Theoretical and experimental study of cutting forces under tool flank interference in ultra-precision diamond milling

In ultra-precision diamond milling (UPDM), the cutting force as an indicator of machining stability deserves to be discussed extensively. However, most studies have focused on the direct material removal under tool rake cutting, not considering tool flank interference in UPDM. In this study, a theoretical and experimental investigation has been conducted to discuss cutting forces under the tool flank interference in UPDM. Firstly, an analysis model of the interference space between tool flank and workpiece was built up to study the tool flank interference positions. Secondly, a kinematic model for the tool cutting motion was constructed to discuss the instantaneous uncut chip thicknesses (IUCT) under the tool flank interference. Moreover, a cutting force model was proposed to reveal the relationship between the cutting forces and IUCT. Finally, a series of milling tests were carried out in UPDM for the validity of the theoretical results. The theoretical and experimental results revealed that the tool flank interference would take place to deform surface generation and induce sudden changes of the cutting forces in UPDM. The tangential cutting force (Ft) reflects the dominant IUCT, and the radial cutting force (Fr) shows an extremely high sensitivity to the IUCT under tool flank interference, while Fr has little correlation with IUCT under tool rake cutting, even negligible. This research work gives a deep insight into the cutting forces with surface generation under the tool flank interference in UPDM.


Nomenclature
F Ati The tangential forces applied to the P i0 F t Tangential cutting forces F Ari The radial forces applied to the P i0 F r Radial cutting forces K Atc Correlation coefficient between F Ati and h Ai F a Axial cutting forces K Arc Correlation coefficient between F Ati and h Ai F x The forces along the x-direction based on the coordinate C e F Bti Tangential forces applied to the tool flank F Bri Radial forces applied to the tool flank F y The forces along the y-direction based on the coordinate C e K Btc Correlation coefficient between F Bti and h Bi K Brc Correlation coefficient between F Bri and h Bi F z The forces along the z-direction based on the coordinate C e n Spindle speed (rpm) v f Feed rate F ti Tangential cutting forces in i section θ The angle of tool rotation F ri Radial cutting forces in i section t Time for tool rotation h A IUCT under the tool rake cutting r i The radius of the projected circle h Ai IUCT under the tool rake cutting in i section z i The distance from the tip top h Bij IUCT under the tool flank interference for microelement in i section r Arc radius of the tool tip w yti Velocity component of w along the y-direction h Bi IUCT under the tool flank interference in i section α ij The angle between the positions of P ij and P i0 v Bt The slope of the h Bt curve h Bj Depth of tool flank interference space 1 3

