Material Removal Rate of Face-Milled Bevel Gears Based on a Ring-Dexel Cutting Simulation

Face milling and face hobbing [1], also known as the single-indexing and continuous-indexing methods, are the most popular and widely used techniques for mass-producing bevel gears. The former method employs a sophisticated nonlinear movement of five axes [2] to fabricate these gears precisely. Modern computer numerical control (CNC) bevel gear-cutting machines can perform all face-milling operations, allowing for precise and efficient production. CNC machines also enable easy adjustment of cutting feed rates through NC code programming, providing flexibility and control over the manufacturing process. Relying solely on the speed planning provided by the machine software may not result in the most efficient solution. Engineers must frequently manually adjust feed rates based on experience to enhance overall cutting efficiency. Because machining power is directly related to material removal rate (MRR) [3], optimizing machining feed rates can be achieved by leveraging MRR data. However, current commercial software cannot accurately predict MRR for face-milling operations. This highlights a gap in the existing software functionality for machines. This issue also emphasizes a need for an advanced tool incorporating predictive MRR models. Therefore, it is essential to develop a fast-cutting simulation method for bevel gear manufacturing on CNC machines. Additionally, an accurate calculation method for MRR is crucial for optimizing the cutting feed rate.

The American company Gleason Works first developed the single-indexing method using a face cutter head in 1913 [1]. Later, the German company Klingelnberg GmbH introduced two continuous-indexing methods: Palloid (1923) and Cyclo-Palloid (1955), based on a tapered hob cutter and a two-part cutter head. These were eventually replaced by the modern face hobbing method, which used a face cutter head with stick blades, developed by the Swiss company Oerlikon. These methods were initially implemented on conventional cradle-type bevel gear cutting machines, which employ specific mechanisms to perform generating processes. Gleason Works developed two primary tooth surface modification methods, modified roll [4] and helical [5], to enhance the gear pair's contact performance. The Litvin research team has extensively studied this topic and published numerous related papers [6–9]. The mathematical models for the auxiliary tooth flank modification movements in manufacturing bevel gears, the helical motion and the cutter tilt methods, are detailed in Refs. [6–8] and Ref. [9], respectively. However, conventional machines often suffer from poor rigidity and accuracy due to complex mechanism designs. Krenzer and Hunkeler [10] of Gleason Works introduced a novel patent in 1991 for a CNC bevel gear-cutting machine. This machine utilized three rectilinear axes positioned orthogonally and three rotational axes to facilitate various gear-cutting motions under numerical control. This patent marked the inception of the CNC bevel gear-cutting machine era. The CNC machine is renowned for its simple yet robust structural design and offers remarkable rigidity and precision. It can execute all essential bevel gear-cutting movements with six degrees-of-freedom, allowing it to replace numerous conventional machines. Consequently, the movement for correcting the tooth surface is no longer confined by mechanism design constraints, as seen in conventional machines. Fong [11] introduced a universal mathematical model for face-milled bevel gears capable of simulating various flank correction motions. Wang and Fong [12] then utilized this mathematical model to propose a fourth-order kinematic synthesis for modified radial motion correction. Fan et al. [13] established fourth-order polynomials for adjusting machine settings to significantly reduce flank errors in manufacturing bevel gears for CNC machines. Shih et al. [14] introduced a universal mathematical model applicable to bevel gears, enabling simulation across various cutting methods for face milling and face hobbing. Subsequently, Hsu et al. [15] utilized this mathematical model to determine the cutting coordinates for a six-axis CNC bevel gear-cutting machine.

