Cutting Force Modeling and Experimental Study for Ball-End Milling of Free-Form Surfaces

In order to improve the machining quality and efficiency and optimize NCmachining programming, based on the existing cutting force models for ball-end, a cutting force predictionmodel of free-form surface for ball-end was established. By analyzing the force of the system during the cutting process, we obtained the expression equation of the instantaneous undeformed chip thickness during the milling process and then determined the rule of the influence of the lead angle and the tilt angle on the instantaneous undeformed chip thickness. It was judged whether the cutter edge microelement is involved in cutting, and the algorithm flow chart is given. After that, the cutting force prediction model of free-form surface for ball-end and pseudocodes for cutting force prediction were given. MATLAB was used to simulate the prediction force model. Finally, through the comparative analysis experiment of the measured cutting force and the simulated cutting force, the experimental results are basically consistent with the theoretical prediction results, which proves that the model established in this paper can accurately predict the change of the cutting force of the ball-end cutter in the process of milling free-form surface, and the error of the cutting force prediction model established in this paper is reduced by 15% compared with the traditional cutting force prediction model.


Introduction
In the aerospace, automotive, marine, and other industrial fields, free-form surface parts occupy an important position [1]. Moreover, search on the cutting force of milling freeform surface parts has played an important role in improving machining accuracy and machining efficiency, improving machine tool utilization, and reducing energy loss, tool wear, and other fields [2]. Establishing the prediction model of cutting force during free-form surface machining and predicting the cutting force during machining can change the cutting parameters and modify the processing technology in time.
is has an important guiding role in improving the above processing phenomenon. However, most of the researches focus on the cutting force in the process of non-ball-end milling, and the research on the cutting force of ball-end milling free-form surface is relatively less [3,4].
In recent years, scholars at domestic and international organizations have carried out related research on the cutting force model for ball-end, mainly including the theoretical model and mechanical models. However, it mainly focuses on plane and regular surfaces, and there is little research on the cutting force model in milling freeform surfaces. e theoretical model is based on the friction angle theory and shear angle theory and using orthogonal cutting or oblique cutting analysis in basic theory to establish the milling force model [5]. Because the theoretical analytical modeling needs to know the exact geometric parameters of the tool and also has a close relationship with the properties of the tool and workpiece materials, there is a lot of simplification in the process of modeling, so the prediction accuracy of theoretical analytical modeling method is low [6].
Mechanical models are more extensive in practical applications. e mechanical model is a simplified model of the theoretical analytical model, which uses cutting force coefficients instead of the interaction influence of shear angle, chip flow angle, and friction angle to the cutting force. Scholars divided the mechanical model into mechanical I type and mechanical II type. Yellowley [7] assumed in mechanical model II that the ratio between shear force and ploughing force is constant and independent of chip thickness. Afterwards, Altinta and Lee [8] confirmed the assumption of Yellowley through experiments: under the general cutting conditions, the size-effect has little effect on the cutting force. As early as the early 1990s, Yang and Park [9] published the research results on the cutting force modeling of ball-end milling for the first time. Kline et al. [10] proposed the method of dividing the end mill into several sections along the axial direction and calculating the cutting force separately. Engin and Altintas [11] established the cutting force model of milling cutter with arbitrary profile. Fontaine et al. [12] established a cutting force prediction model based on bevel cutting and studied the influence of the angle between the workpiece and the tool on the cutting force during milling. Tsai and Liao [13] proposed the machining model under the condition of oblique feed. Kovacic et al. [14] established the milling force model using genetic algorithm. In order to establish the tool displacement model, Wojciechowski et al. [15] of Poznan University of Technology calculated the instantaneous orthogonal cutting force of face milling cutter during milling and selected the cutting length, contact area, effective number of cutting teeth of face milling cutter, and milling cutter position angle as the research object to study their influence on the instantaneous cutting force. Pimenov et al. [16,17] of South Ural National University and others studied the coefficients in Guzeev and Pimenov's cutting force model, obtained the value of k1k2k3 through experiments, and completed the establishment of cutting force model. e instantaneous undeformed chip thickness is an important parameter to calculate the cutting force. Huang et al. [18] proposed a method to calculate the chip thickness in the milling of five-axis ball head and studied the