A Prediction Model of Cutting Force about Ball End Milling for Sculptured Surface

Mou, Wenping; Zhu, Shaowei; Zhu, Menghao; Han, Lei; Jiang, Lei

doi:https://doi.org/10.1155/2020/1389718

Mathematical Problems in Engineering

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Abstract Introduction Conclusions Notation Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 1389718 | https://doi.org/10.1155/2020/1389718

A Prediction Model of Cutting Force about Ball End Milling for Sculptured Surface

Wenping Mou,^1,2Shaowei Zhu,²Menghao Zhu,³Lei Han,³and Lei Jiang³

Academic Editor: Xiaoliang Jin

Received26 Nov 2019

Revised01 Feb 2020

Accepted06 Feb 2020

Published11 Mar 2020

Abstract

Cutting force prediction is very important to optimize machining parameters and monitor machining state. In order to predict cutting force of sculptured surface machining with ball end mill accurately, tool posture, cutting edge, contact state between cutter, and workpiece are studied. Firstly, an instantaneous motion model of ball end mill for sculptured surface is established. The instantaneous milling coordinate system and instantaneous tool coordinate system are defined to describe the position and orientation of tool, and the transformation matrix between coordinate systems is derived. Secondly, by solving three boundaries around engagement of cutter and workpiece, a cutter-workpiece engagement model related to tool posture, milling parameters, and tool path is established. It has good adaptability to the variable tool axis relative to the machining surface. Finally, an algorithm of thickness about an instantaneous undeformed chip is researched, and a prediction model of cutting force is realized with microelement cutting theory. Also, the model is suitable for sculptured surface machining with arbitrary tool posture and feed direction. The accuracy of the proposed prediction model was verified by a series of experiments.

1. Introduction

Ball end mill is an important milling tool. It has good adaptability to surface machining due to the normal orientation of the spherical contour surface pointing to full space. Using the ball end mill for sculpted surface in NC machining is simpler than other tools. Therefore, ball end mill is widely used in the machining of aerospace parts, automotive parts, mold parts, and so on. As an important physical parameter, cutting force directly or indirectly affects wear and deformation of tool, machining efficiency, etc. It is an effective indicator for monitoring the machining process [1]. The evident feature of ball end milling for sculptured surface is that the contact condition between the tool and workpiece varies along the tool path. It changes cutter-workpiece engagement (CWE), which defines the area where cutter and workpiece interact to generate cutting force.

The researches for CWE under different cutting conditions are mainly divided into three types: a solid method based on Boolean operation, Z-map method based on discrete elements, and boundary method based on analytical and numerical calculation. The solid method determines the intersection of the cutter and workpiece by Boolean operation. Larue et al. judged intersection point in flank milling by Boolean operation, and cutting angle was modeled in the machining process by the tool site function [2]. Ju et al. proposed a method of discrete boundary representation based on Boolean operation, which was applied to calculate CWE of each blade in ball end milling [3]. Yang proposed a method to solve CWE based on the ACIS model [4]. Gong and Feng established a triangular grid model of cutter and workpiece, and CWE was solved by Boolean operation [5]. Li and Zhu extracted boundary of the contact region based on the intersection of the cutting edge and workpiece, and a general modeling method of CWE was proposed [6]. The solid method can solve the contact area of the cutter and workpiece with high precision. However, it is necessary to update the entities of the cutter and workpiece to calculate CWE iteratively, which results in low efficiency.

The Z-Map method determines the intersection of the cutter and workpiece by projecting the cutter and workpiece to plane and discretizing it into a set of points by a given direction (usually Z direction). The Z-coordinates of the ray and intersection point between the cutter and workpiece were compared. Kim and Lazoglu et al. calculated the contact area by comparing the Z-Map values during machining [7, 8]. Dongming et al. used the Z-Map method to verify engage section of cutting edge in machining, and a cutting force model of five-axis machining with ball end mill was established [9]. Wei et al. used logical array to improve the Z-Map method for calculating CWE in sculptured surface machining [10]. As a discrete method, it is contradictory between accuracy and efficiency, which needs to be weighed in different applications.

