Modeling Chip-Evacuation Forces and Prediction of Chip-Clogging in Drilling

Transcription

1 Jeffrey C. Mellinger Graduate Research Assistant O. Burak Ozdoganlar Post Doctoral Research Associate Richard E. DeVor Professor Shiv G. Kapoor Professor Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL Modeling Chip-Evacuation Forces and Prediction of Chip-Clogging in Drilling One of the fundamental difficulties of the drilling process is the evacuation of the chips from the drilled hole. As the hole depth increases, the chips tend to cluster together and clog the flutes, causing increased forces, poor hole quality, elevated drill temperatures, and drill breakage. In this paper, a model for chip evacuation has been developed to predict the force and torque arising from the evacuation of the discontinuous chips. The model considers the pressure on a differential chip section being created by the forces required to push the chips out of the hole. The two coefficients of friction required by the model are established via a calibration procedure. The effectiveness of both the calibration and the force models has been assessed via a set of validation experiments. The model can be used to predict the depth when chip-clogging occurs, indicating the need for a pecking cycle, and the depth where the drill experiences an excessive amount of torque, which may result in drill breakage. DOI: / Introduction Chip evacuation represents one of the fundamental difficulties associated with the drilling process because the chips generated at the cutting lips are confined by the hole wall and the drill flute. As the depth of the hole increases, an increased amount of chips fill the flutes, leading to chip-clogging, and eventually to tool breakage 1,2. For deep-hole drilling, pecking periodic drill retraction is one method used to alleviate chip-clogging. While effective for this purpose, pecking greatly reduces productivity. The determination of proper pecking cycles becomes a critical factor in process planning. Therefore, understanding the factors affecting chip evacuation and predicting the occurrence of chip-clogging becomes important, especially when drilling larger depth to diameter ratios. Once the entire drill is engaged i.e., when the depth exceeds the drill-point height, the torque and thrust force reach steadystate values arising from the cutting process. For the drilling processes that produce discontinuous chips, such as those performed on materials of low ductality, e.g., cast aluminum alloys, it has been observed that the torque and thrust continue to increase with increasing depth. As the process progresses, the rate of increase increases until the torque exerted on the drill exceeds its torsional limit, causing drill breakage 1. This phenomenon has been attributed to chip-clogging, and constitutes the major limitation of the drilling process as it increases the forces during the process, increases the drilling temperature, lowers the quality of the hole, and accelerates tool wear and breakage 3. To avoid these adverse effects, researchers have sought to detect and control the initiation of the chip-clogging. Using the fact the cutting forces reflect the development of chip-clogging, control schemes that monitor the cutting forces especially the torque have been employed. These schemes call for the alteration of the cutting conditions to compensate for the torque increase 1,2,4 6, which usually involves decreasing the spindle speed and/or the feed. Another method used to detect the onset of chip-clogging is the use of drill temperatures 7. Although the drill temperatures gradually increase with hole depth when there is no chip-clogging, in the presence of chip-clogging, the rate of the temperature increase has been observed to rapidly grow. The depth where the torque increases more abruptly has been Contributed by the Manufacturing Engineering Division for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received March 2001; revised December Associate Editor: J. Hu. considered as the point of flute clogging and used as a performance criterion for chip-evacuation capability. This provides the means to make comparisons of the chip-evacuation performance of different tooling parameters drill geometry, substrate material, coating and cutting conditions 8. To improve the evacuation of the chips, the mechanism responsible for moving the chips through the flutes must be understood. Although a fair amount of experimental research exists, only a few works have attempted to model chip evacuation. One method to model chip evacuation through the flutes is with a kinematic approach 9. This approach considered the discontinuous chip as a particle, and applied a force balance to this particle when it is moving through the flutes. The differential equation is solved to predict the chip velocity, which can be used as a performance criterion considering that a faster moving chip has a better chance of leaving the drilled hole without initiating clogging. This model can be used to compare the chip velocities of