Mathematical modeling on a novel manufacturing method for roller-gear cams using a whirl-machining process

The whirl-machining process is a precise, efficient, and promising machine process for manufacturing workpieces over the milling process. At the same time, the roller-gear cam has advantages over the other cam-follower system. However, the whirl-machining process has not been applied to manufacture the roller-gear cam. Therefore, this paper proposes a novel method for roller-gear cam manufacturing using a whirl-machining process. Mathematical modeling is described for generating the roller-gear cam, roller, and whirl-milling tool surface. A novel CNC machine and its coordinate system are proposed. Cutting simulations for obtaining the surface topologies and normal deviations are conducted. Globoidal cam with different cylindrical rollers, globoidal cam with a conical roller, and cylindrical cam with a conical roller are taken as case studies, and results are discussed. A virtual cutting simulation was conducted using VERICUT. Some machining examples are shown to verify the benefits of the proposed method in roller-gear cam manufacturing.


Introduction
The whirl-machining process, developed by Burgsmuller (a German company) [1], has several advantages and benefits over the milling process, such as there is no need for coolant (dry machining), less cutting force, final grinding can be eliminated, and nine-times higher production rate [1][2][3][4]. Compared to milling, whirl-machining has a larger cutting area or tool contact path. A comparison of milling and whirl-machining processes can be seen in Fig. 1. The whirl-machining tool is an internal machining tool in which the cutter inserts are radially mounted on the inside of a whirl-machining ring. The lead angle determines the angle of inclination between the tool with the workpiece axis, and the depth of cut is determined by the eccentricity of the workpiece with the whirling ring [5]. A whirl-machining tool removes material from the workpiece and feeds it along the axis to produce a helical form [2]. The whirl-machining tools have been used in various mechanical industries, including aviation, automobiles, machine tools, motion control, implant parts, and medical components [1,5,6]. They have primarily been used to produce cylindrical worms [4], ball screws [1,7,8], lead screws [9,10], and micro screws [11]. On the other hand, whirl-machining tools have not yet been developed and studied for manufacturing roller-gear cams.
As an intermittent-motion mechanism consisting of a globoidal cam and a turret with rollers, the roller-gear cam (RGC) is considered when positioning precision is required for a speed reducer [12][13][14]. A roller-gear cam drive consists of a roller-gear cam and several rollers, as shown in Fig. 2. The rollers are either cylindrical or conical. RGC is widely utilized in many machines and automation-manufacturing devices due to its advantages, such as higher load capacity, lower noise, lower vibration, higher reliability, less space, more accurate, and more compact structure compared to other cam-follower systems and intermittentmotion mechanisms [12,[15][16][17][18].
Globoidal cam geometry is an essential and significant concern in designing and manufacturing an RGC, as it is one of the most complicated cams [12,13,16,17]. Litvin and Fuentes [19] proposed the method for generating the globoidal shape surface. Yan and Chen [16,20] utilized cylindrical and hyperboloid rollers to investigate the geometry design and machining of RGC. Van and Pokorny [17] utilized a cylindrical roller and indexing turret follower for modeling concave globoidal cam. A systematic and complete method for analyzing and simulating the globoidal cam mechanism was presented by Zhang et al. [21]. RGC contact bearing has been established by Lo et al. [18]. Zongyu et al. [22], by considering the clearance, described the dynamics of the RGC system. However, they have not proposed a mathematical modeling method for manufacturing the globoidal cam using the whirl-machining process. Our purpose here is to establish novel mathematical modeling for proposing the tool for manufacturing the globoidal cam. Mathematical modeling methods of tools have been studied in the extant literature, such as milling tool [23][24][25][26], turning tool [27], skiving tool [28], worm-shaped tool [29], barrel-shaped tool [30], form-cutting tool [31], and others [32][33][34][35][36]. A mathematical modeling method of the whirl-machining tool has recently been utilized to simulate double enveloping worms manufacturing [37].
This paper proposes the technique of manufacturing globoidal cams with a whirl-machining tool. A workpiece with a globoidal shape is suitable with the whirl-machining method. The mathematical modeling is conducted based on the theory of gearing. A general mathematical model is investigated for generating the globoidal cam and the roller surfaces. The tool profile is generated based on the lead angle of the roller-gear cam. Computer numerically controlled (CNC) machining is the dominant subtractive manufacturing technology [38]. Therefore, cutting simulation of the whirl-machining process based on a CNC machine is conducted.
Furthermore, the contact lines on the roller-gear cam surface and the tool are achieved. Normal deviations are calculated based on comparing the technical and generated profile of the roller-gear cam. The proposed method can also be utilized for manufacturing RGC of globoidal cam with different cylindrical rollers, globoidal cam with conical rollers, and cylindrical cam with conical rollers. Numerical examples are considered for determining the accuracy and the practicability of the proposed method. A virtual cutting simulation was conducted. where a is half of the profile width; u is profile parameter; is the taper angle; b is tool pitch radius; r is fillet radius; c is the distance between the profile root and the origin point; B is a position vector of a single roller surface; is the circumferential parameter of the roller surface; and A is a position vector of the profile of roller. The axial section of the roller profile can be seen in Fig. 3. When the tool is cylindrical, the value of the taper angle is equal to zero. On the other hand, the taper angle should be more than zero for the conical tool. The roller profile is in the xy-section, while the swivel angle related to the feeding movement of the proposed machine rotates about the y-axis. Therefore, in Eq. (1), the profile is necessary to rotate into an xz-section to fit with the structure and mechanism of the proposed CNC machine.