influence of the main and lateral deflection angles on the chip thickness in the milling of five-axis ball head. Tuysuz et al. [19] proposed a mechanical model to predict the cutting force in three directions during the milling process of ballend cutters by simulating the chip thickness distribution, cutting, and indentation mechanics. Azeem et al. [20] proposed a method for characterizing the thickness of undeformed chips in three-dimensional tool motion and improved the cutting force model for multiaxis ball-end milling. Zhang et al. [21] proposed an accurate model of instantaneous undeformed chip thickness considering tool runout effect. By finding the approximate point of intersection between the reference line of cutting microelement and the actual surface of machined workpiece, the undeformed chip thickness is solved by means of linear iteration. e exact solution of cutter engagement area is the key to calculate the cutting force for the ball-end. ere are four methods for calculating cutter engagement area: the Boolean operation method based on the solid model, the discrete method based on the Z-map model, the cut-in and cut-out method, and the analytical geometry analysis method. e core idea of the Z-map method is discreteness. e Z-map method is robustness and efficiency so that it is widely used in CNC simulation. Lin et al. [22] analyzed the boundary curve of the contact area during plane cutting to obtain the contact area under the fixed tool posture and obtained the contact area under any tool posture through geometric transformation and then extended its analytical expression to a free-form surface.
Wang et al. [23] considered the influence of tool vibration and runout on cutting force when predicting cutting force. rough experiments, the consistency of the results of the proposed predicted force and actual measured force was verified. In China, Ma and Lin [24] modeled the cutting force based on the idea of differential discrete and analyzed the eccentricity in the milling process. According to the milling characteristics of ball-end milling cutter, Shi et al. [25] established the milling force model of ball-end milling cutter. Tan [26] used the idea of axial microsegmentation to divide discrete cutting edges and used geometric analysis to judge the cutting in and cutting out conditions and established a cutting microelement cutting force model. Wei et al. [27,28] improved the cutting force model for the ball-end and the undeformed chip thickness model with the position angle of the cutting edge microelement as parameters, so that it can adapt to the changing cutting geometry conditions in the surface machining, and established a cutting force prediction model of curved surface for ball-end. Guo et al. [29] proposed an analytical algorithm based on the space limited method for cutting edge contact interval for 5axis wide row machining of free-form surface flat end milling cutter and established a milling force prediction model considering tool eccentricity. Wei et al. [27,30] presented an integrated form error compensation approach for ball-end milling of sculptured surface with Z-level contouring tool path.
After analyzing and summarizing the literature at home and abroad, it is found that the research on the cutting force model of ball-end tool milling surface is less and incomplete. e existing models do not include the axial milling force component, so they are incomplete. At the same time, in the analysis of chip geometric parameters, the three-dimensional feed motion of the tool is simplified to a two-dimensional case, which is a general case, which is difficult to be used for the continuous simulation of the three-dimensional machining process of complex surfaces. In addition, in the determination of the important parameter of cutting edge segment involved in milling, most models adopt simple geometric analysis method, and the simulation accuracy is low. erefore, it is necessary to further study the milling force of ball-end tool milling free-form surface.
Although there are many scholars studying ball-nose cutter cutting force modeling, most of them still stay in the plane field of simple rules in low-speed milling, and most of the model errors are large. ere are relatively few studies on high-speed precision machining of surface, especially free surface, which has seriously restricted the actual needs. erefore, it is necessary to further study the cutting force modeling of surface-type parts. Based on Lin et al.'s [22] calculation of contact area and cutting force coefficient, this paper first calculates the instantaneous undeformed thickness in the cutting process and judges whether the microelement of cutting edge participates in cutting, then deduces the cutting force model of free-form surface of ball-nose cutter, and gives the cutting force prediction algorithm. Finally, the correctness of the model is verified by free-form surface and arc surface cutting experiments.

Materials and Methods
In this chapter, a cutting force model of free-form surface for ball-end is established. In the main work, relevant researches and calculations on instantaneous undeformed chip thickness are carried out, to judge whether the microelement cutting edge on the ball head cutter is involved in cutting, so that the cutting force model can be established more accurately. After that, the process and method of establishing the cutting force model are analyzed, and the modeling scheme is studied. en the cutting force model of ball-end milling free-form surface is programmed; MATLAB is used to predict its value and change trend.