Introduction
Ultra-precision machining (UPM) provides a promising surface generation up to sub-micrometric form accuracy and nanometric surface roughness, which has achieved widespread applications into optics, medicine, lighting, and automotive [1,2]. UPM technology mainly includes turning, grinding, and milling [3]; ultra-precision diamond milling (UPDM) has a wide range of applications due to its excellent universal material processing ability and complex three-dimensional shape processing ability [4,5]. The diamond tool is commonly used in UPDM, and usually used in finishing of complex surfaces. The interference area between the tool and the workpiece during cutting is constantly changing; the dynamic characteristics of the cutting are complex [6]. Most of the research on micro-milling is aimed at ball-end milling cutters, which generally have more than two tool edges [7], while the structure of diamond tool is simple and generally asymmetrical, which is rarely studied by researchers. In UPDM, the interference area between tool and workpiece is an important basis for judging the force area. The main methods of solving it include the solid model method, the analytical method, and the discrete differential method. The solid model method is used to perform Boolean operations on the vertices, edges, and faces of the model in simulation to obtain the meshing situation of the tool and the workpiece [8]. The tool edge that participates in milling is the intersection of the NURBS tool edge curve and the workpiece [9]. The model established by identifying the interference area of the tool and workpiece from the tool point of view can be used to simulate multi-process operations such as roughing, semi-finishing, and finishing [10]. Solid modeling can provide accurate simulation and verification, but the increase in geometric primitives in the continuous simulation process leads to higher computational costs. Therefore, the three-dimensional cutting method replaces the method of directly cutting to obtain the interference area of the tool and the workpiece [11], and the method of arc-surface intersection instead of surface-surface intersection is developed and applied [12].
The solid model method is usually used to simulate the cutting process, but if it is a simple linear process, the analytical method is often more efficient and practical [13]. The motion equation of the point on the tool edge is expressed in the local coordinate by the analytical method, and the closed surface of the workpiece is expressed by the point data. After the coordinate transformation, the instantaneous uncut chip thicknesses (IUCT), the milling entry angle, and departure angle can be obtained [14]. Through the analytical simulation algorithm, the interference area between the tool and the workpiece can be calculated, and the interference area can be represented by a closed loop formed by three boundaries [15]. Further, the milling situation of the ballend milling cutter can be divided into the milling plane and the residual surface in the previous step. The interference area between the tool and the workpiece is projected onto the corresponding surface and further analyzed to obtain the analytical expression of the interference area [16].
The discrete differential method can not only simulate the cutting process but also improve the computational efficiency. After discretizing the model of cutting tool, the computational efficiency of applying the logic matrix algorithm to solve the interference area can be increased by 60 times compared with the traditional loop algorithm [17]. Based on the micro-element discrete method, three conditions are proposed to judge whether any point on the cutting edge meshes with the workpiece; it can greatly improve the solution efficiency [18].
Z-map is a typical discrete method that can be effectively used for various calculations on surfaces perpendicular to the Z-axis [19]. Z-map method can be used to analyze the interference areas of the ball-end milling cutter under different milling attitudes, and the interference areas can be obtained by comparing the Z-map data of the tool and the workpiece [20,21]. For the ball-end milling model, the tool can be projected along the three directions to obtain the interference area between the tool and the workpiece more accurately [22]. Researchers usually combine the analytical method and the Z-map method, which can improve the accuracy and the computational efficiency [23].
The tool edge periodically interferences the workpiece in UPDM, the position of the interference area various with time, resulting in cutting forces [24]. Since the speed of spindle is much smaller than the linear speed of the tool edge, the IUCT as the product of the feed per turn and the sine of the rotation angle was first simplified, which is the classical cutting force theory [25]. Then, the cutting force model, which assumes that the cutting force is proportional to the cross-sectional area of the chip, was also proposed to study the variation characteristics of the cutting forces [26]. Currently, the dynamic micro-milling cutting forces model was proposed to identify the dynamics at the tool tip through the mathematical coupling of the experimental dynamics with the analytical dynamics [27]. The cutting force model for predicting general three-dimensional cutting force components [28], the model considering tool run-out, and the model considering chip thickness accumulation have all been proposed for accurate prediction of cutting forces [29,30].
Overall, the IUCT is an important component of theoretical modeling. For the solution of the IUCT, it is necessary to consider not only the tool run-out and the tool nose trajectory [28], but also the impact on the workpiece in the previous turn [31]. However, most studies have used the classical cutting force theory for modeling, which only considers the process of direct material removal under the tool rake cutting in UPDM, while the tool flank interference has not been discussed well. Therefore, this paper focuses on a theoretical and experimental investigation of cutting forces under the tool flank interference in UPDM. Firstly, an analysis model of the tool flank interference was established to reveal the interference interval. Secondly, a kinematic model suitable for the tool cutting motion was constructed to discuss the IUCT under the tool flank. Finally, the cutting force model was constructed to analyze the relationship between the cutting forces and the IUCT, and as compared, many kinds of designed tests were carried out in UPDM for the validity of the theoretical results. The research aims to provide a better understanding of the cutting forces with surface generation under the tool flank interference in UPDM.

Geometric modeling of tool and workpiece
The discrete differential method is used for modeling, and the tool is divided into several sections along z-axis, as shown in Fig. 1a. The ith slice is named as i section, and the projection of the tool rake in the i section is simplified to a point P i0 . The tool flank needs to be divided into microelements; in the i section, the projection of the tool flank is an arc, which is divided into several micro-arcs, and the j micro-arc in the i section is simplified to a point P ij .
Four reference coordinates are established to accurately describe the kinematic trajectories of the tool rake and tool flank, as shown in Fig. 1b.  The cutting trajectories of adjacent cycles in the i section are taken to represent the cutting stable state of the tool, as shown in Fig. 3. It can be observed that both the tool rake and the tool flank are involved in cutting and working together to complete a cutting cycle.