The MRR is critical in determining machining power. It is a constant value during milling, turning, gear hobbing, skiving, and generating grinding and can be easily computed. Lv et al. [3] compared three approaches for predicting machining power consumption in the milling process. One of these approaches involves multiplying the specific cutting energy by the MRR. However, determining the MRR for bevel gear cutting has been much more challenging due to the variable chip geometry during the cutting process. Therefore, an accurate MRR prediction is necessary for bevel gear cutting. Two primary models are utilized in cutting simulation: voxel-based, dexel-based models. The former combines volume and pixel representations, while the latter combines depth and pixel representations. Initially, the voxel-based model was the main computational core [16]. This model divides workpieces into small cubic elements, offering a discretized representation that simplifies simulation. Lorensen and Cline [17] introduced marching cubes for 3D surface construction to enhance surface quality. However, the cutting simulation using the voxel-based model is time-consuming and challenging to integrate into the CNC machine. Consequently, alternative models are being researched to replace the voxel-based model. A one-dimensional dexel model, first introduced by Van Hook [18], is used for real-time simulation in the milling process. This method is widely recognized for its high computation speed. Peng and Zhang [19] proposed a triple-dexel model to improve the surface quality of sculpted workpieces. This enhanced model finds widespread use in cutting simulations for turning, milling, and drilling operations. Recently, Inui et al. [20] utilized the triple-dexel representation for internal gears to simulate the power skiving process, facilitating the prediction of chip geometry during cutting. Habibi and Chen [21] proposed a plane-based model for face hobbing cutting simulation of bevel gears to determine the chip geometry. This approach conducts cutting simulations by discretizing the workpiece into several planes along the gear axis in small increments. Efstathiou and Tapoglou proposed a cutting simulation method for face milling [22] and face hobbing [23], leveraging computer-aided design Boolean operations to determine tooth solid geometry and its cutting chip, as well as to predict cutting forces. However, this cutting simulation uses five-axis milling kinematics instead of real bevel gear cutting kinematics. The WZL of RWTH Aachen University developed two different penetration calculation methods to simulate gear cutting. These two methods model a gear blank as scattered points or planes along the gear axis. Brecher et al. [24] applied the scattered-point penetration calculation method to simulate bevel gear cutting, including Palloid, Cyclo-Palloid, face hobbing, and face milling methods. This enables the determination of chip thickness and cutting length along the tool edge in response to the cutting velocity. Janßen et al. [25] and Kamratowski et al. [26] applied the planar penetration calculation method to simulate gear skiving and bevel gear cutting. This method also determines chip thickness and cutting length. While utilizing a dexel model for cutting simulation is a promising approach, Shih et al. [27] introduced a novel ring-dexel-based model to simulate face-milling cutting. However, this model falls short of accurately representing the geometry of root fillets. Additionally, while machining power is directly linked to the MRR, there is currently no established method for determining MRR specifically for face milling.

This paper aims to develop a calculation method for MRR in face milling based on an improved ring-dexel model. This enhanced model addresses the low display resolution at the tooth root observed in previous research [27]. Additionally, the MRR in face milling can be determined at every cutting position, thereby optimizing the cutting feed rate. The model discretizes the gear workpiece into multiple rings, simplifying the cutting simulation process. Triangular meshes represent the resulting tooth surfaces, enabling precise calculation of the volume of undeformed material removed at specific cutting positions. By incorporating the volume and intended cutting feed rate at each cutting position, MRRs can be precisely assessed. These MRR values can then serve as targets for optimizing cutting feed rates and enhancing machining efficiency. This method is applied to simulate various face-milling processes, including plunging for the ring gear and single-roll generating for the pinion, and to determine their respective MRRs. Experimental results demonstrate that controlling the MRR can effectively reduce machining time. The developed method can be integrated into the CNC machine's human–machine interface, optimizing the gear-cutting feed rate.

The face-milling method involves two operations: generating and forming (plunging). The generating operation is used to manufacture pinions, while the plunging operation is used for ring gears, particularly those with a gear ratio greater than 2.5, to reduce machining time. Given the intermeshing nature of pinion and ring gear, the pinion must rectify any tooth surface errors introduced by the formate method employed on the gear. Additionally, crowning is applied to the pinion tooth surfaces to prevent edge contact. Several dynamic flank modification motions have been developed for the pinion machining, including the modified roll [4], helical motion [5–8], and tilt cutter [9]. References [11,13,14] provide comprehensive mathematical models covering these cutting methods.