Instantaneous Undeformed Chip ickness.
e instantaneous undeformed chip thickness is defined as the distance between the cutting point on the current tool path and the machined workpiece surface along the microelement reference line of the cutting edge [29]. It is an important parameter for calculating the cutting force, which determines the size of the cutting load and is related to the current feed rate per tooth f, the axial position angle κ, and circumferential position angle θ [30]. When studying the changing law of instantaneous undeformed cutting thickness, the main need is to consider the changing law in both horizontal and vertical directions, so the feed direction is equivalent to staying in the horizontal plane. erefore, it is only necessary to analyze the horizontal change of the instantaneous undeformed chip thickness of the cutting microelement, as shown in Figure 1. e thickness of the undeformed chips in the horizontal direction can be expressed by equation (1) [13]. erefore, the instantaneous undeformed chip thickness [9] is (1) f(H) is the chip thickness in horizontal feed direction; f(V) is the chip thickness in vertical feed direction; lead angle is represented by α z ; α c is the tilt angle; κ is the axial position angle, and circumferential position angle is θ. Undeformed chip thickness is f n .

Judgment of the Microelement Cutting Edge Participating in Cutting.
If it is judged that a microelement edge of a ballend cutter participates in cutting when modeling the cutting force, it will inevitably produce cutting force, which will affect the total cutting force. In actual machining process, if it is judged that it does not participate in cutting, the cutting force will be zero, and it will not need to be included in the total cutting force when modeling. erefore, judging whether each microelement edge of the ball-end cutter participates (as shown in Figure 2) in cutting is an important issue in cutting force modeling.
Ding [2] calculated the cutter engagement area and calculated the limit values of the axial position angles κ min and κ max of the cutter engagement area and divided the contact area between κ min and κ max into m-small microelement segments; the cut-in angle θ sti and cut-out angle θ exi corresponding to each microelement segment are calculated. e cut-in angle and cut-out angle of the microelement limit the range of microelement involved in cutting. e geometric diagram of the microelement position angle and the circumferential microelement position angle is shown in Figure 2: R represents the effective cutting radius.
To determine whether the microedge is involved in cutting, the position angle θ of each microedge of the tool needs to be calculated. If θ sti ≤ Ψi ≤ θ exi , the corresponding microedge of the cutter edge participates in cutting. Otherwise, the cutter edge microelement does not participate in cutting. At present, the ball-end cutter widely used in processing is the constant lead helix ball-end cutter, but its helix angle is not constant at the ball-end port. e spiral cutter edge of the head is shown in Figure 3.
In this diagram, ψ is position angle of cutting edge [rad]; ϕ is the lag angle [rad]; and n is the spindle rotational speed [r/min].
After judging whether each microelement on a single tooth participates in cutting separately, the cutting force of each tooth of the tool is calculated. Finally, each angle is calculated by this way so that we can obtain the results of microelement participation in cutting of the entire tool in one cycle. e calculation flow chart is shown in Figure 4.

Cutting Force Modeling.
To model cutting force prediction model of free-form surface for ball-end milling, first we determined the microelement cutting force of the freeform surface cutting edge and then integrated the microelement cutting force along the cutting edge of the ball-end cutter. By this way the cutting force on a cutting edge can be obtained. e cutter edge microelement cutting force prediction model of free-form surface is shown in

Mathematical Problems in Engineering
Substitute the undeformed chip thickness (equation (2) into equation (3)) to obtain In these equation, K ts , K rs , and K es are the circumferential, radial, and axial shear force coefficients; K ts , K re , and K ae are the circumferential, radial, and axial ploughing force coefficients; dF x , dF y , and dF z are the circumferential, radial, and axial microcutting force. e feed per tooth is represented by f c . β 0 is the nominal helix angle. e integral can get the total cutting force of a certain tooth as shown in equation (5). e total cutting force of a cutter tooth during cutting is the total cutting force of the entire ball-end cutter, when precisely cutting a free-form surface. e total cutting force of a cutter tooth during cutting is calculated below; for an N-tooth cutter, the milling forces of all cutter teeth are added and summed to obtain the total milling force of the cutter. Its expression is

