1. Introduction
1.1 Research Background and Significance
Mechanical transmission is a crucial form of power transmission in mechanical systems. It plays a vital role in converting and transmitting the motion of the prime mover to the working machine. Among various mechanical transmissions, gear transmission stands out for its high precision, efficiency, and reliability. It is widely used in different fields such as automotive, aerospace, and industrial machinery. However, traditional cylindrical gears have limitations in achieving variable transmission ratios, which has led to the development and application of non-circular gears. Non-circular gears offer the advantage of variable speed transmission, making them suitable for applications where speed adjustment is required. Nevertheless, the complex tooth surface geometry of non-circular gears poses challenges in their manufacturing. This has spurred research efforts to develop efficient and accurate machining methods for non-circular gears.
1.2 Research Status at Home and Abroad
1.2.1 Research Status of Face Gear and Non-circular Gear Machining
The machining of gears has always been a focus of research in the mechanical field. For face gears and non-circular gears, scholars have proposed various machining methods. For example, Xue Dongbin et al. studied the feasibility of machining straight ruler face gears on a CNC milling machine using the forming method. Li Zhengminqing et al. derived the theoretical tooth surface equation and tool envelope surface equation for face gear shaping and analyzed the interference situation. Ming Xingzu et al. proposed a high-speed milling method for straight ruler face gears using a finger milling cutter. Wang Y et al. proposed a hobbing method, a milling method, and a precision grinding method for face gears, respectively. However, the existing machining methods for face gears and non-circular gears have limitations and cannot be directly applied to the machining of non-circular face gears.
1.2.2 Research Status of Gear Machining Error
Gear machining error has a significant impact on the performance and service life of gears. Scholars have conducted extensive research on the sources and effects of machining errors. Zheng Peng et al. classified the causes of gear machining errors and proposed a new error analysis method. Li Zhengminqing et al. analyzed the influence of processing offset error on the contact characteristics of orthogonal face gears. Zhao Ning et al. verified the correctness of face gear shaping and obtained the influence law of processing errors on the tooth surface accuracy of face gears through simulation. However, there is still a need for further research on the machining error of non-circular face gears.
1.2.3 Research Status of Gear Meshing Characteristics
The meshing characteristics of gears directly affect the transmission performance and reliability of gear pairs. Scholars have used TCA and LTCA technologies to study the meshing characteristics of gears. Litvin and his team made important contributions to the development of TCA and LTCA technologies. He Guoqi et al. simulated the tooth surface contact trajectory and contact area shape of the gear pair composed of cylindrical gears and face gears. Lu Fengxia et al. proposed a contact finite element method based on the meshing characteristics of helical gears. However, the meshing characteristics of non-circular face gears under the influence of machining errors need further in-depth study.
1.3 Research Contents of This Article
This article focuses on the generation principle and meshing characteristics of non-circular face gears machined by disk cutters. The specific research contents are as follows:
- Establish the envelope principle of the involute disk cutter for the non-circular face gear.
- Analyze the influence law of machining error factors on the accuracy of the formed tooth surface of the non-circular face gear.
- Analyze the influence law of machining error factors on the meshing characteristics of the non-circular face gear.
- Establish the motion model of the machine tool and conduct simulation machining.
2. Principle of Generating Non-circular Face Gears with Disk Cutters
2.1 Transmission Mechanism Principle of Non-circular Face Gear
The non-circular face gear transmission mechanism combines the characteristics of non-circular gear and face gear transmissions. It consists of a common cylindrical gear and a non-circular face gear. The small cylindrical gear has fewer teeth than the non-circular face gear, enabling the mechanism to achieve a reduction function. The radius of the non-circular face gear varies with the rotation angle, resulting in a variable output speed and realizing the integrated function of deceleration and variable speed transmission. The transmission process can be equivalent to the pure rolling process between a pitch cylinder and a non-circular curve. The motion relationship between the two gears can be described by the following equations:
2.2 Generation Machining Method of Non-circular Face Gear Based on Disk Cutter
2.2.1 Machining Principle of Disk Cutter for Non-circular Face Gear
The generation machining method of the non-circular face gear based on the disk cutter is derived from the shaping machining principle of the non-circular face gear. The involute disk cutter is used instead of the traditional shaping cutter. The motion of the disk cutter during machining is similar to that of a single tooth of the shaping cutter. The machining process includes cutting motion, generation motion, radial feed motion, and relief motion.
2.2.2 Generation Machining Motion of Non-circular Face Gear Considering Machine Tool Structure
To realize the generation machining of the non-circular face gear, a horizontal five-axis machining center is selected. The machine tool structure includes an L-shaped bed, a tool spindle, slides, a rotary table, and a worktable. The motion of the machine tool axes is coordinated to complete the machining process. The generation motion between the non-circular face gear and the disk cutter is achieved by the rotation of the non-circular face gear around the B-axis and the coordinated movement of the X and Z axes.
2.2.3 Axis Motion Relationship in Generation Principle
Based on the generation principle and machine tool structure, the coordinate transformation matrices between different coordinate systems are established. The transformation matrix from the disk cutter coordinate system to the non-circular face gear coordinate system is obtained by multiplying the transformation matrices of each adjacent coordinate system.
2.3 Model of Formed Tooth Surface of Non-circular Face Gear
2.3.1 Tooth Surface Equation of Involute Disk Cutter
The tooth surface equation of the involute disk cutter is derived based on the tooth surface equation of the shaping cutter. The equation is expressed in terms of parameters such as the base circle radius, feed parameter, and involute spread angle.
