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

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 endmill 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. ,e 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. ,e accuracy of the proposed prediction model was verified by a series of experiments.


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. erefore, 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]. e 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 cutterworkpiece engagement (CWE), which defines the area where cutter and workpiece interact to generate cutting force. e 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. e 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]. e 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. e 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). e 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. e 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]. ree 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. e 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. e 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. e research methods of cutting force modeling are divided into three categories: the theoretical analytic method, finite element (FE) method, and mechanical force method. e 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]. e 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. e 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. e 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). e machining coordinate system (MCS) is established to describe tool posture. e 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. e innovations include the following: 2 Mathematical Problems in Engineering (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) e parameterized expression of the undeformed chip thickness is improved, and it could be applied to the variable axis machining for the sculptured surface e rest of the paper is organized as follows: in Section 2, the motion model of the tool is established and the MCS is defined. e CWE model is established in Section 3. e 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.

Tool Motion State and Coordinate System
e 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. erefore, 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.

e Definition of Tool Posture.
e 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]. (1) Define the instantaneous tool position unit feed vector as corresponding unit axis vector as i W , j W , and k W . e 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.
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 . Figure 3.

MCS to IMCS. e geometric relationship between IMCS and MCS is shown in
Define the angle between Z M and Z W as δ WM , the vector of the intersecting line of coordinate planes X M Y M and X W Y W as s, the angle between s and X W as c 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 (c WM ) is the rotational transfer matrix around Z-axis with angle c WM .

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 c MT between X T and X M could be obtained as follows: Figure 1: Instantaneous position and orientation of tool at time t.  Figure 3: Geometric relationship between the instantaneous end mill coordinate system and MCS. e 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 (c MT ) is the rotational transfer matrix around the Z-axis with c MT .

The CWE Model of Ball End Milling
e 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.
e Swept Profile. e current machining surface is formed by cutter revolve sweeping along the current tool path. e 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 . 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. erefore, the intersecting line could be expressed by a series of discrete points.

e Intersecting Line. As shown in
In the current IMCS, the revolution surface of the cutter at the current CLP could be expressed as follows.
e swept profile in the previous adjacent tool path could be expressed as follows.
x M ′ � 0, By MCS-IMCS transformation, the swept profile in the previous IMCS on the previous adjacent tool path could be transformed to the current IMCS.
e one-dimensional nonlinear equation for θ′ M could be obtained from equations (10), (11), and (12). e numerical method (the Newton-Raphson method) could be applied to solve it. en, the coordinates of intersection points in the current IMCS could be calculated.

e Surface Intersecting Line.
ere are many methods about the sculptured surface machining, such as multilayers cutting and single-layer cutting. ey 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 Maxis to any point on the surface intersecting line) and v (t) .

e Verification of CWE.
In order to verify the accuracy of the proposed CWE model, an experiment was carried out. e parameters of experiment are shown in Table 1, and the workpiece geometry and tool path are shown in Figure 8. e 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. e tool path is the projection of the guiding line on the workpiece surface. e boundary of CWE is simulated by Matlab.  Table 2, which are less than 3%. e former takes approximately 75 s at each CLP, and the latter takes only 0.25 s. e 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. e 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%.

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.

e 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. e line P G Q is parallel to the coordinate plane X T Y T (P G is on the guiding line). e intersection P of line P G Q 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. e circumferential starting angle φ 0 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 . e φ T could be expressed as follows: Since points P and P G have the same coordinates in the Z T -axis, then where β G is the helical angle of the side cutting edge. erefore, the cutting edge could be represented by θ T .  Mathematical Problems in Engineering 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)]. e result of the cutting edge is shown in Figure 12, and the cutting edge is in good agreement with the actual ball end mill. e parameters of the ball end mill are shown in Table 3. 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 M Y M of IMCS. us, the intersection of CWE and cutting edge could be solved by projection.

Boundary Conditions of CWE. As shown in
From (9) and (13) 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. e highest order polynomial could be expressed as follows:    Figure 10: Schematic of the end cutting edge about the ball end mill. 8 Mathematical Problems in Engineering In summary, the cutting edge section that engages the workpiece should satisfy the following conditions:

Calculation of the Engage Section.
e 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) ). e 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.

e 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: dF r � K rc t n db + K re ds, dF a � K ac t n db + K ae ds, dF t � K tc t n db + K te ds, 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.

e Calculation of Microedge.
e width db could be expressed by microaxial position angle dθ T and ball end mill radius R.
e length ds could be solved by the arc length differential formula.
e 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]. e 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 and the spherical normal vector of the end microedge could be expressed as follows: e thickness t n could be solved as follows.

e Instantaneous Cutting
Force. e 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;       Mathematical Problems in Engineering 11

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. e workpiece material is copper alloy ZCuAl8Mn14Fe3Ni2 (tensile strength σ b ≥ 645 MPa, yield strength σ ≥ 280 MPa). e tool is a carbide ball end mill. e experiment parameters are shown in Table 4. e force coefficients were identified by the methods proposed by Wojciechowski [29], which could be expressed as follows:  Tables 5 and 6. e scene of machining process is shown in Figure 14.
e 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. e comparison results are shown in Figures 15 and 16. e 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.

Conclusions
(1) Based on the IMCS and ITCS, a motion model of the ball end mill for the sculptured surface is established. e 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.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.