Modeling of tool cutting motion
In order to accurately find out the dimensional parameters of the interference area, the Z-map method is combined with the analytical method to model the tool cutting motion. The projection of the tool in the i section is shown in Fig. 4; the kinematic equation for M(P i0 ) is expressed as: The modelings are used to discuss the IUCT under the tool rake cutting and the tool flank interference; the programming process is shown in Fig. 6. In the i section, the tool flank becomes a circular arc, and the circular arc is divided into a number of micro-element arcs; final output yields the IUCT under the tool rake cutting (h Ai ) and the IUCT under the tool flank interference (h Bij ) for each section.

Modeling of cutting forces
The instantaneous force equation for the tool rake according to the classical mechanics equation (ignoring the tool coefficient and disregarding the case, where h Ai < h min ) is expressed as: In the process of the tool flank involved in cutting, not each section of micro-arc involved in cutting, the IUCTs are different in each section; in order to simplify the calculation, the IUCTs of micro-element arc in the same section are combined, as shown in Fig. 7. The average IUCT under the tool flank interference and its arc length is positively correlated; the correlation coefficient is defined as l Bi ; the IUCT in the i section is expressed as: Fig. 5 The tool position and interference area in UPDM: a θ = 2π, b 2.5π < θ < 3π, c 3π < θ < 3.5π, and d θ = 4π Assuming that the force on micro-element P ij is proportional to its IUCT, the F Bti and F Bri are expressed as: The equations for the cutting forces in i section are as follows: The equations for tangential cutting force (F t ) and radial cutting force (F r ) in UPDM are expressed as: The feed rate of the tool rake in any sections is v f /n and unrelated to the depth of cut (ap), while the feed rate of the tool flank in any sections is not only related to v f /n but also to the ap, such that: where cosα ij is not related to the selection of i section, so that cosα j = cos α ij , thus:

Milling tests
The equipment used for the UPDM is the Freeform TL ultra-precision single point diamond nano-machine from Precitech, USA. The dynamometer used for the experiments is the Kistler 9119AA1, which is installed on the Freeform TL as shown in Fig. 8a. The cutting force measurement system is shown in Fig. 8b; the cutting forces are finally obtained after a series of processing by the measurement system. The equipment used for surface measurements is the white light interferometer with model Bruke Contour GTX, from USA.
The material of the workpiece used in the experiments is brass, as shown in Fig. 9. The material parameters are shown in Table 1; the cutting parameters are shown in Table 2 with numbers 1 to 10. In order to visualize the effect of the tool rake and the tool flank on the machined surface during cutting, the experiment with number 11 in Table 2 was completed, which the v f /n and ap are several times greater than the other groups.

Extraction of cutting forces
The force obtained from the dynamometer is defined based on the coordinate C e , while the modeling is based on coordinate C d ; thus, a transformation of the force is required, and the specific transformation process is shown in Fig. 10. The overall equation is: After transformation, the F r , F t , and F a are expressed as:

