Abstract:
Based on the principle of gear shaping, this paper utilizes VERICUT to simulate the gear shaping process for non-circular gears. Firstly, the tooth profile of the non-circular gear is designed according to the tooth profile envelope principle, and a theoretical model of the non-circular gear is established. Secondly, the four-axis CNC gear shaper model, gear shaping cutter, and gear blank are constructed in Creo and imported into VERICUT. The corresponding NC program for gear shaping is written, and the gear shaping simulation is carried out. Finally, the non-circular gear obtained from the simulation is automatically compared with the design model to verify the feasibility of the numerical control gear shaping simulation method. This provides convenience for the actual gear shaping process of non-circular gears, reduces production costs, shortens the cycle time from design to machining, and thereby enhances the production efficiency of non-circular gears.
1. Introduction
Non-circular gears, as a new type of gear, inherit the transmission characteristics of ordinary cylindrical gears and possess advantages such as a compact structure and the ability to achieve variable ratio transmission. They are widely used in printing presses, automatic machines, flow meters, and hydraulic motors [1]. However, due to the variety and complexity of non-circular gear shapes, with each tooth being unique, their machining presents certain difficulties. The main machining methods for non-circular gears include wire electrical discharge machining (WEDM), gear shaping, and hobbing. Among them, gear shaping is currently the most widely used gear machining method, known for its high precision, efficiency, wide processing range, and strong versatility [2]. At this stage, three-axis linkage CNC gear shapers are primarily used for machining non-circular gears, which can only achieve tool retraction along the centerline direction. This method of retraction may lead to tool interference issues, which can significantly reduce gear shaping efficiency to avoid interference [3].
With the continuous improvement of industrial equipment standards, increasingly higher requirements are placed on the machining efficiency and accuracy of non-circular gears. Meanwhile, the performance of CNC machining machines has become more powerful, and more advanced four-axis linkage CNC gear shapers have emerged as the future development trend. Four-axis linkage CNC gear shapers can not only achieve four-axis normal direction tool retraction machining but also realize four-axis oblique direction tool retraction machining and three-axis centerline tool retraction machining [4].
The rapid development of CNC technology and computer simulation technology has greatly promoted the development of simulation processing technology for non-circular gears [5]. The following is a summary of relevant research on simulation processing for non-circular gears: Yang Yanfang et al. [6] realized the virtual gear shaping process simulation for non-circular gears based on Visual Studio 2008 combined with the OSG graphics library; Zheng Fangyan [7] achieved automated programming and dynamic envelope simulation for the machining of non-circular gears based on VC++ and MATLAB; Li Jiangang et al. [8] conducted CNC machining simulation for non-circular gear grinding based on VERICUT; Zhao Ning et al. [9] performed simulation processing for non-orthogonal face gears based on the gear shaping principle using VERICUT software. The CNC machining simulation software VERICUT can realistically simulate the actual machining process, verify the correctness of NC codes, and accurately reflect the machining situation of the workpiece (such as overcutting, undercutting, etc.), which can better replace trial cutting, avoid mistakes in the design and machining of non-circular gears, reduce production costs, and shorten the cycle from design to production.
The above literature mainly relies on the Boolean operation functions in existing graphics libraries to program and implement independent graphics simulation systems for the simulation processing of non-circular gears. However, there is relatively little research on the simulation of CNC gear shaping for non-circular gears. In this paper, we first establish a theoretical design model for non-circular gears and build a four-axis linkage CNC gear shaper model. Then, we utilize VERICUT to perform CNC gear shaping simulation processing for non-circular gears. Finally, we automatically compare the non-circular gear model obtained from gear shaping with the design model to verify the correctness of the gear shaping process.
2. Non-circular Gear Machining Principle and Tooth Surface Equation
2.1 Non-circular Gear Shaping Principle
The principle of non-circular gear shaping is that the pitch circle of the gear shaping cutter performs pure rolling in the normal direction on the pitch curve of the non-circular gear, ensuring that the arc length rolled over by the pitch circle of the gear shaping cutter on the pitch curve of the gear blank is equal to the arc length rotated by the gear shaping cutter around its own rotation center C2 axis. The tooth profile of the non-circular gear is enveloped by the tooth profile of the gear shaping cutter
During the non-circular gear shaping process, the gear shaping cutter and gear blank involve six basic motions:
(1) Main motion of gear shaping: reciprocating cutting motion of the gear shaping cutter along the Z-axis direction;
(2) Circular feed motion of the gear shaping cutter: rotational motion around the C2 axis;
(3) Tooth division motion: rotational motion of the gear blank around the C1 axis;
(4) Relative position adjustment motion: linear motion of the gear shaping cutter and gear blank along the X and Y axes;
(5) Feed motion of the gear shaping cutter: feed motion of the gear shaping cutter in the tooth depth direction;
(6) Tool retraction motion: retraction of the gear shaping cutter along the normal direction of the pitch curve.
