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Prediction of cutting force is one of the most important steps toward accurate simulation of each machining process. Predicted cutting forces can be used to evaluate the required power and torque of machine tool, dimensional accuracy of workpiece, and vibratory characteristics of the whole machining system. Also, it can be applied to design an appropriate structure for the machine tools to satisfy specific purposes. According to the mechanistic approach, cutting force components are functions of chip thickness, width of cut, and cutting constants. Chip thickness and width of cut are geometric parameters and they depend on geometry of cutter and geometry of engagement between cutting tool and workpiece. In contrast, the cutting constants are functions of tool and workpiece materials as well as tool geometry and cutting conditions; therefore each force model must be calibrated in order to identify the cutting coefficients.
Two different calibration approaches for mechanistic force modelling have been reported in the earlier researches. The first approach is the unified method which relies on an experimentally established orthogonal cutting database. Budak, Altintas and Armarego (1996) presented a force model for the flat end milling, in which the force coefficients were obtained using orthogonal to oblique transformation. Earlier, Yang and Park (1991) developed a force model for ball-end milling. They obtained cutting coefficients from orthogonal end turning. Altintas and Lee (1996), Merdol and Altintas (2004), and Engin and Altintas (2001) all evaluated the varying rake face friction, pressure distribution, and the chip flow angle in peripheral milling in order to provide accurate cutting force predictions. Larue and Anselmetti (2003) and Altintas (2000) have demonstrated that the milling force coefficients could be obtained from orthogonal cutting tests with oblique cutting analysis and transformation.
The second approach is direct calibration method which determines the milling force coefficients directly from milling tests for the specific cutter-part combination. Kline and DeVor (1983) applied this approach to develop a mechanistic cutting force model for flat-end milling. Azeem, Feng and Wang (2004) employed a simplified approach and developed a rigid mechanistic cutting force model for the complicated ball-end milling process. With respect to the ball-end milling force models, Wang and Zheng (2002) presented a work based on decomposing the elemental cutting forces into shearing and ploughing components in ball-end milling. Meng et al. (2004) conducted a series of face milling experiments to study the relationship of the derived cutting force coefficients with the relevant cutting parameters from the measured forces. Jayaram, Kapoor and DeVor (2001) estimated the cutting force coefficients for face milling applications with multiple inserts from the Fourier transform of the measured force signals at the zero frequency interval. All of the research studies referenced above require a large number of calibration test cuts to determine the empirical cutting force coefficients. The average measured forces were mostly used in the calibration processes.
Shin and Waters (1998) proposed an effective procedure to reduce the required number of experiments in determining the cutting force coefficients for face milling inserts using the effective chip thickness to generalize the effects of feed per tooth and axial depth of cut. Yun and Cho (2000) determined the cutting force coefficients for flat end milling based on the synchronization of one reference cutting test with the measured force signals. The cutting mechanics parameters were kept constant and the size effect in metal cutting was not considered.
Due to complicated geometry of cutting edge, direct calibration of the milling force models still requires a large number of calibration cuts. A new approach is proposed in this paper to determine the cutting force coefficients for milling processes from only a single test cut. Instantaneous cutting forces are used instead of average forces to obtain the empirical force coefficients. This single experiment significantly reduces the time and effort needed to determine the coefficients in order to cover a wide range of cutting conditions. In the next section, a brief description of the geometric model for serrated tapered ball-end mill is presented followed by detailed description of the calibration procedure and the experimental work. Finally the measured forces from the verification test cuts with different cutting conditions are applied to calibrate cutting force coefficients and validate the proposed method.
2 Geometric modeling of serrated tapered ball-end mill
Prediction of the cutting force requires identification of instantaneous chip thickness and cutting edge geometry. Taper ball-end mills have a varying geometry along the contact length between the tool and workpiece. The geometry of the taper ball-end mill is shown in Figure 1. The tool has a cutter radius . The tool envelope surface consists of a hemisphere surface and a cone surface which is a function of the half-apex taper angle , as shown in figure 1.
In equation 1, , , and are the tool radius at the ball end part, tool radius at taper part, and variation of radius due to serrations along the axis of the cutter, respectively. and are defined as:
where, is the ball radius and is the radius of the ball part at elevation .
