This paper presents a simulation analysis method for the instantaneous cutter-workpiece engagement (CWE) area during multi-axis milling of cylindrical spur gears using ball-end cutters, based on three-dimensional solid modeling technology. A gear milling motion model is established to accurately describe the motion relationship between the cutter and the workpiece. Based on the precise model of the gear tooth profile, the tool path planning for the ball-end cutter is realized using the constant scallop height method. The updated workpiece model is obtained by Boolean subtraction between the tool swept volume and the process workpiece model. The CWE area is identified from the updated workpiece geometric model according to the spherical geometric characteristics of the CWE area. The CWE area angle interval is obtained by slicing the CWE area with planes parallel to the tool axis. Cutting experiments show that the simulated CWE results are consistent with the actual machining results, verifying the validity and accuracy of the simulation calculations. The accurate extraction of the CWE area between the ball-end cutter and the workpiece lays a foundation for precise predictions of physical quantities such as gear milling forces and tool wear.
1. Introduction
The multi-axis milling of gears using universal five-axis machining centers is widely applied in areas such as rapid prototyping of new gears, personalized machining of large gears, and gear repair due to its high flexibility [1-2]. Currently, research on multi-axis gear milling primarily focuses on tool path planning. Guo Erguo et al. [3] proposed a method for envelope milling of involute cylindrical gears using end mills, establishing a matching relationship between tool path planning and tooth surface accuracy characteristics. GUO et al. [4] derived a tooth profile error model for gear milling with side milling cutters, studying the influence of tool feed strategies and gear geometric parameters on tooth profile errors. SHI et al. [5] proposed a roughness-constrained tool path planning method for cylindrical gear milling. ALVAREZ et al. [6] completed spiral bevel gear milling using different machining strategies, optimizing the tool path through error analysis. The aforementioned tool path planning studies demonstrate the constraint relationship between tooth surface accuracy and processing parameters from a geometric perspective.
However, in actual machining, the determination of processing parameters is also affected by the physical manifestation of cutting forces. Inappropriate selection of processing parameters often leads to a sharp increase in milling forces, accelerated tool wear, and failure to obtain satisfactory gear surface accuracy and dimensional accuracy. Therefore, achieving precise prediction of cutting forces during gear milling is the basis for determining and optimizing gear milling parameters. The calculation of cutting forces requires an accurate representation of the CWE area. Currently, CWE area calculation is mainly divided into three methods: the solid method, the discrete method, and the analytical method.
The solid method determines the CWE area through Boolean operations between the tool and workpiece solid models. YANG et al. [7] used the possible engagement surface of the tool rotating surface to trim the tool swept volume in NX software to obtain the CWE area. BOZ et al. [8] utilized the Parasolid solid modeling engine to obtain the CWE area by trimming the intersection surface between the tool swept volume and the workpiece. Zhi Junjie et al. [9] extracted the CWE area from the updated solid geometric model based on NX secondary development technology.
The discrete method determines the CWE area by judging whether the discretized tool and workpiece intersect. Dong Yongheng et al. [10] identified the instantaneous engagement state during ball-end cutter machining based on an improved Z-MAP method. WEI et al. [11] improved the Z-MAP method by introducing the identification function of logical arrays, improving the calculation efficiency of the tool contact area. QIN et al. [12] introduced an improved Z-MAP method by drawing on the analytical model of the CWE area in multi-axis milling, which only requires updating the discrete model of the tool during the CWE area calculation process, achieving higher calculation accuracy while reducing calculation time. The analytical method represents the CWE area using spatial curves and then solves it through dimensionality reduction methods. Wei Zhaocheng et al. [13-14] simplified the problem of solving the boundary curves of the CWE area in multi-axis milling, respectively, and proposed a CWE area calculation method based on semi-analytical modeling. Each tooth profile envelope milling can be regarded as a three-axis milling process compared to complex curved surface milling with varying tool feed directions, and the number of tool paths is relatively small [1]. Among the aforementioned three CWE area calculation methods, the solid method has high calculation accuracy, is suitable for three-axis processing, and its calculation efficiency can meet simulation requirements. The discrete method has high calculation efficiency, but the discretization of the workpiece and tool often leads to lower calculation accuracy, making it difficult to balance the contradiction between calculation efficiency and accuracy. The analytical method has high calculation efficiency under certain calculation accuracy requirements. However, for spur gear milling processes, it involves complex intersection calculations between the involute surface representing the workpiece surface and the spherical surface representing the tool swept surface, leading to a cumbersome calculation process and poor adaptability to different working conditions. Therefore, for cylindrical spur gear milling, the solid method is adopted to extract the CWE area, which can obtain high calculation accuracy while meeting the simulation calculation efficiency, while avoiding the precision defects of the discrete method and the complex intersection problem between the involute surface and the spherical surface in the analytical method.