Cutting Force Prediction Algorithm.
In order to represent the cutting force prediction algorithm, the pseudocode for the prediction of the cutting force prediction model of free-form surface for ball-end is given in Table 1. e user inputs cutting conditions, tool geometry parameters, tool posture, cutting force coefficients, axial position angle Axial microelement steps n � 360/∆ang Angle rotation value in one cycle j � 1 to n Calculate the cutting force value for the entire cycle Calculate all cutting microcutting forces Initialized cutting force value k � 1 to N Calculate the cutting force of all cutting edges If the cutter edge microelement participates in cutting f n � f c · sin κ · sinθ 2 Chip thickness at this cutting point Calculate microcutting forces in tangential, radial, and axial directions Matrix transformation of tangential, radial, and axial microelement cutting forces is performed to obtain the cutting force in the tool coordinate system, and all microelement cutting forces are summed to obtain the total cutting force Else Next i Mathematical Problems in Engineering intervals, and angle intervals within a cycle. After the program runs, the cutting force in a cycle of the entire cutter tooth is output.

Experimental Conditions and Equipment.
In the process of verifying the accuracy of the free-form cutting force model of the ball cutter proposed in this paper, two sets of surface cutting experiments were carried out. Except that the shape of the curved surface to be cut is different in the experiment, the tools, specimen materials, and dynamometers used are the same. e equipment used in the whole experiment is as follows: (1) Machine tool: Mikron hsm700 is used for the experiment (2) Dynamometer: Kistler 9257b three-way dynamic piezoelectric dynamometer (3) Data acquisition system: Kistler 5019b charge amplifier, dewesoft3010 data acquisition and processing system, Lenovo E430 notebook computer (4) Sampling frequency: 20000 Hz e schematic diagram of the experimental data acquisition system is shown in Figure 5, and the machine tool used in the experiment is shown in Figure 6. e material used is TC4. Its chemical composition is shown in Table 2. In the comparative experiment, one set of cut surfaces is a free-form surface, and the other set is a cylindrical surface with a radius of 10 mm. e cutting parameters are shown in Table 3.
It can be seen from Table 4 that the variation range of the lead angle and the tilt angle between adjacent tool positions on the tool path is small, so the cutter engagement area is basically the same, and the actual measured cutting force is not much different. So we just select a point on each tool path, calculate the cutting force of its predicted force, and compare it with the actual cutting force to verify the correctness of the cutting force prediction model of free-form surface for ball-end proposed in this paper. Based on this, this paper selected the No. 2 tool position in Figure 7 to predict the cutting force and compared it with the actual cutting force, as shown in Figure 7. Figure 8(a) shows the change of the cutting force in the X direction during one revolution of the tool. Figure 8(b) shows the change of the cutting force in the Y direction during one revolution of the tool. Figure 8(c) indicates the change of the cutting force in the Z direction during one revolution of the tool. Figure 8(d) shows the change of the overall cutting force during one revolution of the tool.
It can be seen from Figure 8 that there is an error in the peak values of the predicted force curve and the actual cutting force curve in the X and Z directions. e measured cutting force is larger. However, the predicted cutting force and the predicted cutting force in the Y direction are realistic. In order to quantitatively analyze and predict the deviation of the cutting force, a calculation for the deviation is performed. e ratio of the difference between the peak predicted force and the measured peak force and the measured force is taken as the deviation. rough calculation, the average deviation in the X direction is 27.8%, the average deviation in the Y direction is 9.0%, the average deviation in the Z direction is 23.9%, and the deviation of the total force is 17.6%.