Generation of the profile and surface of the whirl-machining tool
The whirl-machining process is commonly performed to machine the workpiece in the combination of the rotation of the tool and the workpiece. In the whirl-machining, the lead of the workpiece is controlled by the inclination of the whirlmachining tool concerning the workpiece axis and the feed rate of the whirl-machining tool. The roller-gear cam is identical to the shape of the globoidal worm. Based on the theory of gearing by Litvin [19], the machining method of the roller-gear cam can be seen in Fig. 4, which is identical to the generation of globoidal worms. The whirl-machining tool is tilted to generate the lead angle of the thread and is placed eccentric to the workpiece. Therefore, the whirl-machining tool profile is provided from the roller profile, which is projected with an angle the same as the lead angle of the roller-gear cam. The relative position of the roller profile and whirl-machining tool profile can be seen in Fig. 5. The equations of the whirling tool profile can be provided by Eq. (3). The position vector and the unit normal vector for the whirling tool surface is developed using Eqs. (4) and (5), respectively. The whirl-machining tool surface can be seen in Fig. 6. The 3D modeling design of the tool can be seen in Fig. 7 by considering relief and rake angles: ; is the circumferential parameter of the whirl-machining tool surface; D is a position vector of the whirl-machining tool surface; and n D is the unit normal vector of the whirl-machining tool surface. The whirl-machining tool profile is in the xy-section. At the same time, the swivel angle related to the feeding movement of the proposed machine rotates about the y-axis. Therefore, the profile is necessary to rotate into an xz-section to fit with the structure and mechanism of the proposed CNC machine.

Generation of the roller-gear cam surface
A method to generate the roller-gear cam surface has been presented in previous studies [12,13,21]. However, the shape of the roller-gear cam in previous studies differs from this paper. Therefore, this paper utilizes the roller profile to generate the cutting tool for generating the roller-gear cam surface. The coordinate system of the cutting tool should be defined before deriving the surface equation of the roller-gear cam. The roller-gear cam surface can be described, based on rigid body transformations, as the swept surfaces of the cutting tool. In terms of studying the motion of a rigid body associated with a different coordinate system, the coordinate transformation is suitable for studying it. The surface equation for the cutting tool is expressed by: where E is a position vector of the cutting tool surface; is the distance between the rollergear cam and the cutting tool; is the setting angle of the cutting tool; is the surface parameter of the cutting tool; and E is a unit normal vector of the cutting tool.
According to the theory of gearing [19], the roller-gear cam surface equation can be achieved by considering the locus of the cutting tool surface [13]. The equation can be represented in the roller-gear cam coordinate system and the meshing equation of the roller-gear cam with the cutting tool.
The relative motion between the cutting tool surface and the roller-gear cam can be seen in Fig. 8. Coordinate systems S E (x E , y E , z E ) and S F (x F , y F , z F ) correspond to the cutting tool surface and the roller-gear cam, respectively. Therefore, the coordinate system S 1 (x 1 , y 1 , z 1 ) and S 2 x 2 , y 2 , z 2 are connected rigidly to the cutting tool surface and the roller-gear cam, respectively.
The coordinate system S 1 x 1 , y 1 , z 1 of the cutting tool rotates about the y 1 axis Through an angle . The coordinate system S 2 x 2 , y 2 , z 2 of the roller-gear cam rotates about the z 2 axis through an angle . The relationship between the two angles is = ∕z , where z is the number of the roller.
x i cos 2 sin + x i sin 2 sin + x i z i cos sin The position vector for the trajectory of the cutting tool surface, represented in the coordinate system n Ex cos cos + n Ez cos sin + n Ey sin n Ey cos − n Ex cos sin − n Ez sin sin surface locus; f F is the meshing equation; is the rotation angle of the coordinate system S F ; is the rotation angle of the coordinate system S E ; and C 2 is the center distance between the coordinate system S F and the coordinate system S E , which equals the total distance between the roller and the roller-gear cam at the throat. Furthermore, by simultaneously considering Eqs. (8) and (10), the mathematical model of the surface of the roller-gear cam can be obtained.
The roller-gear cam surface and roller surface can be seen in Fig. 9, where the clearance and the backlash are not considered.