Fong and Shih's research team has developed bevel gear-cutting methods for over a decade [11,14,15]. The face-milling cutting tool features numerous blade groups, each consisting of inside and outside blades. The inside blades produce the convex surfaces, while the outside blades produce the concave surfaces. These blades are strategically positioned along the pitch circle on a plane, as depicted in Fig. 1. The blade profile typically has at least a straight line (I) and a fillet (II). The straight line is sometimes substituted with a circular arc to incorporate profile crowning. Two cutter parameters need to be given: the cutter radius $r0$ and the fillet radius $ρb$⁠.

Figure 2 shows a universal cradle-type bevel gear-cutting generator that has been used for decades. This versatile tool has been proven to have the ability to simulate all bevel-gear cutting methods, as documented in published papers [11,14,15]. It is a virtual machine and has nine machine settings, including i, j, $SR$⁠, $θc$⁠, $Em$⁠, $ΔB$⁠, $ΔA$⁠, $γm$⁠, and $Ra$⁠. The cutter head and gear are fixedly connected to coordinate systems $St$ and $S1$⁠. This arrangement ensures a stable and synchronized connection between these components during the machining of bevel gears.

Machining bevel gears consists of two primary operations: plunging and generating, as depicted in Fig. 3. In the plunging operation, the tool advances gradually by the cutting depth (⁠$s$⁠) while the generating angle is fixed. On the other hand, in the generating operation, the tool progressively advances from the gear toe to heel by the generating angle (⁠$ϕc$⁠) while the cutting depth is fixed. The cutter head and workpiece gear are moved through the coordinate transformation matrix $M1t(s,ϕc)$⁠, which depends on two parameters: s and $ϕc$⁠. Hence, by manipulating these two parameters, the cutting positions can be governed to simulate diverse bevel gear processing operations, encompassing plunging, single-rolling, double-rolling generating, and beyond.

A rapid-cutting simulation method has been developed over the past decade [18–20]. This simulation utilizes a dexel model, which discretizes a virtual workpiece into rays. A sequence of spatial points defines the tool path. The tool moves along this path and intersects with the virtual workpiece, allowing material removal between them. With the high-speed operation, the tool's cutting process on the workpiece can be visualized in real-time in the simulation. Consequently, it is commonly utilized in milling process simulations, as illustrated in Fig. 4. However, this method has yet to be applied in manufacturing bevel gears. Considering the axis symmetry of the gear workpiece, a novel ring-dexel-based model is proposed to simulate gear cutting effectively.

Figure 5 illustrates our developed cutting simulation of a left-hand face-milled bevel gear using a generating operation. Initially, the gear blank is discretized into multiple three-dimensional rings. The cutting depth s is set to a specified value (for instance, setting it to 0.5 mm for roughing while setting it to 0.0 mm for finishing). The tool moves with a generating motion from the toe (⁠$ϕct$⁠) to the heel (⁠$ϕct$⁠), governed by a range of generating angles segmented into cutting positions, where $Δϕc$ represents an increment. When cutting a left-hand gear, the angle $ϕct$ is larger than $ϕch$⁠; when cutting a right-hand gear, the angle $ϕct$ is smaller than $ϕch$⁠. First, for cutting the first tooth (⁠$i=1$⁠), the tool advances toward the gear workpiece depending on the cutting position (⁠$s$ and $ϕc$⁠), guided by the motion matrix $M1t$ shown in Eq. (2). Subsequently, intersection points (IPs) on the tool surface with the blank rings are calculated, enabling the determination of cutting points (CPs) on the tooth surface. Finally, the tooth surface geometry is constructed by triangular meshes based on these cutting points. Repeat this process until all cutting positions for the first tooth are completed. The tooth surfaces from the second tooth to the last tooth at each cutting position are generated by replicating and rotating the cutting points obtained from the first tooth to enhance the simulation speed.