During the free-form cutting process, the change of the lead angle and the tilt angle of all the tool positions on the entire tool position path is small; it is not very accurate to show the rule that the cutting force changes when the cutter engagement area changes greatly. erefore, this paper has carried out circular arc with a radius of 10 mm cutting experiment. Table 5 shows the lead angle and the tilt angle corresponding to each tool point of the circular arc cutting experiment. e change diagram of the cutter engagement area at each tool point is shown in Figure 9. e change rule of the maximum value and minimum value of the predicted cutting force and the actual measured cutting force during the entire cutting process is shown in Figure 10 As can be seen from Figure 10, the minimum value predicted by modeling is 0, that is, the cutting force when the cutting edge of the tool does not participate in cutting. e maximum value is the maximum value of the cutting force of the tool at each tool location. In order to clearly explain the change law of cutting force on the tool path, the curve is fitted and the maximum curve in Figure 11 is obtained. e predicted cutting force change trend is basically consistent with the measured cutting force change trend.
Although the experimental results of the above experiments show that the cutting force model established in this paper is basically consistent with the experimental results, it is not yet able to prove the advanced nature of the model established in this paper. erefore, the comparative experiment between the traditional cutting force model for inclined plane and the cutting force model established in this paper is done. e inclined plane angle is 60°, as shown in Figure 11. e predicted force of the traditional predicted force model and the cutting force model in this paper are compared with the measured force, and Figure 12 can be obtained. Figures 12(a) It can be seen from Figure 12 that the cutting force predicted by the traditional cutting force model and the cutting force predicted by the cutting force model in this paper are in good agreement with the actually measured cutting force. Especially at the speed of 8000 r/min, the peak value of cutting force is obvious, and the measured cutting force is basically stable and small without cutter teeth.
In order to calculate the deviation between the predicted cutting force and the actual cutting force, the root mean square error is introduced to quantitatively analyze the consistency between the predicted cutting force and the measured cutting force with different cutting force coefficients.
rough the measured cutting force data, the 8 Mathematical Problems in Engineering maximum cutting force of each cutter tooth in the cutting process can be calculated, and the calculation formula is as follows: where F xi_max , F zi_max are the measured maximum cutting force of the ith cutter tooth in X and Z directions, F yi_min is the measured minimum cutting force of the ith cutter tooth in the Y direction, and j is the number of teeth of the cutter. e root mean square error is calculated as follows: where F x_th , F z_th are the predicted maximum cutting force in the X and Z directions and F y_th is the predicted minimum cutting force in the Y direction. According to the above calculation model, the root mean square error results can be obtained, as shown in Figure 11. Among them δ_all is the root mean square error obtained by applying the traditional cutting force model, and δ_sig is the root mean square error calculated by using the prediction results of the cutting force model proposed in this paper.     It can be seen from Figure 13 that when the speed is 5000 r/min, δ_all is 13.4%, and δ_sig was 11.9%. When the speed is 8000 r/min, δ_all is 26.4%, and δ_sig was 11.4%. At the same speed, δ_sig is less than δ_all. In addition, as the speed increases, δ_all increment is large, while the change of δ_sig is very small.
Only one comparative experiment cannot verify the correctness of the calibration method of tooth cutting force coefficient. erefore, two groups of verification experiments are carried out on the basis of this experiment. e cutting parameters of rotating speed n � 5000 r/min are feed rate per tooth FC � 0.04 mm, cutting depth AP � 0.4 mm, and row width � 0.8 mm. e cutting parameters of rotating speed n � 8000 r/min are feed rate per tooth FC � 0.05 mm, cutting depth AP � 0.4 mm, and row width � 0.8 mm. e experimental results are shown in Figure 14, and the root mean square error results are shown in Figure 15.
It can be seen from Figure 14 that the predicted cutting forces in the two groups of experiments verified are also in good agreement with the measured cutting forces. When the rotating speed changes from 5000 r/min to 8000 r/min, the oscillating cutting force in three directions of X, Y, and Z becomes larger when cutting without cutter teeth. It can be seen from Figure 15 that when the speed is 5000 r/min, δ_all is 22.6%, and δ_sig was 22.8%. When the speed is 8000 r/min, δ_all is 32.4%, and δ_sig was 22.7%. As the speed increases, δ_all increment is large, while δ_sig is basically unchanged.