different drill geometries and cutting conditions, however, it cannot be used to predict the onset of chip-clogging, since the model is independent of depth. Chip velocity is difficult to measure experimentally, so only qualitative validation experiments were performed to qualitatively verify the chip velocity model. The objective of this work is to develop a model that is capable of predicting the chip-evacuation force and torque in drilling with discontinuous chips. The purpose of the chip-evacuation model is to establish a relationship between the hole depth and the chipevacuation force/torque as a function of the process parameters and flute geometry. The model can be used to find the depth where the drill is susceptible to drill breakage, and the location where a pecking cycle is warranted because of the development of flute clogging. 2 Modeling Chip Evacuation The model presented here is concerned with drilling processes that produce discontinuous chips of considerably smaller dimensions when compared to the flute cross-sectional area, such as those produced when drilling cast aluminum alloys or cast iron. Thus, the chips can be classified as granular solids, which are characterized as a group of particles that are of roughly the same size 10. Granular solids have characteristics of both fluids and solids. They occupy the shape of the container in which they fill, exert pressure on the container boundaries, and flow through openings like fluids. However, like solids, they possess cohesive strength, are capable of having nonisotropic stress distributions, Journal of Manufacturing Science and Engineering AUGUST 2002, Vol. 124 Õ 605 Copyright 2002 by ASME

2 Fig. 3 Straight-fluted drill during cutting Fig. 1 Typical plot of the torque varying with depth and have shear stresses that are proportional to the normal stress 10. The model presented here, in parallel to the properties of the granular solids, assumes that the chips 1 exert pressure on the flute surface of the drill body and hole walls, 2 flow through and fill the flutes, 3 remain in contact with each other, 4 exhibit nonisotropic stress distributions, and 5 have friction forces proportional to the normal forces. Modeling the movement of granular solids has been investigated in the solids conveying zone of a screw extruder that is used for polymer processing The techniques employed in solids conveying to model the movement of granular solids will serve as the foundation for the chip-evacuation model presented in this paper. In the following analysis, a straight-fluted drill is used to reduce the geometric complexity that arises from the helix angle of twist drills. Straight-fluted drills are generally used in horizontal drilling applications for materials that produce broken chips such as cast iron or cast aluminum. These drills are chosen to produce better roundness and straightness properties than those of twist drills for moderate to high depth-to-diameter ratios. 2.1 Cutting and Chip-Evacuation Forces. The force increase experienced during drilling as a function of hole depth can be considered as representative of the chip-evacuation performance. Figures 1 and 2 illustrate how the torque and thrust change with the nondimensional depth-to-diameter ratio for a typical drilling process that exhibits discontinuous chips. The initial rapid increase of the forces reflects increasing cutting forces due to the engagement of the drill point. Once the drill point is fully engaged, the thrust and torque reflect those that are required for chip formation, which will be referred to here as the cutting force and cutting torque, respectively. It can be observed from the figures that the thrust and the torque keep increasing after this point. Considering the cutting forces to be constant, the force increase can be attributed solely to the higher force required to move the increasing amount of chips that fill the flutes. It is natural, then, to refer to this force as the chip-evacuation force, and associated torque as the chip-evacuation torque. This chip-evacuation force acts in the direction of the drill helix angle. At some depth, the chip-evacuation force and torque begin to experience an increasing rate of increase, which will be referred to as the critical depth. This is the depth that defines the onset of chip-clogging, and if the process continues, the excessive chip-evacuation force and torque can cause the torsional capacity of the drill to be exceeded, causing drill breakage. 2.2 Chip-Evacuation Force Model. The behavior of the chip-evacuation force can be explained by considering the pressure distribution and associated force balance on a section of chips in the flute. Figure 3 shows the straight-fluted drill in the workpiece and the differential chip section used for the force balance. The drill cross-section can be seen in Fig. 4, where the shaded region represents the half flute area. An enlarged view of the differential chip section and associated forces are shown in Fig. 5. The differential chip section has three surfaces on the flute and one surface on the hole wall. In these figures, A 0 is the crosssectional area of the flute, z is the distance from the workpiece surface to the current location of the drill point, and is the Fig. 2 Typical plot of the thrust force varying with depth Fig. 4 Flute cross-section 606 Õ Vol. 124, AUGUST 2002 Transactions of the ASME