The produced tooth surface geometry is constructed by cutting points on the tooth surface. These points are determined from the intersection points on the locus of the cutting edge with the workpiece rings at each cutting position. As depicted in Fig. 6, the gear blank is discretized into multiple rings arranged sequentially. These rings are located on lines extending from the toe to the heel of the gear blank. Each line consists of multiple points spanning from the top to the root, facilitating the generation of rays. Each ring is represented in $Sd$ by rotating the two-dimensional topographic points $rp(xp,yp)$ around the $x$-axis, as described in the following manner:

Figure 7 illustrates the intersection points at a specific cutting position in two and three dimensions. The intersection points on the rings must be determined one by one along each blank line. To enhance the resolution of the tooth root simulation, an alternative strategy is employed for handling the intersection points at the tooth root. Specifically, the intersection points are categorized into two groups: those belonging to the tool's straight-lined part (I), which is determined from the rays, and those belonging to the tool's fillet part (II), which is determined from the defined blank lines. The tool surface is transformed from $St$ to $S1$ based on the provided cutting parameters: cutting depth (⁠$s$⁠) and generating angle (⁠$ϕc$⁠). This tool locus $r1$ is expressed by Eq. (4); it has two variables u and $β$⁠.

Before determining the intersection points in part I, it is important to identify which ring intersects with the edge swept surface belonging to the straight-lined part. To achieve this, a length ratio is applied. In Fig. 8, the ring intersects with the straight-lined part when situated on a blank line between the gear blank's top line and the tool's fillet upper boundary. The points of the rings are positioned on the blank line with two endpoints: $p1(xp1,yp1)$ and $p2(xp2,yp2)$⁠. The length ratio t of each ring point on the line can be computed by Eq. (5). This ratio helps in determining the precise location of the intersection point on the line segment.

The intersection points in part II are determined based on the blank lines. Initially, several curves on the fillet surface are established by substituting a specified set of variables, denoted as u, into Eq. (4). These u values are generated by subdividing the interval of the fillet curve variable u. The number of u values determines the resolution of the tooth root simulation. Subsequently, these curves are projected onto the $x−y$ plane as described in Eq. (7). Finally, the intersection points are determined using Eq. (8). Figure 9 illustrates the intersection points on each section over the gear toe to heel at the specific cutting position. Solid points denote intersections on the straight edge on the left side, while hollow points signify intersections on the fillet edge.

The cutting points, arranged from the gear toe to the heel, construct the tooth surfaces. These cutting points are determined through the intersection points. Figure 10 illustrates the process of determining these cutting points. The cutting points on each line distributed on the gear blank must be obtained individually. They are determined separately for the convex and concave flanks. There are two modes for face milling: plunging and generating processes. In the plunging process, the tool penetrates from the tooth tip to the root through control of the cutting depth s (see Fig. 10(a)). The new cutting points replicate the intersection points at each cutting position.

To determine the cutting points for the second scenario, the bounding box method is used to determine whether a point is inside or outside a polygon. In Fig. 10(c), the cutting points from the previous step and the intersection points from this step create bounding boxes for Polygon I and II, respectively. If the intersection points lie outside Polygon I and the cutting points lie outside Polygon II, they are designated as the cutting points.

Those determined cutting points are used to construct triangle meshes, which represent the surface geometry of the 3D gear. As depicted in Fig. 11, the surfaces are individually delineated for the tooth surface, toe, heel, top, upper plane, and lower plane. Increasing the number of cutting points will improve the display resolution of the triangular mesh surfaces.