Figure 1 Geometry of serrated tapered ball-end mill: (a) Definition of the geometric features and (b) The 3-axis tool coordinate system
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In figure 1(b), and are axial and radial depths of cut. Coordinates of a point along the cutting edge, cutting force components acting on point and geometry of tool tip similar to that of Altintas and Lee (1995, 1996), Merdol and Altintas (2004), and Ehmann et al. (1997) are presented in figure 2. In figure 2, represents the tangent direction to the first flute at the tool tip, and is perpendicular to . The cutter rotation angle is measured clockwise between axis and the direction of . For a cutting edge point with distance from the tool tip, the lag angle is defined as the angle between and the direction of .
Figure 2 Coordinates of Point along the cutting: (a) Isometric view of point and cutting force components acting on point and (b) Tool tip geometry
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The lag angle is determined as:
It should be noted that in a multiple flute ball-end mill, some of the flutes may not be continuous around the tool tip due to the sharpening process. As shown in Figure 2(b), for a ball-end mill tool with five flutes, the starting point for some of the flutes has a radial deviation from the tool tip. After defining the geometry of cutting edge, the angular position of a cutting edge point on the flute at height , measured from axis in clockwise direction as shown in Figure (2b), is given by:
where, is the number of flutes, is flute index and is obtained from Equation 3. The local coordinates of the cutting edge point can then be determined by:
More specific geometric definition can be found in Altintas and Lee (1995, 1996), Merdol and Altintas (2004), and Ehman et al. (1997).
3 Modeling of instantaneous uncut chip thickness
The chip load model determines the instantaneous undeformed chip thickness distribution along the cutting edges. Figure 3 demonstrates the progression of the undeformed chip thickness for a general three-axis milling situation. The uncut chip thickness is defined as the distance between the path generated by the current intersecting tooth and the exposed workpiece surface generated by the previously passing tooth. In three dimensional machining, as depicted in Figure 1(b) and 2(a), represents the current tool position and orientation, represents the previous tool position and orientation (before one feed per tooth), and represents the elemental cutting edge position determined from equation 5. The undeformed chip thickness is the distance between and .
Figure 3 Illustration of current and previous tool position to create an instantaneous undeformed chip thickness
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Since the process faults (run-out), flute deviation, and flute chipping/breakage could alter the undeformed chip thickness in end milling, the equations developed by Kline and DeVor (1983), and Sutherland and DeVor (1986) are extended here. Assuming that represents the parallel axis offset run-out, is the locating angle for , is the axis tilt angle, and is the locating angle for , the equation for determining the undeformed chip thickness under the presence of process faults is:
where the angle is given by:
and the effective radius for each disc is defined as:
where, is the nominal radius of the cutter, is the lag angle determined from Equation 3, is effective length of the cutter and is the individual flute deviation. After the undeformed chip thickness is determined for each engaged elemental cutting edge, the differential cutting forces can be computed by assuming an oblique cutting process.
4 Formulation of a mechanistic cutting force model
Using three different perspectives of the 3-axis coordinate system of , , and as illustrated in Figure 1, a taper ball-end milling cutter can be divided into a finite number of disc elements along the z-axis. The total force components along , , and acting on a flute at a particular instant are obtained by numerically integrating the force components acting on each individual disc element. The total or resultant force acting on the cutter/workpiece interface at any given time during machining is determined by summing the independent forces of all the flutes engaged in cutting at any time instant.
The cutting force acting on the rake surface of a disc element is divided into two orthogonal components: the normal pressure force , and the frictional force . These can be obtained using the following equations:
where, is the differential normal cutting force and is the differential friction force. is perpendicular to the rake face and is along the direction of chip flow. The specific normal cutting coefficient is , while represents the friction coefficient. The parameters and are experimentally determined and are mainly dependent on the local uncut chip thickness, cutting velocity and normal rake angle, according to the work of Yucesan and Altintas (1996) and Chandrasekharan, Kapoor and DeVor (1997). The uncut chip thickness for each engaged elemental cutting edge is mentioned in Equation 6. The elemental oblique cutting process has a cutting width of . The cutting velocity is given by where is the angular velocity of spindle rotation. In mechanistic modeling approach, the differential cutting forces on the rake face are assumed to be proportional to the chip load. The elemental cutting forces on the rake face may be transformed to the local tool coordinate system to obtain the elemental tangential, radial, and axial forces , , and (the directions of these forces are shown in Figure (2a)) respectively. The transformation is given by Kapoor et al. (1998):
where is inclination angle equal to the local helix angle, is normal rake angle, and is chip flow angle that can be related to the local helix angle as:
In equation 13, is the chip flow coefficient which can be determined from the experimental data. The differential cutting forces on the global workpiece coordinate system are then determined by rotations through:
where the orientation is a function of the distance and angle to point followed by and which are minimum and maximum limits of chip thickness area indicated in one full rotation , and is transformation matrix.