IUCT under the tool flank
In a complete cycle, the interval under the tool rake cutting is not 0.5π to 1.5π, as shown in Fig. 11. The v f /n is positively correlated with the interval, but the ap is negatively correlated with the interval. The position where the peak IUCT under the tool rake cutting is not π, because the component w yti of the rotational speed w along the y ti direction is 0 when P i0 rotates to position π, as shown in Fig. 4. After that, w yti increases gradually along −y ti , but the IUCT is still increasing in the area where w yti < v f ; thus, v f is positively correlated with the position of the peak IUCT under the tool rake cutting. From curves (2), (3), Fig. 8 The equipment for cutting and measuring: a installation of the workpiece and the dynamometer and b the cutting force measurement system  and (4), it can be concluded that the peak IUCT under the tool rake cutting is entirely dependent on v f /n. In Figs. 12 and 13, it can be found that the IUCT under the tool flank interference is positively correlated with the v f /n and ap, because v f /n is positively correlated with h Bj , and ap is positively correlated with l B ; according to Eq. 15, v f /n and ap are positively correlated with h Bt and h Br . Similarly, comparing the peaks of curves (2) and (7), it is clear that the IUCT is positively correlated with v f /n. Therefore, the IUCT under the tool flank interference is determined by the feed per tooth and the depth of cut in UPDM.
In order to accurately analyze the trend of IUCT in a complete cycle, curve 8 was selected for further analysis, and the interference interval was divided by radians, as shown in Fig. 14. From ω 2 to φ 1 , there is neither direct material removal under the tool rake cutting nor the tool flank interference. At position φ 1 , the tool rake enters the cutting area, while the tool flank does not. At position ω 1 , the l B and h Bj start increasing from zero due to the tool  From σ 4 to φ 2 , l B remains constant; the h Bt and h Br enter a downward trend because h Bj starts to decrease. At the position φ 2 , there is no direct material removal under tool rake cutting at all, while the tool flank interference still exists. From φ 2 to ω 2 , l B enters a downward trend and h Bj is still decreases; that is, the h Bt and h Br decrease rapidly without any slowing down areas; the reduction rate is slightly larger than the previous interval (from σ 4 to φ 2 ), and v Bt fluctuations are also at a higher level. It should be noted that at the position 2π, h Bt is still not reduced to 0 because the tool flank interference still exists. Until to position ω 2 , the tool flank interference disappears.
In order to observe more intuitively the milling state of the tool during a whole cycle, the original surface morphology of the machined workpiece in experiment 11 was extracted, as shown in Figs. 15a and 16a. The marked area 1 is the interference area between the tool flank and the workpiece; the surface of the workpiece is clearly marked by frictional extrusion, while in the direct material removal area 2, the surface is smooth, which is especially clear in the coloring map in Fig. 15b. Figure 16b shows the contour map, which can clearly show the direct material removal trajectories under tool rake cutting and the tool flank interference trajectories; the i section is taken for description, and the approximate trajectories are marked. The figures show that the initial position of the direct material removal area is φ 1 instead of 0.5π and the end position of the direct material removal area is φ 2 instead of 1.5π. The tool flank interference trajectories always appear after the direct material removal trajectories under tool rake cutting. The position of the maximum h A is φ 3 instead of π. According to Fig. 7 and Eq. 7, the area enclosed by the trajectories of the P i0 , P ij , and the angle α ij is the IUCT under the tool flank in i section; it can be more intuitively seen that σ 4 is the position of the maximum IUCT under the tool flank. The analysis reveals that the results of the surface measurements are consistent with the simulation.