A face-milled hypoid gear pair is used as an example to illustrate cutting simulation for both the generating and plunging methods. The pair consists of a pinion with 13 teeth and a ring gear with 41 teeth. The fundamental design parameters are outlined in Table 1. The manufacturing parameters, including the tool, workpiece, and machine settings, are computed from Gleason's calculation table [5]. The results are organized in Tables 1 and 2. In this scenario, a helical motion is employed to adjust the tool cutting depth during generating, thereby modifying the pinion's tooth surface. The length unit used in the tables is millimeters.

The developed cutting simulation method is broadly applicable, encompassing various processes such as plunging, single-roll, and double-roll generating methods. Table 3 depicts the CNC machine's nonlinear coordinates for cutting the pinion and ring gear, functions of s and $ϕc$⁠, as Eqs. (14) and (15). An example demonstrates the machining of a left-hand pinion using a single-roll generating process. Figure 13 illustrates the simulation of intersection points on the tool surface and cutting points on the tooth surface, alongside the pinion surface geometry represented by triangle meshes. This cutting method is divided into two stages: rough plunging (P) and finish generating (G). During rough plunging, the tool advances inward from a cutting depth of 4.688 mm to 0.500 mm at the toe position (ϕ_c = 15.463 deg), then transitions to the finish position (s = 0.000 mm) and proceeds to cut the pinion from the toe (ϕ_c = 15.463 deg) to the heel (ϕ_c = −15.111 deg), as illustrated in Fig. 14. This sequential process repeats for each tooth, advancing to the next tooth for cutting until all teeth are processed.

In the alternate example, the ring gear is manufactured using the plunging method, where the tool enters from the middle of the tooth at the generating angle of $ϕc=0.000deg$⁠. Figure 15 illustrates the cutting simulation of its first tooth, with the cutting depth gradually decreasing from 5.990 mm to 0.000 mm.

The topographic errors of the simulated tooth surface were evaluated by comparing with the theoretical one. Figure 16 illustrates their results for both the pinion and ring gear. The maximum surface errors measure +4.4 μm and +1.3 μm, while the tooth thickness errors are −0.09 μm and +0.05 μm, respectively. The simulation errors indicate minimal discrepancies, confirming the accuracy of the mathematical model.

Figure 17 illustrates the material removal while producing a pinion using solidworks, covering the rough plunging and finish generating processes. The corresponding volumes, obtained from both the cutting simulation and solidworks, are presented in Table 4 and visually represented in Fig. 18. These visualizations offer detailed information on the volumes of material removed at each of the 40 cutting positions. A comparison between the simulation and solidworks outputs demonstrates a close correspondence, with differences amounting to less than 1% after the second cutting position. This high level of agreement confirms the simulation's reliability in accurately modeling the material removal process.

To clearly explain how to determine the MRR, Table 5 offers detailed data for the pinion during the generating process. The five cutting coordinates are computed from Table 3 by utilizing the specified cutting position (⁠$s$ and $ϕc$⁠). Subsequently, the MRR can be calculated by considering the distance between two consecutive cutting positions and their respective time consumption. Figure 19 illustrates the pinion's MRR during the rough plunging and finish generating processes. Analysis of the figure reveals that the MRR is relatively small at the initial position during rough plunging machining and at the beginning and end positions during finish generating machining. Consequently, accelerating the cutting speed at the lower MRR can effectively enhance processing efficiency.

Figure 20 depicts the material removal while producing a ring gear using solidworks, encompassing the plunging process. The corresponding volumes, extracted from the cutting simulation and solidworks, are tabulated in Table 6 and graphically represented in Fig. 21(a). Their differences are less than 1% after the second cutting position. Figure 21(b) illustrates the ring gear's MRR during the plunging operation. The figure reveals that MRR is relatively small at the initial position, suggesting that its feed rate can be increased.