Based on the analysis of geometric relationship between the cutter and workpiece, the boundary method was proposed to describe CWE by mathematical expression. Gupta et al. proposed an analytical algorithm for CWE of 2.5D milling (the vector of tool axis is fixed.) [11]. Ozturk et al. calculated boundary of CWE for 2.5D milling by ball end mill analytically, and the method was proved to be more efficient than the Z-Map method for CWE [12, 13]. But, the abovementioned methods are difficult to apply to 3D milling (the vector of tool axis is varied during machining). Sun and Guo used the Newton iteration method to calculate the boundary of CWE between the cutter and workpiece [14]. Kiswanto et al. established an analytical semifinished tangent region model based on surface profile [15]. Wei et al. discretized sculptured surface into a series of microelement planes. The microelement contact region of 2.5D milling by ball end mill was constructed, and the contact region of 3D milling was obtained by rotation transformation [16, 17]. However, the abovementioned methods are not suitable for sculptured surface machining. Error could be produced when the curvature of surface or cutting depth are large. Zhu et al. proposed a 3D machining contact region model for fillet milling [18]. Three spatial curves are applied to describe the boundary of CWE, but it does not consider the condition about the curved tool path.

By comparing the abovementioned studies, the solid method has the highest accuracy, but its calculation efficiency is low. The Z-Map method has a contradiction between accuracy and efficiency. In actual applications, the size of divided mesh needs to be adjusted according to the priority of accuracy and efficiency, and this adjustment process is usually very tedious. The existing boundary method is well applied in three-axis curved surface machining, but it has limitation in variable axis machining with variable cutting conditions.

CWE is the prerequisite for predicting cutting force. Furthermore, how to calculate the cutting force from CWE is another critical part. The research methods of cutting force modeling are divided into three categories: the theoretical analytic method, finite element (FE) method, and mechanical force method. The theoretical analysis method is based on metal cutting theory, involving multifield coupling effects such as force, heat, strain, and so on. According to the orthogonal cutting model proposed by Merchant, cutting deformation is occurred on a single shear plane only, and shear angle is calculated based on the principle of minimum energy [19]. Lee and Shaffer proposed a slip line theory considering the workpiece material as ideal plastomer, and a cutting force model was established [20]. Oxley et al. researched flow stress of material during cutting and effect of work hardening to complete the cutting force model [21]. Moufki et al. proposed an algorithm for flow stress of bevel cutting based on thermal coupling properties of material [22]. The theoretical model of cutting force is very complicated, and it is necessary to consider many factors, such as material property, cutting temperature, and cutting deformation. It is related to the factors about elastic mechanics, thermodynamics, and tribology. In order to reduce the difficulty of modeling, many simplifications and assumptions are researched, which leads to lower accuracy and smaller application range.

The FE method simulates the distribution of each physical field and deformation process, which could deal with the complex physics coupling effect effectively. At present, some commercial FE softwares (such as ABQUS, ANSYS, and AdvantEdge) have been applied to 2D and 3D milling. However, the FE model is very complicated, and calculation load is huge.

Mechanical method applies cutting coefficient to represent different geometry and physical parameters of cutting. The model is simple, and its adaptability is strong. Lamikiz et al. expressed shear force coefficients as polynomial, and a cutting coefficient identification model was proposed for ball end milling [23]. Lee and Altintaş transferred the parameters that were obtained from orthogonal cutting experiments to classical oblique cutting to predict cutting force of ball end milling [24]. Cao et al. proposed a cutting force model considering the inclination of tool axis [25]. Luo et al. established a cutting force model considering the influence of cutting edge at the apex, and the variation of CWE was analyzed with the feed path and contour of workpiece during machining [26]. Lin et al. calculated the cutting force coefficients of ball end mill based on average single-tooth cutting force [27]. However, the abovementioned research studies are mainly aimed at slab milling with constant cutting conditions.

In this paper, a new CWE model considering arbitrary feed direction of tool path is proposed and the cutting force model is established by the mechanical method. Firstly, the vector of the feed path and normal vector of the sculptured surface are calculated from cutter location points (CLP). The machining coordinate system (MCS) is established to describe tool posture. The contact condition between the cutter and workpiece is analyzed to establish the boundary of CWE. In addition, the engage section of cutting edge is determined by plane projection of CWE and cutting edge. Finally, based on the classical oblique cutting theory and mechanical method, a prediction model of cutting force about sculptured surface machining is established. The innovations include the following:(1)A modeling method for the CWE of sculptured surface milling is proposed, which considers the tool axis vector and feed direction(2)A projection dimensionality method is used to simplify the solving process of the instantaneous engage section between the cutting edge and the workpiece(3)The parameterized expression of the undeformed chip thickness is improved, and it could be applied to the variable axis machining for the sculptured surface

The rest of the paper is organized as follows: in Section 2, the motion model of the tool is established and the MCS is defined. The CWE model is established in Section 3. The blade section of instantaneous engaging with the workpiece is obtained in Section 4. In Section 5, the cutting force model is established. A series of experiments is presented in Section 6 to verify the model. Finally, some conclusions are drawn in Section 7.