Cutting forces
The F t and F r are shown in Figs. 17 and 18. According to curves (1), (3), and (5), the F t and F r are positively correlated with the v f /n. According to curves (2), (3), and (4), because ap is positively correlated with h Bt and h Br , according to Eq. 26, the F t and F r are positively correlated with ap, while it is important to note that the trend of h A is almost the same at a constant v f /n. The trend of F t (or F r ) is approximately overlapping under the same ap and v f /n (the curves cannot overlap exactly due to noise interference in the experiment), as shown in Fig. 19, because the trend of h A , h Bt , and h Br is completely overlapping; according to Eq. 14, F t (or F r ) should have the same trend in theory. Thus, the cutting forces are determined by feed per tooth and depth of cut, and it is dominated by IUCT under the flank interference when ap affects the cutting forces. The results show the correctness in constructing the theoretical model of the tool flank interference.
In order to better observe the cutting forces' detail characteristics, the cutting forces of one cycle are extracted for data comparison. Since the peaks of F t and F r in each cycle are not approximately symmetric, indicating their low The cutting forces should be zero in the interval from ω 2 to φ 1 , but milling vibration, tool wear, and uneven workpiece shape cause the experimental data to be noisy from the forces, as shown in Fig. 20. From φ 1 to ω 1 , h A dominates the rise of F t , but F r remains constant, indicating that F t is sensitive to h A , while F r is not. In the interval from ω 1 to σ 1 , F t does not follow the h Bt rises rapidly; instead, there are oscillations appearing when the growth rate of h A slows down, and F t continues to be dominated by h A . However, F r follows h Br increases in   the interval from ω 1 to σ 1 , indicating that F r is sensitive to h Br , and F r is dominated by h B . From σ 1 to σ 2 , F t and F r follow the trend of h Bt and h Br , h Br increases sharply in the interval and induces sudden changes of F r , and the cutting forces are dominated by h B . From σ 2 to σ 3 , the cutting forces are still dominated by h B ; h Bt increases sharply and induces sudden changes of F t . It is important to note that the increase rate of F t is unchanged compared with the previous interval (σ 1 to σ 2 ) and is not consistent with the change of h Bt , because h A is in a decreasing trend, which affects F t . However, due to the insensitivity of F r to h A , F r is consistent with the change of h Br .
From σ 3 to σ 5 , F t and F r follow the trend of h Bt and h Br , respectively, where the cutting forces are still dominated by the h B . Note that h A has decreased to 0 at position φ 2 , but it does not appear that h A has affected the trend of F t and F r . At position σ 5 , F t has dropped to 0, which is inconsistent with the trend of h Bt at this position; it indicates that F t is extremely insensitive to the small h B . However, F r is still in a decreasing trend after σ 5 , which is consistent with the trend of h Br , indicating that F r is extremely sensitive to h B .
In order to investigate the applicability of cutting forces' intervals and trends, the variations of cutting forces are analyzed when the v f /n is increased. The intervals and trends are consistent with the above analysis, as shown in Fig. 21. When the ap is increased, the intervals of F t are consistent with the above analysis, while the positions of σ 1 , σ 2 , σ 3 , and σ 4 of F r undergo different degrees of forward shift, and h Br induces multiple sudden changes of the F r , as shown in Fig. 22. The positions σ 1 and σ 2 are the starting points of the violent fluctuation of v Br , and the positions σ 3 and σ 4 are the starting point and ending point of the gentle fluctuation interval of h Br , respectively. F r always follows the fluctuation of v Br to respond quickly, while the response of h Br is lagged, which is because the sensitivity of F r to h Br is positively correlated with the ap, leading to the fact that F r always shows the interference relationship between the tool flank and the workpiece before h Br .

Conclusion
This study focused on the analysis of cutting forces in ultraprecision diamond milling (UPDM), considering the specific phenomena occurring in the case of interference between the tool flank and the workpiece. The investigations conducted involved the evaluation of instantaneous uncut chip thickness (IUCT), as well as the cutting forces under the tool flank interference. The proposed model was used to analyze the relationship between the cutting forces and the IUCT, and many kinds of designed tests were carried out for the validity of the theoretical results. The following conclusions were drawn: (1) The combined action of the tool rake cutting and the tool flank interference would affect the surface generation. The tool flank interference would take place to deform surface generation in UPDM. The material was removed by squeezing under the tool flank interference, and workpiece surface is clearly marked by frictional extrusion, while in the direct material removal area, the surface is smooth. (2) The tool flank interference would induce sudden changes of the IUCT and further affect cutting forces. The sudden changes of F r occur before that of F t and are more obvious, because F r is not sensitive to the IUCT under the tool rake cutting, and it is consistent with the trend of the IUCT under the tool flank interference, while F t is affected by the IUCT under the tool rake cutting. cutting forces. In other cases, the IUCT under the flank interference determined the cutting forces. F t could better reflect the trend of determined IUCT, while F r had little correlation with the IUCT under the tool rake cutting, even negligible. However, F r showed extremely high sensitivity to the IUCT under the tool flank interference, and even revealed the degree of the tool flank interference before h Br .
Author contributions All authors contributed to the study conception and design. Conceptualization, data collection, and analysis were performed by Tongke Liu. Investigation and software were performed by Wei Peng. Material preparation and formal analysis were performed by Zhiwen Xiong. Resources, validation, and writing-review and editing were performed by Shaojian Zhang. The first draft of the manuscript was written by Tongke Liu, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Funding
The work was partially supported by the National Natural Science Foundation of China (grant no. 51405217).
Data availability All the data have been presented in the manuscript.
Code availability Not applicable.

Declarations
Ethical approval The paper follows the guidelines of the Committee on Publication Ethics (COPE).

Consent to participate
The authors declare that they all consent to participate this research.