The total cutting forces are thus the summation of differential cutting forces for all those engaged cutting edge elements. The above mentioned force model can be used to predict the instantaneous cutting force along the axial depth of cut and angle of tool rotation. The proposed approach is based on discretizing the cutting tool along the axial depth of cut and for each 1 degree of tool rotation. The tool/workpiece instantaneous contact geometry is first determined, and then the instantaneous cutting force for each degree of rotation will be calculated as a function of force coefficients . The measured cutting force components are then correlated to the calculated force components to predict the force coefficients .
5 Calibration approach
The cutting force coefficients that are used in equations 10 and 11 for taper ball-end milling are namely and . These coefficients need to be determined in order to calculate the cutting forces using the developed cutting force model. Direct calibration of this mechanistic cutting force model was based on average forces and a large number of experiments for accurate model calibration based on El-Mounayri et al. (1997), Zhu, Kapoor and Devor (2001), and Azeem, Feng and Wang (2004). The two domain representations were employed in which the cutting edge elements on the cylindrical coordinates were geometrically the same. These also exhibited uniform cutting characteristics, while those on the ball-end section were geometrically different from each other and this resulted in varying cutting characteristics. The size effect parameters were always assumed to be constant for the specific cutter-part combination. Large number of tests with different feed, speed, and rake angle are needed to perform in this case. The present work aims at reducing the large number of experiments required for accurate model calibration of the taper ball-end milling process. Only a few experiments with different emerging ratios and axial depth of cuts are needed to determine the cutting force coefficients. Those predicted coefficients are valid over a wide range of cutting conditions. The cutting force coefficients and are calculated as lumped discrete values along the cutter axis whereas the size effect parameters can be calculated as constants.
Instantaneous cutting forces are to be measured and used for the numerical calibration procedure. Since the instantaneous cutting force is used, the tangential and radial cutting force coefficients cannot be decoupled. Usually, previous calibration methods (Azeem, Feng and Wang, 2004), include the instantaneous effect of the tool rotational speed, one cutting edge is engaged in cutting at any particular cutter orientation during the half slot cut. In the present work a cutting edge initially engages the work material from the cutter free end and as the cutter rotates, the upper portion of the cutting edge is gradually engaged until it reaches the tapered section, i.e., the selected axial depth of cut. Once the full depth is engaged with the cutting edge, the engagement continues until the cutting edge gradually starts disengaging from the workpiece. Similar to the gradual engagement phase, the disengagement phase also starts from the cutter free end. The specific cutter orientations for the engagement and disengagement phases depend on the cutter helix angle and the associated lag angles. The lumped discrete values of the cutting mechanics parameters are determined by dividing the cutting edge along the axial depth of cut into a finite number of discs. These discs are used to calculate parameters such as the lag angle for the cutting edge segment of each corresponding disc. Lag angle calculations are based on the distance of each disc from the cutter free end or non-engaged tooth, the total number of flutes, and the helix angle of the tool. This angle gives the cutter orientation at which the cutting edge segment is first completely engaged with the workpiece. The first disc at the cutter free end starts engaging the workpiece and the remaining discs are gradually engaged until all the discs are engaged simultaneously. In the present work, the tool was separated into discrete segments that have a height of 0.01 times axial depth of cut. Each segment width is divided into small cutting elements with an angle of 1 degree. The elemental cutting forces are calculated for all the cutting elements of each disc and the sum of these elemental forces represents the total force acting on the corresponding segment. Using equations 10 and 11, the elemental tangential and radial cutting forces for the first cutting element on the first disc can be expressed as:
where and are the lumped discrete values of and for each angle , and is the undeformed chip thickness at this instant at the cutter orientation angle . The elemental tangential and radial cutting force components are resolved into the , , and workpiece global directions as in equation 14 for each angle . The cutting force coefficients and are taken as constant lumped parameter values for each angle of rotation but vary from one angle to another. The , , and cutting forces acting on the cutter can be formulated for each cutter orientation angle when the corresponding cutting edge segment is first completely engaged with the workpiece. Equating these force expressions with the measured instantaneous cutting force data, the resulting force equation can be expressed in a matrix form as:
where, , , and are the measured instantaneous cutting forces in the , , and directions at the specific cutter orientation angles corresponding to the first complete engagement of each segment. The decoupling of the lumped discrete values of and for each segment is clearly shown by the matrix in equation 18. This greatly facilitates the numerical procedure for best fitting of the cutting force coefficients and enables them to be accurately determined from the instantaneous cutting force data. The present direct calibration approach uses instantaneous cutting forces to determine the empirical cutting force coefficients. Accurate representation of the helical cutting edge profile is essential when dealing with the instantaneous forces. The cutting edge profile on the taper cylindrical part of a ball-end mill is not the same as that on a flat end mill and it can be arbitrary along the whole edge starting from the ball tip, and thus, varies from cutter to cutter. A typical design is to project the helical cutting edges extended from the taper part onto the hemisphere of the ball part. For a taper ball-end mill with a constant helix angle , the lag angle at a distance from the cutter free end can be expressed as in equation 3. In addition, the instantaneous chip flow angle at any rotational angle can be determined from the ratios of the measured forces for an arbitrary design of the cutting edge profile. A coordinate measurement machine (CMM) can be used to trace and establish the cutting edge profile. The associated procedure includes measuring the coordinates of cutting edge points at uniform intervals along the cutter axis.