2. Tool Motion State and Coordinate System

The CWE and cutting edge could be expressed in different coordinate systems, respectively, but the intersection between them needs to be calculated in a same coordinate system. Therefore, a universal coordinate system transformation method is essential. In this paper, the vectors of tool and tool path are used to define the corresponding coordinate systems at each CLP. Also, the transformation relation is deduced between them for sculptured surface machining.

2.1. The Definition of Tool Posture

The position and orientation of the tool are determined in MCS. As shown in Figure 1, define the serial number of tool position in NC program as i, the instantaneous tool position as P_L(t), the instantaneous tool contact point as P_C(t), and the instantaneous unit vector of tool axis as u_(t) in MCS at time t. P_L(t), P_C(t), and u_(t) could be obtained by the interpolation of adjacent two tool positions as follows [4].

Define the instantaneous tool position unit feed vector as v_(t) at time t in MCS, so

2.2. The Definition of MCS

Define MCS as O_W-X_WY_WZ_W, corresponding unit axis vector as i_W, j_W, and k_W. The instantaneous milling coordinate system (IMCS) and instantaneous tool coordinate system (ITCS) are established to describe the contact relationship between the ball end mill and workpiece for sculptured surface machining, as shown in Figure 2.

2.2.1. IMCS

Define IMCS as O_M-X_MY_MZ_M. The instantaneous tool position is set as the coordinate origin O_M. X_M axis is parallel to the instantaneous feed vector v_(t), and Z_M axis is parallel to the instantaneous normal vector n_(t) of the sculptured surface. Define the unit axis vector of IMCS as i_M, j_M, and k_M.

2.2.2. ITCS

Define ITCS as O_T-X_TY_TZ_T. Set the instantaneous tool position as coordinate origin O_T. Z_T axis is parallel to the instantaneous tool axis vector u_(t), and X_T axis is perpendicular to Z_M and Z_T. Define the unit axis vector of ITCS as i_T, j_T, and k_T.

2.3. The Transformations of Coordinate Systems

2.3.1. MCS to IMCS

The geometric relationship between IMCS and MCS is shown in Figure 3.

Define the angle between Z_M and Z_W as δ_WM, the vector of the intersecting line of coordinate planes X_MY_M and X_WY_W as s, the angle between s and X_W as γ_WM, and the angle between s and X_M as ε_WM.

Set the coordinates of instantaneous tool position in MCS as P_L(t)(x, y, z), and then IMCS could be obtained from MCS with transformation matrixes:where T_WM is the translation transfer matrix; R_Z(ε_WM) is the rotational transfer matrix around the Z-axis with angle ε_WM; R_X(δ_WM) is the rotational transfer matrix around X-axis with angle δ_WM; R_Z(γ_WM) is the rotational transfer matrix around Z-axis with angle γ_WM.

2.3.2. IMCS to ITCS

According to the geometric relationship between ITCS and IMCS, the angle δ_MT between Z_T and Z_M and the angle γ_MT between X_T and X_M could be obtained as follows:

The transformation matrix M_MT from IMCS to ITCS could be expressed as follows:where R_X(δ_MT) is the rotational transfer matrix around the X-axis with δ_MT; R_Z(γ_MT) is the rotational transfer matrix around the Z-axis with γ_MT.

3. The CWE Model of Ball End Milling

The schematic of CWE in sculptured milling is shown in Figure 4, which is surrounded by three boundaries: the swept profile between the current tool path sweep surface and cutter revolution surface, the intersection of the previous adjacent tool path sweep surface and current cutter revolution surface, and the intersection of the tool revolution surface and unmachined surface.

3.1. The Swept Profile

The current machining surface is formed by cutter revolve sweeping along the current tool path. The swept profile is the tangent line between the cutter revolution surface and current machining surface, which is perpendicular to the tool path direction, as shown in Figure 5.

In IMCS, the swept profile could be expressed as follows:where θ_M is the angle between the line connecting the origin of IMCS to any point on the swept profile and Y_M.

3.2. The Intersecting Line

As shown in Figure 6, the points, which are the intersection of the current cutter revolution surface and the swept profile on the previous adjacent tool path, form the intersecting line. Therefore, the intersecting line could be expressed by a series of discrete points.

In the current IMCS, the revolution surface of the cutter at the current CLP could be expressed as follows.