3 Fig. 5 Force balance on a differential section of chips direction of the drill rotation. The dimensionless depth, z, is used in the analysis and is related to the depth by the following transformation z z D, (1) where D is the drill diameter. To determine the chip-evacuation force per flute, F c, a force balance on the chip section in Fig. 5 is performed in the axial direction resulting in 0 F c F c df c F ff F hf F rf F fwi, (2) where F ff, F hf, and F rf are the axial friction forces from the flute face, flute heel, and flute root, respectively, and F fwi is the friction force resulting from the interaction between the chips and the hole wall. In the force balance, the inertial effects are assumed to be negligible. The friction force from the wall F wf does not appear in Eq. 2 because it does not act in the axial direction. It is taken to be in the tangential direction considering the fact that the magnitude of the tangential velocity, which results from the rotational speed of the drill, is several orders of magnitude greater than the axial velocity, which results from the feed rate. The chipevacuation force, F c, causes an axial pressure, P, in the differential section, which is aligned with the chip-evacuation force for the straight-fluted drill. The chip-evacuation force is the product of the axial pressure and the flute cross-sectional area, viz., F c A 0 P. (3) The axial pressure compresses the differential chip section, forcing the chips to expand laterally in the plane of the flute crosssection. Since the chips are contained on all sides of the cross section from the flute and hole wall, a lateral pressure develops on the chips in the plane of the flute cross-section. This lateral pressure is considered to be proportional to the axial pressure via a constant k. Thus, the lateral pressure can be given as kp. There will be a tendency for this pressure to vary somewhat; however, following the methodology employed for solids conveying, the lateral pressure is assumed uniform. The normal forces on the outer surfaces of the differential section can then be calculated by multiplying the lateral pressure by the affected area. The resulting normal forces on the chip section are: F fn kps f Ddz, (4) F hn kps h Ddz, (5) F rn kps r Ddz, (6) and F wn kps w Ddz, (7) where F fn, F hn, F rn, and F wn are the normal forces from the flute face, flute heel, flute root, and hole wall, respectively. S f, S h, S r, and S w are the lengths of contact and are defined in Fig. 4. The friction forces at each surface, which act in the opposite direction of the relative velocity between the contacting surfaces, are related to the normal force at that surface by a coefficient of friction. The friction forces on the chip section from the flute are: F ff f kps f Ddz, (8) F hf f kps h Ddz, (9) and F rf f kps r Ddz, (10) while the friction force on the chip section from the hole wall is F wf w kps w Ddz, (11) where f is the coefficient of friction between the chips and flute and w is the coefficient of friction between the chips and hole wall. F wf is the friction force from the hole wall. When calculating the normal force from the hole wall, the effective area, that is the area component of the wall interface perpendicular to the normal force, must be considered. The concept of effective area can be explained by considering the section between the shaded plane and the hole wall Fig. 5 interface as a separate entity. For this entity, the sum of the forces on the Journal of Manufacturing Science and Engineering AUGUST 2002, Vol. 124 Õ 607

4 Fig. 6 Calibration procedure flow chart shaded plane must be equivalent to those on the hole wall to satisfy the equilibrium. Since the normal force F wn is considered to have a line of action passing through the center of the drill, the shaded area shown in Fig. 5 is taken as the effective area with the length S w. The resulting friction force from the wall is taken to be in the tangential direction as previously discussed. The friction force from the hole wall, F wf, causes the chips to be pushed against the flute face, thus increasing the normal force on the flute face. While this would have an impact on the normal forces developing on the flute heel and rake, it is assumed to be negligible. This normal force creates a friction force, F fwi, and can be expressed as F fwi f F wf sin, (12) where is the angle between the flute face and the shaded plane. Substituting the appropriate forces into Eq. 2 results in 0 A 0 P A 0 P dp f kps f Ddz f kps h Ddz f kps r Ddz f w kps w D sin dz, (13) which, after simplifying, becomes dp kd S P A f f S h f S r f S w w f sin dz. (14) 0 Defining B as B S f f S h f S r f S w w f sin, (15) and integrating Eq. 14, the