6-1 Cutting tool and workpiece material
The calibration tests were performed using a three-axis horizontal CNC milling machining center. The workpiece material of Titanium (Ti4Al6V) and a right-handed five-flute carbide taper ball-end mill of diameter 12.7mm were chosen for the experiments. The taper ball-end mill has a taper angle of 5 degrees, normal rake angle of 9 degrees on the tapered section and a normal rake angle of 2 degrees on the ball part of cutting edge. The helix angle on the cylindrical part of cutting edge is constant and equal to 26 degrees. Figures (4a) and (4b) demonstrate the SEM image of tool tip and serrated cutting edge. The images were obtained by a JEOL-6400 scanning electron microscope (SEM).
Figure 4 (a) SEM image of ball-end mill tool tip and (b) cutting edge geometry
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As it can be seen from figure 4(a), four flutes are not continuous near the tool tip. The distances between their starting points and the tool tip which are presented by , , and can be found in table 1.
Table 1 Measured distances for ball-end tool tip geometry
The height profile was obtained with a JEOL-733 electron microprobe. Figure 5 shows the obtained profile for cutting edge at the tapered part of the tool. This profile was used to predict actual area of contact when calculating the instantaneous cutting force.
Figure 5 Graphical representation of serrated profile of a section of the cutter
6-2- Cutting Conditions
Multiple tests were conducted with coolant, at spindle speeds ranging from 150-700 rpm. A table feed rate of 0.0254-0.127 mm/tooth was employed to reduce the machine vibration. The instantaneous cutting forces in the , , and directions were measured with a Kistler 9255B dynamometer at a sampling rate of 707 samples per second to achieve an average of 78 data points per revolution for a minimum of 15 complete cutter rotation cycles of cutting force data.
Figure 6 Machining test with varying parameter changes during a single pass
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Multiple tests were designed; Figure 6 represents one of these tests during the design phase. This test encompasses four different segments, each with a changing variable. The first being an increasing axial depth of cut, stage two and three having a varying feed rate and spindle speed, respectively, while in the fourth, an increasing radial depth of cut is sought after. Previously presented by Merdol and Altintas (2004), a serrated undulated profile on cutters resulted in a more stable roughing operation to avoid chatter vibrations that cause tools to fail under varying loading conditions. The proposed model by Merdol and Altintas (2004) has been integrated to our comprehensive simulation program and validated experimentally in milling Titanium using the cutting parameters listed in table 2.
Table 2 Cutting conditions for step cut tests at an ADOC of 10.5 mm and a RDOC of 2 mm.
Spindle Speed (rpm)
0.0254- 0.0508- 0.0762- 0.1016- 0.127
Sections of data were taken from figure 7, in intervals of the beginning, middle and end of each feed rate then combined together to achieve an accurate average for each section of the plot.
Figure 7 Measured force components and over a range of feed rates (0.0254-0.127 mm/tooth) at constant spindle speed of 150 rpm.
In figure 7 the averages have been mathematically filtered using MATLAB to eliminate any frequency greater than that of the machining frequency average of 45 Hz. This reduces the effects of any exterior frequencies that would possibly alter the recorded signal. Once this data is separated and filtered, the information is put into a text file and used as an input file for the calibration model. The tool geometry is also entered into a separate file and put into the calibration model. The model results are shown in table 3.
Table 3 Results of model calibration
The top row of in table 3 shows the highest and lowest amounts of chip thickness , highest and lowest velocities , and highest and lowest rake angles. These parameters are varied along the cutting edge due to taper angel and serration. Variation of uncut (undeformed) chip thickness in one rotation of cutting tool for two different spindle speeds is illustrated in figure 8.