Coordinate system and machine setting
The whirl-machining tool, powered by a motor and rotates at high speed, performs the cutting motion in the whirl-machining process [2]. Four motions occur in the process: the rotation of the workpiece, the rotation of the whirl-machining tool, the feeding motion of the whirl-machining tool, and the radial translation of the tools [8]. The coordinate system and motion diagram based on the CNC machine for this study can be seen in Fig. 10.
The machine consists of three sliding axes (X, Y, and Z) and four rotation axes (A, B, C, and D). The x-axis is the axis of axial sliding of the whirl-machining tool; the y-axis is the axis of tangential sliding of the whirl-machining tool; and the z-axis is the axis of radial sliding of the whirl-machining tool. A is the feeding spindle axis; B is the tool rotation axis; C is the tool spindle axis; and D is the workpiece spindle axis. A, C, and D are motions to generate the roller-gear cam surface; while X, Y, Z, and B control the whirl-machining tool position.
The coordinate system in Fig. 10 is utilized for generating the coordinate transformation matrix GD of the roller-gear cam coordinate system S G , which is transformed from a whirlmachining tool coordinate system S D on a CNC machine as follows:  where GD is the transformation matrix from the coordinate system of the whirl-machining tool S D to the coordinate system of the roller-gear cam S G based on the proposed CNC machine; is the rotation angle of the roller-gear cam spindle (D); is an angle of the rotation feeding (A); is the whirl-machining tool setting angle about the x-axis (B); C 3 is the total distance between feeding rotation center and whirl-machining tool; C x is the sliding feed of the x-axis/ horizontal slide (1); C y is the sliding feed of the y-axis/vertical slide (Y); and C z is the sliding feed of the z-axis/whirling arm slide (Z). The mechanical setup of each axis can be expressed as follows: The coordinate system in Fig. 10 is a machining coordinate system of the whirl-machining tool, which can be utilized to conduct a cutting simulation of a roller-gear cam with the whirlmachining tool in the proposed multi-axis CNC machine. The equation of the whirl-machining tool trajectory and its unit normal vector can be obtained as follows: where G is the position vector of whirl tool machining locus; G is the unit normal vector of whirl tool locus; and GD is the 3 × 3 upper left submatrix of GD . Contact point can be obtained when the normal vectors are perpendicular to the direction of the relative velocity from two curves of related motions. Furthermore, the meshing equation can be obtained as follows: , h 1 = −C x cos + C 3 cos cos + C y sin + x D cos cos ,

Normal deviation in the roller-gear cam surface
The proposed mathematical model is utilized to calculate the points on the roller-gear cam surface, which is generated by the whirl-machining tool. The original theoretical roller-gear cam surface generates a datum surface for determining the deviation and cutting precision in the normal direction. Figure 11 shows that the roller-gear cam surface is divided into seven axial profile sections. The sections are based on the input value of . The range input value is from 3π to − 3π. The cutting simulation of the whirl-machining tool provides cutting points or generated points on these sections. A single profile on one section consists of 160 points. The normal deviation for each point is calculated as: where F is the normal deviation, and F is the unit normal vector of the roller-gear cam surface. A zero normal deviation on the datum surface can be seen in Fig. 11 at the point V i , where F is the vector of its position, and F is the vector of its unit normal. A point W i is a point that is not on the datum surface; this point is labeled as the generated point. From the point W i to the V i , the minimum distance must be in the same direction as the normal vector. A surface topology coordinate was proposed to compare the distance between the datum and generated roller-gear cam. The distances are labeled as normal deviation. Coordinate of the surface topology was developed into Points, c , z c .