Cutting experiments are conducted using our developed five-axis CNC bevel-gear cutting machine. Solid cutters made of HSS M2 material with TiN coating were utilized for the wet cutting (cooling lubricant: Daphne Super Multi Oil #2) of both the pinion and ring gear, composed of S50C material with a hardness of 200HB. The tool cutting speed was set at 50 m/min (140 rpm). Figure 22 illustrates the production process for both gears. The tooth surface errors of this gear pair were subsequently examined by a Klingelnberg P40 gear measuring center. The topographic deviations are depicted in Fig. 23, indicating maximum flank errors of +6.2 μm and +4.4 μm for the pinion and ring gear, respectively. Additionally, tooth thickness errors were found to be −19 μm for the pinion and +1 μm for the ring gear.

The experimental machine is equipped with the Siemens 840D sl controller, which allows the torques and positions of the axes to be monitored and acquired using the controller's Diagnostics/Trace function. The pinion and ring gear underwent rough and finish processes, with the pinion undergoing rough plunge and finish generating processes, while the ring gear underwent rough and finish plunging processes. The MRRs and the tool torques of their rough processes were recorded in detail, as shown in Figs. 24 and 25, respectively. In these figures, tool torque positively correlated to the MRR. Therefore, the tool torque can be predicted based on the calculated MRR. Two cutting experiments were conducted for each gear (pinion and gear). The initial MRR was determined from the feed rates provided by the CNC machine, while the improved MRR was derived from the target tool torque. The MRR is increased to enhance the cutting feed rate for both the pinion and ring gear, as shown in Figs. 24(b) and 25(b). Consequently, the machining times for the pinion and ring gear were reduced by 44% and 18%, respectively. The total machining times for the pinion and ring gear were 3.5 min and 4 min, respectively. These experiments demonstrate that MRRs can be effectively utilized to optimize the processing efficiency of CNC bevel gear-cutting machines.

The paper introduces a novel ring-dexel-based model for the face milling simulation of bevel gears, focusing on calculating the MRR. This model provides the flexibility to adjust cutting depth and generating angle, allowing for the simulation of various face-milling processes, such as plunging and single-roll generating processes. Using triangle meshes, the simulation accurately constructs tooth surfaces and facilitates the calculation of irregular material removal during different machining operations. The MRR for producing the pinion and ring gear is determined based on the volume of material removed and the cutting feed rate. This relationship enables the optimization of cutting feed rates according to predefined MRR criteria. Cutting experiments were conducted. Machining times for the pinion and ring gear are significantly reduced by 44% and 18%. Additionally, the model is feasible for integrating cutting simulation into human–machine interface software for CNC bevel gear-cutting machines. This integration can enhance the overall efficiency of bevel gear production.

The authors thank the National Science and Technology Council of the Republic of China (ROC) for its financial support. This work was partially conducted under Contract No. MSTC 111-2221-E-011-098. Special thanks to Dr. Bor-Tyng Sheen and Mr. Jing Guo for their valuable efforts in revising the paper.

There are no conflicts of interest.

No data, models, or code were generated or used for this paper.

$s$ =

cutting position parameter 1: cutting depth

$t$ =

length ratio

$r1$ =

locus of the cutting tool in the coordinate system $S1$ on the universal machine

$rd$ =

position vector of the gear blank ring in the coordinate system $Sd$

$rt$ =

position vector of the cutting tool in the coordinate system $St$

$Mij$ =

homogeneous transformation matrix from coordinate system $Sj$ to coordinate system $Si$

$cx,cy,cz$ =

translating coordinates for the CNC bevel gear cutting machine $i,j,θc,SR,Em$

$u$⁠, $β$ =

parameters of the tool surface

$ΔA,ΔB,γm,Ra$ =

machine settings of the universal cradle-type bevel-gear cutting machine

$ϕ1$ =

rotation angle of the work gear

$ϕc$ =

cutting position parameter 2: generating angle

$ψa$⁠, $ψb$⁠, $ψc$ =

rotation angles for the CNC bevel gear cutting machine