pressure in the flute can be expressed as P z P 0 e kbd/a 0 z, (16) where P 0 is the initial axial pressure at z 0. Equation 16 defines how the axial pressure on the chips changes from the workpiece surface to the drill point as a function of flute geometry and two coefficients of friction, f and w. Assuming that the cross-sectional area remains constant throughout the flute, the chip-evacuation force at any depth can be determined by multiplying Eq. 16 by the area A 0 to give F c z F c 0 e kbd/a 0 z, (17) where F c (0) is the initial chip-evacuation force at z 0. The change in torque due to the chip-evacuation force, which will be defined as the chip-evacuation torque, can be calculated from the friction force on the wall, F wf as dm RF wf, (18) where R is the radius of the drill. Substituting Eq. 11 for F wf and Eq. 16 for the pressure, P, the chip-evacuation torque per flute can be found as zr w ks w D F c 0 e M 0 kbd/a 0 d, (19) A 0 and integrating results in M R ws w F c 0 e kbd/a 0 z 1. (20) B 3 Model Calibration Procedure Coefficients of friction depend on many factors, such as velocity between the contacting surfaces, temperature, surface roughness, and applied load 15. These parameters are affected by the cutting conditions of the process, since they dictate the velocities of the contacting surfaces, temperatures, and chip thickness. Therefore, it is expected that the coefficients of friction are a function of the cutting conditions, which can be determined via a set of calibration experiments and associated regression coefficients of the calibration equations. Here, a power law model 16 is proposed relating these process parameters to the coefficients of friction, viz., ln f a 0 a 1 ln f a 2 ln N a 3 ln f ln N, (21) ln w b 0 b 1 ln f b 2 ln N b 3 ln f ln N, (22) where f is the feed and N is the spindle speed. In order to make the coefficients dimensionless, the parameters can be coded between 1 and 1. The coefficients a " and b " for a given workpiece material and tooling combination can be determined via a 2 2 factorial design of experiments with feed and spindle speed as the variables. A replicated factorial design can be used to obtain the statistical significance of the coefficients and develop a confidence interval on the predicted coefficients of friction. The calibration procedure is described in the flow chart in Fig. 6. The coefficients of friction for each experiment can be found from the measured chip-evacuation force and torque via a nonlinear least-squares curve fit as min z F Model F Exp 2 M Model M Exp R 2, (23) f w, (24) where F Model and M Model are defined in Eqs. 17 and 20, and F Exp and M Exp are the chip-evacuation force and torque established from the experimentation, respectively. The chipevacuation torque residual is divided by the radius to make the two residuals have the same dimensions. The last step is to perform a linear regression to determine a " and b " from Eqs. 21 and 22. Table 1 Experimental design for model calibration 608 Õ Vol. 124, AUGUST 2002 Transactions of the ASME

5 Table 2 Coefficients of friction for the calibration experiments Fig. 7 Chip-evacuation force for calibration test 1 4 Experimentation for Calibration and Validation Calibration and validation experiments have been conducted to validate the model presented above. Experiments were performed on a Mori Seiki SH-400 Horizontal Machining Center. A Kistler component drilling dynamometer was used to collect the forces. Flood coolant Master Chemical Trim-Sol was used in all the experiments. The workpiece material was aluminum 356-T6. A mm diameter two-fluted solid carbide drill was used with a zero degree helix angle. 4.1 Calibration Experiments. The first step in calibrating the chip-evacuation model is to identify the flute geometry inputs to the model. The cross-section of the flute, shown in Fig. 4, was measured with a Precision Twist Drill, Co. Tool Analyzer model 560. The flute area is mm 2. The lengths of contact on the hole wall, S w, is mm. The length of contact on the flute face, flute heel, and flute root are mm, mm, and mm, respectively. The angle,, is 37.8 deg. The constant k, which is the ratio between the lateral and axial pressures, is a material property that has received a limited amount of research. In the polymer literature, determination of this property remains poorly understood 17. To determine this Fig. 8 Chip-evacuation torque for calibration test 1 Table 3 Friction model coefficients Fig. 9 Chip-evacuation force for calibration test 4 Fig. 11 Fitted chip-evacuation forces for the calibration experiments Table 4 Cutting conditions for the validation experiments Fig. 10 Chip-evacuation torque for calibration test 4 Journal of Manufacturing Science and Engineering AUGUST 2002, Vol. 124 Õ 609