2.2 Tooth Surface Equation of Non-circular Gear
The design methods for the tooth profile of non-circular gears are mainly divided into two categories: involute unfolding method and generative method [10]. This paper adopts the involute unfolding method.
The pitch curve equation of the non-circular gear is:
r = A(1 – k²) / (1 – k cos φ) (1)
where: A is the semi-major axis radius of the ellipse; k is the eccentricity.
The vector equation of the tooth profile of the non-circular gear is:
r_f = r_g + a_n (2)
where: r_f is the radial vector of any point on the tooth profile; r_g is the radial vector at the intersection of the normal line of the tooth profile point and the pitch curve; a_n is the vector with a direction consistent with the normal direction of the tooth profile and a length equal to the distance between the pitch curve and the tooth profile.
The tooth profile of the non-circular gear is divided into points above and below the pitch curve.
For points on the tooth profile above the pitch curve, the angle between the vector a_n and the polar axis is θ – μ + α_μ for the right tooth profile and μ – θ + α_μ for the left tooth profile.
The equation for the right tooth profile is:
x_R = r_g cos θ + a_n cos(θ – μ + α_μ)
y_R = r_g sin θ + a_n sin(θ – μ + α_μ) (3)
The equation for the left tooth profile is:
x_L = r_g cos θ + a’_n’ cos(μ – θ + α_μ)
y_L = r_g sin θ + a’_n’ sin(μ – θ + α_μ) (4)
For points on the tooth profile below the pitch curve, similar equations can be derived.
From Equations (3) to (6), the three-dimensional tooth surface equations for the non-circular gear can be obtained as follows:
Right tooth surface equation:
x_R = r_g cos θ + a_n cos(θ – μ + α_μ)
y_R = r_g sin θ + a_n sin(θ – μ + α_μ) (7)
Left tooth surface equation:
x_L = r_g cos θ + a’_n’ cos(μ – θ + α_μ)
y_L = r_g sin θ + a’_n’ sin(μ – θ + α_μ) (8)
Based on the tooth surface equation of the non-circular gear, the three-dimensional design model of the non-circular gear is obtained using 2D and 3D design software.
3. CNC Gear Shaping Simulation Processing for Non-circular Gears
In this paper, we first build a four-axis linkage CNC gear shaper model in VERICUT. Next, we complete the settings for system parameters, gear shaping cutters, blanks, designs (theoretical 3D models), and other components. Finally, we import the G-code for gear shaping and perform simulation processing on non-circular gears, mimicking the four-axis linkage CNC gear shaping process. The basic parameters of the non-circular gear workpiece are outlined in Table 1.
3.1 Design of Gear Shaping Cutter and Gear Blank
Based on the relevant parameters of the gear shaping cutter, a 2D design software is utilized to design the tooth profile curve of the gear shaping cutter. Theoretically, the gear blank for a non-circular gear corresponds to the tooth tip curve of the non-circular gear, which is the normal equidistant line of the pitch curve of the non-circular gear [3]. Using 2D design software, the rough casting of the non-circular gear is designed.
These designed tooth profile and blank diagrams of the gear shaping cutter are imported into Creo to establish 3D models and saved as .stl files.
3.2 Four-Axis Linkage CNC Gear Shaper Model
Firstly, based on the analysis of the gear shaping motion of non-circular gears and the structure of existing gear shapers mentioned earlier, a 3D model of the four-axis linkage CNC gear shaper is designed in Creo, and each component is saved as an .stl file. Subsequently, in the VERICUT project tree, the dependency relationships between the components are configured. Finally, the .stl files of each part of the gear shaper are imported into their corresponding positions, and fixtures, blanks, designs, and other components are added to complete the setup of the gear shaper model.
3.3 NC Programming
VERICUT’s control system library includes numerous control systems for users to choose from. In this paper, the SINUMERIK840D control system is selected as the control system for the gear shaper. Based on the equal arc-length processing strategy for the gear blank and the geometric relationship when retracting along the normal to the pitch curve, the pitch curve of the non-circular gear is divided into equal arc-length segments, and the corresponding polar angles are calculated inversely. With the polar angle as the variable, MATLAB programming is employed to calculate the position coordinates of each motion axis during the gear shaping process, thereby obtaining the G-code for gear shaping of non-circular gears.
By following the aforementioned process, the NC program is prepared, and it is ready to be utilized within the simulation environment of VERICUT.