Numerical examples
Due to its high productivity and quality characteristics, the whirl-machining process is widely used in the machining of helical shape workpieces, where the cutting motion in the process is performed by the tool ring, which is powered by a motor and rotated at high speed [1,2]. Compared to turning, whirling has advantages in surface finish, tool wear, and chip control because the material is removed in a small volume at a high cutting speed [11].
The numerical examples validate the proposed mathematical model for whirling a roller-gear cam. Based on a theoretical point of view, the generated roller-gear cam surface should coincide with the datum of the roller-gear cam surface. However, errors may occur due to the curve fitting and numerical approaches [29]. In this case, a whirl-machining tool was utilized to machine a roller-gear cam of RGC drive to observe if the proposed mathematical model could attain the requisite numerical accuracy. Design parameters in the examples can be seen in Table 1. The 3D coordinate system of the cutting line is listed in Table 2 (16 points from 160 points). The 2D coordinate system of the generated rollergear cam profile at the axial section is listed in Table 3 (16 points from 160 points).
As shown in Fig. 12, the datum roller-gear cam and whirlmachining tool are assembled and simulated. The whirlmachining tool surface is located at the relative position of the roller-gear cam, and it fits the roller-gear cam surface. The proposed mathematical model calculates a single cutting line and locus of cutting lines (200). The cutting line is projected as a collection of instant contact points between the surfaces of the roller-gear cam and the whirl-machining tool, although it may more accurately reflect the machining result on the machined roller-gear cam surface. The cutting simulation result shows that the generated roller-gear cam profile, calculated by the simulation, closely fits along the datum roller-gear cam surface.

Normal deviations for globoidal roller-gear cam with cylindrical roller
The type of roller-gear cam is globoidal, while the roller is cylindrical. The roller-gear cam has a lead angle of 10° and a length of 44 mm. The tool has a pitch radius of 40 mm, and a root length of 9 and 5 mm. The distance between the tool root and the origin point is 37 mm. The center distance between the roller-gear cam and the tool is 100 mm, the center distance between the roller-gear cam and the roller is 100, and the center distance between the tool and the rotational feeding center is 120 mm. The roller has a number of 50 pieces. The pitch radius of the whirl-machining tool is 40 mm. The normal deviations in Fig. 13 were obtained by comparing the generated roller-gear cam profile and the datum roller-gear cam profile. Figures 14 and 15 show the surface topology of two different roller-gear cam surfaces. In this case, the simulations are conducted with two different lengths of tool roots. In order to further understand the precision of the machined roller-gear cam surface, each sectional profile of the machined roller-gear cam is compared with the datum roller-gear cam surface in the normal deviation; thus, the surface topology can be achieved as shown in Figs. 14 and 15. The figures show the maximal deviations for two different roller-gear cams with the value of 2.1 × 10 −11 µm and − 2.3 × 10 −11 µm, respectively. As a result, the proposed mathematical model is validated, and the whirl-machining process of the roller-gear cam is proved to be practical for different roller-gear cam geometries in this example.

Globoidal roller-gear cam with conical roller and conical roller with cylindrical roller-gear cam
The machinery equipment industry has widely utilized the whirlmachining process to produce helical shape workpieces from difficult-to-machine material for transmissions [3]. The goal of whirling process simulation is to have better knowledge of the surface generation process and roller-gear cam geometry. In order to spread the accuracy of the proposed model, in this   section, the type of roller-gear cam is globoidal with the conical roller shape utilized. The conical shape of the roller can be achieved by giving a value in the taper angle equation. The simulation result for a globoidal roller-gear cam with a conical roller can be seen in Fig. 16. The maximum normal deviation that can be achieved is 2.3 × 10 −11 µm (Fig. 17). The same as the cylindrical roller, the normal deviations of the conical roller approaches zero. The result from the cutting simulation shows that RGC with globoidal roller-gear cam and conical roller is able to be manufactured using the proposed method.
Whirl-machining process is typically utilized to manufacture workpieces by combining tool and workpiece rotations. There are four types of rotational and translational motions in the process: workpiece rotation motion, tool rotation motion, tool axial feed motion, and tool radial translation motion [5,8]. In addition, in this section, the type of roller-gear cam is cylindrical, while the roller is conical. Cylindrical roller-gear cam mechanisms are widely used in machinery applications such as rotary indexing tables, packing machines, knitting machines, and elevators, which have smaller sizes and higher driving torque than other cam mechanisms. The surface topology of the cylindrical roller-gear cam from the cutting simulation can be seen in Fig. 18. The maximum value of normal deviation is − 1.4 × 10 −11 µm (Fig. 19). The result shows that the whirlmilling process is able to manufacture cylindrical roller-gear cam precisely.