2.3.2 Working Tooth Surface Equation of Non-circular Face Gear
According to the meshing theory, the meshing equation between the disk cutter and the non-circular face gear is established. By solving the meshing equation and combining the coordinate transformation matrices, the tooth surface equation of the non-circular face gear is obtained.
2.3.3 Transition Surface of Non-circular Face Gear
The transition surface of the non-circular face gear is formed by the tip of the disk cutter. By substituting the relevant parameters into the tooth surface equation, the equation of the transition surface can be obtained. The transition curve is the intersection line of the working tooth surface and the transition surface.
2.4 Three-dimensional Model of Formed Tooth Profile of Non-circular Face Gear
Based on the obtained tooth surface equations, a three-dimensional model of the non-circular face gear is established using data analysis and modeling software. The model visually shows the shape and characteristics of the non-circular face gear tooth profile.
3. Model of Formed Tooth Surface with Machining Error and Deviation Distribution Characteristics
3.1 Solution of Formed Tooth Surface Model with Machining Error
3.1.1 Offset Error Analysis
During the machining process, offset errors may occur due to factors such as fixture accuracy and clamping methods. These errors cause the position of the disk cutter relative to the gear blank to deviate, affecting the accuracy of the formed tooth surface. The transformation matrix considering the offset error is established.
3.1.2 Translation Axis Error Analysis
The motion accuracy of the machine tool translation axes directly affects the accuracy of the machined workpiece. In the X and Z axes directions, errors can cause deviations in the relative position between the tool and the workpiece, thereby affecting the tooth surface accuracy. The transformation matrices considering the X and Z axis errors are established.
3.1.3 Swing Axis Error Analysis
The generation motion between the non-circular face gear and the disk cutter requires the rotation of the B and C axes. The swing error of the B axis has a significant impact on the accuracy of the formed tooth surface. The transformation matrix considering the swing error of the B axis is established.
3.2 Tooth Surface Deviation Analysis Method
The tooth surface deviation is analyzed using the rotation projection method. The tooth surface points are projected onto a plane, and the deviation between the error tooth surface and the error-free tooth surface is represented by the vector between the projection points. The tooth surface error cloud chart is used to visually show the distribution of tooth surface deviation.
3.3 Influence of Machining Error on Tooth Surface Deviation
3.3.1 Tooth Surface Deviation Analysis with Offset Error
When an offset error exists, the left and right tooth surfaces of the non-circular face gear will shift uniformly in a certain direction. The deviation values and distribution trends on different tooth surfaces are analyzed.
3.3.2 Tooth Surface Deviation Analysis with X-axis Error
The X-axis error causes the deviation direction of the tooth surface to change periodically with the change of the tooth surface radius. The deviation characteristics on different tooth surfaces are analyzed.
3.3.3 Tooth Surface Deviation Analysis with Z-axis Error
Similar to the X-axis error, the Z-axis error also causes the deviation direction of the tooth surface to change periodically. The deviation characteristics on different tooth surfaces are analyzed.
3.3.4 Tooth Surface Deviation Analysis with Swing Axis Error
The swing axis error causes the left tooth surface to shift outward and the right tooth surface to shift inward. The deviation values and distribution trends on different tooth surfaces are analyzed.
3.4 Summary of This Chapter
This chapter analyzes the sources of machining errors that cause tooth surface deviation of non-circular face gears, establishes error tooth surface models, and analyzes the influence laws of different error factors on tooth surface deviation through error cloud charts.
4. Meshing Characteristics of Non-circular Face Gear Tooth Surface with Machining Error
4.1 Meshing Trajectory of Non-circular Face Gear Transmission
4.1.1 Point Contact Mathematical Model of Non-circular Face Gear Transmission
To avoid load imbalance during the meshing process of the non-circular face gear pair, the number of teeth of the small cylindrical gear is selected to be 1 – 3 less than that of the shaping cutter. The point contact mathematical model between the non-circular face gear and the small cylindrical gear is established based on the meshing theory. The position vectors and normal vectors of the two gears at the contact point are related by equations.
4.1.2 Solution of Point Contact Mathematical Model
By substituting the tooth surface equations and normal vector equations of the two gears into the meshing equations, a system of equations with multiple unknowns is obtained. By solving this system of equations, the meshing points on the tooth surface can be determined, and the meshing trajectory can be obtained.
4.1.3 Visualization of Contact Trajectory
Using MATLAB software, the meshing points on the tooth surface are calculated and visualized. The meshing trajectory on a single tooth surface is shown, and the distribution characteristics of the meshing trajectory on different tooth surfaces are analyzed.
4.2 Influence of Machining Error on Meshing Characteristics
4.2.1 Influence of Offset Error on Meshing Characteristics
When an offset error exists, the contact trajectory on the tooth surface will shift in the radial direction. The shift direction and magnitude are related to the sign and value of the offset error.
4.2.2 Influence of X-axis Error on Meshing Characteristics
The X-axis error causes the contact trajectory on the tooth surface to change periodically. The change law and characteristics are analyzed.
4.2.3 Influence of Z-axis Error on Meshing Characteristics
Similar to the X-axis error, the Z-axis error also causes the contact trajectory on the tooth surface to change periodically. The change law and characteristics are analyzed.