The swept profile in the previous adjacent tool path could be expressed as follows.

By MCS-IMCS transformation, the swept profile in the previous IMCS on the previous adjacent tool path could be transformed to the current IMCS.

The one-dimensional nonlinear equation for θ′_M could be obtained from equations (10), (11), and (12). The numerical method (the Newton–Raphson method) could be applied to solve it. Then, the coordinates of intersection points in the current IMCS could be calculated.

3.3. The Surface Intersecting Line

There are many methods about the sculptured surface machining, such as multilayers cutting and single-layer cutting. They result in two cases of the unmachined surface, one is the previous machined surface and the other is the surface of workpiece blank. For the former, the surface intersecting line is formed by current cutter revolve and the previous machined surface, which can be solved in the same way in Section 3.2. However, because the tool paths in two cutting layers may be different, it is difficult to be solved. For the latter, the uncertain workpiece blank makes it difficult to describe the unmachined surface. In order to improve the calculation efficiency, an offset plane is defined to substitute the two types of unmachined surface. As shown in Figure 7, the unmachined surface is simplified as the plane that offset a_p from the tangent plane at P_C(t).where φ_M is the angle between the line (it connects the Z_M-axis to any point on the surface intersecting line) and v_(t).

3.4. The Verification of CWE

In order to verify the accuracy of the proposed CWE model, an experiment was carried out. The parameters of experiment are shown in Table 1, and the workpiece geometry and tool path are shown in Figure 8. The guiding line is an arc (radius is 208 mm) located in the bottom plane of workpiece, and the workpiece entity is generated by the “scanning and mixing” instruction of PTC Creo. The tool path is the projection of the guiding line on the workpiece surface. The boundary of CWE is simulated by Matlab. Based on the Boolean operation function of PTC Creo, the solid model of CWE is established. The simulations of the two models are all performed on the same computer (Intel(R) Celeron(R) CPU G1840 @ 2.80 GHz, RAM 8 GB). 16 CLPs are selected equidistantly, and the shape and size of CWE obtained by two ways are almost the same.

Numerous literatures show that the solid model has very high accuracy. Therefore, the results of the solid model are used as the standard value to analyze the errors of the boundary model. In the solid model, the area of CWE can be extracted by the software directly. Also, in the boundary model, the area can be obtained by the Monte Carlo method. The errors of the boundary model at different CLPs are shown in Table 2, which are less than 3%. The former takes approximately 75 s at each CLP, and the latter takes only 0.25 s. The results show that the proposed boundary model has high computational efficiency and accuracy.

When solving the surface intersecting line of CWE, unmachined surface is simplified to a plane, which shows that the curvature of the workpiece surface has some influence on the accuracy of the boundary model. The accuracy is influenced by the curvature radius r of the workpiece surface and the radius R of ball end mill, as shown in Figure 9. When R/r < 0.12, the error is within 5%.

4. The Engage Section of Cutting Edge in CWE

In machining, only part of the blade can engage with the workpiece. Obtaining the instantaneous engage section of the cutting edge is critical for calculating the cutting force.

4.1. The Model of End Cutting Edge About the Ball End Mill

As shown in Figure 10, the center of the ball coincides with the origin of ITCS, and the tool axis coincides with Z_T. Define pitch helix on the cylindrical surface coaxial with Z_T as guiding line that forms the cutting edge. The line P_GQ is parallel to the coordinate plane X_TY_T (P_G is on the guiding line). The intersection P of line P_GQ and spherical surface is a point of the cutting edge.

Since the cutting edge is located on the spherical surface, the end cutting edge could be expressed as follows:where R is the radius of the ball end mill; θ_T is the axial position angle of the microedge; and φ_T is the circumferential position angle of the microedge.

The circumferential starting angle φ₀ defines the starting point of the cutting edge. Define the circumferential angle of microelement relative to the starting point as the circumferential offset angle ∆φ_T. The φ_T could be expressed as follows:

Since points P and P_G have the same coordinates in the Z_T-axis, thenwhere β_G is the helical angle of the side cutting edge.

Therefore, the cutting edge could be represented by θ_T.

Some cutting edges do not cross the tool center at the apex. Define the distance from the edge to the center in the top view of the ball end mill as h, as shown in Figure 11.

For the cutting edge not passing the tool axis, the range of θ_T is within [0, arccos(h/R)]. The result of the cutting edge is shown in Figure 12, and the cutting edge is in good agreement with the actual ball end mill. The parameters of the ball end mill are shown in Table 3.