6 Table 5 Results of the validation experiments ratio, specially designed compaction cells have been used 17, which have been employed for a selected set of solid polymers. In this work, the ratio of the lateral and axial pressures was found to be strongly dependent on temperature and vary between for different materials. Determination of this property for metals has only been done with metal powders, where initial discoveries found that the ratio takes the value of Poisson s ratio at low axial pressures 18. Following this work on granular powders, k is assumed to be 0.33, which is the value of Poisson s ratio for aluminum 19. Table 1 provides the cutting conditions for the calibration experiments. For each case, the chip-evacuation force and torque were calculated by subtracting the cutting force and cutting torque from the total observed data, respectively, and dividing by the number of flutes. Figures 7 10 show the chip-evacuation forces and torques obtained from two of the four calibration experiments. In the figures, the abrupt increase in the force and torque is recognized as the onset of chip-clogging. Further, the variation of the chip-evacuation force and torque increase after the onset of chipclogging. Once the chip-evacuation force and the torque are determined, the coefficients of friction values can be found from the least-squares method given in Eq. 23 and are given in Table 2. Using the coefficient of friction values of Table 2, a linear regression procedure was followed to find the coefficients of the friction model given in Eqs. 21 and 22. The results are shown in Table 3. From the replicated calibration tests, a confidence interval can be established to ascertain the variation of the coefficients of friction. At a 95% confidence, the interval for f is , and w has an interval of The confidence intervals demonstrate that the cutting conditions do affect the coefficients of friction significantly. To observe the effectiveness of Fig. 12 Experimental and predicted chip-evacuation force for validation test 5 Fig. 14 Experimental and predicted chip-evacuation force for validation test 9 Fig. 13 Experimental and predicted chip-evacuation torque for validation test 5 Fig. 15 Experimental and predicted chip-evacuation torque for validation test Õ Vol. 124, AUGUST 2002 Transactions of the ASME

7 Table 6 Coefficients for the mechanistic force model this procedure, these coefficients of friction were used to calculate the chip-evacuation force and torque from Eqs. 17 and 20. Figure 11 shows the fitted chip-evacuation force curve for each calibration test and visually reinforces the variation of the coefficients of friction with the cutting conditions. 4.2 Validation Experiments. A set of validation experiments were conducted to assess the effectiveness of the chipevacuation model. The cutting conditions were chosen within the calibration range as seen in Table 4. First, an assessment of the calibration model is made by comparing the predicted and experimental coefficients of friction. The predicted coefficients of friction are calculated from the friction models in Eqs. 21 and 22 with the friction model coefficients given in Table 3. The experimental coefficients of friction are determined in the same manner as the calibration experiments by using the nonlinear least-squares method given in Eq. 23. The results are given in Table 5, with the percent errors. The average error for the coefficient of friction, f, is 4.09%. The coefficient of friction, w, has an average error of 8.20%. The chip-evacuation model in Eqs. 17 and 20 was then used to predict the chip-evacuation force and torque. Figures illustrate the experimental and predicted chip-evacuation force and torque for two of the validation experiments. To determine the effectiveness of the chip-evacuation model in capturing the trends of the experimental data, a normalized error was used to quantify the quality of the curves. The normalized error is defined as Normalized Error X Model X Exp 2 2, (25) X Exp where X represents either the chip-evacuation force, F, orthe chip-evacuation torque, M, and X Exp is from the experimental data and X Model is from the chip-evacuation model. The calculated normalized errors for the validation experiments are in Table 5. For each validation experiment, the normalized error is determined for both the chip-evacuation force and torque. The average error for the chip-evacuation torque and force was seen to be 15.75% and 14.50%, respectively, for validation tests 1 through 8. It should be noted in examining Figs. 14 and 15 for validation test 9 that the chip-evacuation process becomes much more erratic in nature past the point of chip-clogging. However, it is clear from Figs that, even in the worst-case scenario, the chip-evacuation model predicts the evacuation force and torque quite well to the onset of chip-clogging, which is the primary point of interest for deep-hole drilling. 