The conjugation of the roller and the roller-gear cam
In this section, generated roller profile was obtained from the trajectory of the roller-gear cam (Fig. 20). The red lines are the roller profiles and the blue lines are the locus of the roller-gear cam. Compared with conventional workpiece machining processes (grinding, turning, and milling), whirl-machining processes produce a high material removal rate and minimize cutting forces due to their smooth and tangential cutting motions. At the same time, it is regarded as an advanced cutting process that has gradually gained popularity [9,10]. The basis for the development of differential geometry is a theory of surface. In the gear transmission field, the theory is known as the geometry theory of conjugate surfaces. Conjugation of two surfaces is explained as surfaces that keep continuous and tangent contact. The total power and motion qualities of a gear drive are substantially influenced by the operational performance of conjugate surfaces. Therefore, the conjugation between both rollergear cam and roller is achieved.

Virtual cutting verification
The virtual cutting simulation was conducted to avoid machine collisions, reduce execution times [39], and reduce manufacturing costs. Simulation of the whirl-machining process on the roller-gear cam using the VERICUT (by CGTECH) is conducted. The whirl-machining tool profile, generated from the proposed model using mathematical modeling from numerical example 2, is imported into CAD software to construct a 3D tool model. The roller-gear cam and the proposed machine structure are generated into a CAD model. Furthermore, the tool, the roller-gear cam, and the machine structure are imported into VERICUT. The proposed machine structure, mechanical  Locus of the roller-gear cam Generated profile of roller setup settings, and the coordinate system in Fig. 10 are utilized to create the G-code. The virtual cutting process can be seen in Fig. 21. The results can be seen in Fig. 22. Mathematical modeling ( Fig. 22(a)) and virtual cutting simulation ( Fig. 22(b)) results of the roller-gear cam are compared. The roller-gear cam shape of both results presents an identical result, which is globoidal. The globoidal shape occurs in order to the motion of the swivel angle in the proposed machine. The results show that the mathematical model and virtual machining cutting simulations confirm the kinematics relationship on tool-workpiece interaction. Consequently, the tool design is fully functional. Furthermore, the result of the virtual cutting simulation can be utilized for validating comparison [40] with experimental data.

Conclusion
In this paper, the main goal of the current study was to propose the technique of manufacturing a roller-gear cam with a whirlmachining tool. Mathematical expressions for roller-gear cam surface, roller surface, and whirl-machining tool are derived. The mathematical model is based on the theory of gearing, differential geometry, and coordinate transformation matrixes. Basic equations tool coordinates and direction for obtaining the mechanical setup parameter are derived. Several numerical examples are conducted to verify the model. The findings indicate that the results obtained from the cutting simulation are compared with theoretical data. The study has shown that the normal deviations approach zero. A virtual cutting simulation was conducted and showed a good result. The model is also used to study the conjugation on roller-gear cam and rollers. The study contributes to our understanding that the proposed method can be applied effectively for manufacturing roller-gear cam by utilizing the whirl-machining process. The absence of experimental investigation and cutting force limited this study. Therefore, further machining experimentation, meshing test investigation, and analyzing the cutting forces are strongly recommended.
Author contribution Moeso Andrianto constructed the research design, accomplished the analytical simulation, conducted the virtual simulation, and composed the manuscript. Yu-Ren Wu earned the funding and directed the research implementation, whereas Achmad Arifin supervised the analytical and virtual simulation. All the authors worked concurrently to proofread and structure the submission.
Funding This research was supported by the Ministry of Science and Technology in Taiwan, project number MOST 109-2221-E-008-004-MY2.

Machined roller-gear cam
The whirl-machining tool