Analysis and Discussion of Experimental Results.
Observing Figures 7 and 9, it can be seen that when the ballend cutter cuts free-form surface parts, the cutter engagement area is related with change of the shape of the free-form surface at the tool point. e smaller the absolute value of the lead angle and the tilt angle are, the closer the cutter engagement area is to the point of the tool tip, and, conversely, the farther the cutter engagement area is from the point of the knife point.
It can be seen from some of the graphs in Figures 8 and  10 that there are oscillation signals in the measured cutting force. It can be seen that the oscillation signal decreases with the decrease of the lead angle; the main reason is that the stiffness of the tool becomes better and the cutting force in the circumferential direction becomes smaller during the process of the lead angle from large to small, and the stability of the system becomes better. As for the situation where the cutting force is relatively large at the initial cutting and at the end of the cutting, it is because the cutting state at the beginning of the cutting and the final cutting is an unstable cutting state, which is not within the scope of this article.
is paper studies the cutting force under stable cutting state. In the process of just cutting in and finally cutting out, the cutting force is actually a sudden process, which is greater than the stable cutting force. e cutting forces calculated in this paper are all cutting forces under stable cutting state. In order to improve the unstable cutting F C 1 2 condition, before the cutting experiment, firstly the tool is slotted along the normal direction of the workpiece in the tool cut-in area and tool cut-out area, which can greatly reduce the cutting in and cutting out process and prolong the time of stable cutting state. However, the whole process from the beginning of cutting to the completion of cutting is in a stable cutting state. rough the analysis of the above chart, the following facts can be basically obtained. It can be seen from the measured cutting force curve that the peak value of the measured cutting force is obvious, the number of cutter teeth is clear, and the vibration value is relatively small. e measured data under the conditions of high speed, small cutting depth, and small total cutting force is enough to show that the measured cutting force is accurate. Secondly, the predicted cutting force is basically consistent with the measured cutting force. ere is a small error in some parts, which is mainly due to the limitation of the defined cutting force coefficient. Using the cutting force coefficient calibration method cited in this paper, because the calculated cutting  contact area is small, the calculated cutting force coefficient curve is accurate near a fixed axial position angle, but the cutting force coefficient at other position angles is relatively inaccurate; for other position angles, the cutting force coefficient is relatively inaccurate. en when the change range of contact area is large, there will be a small error between the prediction cutting force and the measure cutting force. After eliminating the above interference factors and other uncontrollable factors, the results of the predicted cutting force and the measured cutting force in this paper are in good agreement with small errors. is shows that the established model can more accurately predict the actual cutting force of free-form surface parts in the precision cutting process. Under the condition of ball-end tool finish milling, compared with the traditional cutting force model, the error between the predicted cutting force and the measured cutting force is small, especially in high-speed milling, the error is much less than the cutting force predicted by the traditional cutting model, the prediction accuracy is high, and it is more in line with the actual cutting state.
Although the cutting force model established in this paper has high prediction accuracy in high-speed milling, it still has some limitations. And although the research on ballnose cutter cutting force modeling has been mature, there is no commercialized cutting force prediction software. Because there is little research on unstable factors in cutting process, in most circumstances, the actual cutting state is not in the ideal cutting state. For example, first in this paper the cutting force coefficient calibrated is accurate at the axial position angle corresponding to the contact area because of the small cutting depth, while the relative accuracy in other areas is low. Secondly, the cutting force established in this paper is established without considering the stiffness of the tool and the cutting vibration, etc. However, in multiaxis machining, the stability and vibration amplitude of cutting system will change due to the change of tool axis and stiffness of tool, which will directly or indirectly affect the cutting force. Finally, the cutting force under stable cutting state is studied in this paper. ere is no research on cutting force under unstable cutting conditions such as first-cutter cutting, cut-in, and cut-out. In order to make the research on cutting force more complete, the cutting force under unstable cutting state mentioned above can be considered in further research.

Conclusion
In this paper, a calculation model of instantaneous undistorted cutting thickness under variable cutting conditions is derived, and the participation of cutting edge element in cutting at a certain cutting time is judged, and a free-form surface cutting force model of ball-nose cutter is obtained. rough free-form surface and arc cutting experiments, it is proved that the predicted cutting force and actual cutting force are in good agreement both at a cutter point and on the whole cutting path; it is verified that the established cutting force model is accurate. However, at the same time, the modeling in this paper also has some limitations, mainly reflected in the less research on unstable factors in cutting process, so the research on cutting force of ball-nose cutter milling free-form surface under unstable state should be carried out more deeply afterwards.
In order to prove the advancement of the model established in this paper, a series of inclined face comparison experiments were carried out. e experimental results show that the error of the cutting force prediction model established in this paper for ball-nose cutter milling free surface is reduced by 15% compared with the traditional cutting force model, and it is more suitable for high-speed and precise milling. It provides reliable and effective guidance for reducing errors and chatter in actual machining.

Data Availability
e data used to support the findings of this study are included within the article.

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