5 Determination of the Critical Depth in Drilling The chip-evacuation model can be used to determine two depths of interest in drilling: the depth that can be drilled prior to subjecting the drill to a torque limit based on the drill breakage torque and a desired factor of safety, and the depth at which chipclogging begins to occur indicating the need for a pecking cycle. These two critical depths can be used jointly during process planning. For a given specified depth, the chip-evacuation model can be used to determine if the hole can be drilled in one pass without exceeding the chosen torque limit. If the hole cannot be drilled without violating the limit, the depth at which to perform a pecking cycle can be determined. This depth is determined from chipclogging, since once chip-clogging becomes substantial, pecking will be ineffective in removing the chips. 5.1 Drill Breakage-Based Criterion. The failure mode breakage of drills is torsional 20. Based on this consideration, researchers 20,21 have developed various criteria that rely on the breaking torque of the drill, M b, and a desired factor of safety, FS. Thus, the depth of the hole that can be drilled prior to violating this threshold torque, M th, can be determined, and will be defined as the threshold depth z t. The threshold torque is defined as M th M b /FS. Drill breakage results from the total torque exerted on the drill. Thus, in addition to the chip-evacuation torque, the cutting torque, M cut, is also required for the prediction of the threshold depth. Therefore, the total torque on a sharp drill can be written as M total M cut n f M, (26) where n f is the number of flutes. The chip-evacuation torque per flute, M, can be determined from the chip-evacuation model in Eq. 20. At the threshold depth, the total torque on the drill will be M th M cut n f M z t, (27) where M(z t ) represents the chip-evacuation torque per flute evaluated at the threshold depth. After substituting the chipevacuation torque from Eq. 20, the threshold depth becomes z t A 0 kbd ln B M n f R w S th M cut w 1. (28) In this study, the cutting torque, M cut, at each cutting condition is predicted with the use of the mechanistic model developed in 16. This model divides the cutting edge into small elements. For each element, the forces on the rake face, the normal and friction forces, are determined by multiplying the elemental chip areas with specific force coefficients, which can be given as ln K n a 0 a 1 ln t c a 2 ln V a 3 ln 1 sin n a 4 ln t c ln V (29) Table 7 Experimental and predicted cutting torques and threshold depths Journal of Manufacturing Science and Engineering AUGUST 2002, Vol. 124 Õ 611

8 Fig. 16 test 1 Experimental and predicted total torque for calibration Fig. 18 Experimental and predicted chip-evacuation force for calibration test 1 ln K f b 0 b 1 ln t c b 2 ln V b 3 ln 1 sin n b 4 ln t c ln V, (30) where K n and K f are the normal and frictional force constants, respectively, t c is the elemental chip thickness, V is the elemental cutting surface speed, and n is the elemental normal rake angle. The elemental torque can then be calculated by coordinate transformation and the torque can be found by summing all the elemental torques. From validation experiments, the mechanistic model predicted the drilling forces with errors less than 10%. The same calibration experiments used for the chip-evacuation model are used to determine the mechanistic force model specific force coefficients. Therefore, the validity range of feeds and speeds for the mechanistic force model is mm/rev and rpm, respectively. The resulting values of the specific force coefficients can be seen in Table 6. The coefficients a 3 and b 3 in Table 6 deal with the normal rake angle of the cutting element, and therefore take the value of zero because the straightfluted drill used here has a constant zero degree rake angle along the cutting lips. The predicted cutting torque, via the model of 16, and experimental cutting torque are within 12%, as shown in Table 7. In this analysis, the breaking torque of the drills must be determined. For the purpose of demonstration here, a drill was deliberately broken during an experiment, resulting in the breaking torque of 175 Ncm. In an actual application, two or more drills might be broken to obtain a more precise estimate. This value could be determined analytically as well. A factor of safety of 5 is chosen here 20, which makes the threshold torque 35 Ncm. Figures 16 and 17 show the total torque measured from the experiments Calibration Test 1 and Validation Test 2 in Tables 1 and 4, respectively and those predicted by the model in Eq. 26, along with the threshold torque and the location of the threshold depth. The experimental and predicted threshold depths are also provided in Table 7. The predicted threshold depth, z t, is calculated from Eq. 28, using the threshold torque, cutting torque, and the predicted coefficients of friction. The experimental threshold depth is determined when the mean experimental torque exceeds the threshold torque. As seen in Table 7, the average error between the predicted and experimental threshold depth is 5.73%. 5.2 Flute-Clogging Based Criterion. When drilling holes with large depth-to-diameter ratios, pecking cycles periodic drill retractions are generally used. The goal of the pecking cycle is to clear the chips from the flutes. After a certain depth, which has been defined here as the critical depth, chips may become so severely compacted that pecking will be ineffective. When chipclogging occurs, the forces begin to rapidly increase, thus the gradient of the chip-evacuation force can be used as the criterion to determine the critical depth. The gradient of the modeled chip-evacuation force is obtained by differentiating Eq. 17 as df c kbd e kbd/a 0 z. (31) dz A 0 Therefore, the critical depth, z*, can be given as z* A 0 kbd ln A 0 df c * kbd dz, (32) Fig. 17 test 2 Experimental and predicted total torque for validation Fig. 19 Experimental and predicted chip-evacuation force for calibration test Õ Vol. 124, AUGUST 2002 Transactions of the ASME