Figure 8 Instantaneous uncut chip thickness in a complete rotation for two different spindle speed: (a) 150 rpm and (b) 700 rpm
Unique tool geometry made from different flute grindings commonly create cutter run-out and as a result, unique profiles. The surface generated by the flutes at each time instant is calculated and then subtracted from the previously cut surface in the radial direction where the chip thickness is defined. As a result, the kinematics model allows the accurate identification of chip thickness and surface finish generated. The simulation model considers the true kinematics of milling as previously presented by Engin and Altintas (2001) and Merdol and Altintas (2004).
Cutting edge force coefficients for the tangential and normal directions are identified, while the axial direction for coefficients are known to be very small in oblique cutting and can be taken as zero. At some locations and often at larger radial depth of cuts, teeth share the total amount of chip to be removed in one revolution of the cutter. However, at often much lower radial depth of cuts, only one tooth removes material while others do not even touch the workpiece. In the presented cases in figure 8(a) and 8(b), it can be clearly seen that all teeth are in contact with the workpiece and each flute has a different run-out. Comparing Figure 8(a) with Figure 8(b), the alternating speeds of 150-700 have no significant impact on uncut chip thickness. Assuming an average value for the cutting force coefficients and fitting them to experimental data using a linear regression analysis, each coefficient is expressed as an exponentially decaying function of chip thickness and cutting speed. The coefficients having constant feed per tooth are then combined and averaged for the speeds of 150, 300, 540, and 700 rpm, respectively. Figures 9 and 10 show the results of the calibration procedure for cutting coefficients. From the calibration data, the decaying trend of average coefficients is apparent with increasing velocity (m/min). This work shows a consistent correlation with the work of Bailey et al. (2002). Figures 9 and 10 demonstrate the trends of and for two different feeds (0.0254 mm/tooth and 0.127 mm/tooth) respectively.
Figure 9 A decreasing trend of and coefficients at a feed of 0.0254 mm/tooth while cutting speed increases
Figure 10 A decreasing trend of and coefficients at a feed of 0.127 mm/tooth while cutting speed increases.
At lower feeds (figure 9) the fluctuation in the normal cutting force coefficient is remarkable. Poor machinability of titanium and large axial depth of cuts are proven to produce smaller chip loads and this is often the cause of this fluctuation. Comparing figures 9 and 10, there is greater accuracy in the profile of data when higher feed (0.127 mm/tooth) is applied. A more distinguishable decay of force coefficients with increasing in velocity can be seen in Figure 10.
As it has been illustrated in figures 11 and 12, and are highly dependent of the increase and decrease of spindle speeds. Coefficients and decrease rapidly by increasing spindle speed.
Figure 11 Effect of chip thickness on at different speeds (feed = 0.127 mm/tooth)
Figure 12 Effect of chip thickness on at different speeds (feed = 0.127 mm/tooth)
Generally, the coefficients remain constant for one revolution of cutter. The cutting condition is the major factor affecting the uncut chip thickness, but the cutting coefficients do not vary according to the cutting condition. Effects of the cutting condition have to be reflected in uncut chip thickness and not in cutting force coefficients. In figure 13, however, it can be seen that departs slightly from that constant level in several regions. This represents a size effect, whereby specific cutting forces become large at a low uncut chip thickness (Armarego and Brown, 1969). Figure 13 also shows that becomes large at a low uncut chip thickness, which again shows the size effect. To avoid complexity, size effects were not considered in this study.
Figure 13 Instantaneous uncut chip thickness, and values at 150 rpm
As stated by Yun and Cho (2000), the discrepancies are caused by size effects that occur at low uncut chip thicknesses. At the peak regions, it is clear that the predicted forces are slightly larger than those measured, since tool wear was not considered. The measured peak values for all the forces are small, owing to the effect of tool wear.
A new method of estimating the cutting force coefficients from a single cutting test was presented. Instantaneous cutting forces are to be used instead of average forces to calibrate the empirical force coefficients. The method involves the calculation of uncut chip thickness and synchronization of the cutting coefficient forces. The effectiveness of the proposed method was verified by comparing the estimated cutting coefficient forces with the model calibrated values. The simulation model presented in this paper was reformulated so that the cutting force coefficients account for the effects of feed rate, cutting speed, and a complex cutting edge design. Experimental results were presented for the calibration procedure. This method predicts three-dimensional cutting force coefficient components with the same accuracy as the existing methods.