4.2. The Engage Section

4.2.1. Boundary Conditions of CWE

As shown in Figure 13, CWE is a closed local spherical area and intersects the cutting edge at two points. Both cutting edge and cutter turning surface have unique projection on the coordinate plane X_MY_M of IMCS. Thus, the intersection of CWE and cutting edge could be solved by projection.

From (9) and (13)fd13, the projection function f_SP of sweep profile and f_SIL of surface intersecting line on the coordinate plane X_MY_M could be expressed as follows:

Also, the projection function f_IL of the intersecting line could be expressed as a polynomial, which could be obtained by fitting the discrete points on the intersection with the Newton interpolation method. Considering the stability and accuracy of interpolation, no more than 7 discrete points on the intersection are selected for fitting the curve. Also, the order of polynomial obtained by the Newton interpolation method is less than the number of interpolation nodes. The highest order polynomial could be expressed as follows:

In summary, the cutting edge section that engages the workpiece should satisfy the following conditions:

4.2.2. Calculation of the Engage Section

The axial position angles at the intersection points of the CWE boundary and cutting edge are applied to indicate the engage section, which is in the range (θ_T(st), θ_T(end)). The steps are as follows: Step 1: calculate the coordinates of all the microedges in ITCS by equation (17) with step ∆θ_T. Step 2: transform the microedge coordinates from ITCS into IMCS by M_MT. Step 3: search the coordinates of microedges in IMCS base on the boundary condition equation (21). Moreover, find out the microedges which adjacent to the boundary of CWE (two adjacent microedges that are inside and outside CWE, respectively). Step 4: search the microedges that are obtained in Step 3 with the dichotomy method, and find out the intersection of the CWE and cutting edge. Step 5: calculate the axial position angles of the intersection points.

5. The Prediction Model of Cutting Force

5.1. The Cutting Force Model

According to the Armarego oblique angle microelement cutting force model [28], the cutting force of the microedge involved in CWE could be expressed as follows:where dF_r, dF_a, and dF_t are the radial, axial, and tangential forces of the microedge cutting edge; K_rc, K_ac, and K_tc are the shear coefficients; K_re, K_ae, and K_te are the blade force coefficients; t_n is the thickness of undeformed chip; db is the projection width of the microedge on the generatrix; and ds is the projection length of the microedge on the generatrix.

5.2. The Calculation of Microedge

The width db could be expressed by microaxial position angle dθ_T and ball end mill radius R.

The length ds could be solved by the arc length differential formula.

The thickness t_n is a key parameter in the bevel cutting model, which is the projection of feed per tooth in the normal direction of the sphere [17]. The tool feed direction is consistent with the X_M-axis of IMCS, and then the feed vector could be expressed as follows:where f_t is the feed per tooth.

Convert v_M to ITCS, thenand the spherical normal vector of the end microedge could be expressed as follows:

The thickness t_n could be solved as follows.

5.3. The Instantaneous Cutting Force

The cutting force of the microedge is not parallel to the axes of ITCS, which could be decomposed by the following equation:where dF_xT is the component of the microcutting force on the X_T axis; dF_yT is the component of the microcutting force on the Y_T axis; and dF_zT is the component of the microcutting force on the Z_T axis.

Also, the instantaneous cutting force could be obtained as follows:where F_xT is the component of the cutting force on the X_T-axis; F_yT is the component of the cutting force on the Y_T-axis; and F_zT is the component of the cutting force on the Z_T-axis.

6. Verification of the Cutting Force Model

In order to verify of cutting force model, a series of machining experiments for turbine blade were carried out with DMG DMU 100 mono Block five-axis machining tool. The workpiece material is copper alloy ZCuAl8Mn14Fe3Ni2 (tensile strength σ_b ≥ 645 MPa, yield strength σ ≥ 280 MPa). The tool is a carbide ball end mill. The experiment parameters are shown in Table 4.

The force coefficients were identified by the methods proposed by Wojciechowski [29], which could be expressed as follows:

Some CLPs, tool axis vectors, feed vectors, and normal vectors in MCS are shown in Tables 5 and 6. The scene of machining process is shown in Figure 14.

The cutting force was acquired by a Kistler 9272 dynamometer and 5070A charge amplifier at 10 kHz. Combined with the calibration coefficients, cutting force was predicted by Matlab. The comparison results are shown in Figures 15 and 16. The trend of predicted value and the measured value has good consistency. Figure 16 shows that there is some noise in the measured data. Factors that may contribute to this effect include tool deformation, cutting vibration, and random measurement error of sensor. Table 7 shows that the errors of the predicted cutting force are less than 20%. Considering the unstable cutting conditions, the errors are within the acceptable range.