9 Table 8 Observed critical depths and gradients for the calibration experiments where df c */dz is the gradient of the chip-evacuation force at which the critical depth occurs. The selection of df c */dz can be made by analyzing the calibration data. From each calibration experiment, the depth where the chip-evacuation force rapidly increases can be identified as the observed critical depth, as seen in Figs. 18 and 19. This depth is chosen by observing the experimental data and selecting the depth at which the forces begin to rapidly increase. While somewhat subjective, this technique is shown here to be effective. Substituting this observed critical depth in Eq. 31 for z, the gradient of the chip-evacuation force at this depth can be found from the model. This gradient is illustrated in Figs. 18 and 19. Table 8 provides the observed critical depths and associated gradients determined from the calibration experiments of Table 1. Using the gradients thusly obtained, an equation to predict the gradient for different cutting conditions can be developed. For the sake of simplicity, a linear relationship is used, which resulted in the following equation, df c * f 1.189N 3.168fN. (33) dz Table 9 provides the predicted gradient and the observed and predicted critical depths for the validation experiments. The predicted gradient, df c */dz, is calculated from Eq. 33 with the corresponding coded cutting conditions. Using the predicted gradient and coefficients of friction for each validation experiment of Table 4, the predicted critical depth can be obtained from Eq. 32. The observed critical depth is determined in the same manner as in the calibration experiments. It can be seen from the difference between the observations and predictions that the critical depth can almost be predicted within a depth less than one drill radius and has an average error of 4.60%. 6 Conclusions The specific conclusion of this work are: 1 A chip-evacuation model that predicts the chip-evacuation force and torque has been developed for drilling processes with discontinuous chips. The model is based on the force balance on a differential chip section, and considers straight-fluted drills, which are common for deep-hole drilling processes. Table 9 Observed and predicted critical depths for the validation experiments 2 A calibration procedure has been proposed to determine the two coefficients of friction required by the model. The friction models considered are power functions with feed and spindle speed as the variables. 3 A set of calibration and validation experiments has been completed to assess the effectiveness of both the calibration procedure and the chip-evacuation model. It has been seen that the calibration model is capable of accurately predicting the coefficients of friction in the applicable range. The coefficient of friction, f is predicted with an average error of 4.09%, while the coefficient of friction, w has an average error of 8.20%. A normalized error criterion is used to compare the predicted and experimental chip-evacuation forces and torques. For the validation experiments, the chip-evacuation force and torque had an average normalized error of 14.50% and 15.75%, respectively. 4 Two drilling critical depths are predicted by the chipevacuation model using a torque-limit based criterion, and a chipclogging criterion. The former imposes an upper limit on the torque exerted on the drill with a factor of safety to prevent breakage, and the latter specifies a limit on the slope of the chipevacuation forces to determine the pecking depth. Validation experiments have been used to prove the effectiveness of the chipevacuation model in predicting these depths. The torque-limit based critical depth has an average error of 5.73%, while the chip-clogging critical depth is predicted with an average error of 4.60%. Acknowledgment The authors wish to state their appreciation to the National Science Foundation Industry/University Cooperative Center for Machine Tool Systems Research and Delphi Automotive Systems Co. for support of this research. The authors many conversations with Dr. R. Khetan of Delphi Automotive Systems Co. are greatly appreciated. The authors are grateful to Mr. Shiva Kalidas and Mr. Mike Vogler for their valuable comments and suggestions. References 1 Kavaratzis, Y., and Maiden, J. 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