then translated and rotated to the coordinate system \(O_{m1}X_{m1}Y_{m1}Z_{m1}\), and finally translated to the coordinate system \(O_wX_wY_wZ_w\). This transformation can be described using a spatial coordinate transformation, where \(T\) represents the translation transformation matrix and \(R\) represents the rotation transformation matrix.

 

 

3.3 Implementation of five-axis post-processing application program

The post-processing procedure is illustrated in Figure 9. First, the system reads the tool position data from the tool position file line by line. Next, it checks whether the end of the file has been reached. If the file is not finished, the code is converted into the corresponding recognition keyword for the read tool position data. Following this, the rotation angle of the rotary axis and the movement distance of the translation axis are calculated using equations (9) to (11), based on the machine tool’s motion characteristics. Finally, the NC program is generated and output according to the program format required by the NC system. The process then returns to reading the tool position file. If the end of the file is reached, the post-processing concludes.

Based on the five-axis machine tool post-processing algorithm described earlier, a five-axis post-processing application was developed using Visual Studio as the development platform and C++ as the programming language. The application features a user-friendly interface, as illustrated in Figure 10, and various code files are displayed in Figure 11.

The post-processing application is used to post-process the tool location file obtained in the pre-processing. Some data in the output NC program are shown in Table 3.

PART 4 Five-axis machining simulation of plane secondary enveloping toroidal worm

In the CNC machining process, it is essential to conduct strict inspection after generating the CNC program to ensure the final product’s quality. However, there are inherent risks associated with operating machine tools in actual production environments. If issues arise during the machining process, they can lead to significant damage to the equipment and even pose safety hazards. Therefore, performing a comprehensive simulation inspection using virtual machining simulation technology is crucial.

In UG software, the built-in swing head rotary machine tool is selected, and parameters such as the stroke limits for each axis, the machine tool’s zero point, and the machining reference coordinate system are set. During the machining simulation, a rod milling cutter is used with specifications including a diameter of 10 mm, a tool length of 70 mm, a groove length of 56 mm, and a cutting edge length of 50 mm. The tool structure is illustrated in Figure 12.

 

The tool model and the worm semi-finishing model are imported into the simulation environment, where the NC program generated through the aforementioned post-processing is executed. This allows for a five-axis machining simulation of the plane secondary enveloping toroidal worm, as illustrated in Figure 13. This machining simulation effectively mitigates the impact of factors such as machine tool errors and tool wear that may occur in actual machining CNC process, leading to more optimal machining conditions. Moreover, this simulation method serves as an efficient means to verify the accuracy of the previously mentioned algorithm.

 

During the machining simulation, the software’s “collision detection” function was activated, and the option to “stop when a collision occurs” was selected. Throughout the simulation, the machining process continued without interruption, despite potential interference issues such as collisions. This indicates that the approach described in this paper is both safe and reliable.

 

PART 5 Tooth surface accuracy evaluation of machining simulation model

To further verify the accuracy of the machining toroidal worm theory, it is essential to assess the tooth surface precision of the machining simulation model. The tooth surface deviation method is employed for this analysis. This method characterizes tooth surface accuracy by measuring the deviation value of each point on the theoretical tooth surface along its normal direction to the machined tooth surface, as illustrated in Figure 14.

 

In SolidWorks software, six tooth surface points are evenly distributed across the theoretical tooth surface. These points are randomly selected and assembled with the machining simulation of a toroidal worm. To ensure proper alignment, constraints are applied so that the central axes and the coordinate origins coincide, allowing one of the theoretical tooth surface points to align with the aerospace CNC machining simulation tooth surface, as illustrated in Figure 15. The deviation values of the remaining five theoretical tooth surface points are measured along their normal directions relative to the machining tooth surface, with the measurement results presented in Table 4.

 

Table 4 shows that there is a minor discrepancy between the theoretical tooth surface and the processed simulated tooth surface. However, this deviation meets the accuracy requirements for the secondary enveloping toroidal worm, and the error is within acceptable limits for engineering applications.

 

PART 6 Conclusion

This paper presents a tool path planning algorithm based on precise solid modeling of a planar secondary enveloping toroidal worm. The post-processing algorithm is developed by considering the motion relationships between the axes of a five-axis machine tool. A corresponding post-processing program is created, and five-axis machining simulations are conducted using specialized software to verify the algorithm’s accuracy and assess the tooth surface precision. The final machining results meet the required standards.

Furthermore, utilizing a low-cost stick milling cutter for the precise milling of the planar secondary enveloping toroidal worm on a five-axis machine tool enhances machining accuracy, reduces costs, and improves overall machining efficiency. This approach also offers a viable platform for the digital manufacturing of planar secondary enveloping toroidal worms.

 

 

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