(a)

(b)

(a)

(b)

7. Conclusions

(1)Based on the IMCS and ITCS, a motion model of the ball end mill for the sculptured surface is established. The motion state and the contact relationship between the cutter and workpiece could be described in a quantitative way.(2)By solving three boundaries around the engagement of the cutter and workpiece, a CWE model is obtained. It has good adaptability for the variable tool axis relative to the machining surface.(3)A prediction model of cutting force about ball end milling for a sculptured surface is constructed. It is suitable for sculptured surface machining with arbitrary tool posture and feed direction.(4)The experiment results prove the effectiveness and validness of the proposed CWE model and cutting force prediction model.

Notation

P_L(t):	Instantaneous tool position at time t
P_C(t):	Instantaneous tool contact point at time t
u_(t):	Instantaneous unit vector of tool axis at time t
v_(t):	Instantaneous tool position unit feed vector at time t
n_(t):	Instantaneous normal vector at time t
i:	Serial number of the tool position in the NC program
t:	Arbitrary time
t_i:	Time of tool position i in NC program
O_W-X_WY_WZ_W:	Machining coordinate system(MCS)
O_M-X_MY_MZ_M:	Instantaneous milling coordinate system(IMCS)
O_T-X_TY_TZ_T:	Instantaneous tool coordinate system(ITCS)
i_W, j_W, k_W:	Unit axis vector of MCS
i_M, j_M, k_M:	Unit axis vector of IMCS
i_T, j_T, k_T:	Unit axis vector of ITCS
δ_WM:	Angle between Z_M and Z_W
s:	Vector of the intersection of coordinate planes X_MY_M and X_WY_W
γ_WM:	Angle between s and X_W
ε_WM:	Angle between s and X_M
P_L(t)(x, y, z):	Coordinate of the instantaneous tool position in MCS
M_WM:	Transformation matrixes for MCS to IMCS
T:	Translation transfer matrix
R_Z:	Rotational transfer matrix around the Z-axis
R_X:	Rotational transfer matrix around the X-axis
M_MT:	Transformation matrixes for IMCS to ITCS
δ_MT:	Angle between Z_T and Z_M
γ_MT:	Angle between X_T and X_M
R:	Radius of the ball end mill
θ_T:	Axial position angle of the microedge
φ₀:	Circumferential starting angle
φ_T:	Circumferential position angle of the microedge
∆φ_T:	Circumferential offset angle
β_G:	Helical angle of the side cutting edge
h:	Distance of the tooth offset the center from the top view of the ball end mill
θ_M:	Angle between the line connecting the origin of IMCS to any point on the swept profile and Y_M
θ′_M:	Angle between the line connecting the origin of previous IMCS to any point on the swept profile and Y_M
P_i:	Characteristic point of the control curve
N_i,k:	k-order B-spline basis function
φ_M:	Angle between the line connecting the Z_M-axis to any point on the surface intersection and _(t)
∆θ_T:	Axial position angle step size
dF_r:	Radial force of the microedge cutting edge
dF_a:	Axial force of the microedge cutting edge
dF_t:	Tangential force of the microedge cutting edge
dF_xT:	The component of the microcutting force on the X_T-axis of ITCS
dF_yT:	The component of the microcutting force on the Y_T-axis of ITCS
dF_zT:	The component of the microcutting force on the Z_T-axis of ITCS
F_xT:	The component of the cutting force on the X_T-axis of ITCS
F_yT:	The component of the cutting force on the Y_T-axis of ITCS
F_zT:	The component of the cutting force on the Z_T-axis of ITCS
K_rc:	Radial shear coefficients
K_ac:	Axial shear coefficients
K_tc:	Tangential shear coefficients
K_re:	Radial blade force coefficients
K_ae:	Axial blade force coefficients
K_te:	Tangential blade force coefficients
t_n:	Thickness of undeformed chip
db:	Width of the microedge
dθ_T:	Axial position angle
ds:	Length of the microedge
:	Feed vector in IMCS
f_t:	Feed per tooth
:	Feed vector in ITCS
t_n(θ_T):	Instantaneous undeformed chip thickness at θ_T

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Special Fund of High-end CNC Machine Tools and Basic Manufacturing Equipment (2017ZX04002001), China.

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Copyright © 2020 Wenping Mou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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