In summary, this paper takes spur gears as the object and realizes the simulation analysis of the CWE area between the tool and the workpiece during the milling process based on the solid modeling method. Firstly, based on the gear milling motion model and the mathematical model of the workpiece tooth profile to be processed, the gear milling tool path is planned using the constant scallop height method. Secondly, the tool swept volume is constructed, and the workpiece geometric model is updated through Boolean operations between the swept volume and the gear workpiece geometric model. The CWE area during gear milling is identified on the updated gear blank geometric solid model, and the entry and exit angles are determined. Finally, the effectiveness and accuracy of the proposed method are verified by comparing the CWE area results after actual machining with the simulation results.
2. Mathematical Model of Gear Milling Process
2.1 Coordinate System Definition
To accurately describe the relative position relationship between the tool and the workpiece during the gear milling process, a coordinate system as shown in Figure 1 is established.
During spur gear milling, all teeth have the same milling strategy. In the milling process of a single tooth, the milling cutter moves the tool in the tooth height direction with air cutting, then feeds in the tooth width direction, i.e., along the gear axis, to remove material by milling, repeating the tooth height direction tool movement and tooth width direction cutting until the entire tooth surface is machined. The coordinate systems defined in the coordinate system are:
oW-xWyWzW (abbreviated as {SW}) is the workpiece coordinate system, which is the reference coordinate system for describing the workpiece geometric model and tool motion. It is fixedly connected to the gear, with its origin oW located at the rotational center of the upper end surface of the gear, the coordinate axis oWyW coinciding with the centerline of the tooth profile of the 0th tooth, and the coordinate axis oWzW coinciding with the gear axis.
oF-xFyFzF (abbreviated as {SF}) is the feed coordinate system describing the tool feed motion. The origin oF is located at the center of the ball-end cutter, and its position in {SW} is the tool location point CLk,i (where the subscripts k and i represent the kth tool path and the ith tool location point on it). The coordinate axis oFzF is in the direction of the normal nk of the ideal involute surface. From the geometric characteristics of the involute surface, during straight-line feeding along path k, oFzF has the same normal nk at any tool location point CLk,i. The angle between oFzF and oWzW is βk, which is the coordinate transformation angle describing the relationship between {SF} and {SW}. The coordinate axis oFxF is parallel to the gear axis and in the same direction as the coordinate axis oWzW, i.e., parallel to the tool feed direction, and {SF} conforms to the right-hand rule.
oT-xTyTzT (abbreviated as {ST}) is the tool coordinate system with its origin oT fixed at the center of the tool, coinciding with oF. The coordinate axis oTzT coincides with the tool axis, with the direction away from the workpiece surface being positive. {ST} can be represented as the coordinate system constructed by rotating {SF} around its own axis oFxF (counterclockwise is positive, clockwise is negative) with a rotation angle λk, which is the angle between oFzF and oTxT, defined as the tool yaw angle. According to the definitions of {ST} and {SF}, the coordinate plane yFzF is parallel to the coordinate plane yTzT and also parallel to the end face of the spur gear, defining the tool tilt angle for multi-axis machining as 0.
(a) Axonometric Drawing (b) Right View (c) Top View
Figure 1. Reference Coordinate System for Spur Gear Milling Motion
2.2 Transformation Matrix
The entry and exit angles describing the CWE area are in the tool coordinate system {ST}, but their calculation process needs to be performed in the workpiece coordinate system {SW}, so the transformation relationship between the two needs to be Of course, to continue from where we last left off, let’s assume the previous output ended with a sentence like this:
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