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Mechanics and dynamics of drilling Roukema, Jochem Christiaan 2006

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M E C H A N I C S A N D D Y N A M I C S O F D R I L L I N G by JOCHEM CHRISTIAAN R O U K E M A M.Sc. Delft University of Technology, Delft, The Netherlands, 2000 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T O F T H E REQUIREMENTS F OR T H E D E G R E E OF D O C T O R OF PHILOSOPHY in The Faculty of Graduate Studies (Mechanical Engineering) T H E UNIVERSITY OF BRITISH COLUMBIA December, 2006 © Jochem Christiaan Roukema, 2006 Abstract This thesis presents the mathematical modeling of drilling mechanics and dynamics in order to improve hole shape accuracy, optimize the drill tool geometry and drilling operations. The thesis presents prediction of cutting forces, torque, power, vibrations and hole shape as a function of drill edge geometry, work material dependent cutting coefficients, drill structure and drilling conditions. The forces and torque are expressed as a function of chip load distribution along the cutting edge and cutting force coefficients. As the drill rotates and thrusts into the mate-rial, it experiences torsional, lateral and axial vibrations. The coordinates of the cutting edges, which generate the cut surface, are predicted by applying cutting forces and torque on the drill's structural dynamic model. The generated surface is digitized, and the chip distribution along the flute is calculated by subtracting it from the surface generated by the previous flute. Hence, the exact model of drilling kinematics and structural dynamics are considered, which leads to inte-grated simulation of static, dynamic and regenerative chatter vibrations of the drilling process and generated hole surface. The model is also used to investigate the mechanism of whirling vibra-tions and the hole wall formation by imposing commonly observed whirling motion on the drill. The simulation shows good similarity with experimentally measured cutting forces and hole geometries. Although it is computationally costly, the numerical model of the drilling process considers the full physical model with true kinematics, dynamics and nonlinearities such as cutting coefficients and tool jumping out of cut due to excessive vibrations. As an alternative, an analytical frequency domain stability analysis for drilling is proposed for efficient generation of stability charts. The stability lobes predicted by the numerical and linear frequency domain models agreed well. Although the models agreed well with the experimental results published in the literature, a sig-ii 1 nificant discrepancy is observed at practical drilling speeds where the high frequency modes of the drill led to chatter. It is shown that the unmodeled rubbing of the drill's flank with the wavy surface finish and chisel edge contact, i.e. process damping, remains a fundamental challenge in further research. , iii Table of Contents Abstract ii Table of Contents iv List of Tables vii List of Figures viii Nomenclature xiii Acknowledgements xx 1. Introduction 1 2. Literature Review 5 2.1.Overview 5 2.2. Drill geometry 5 2.3. Mechanics of drilling 7 2.3.1. Mechanistic cutting force models 9 2.3.2. Cutting mechanics force model 11 2.4. Dynamics of drilling 12 2.4.1. Torsional-axial chatter vibrations in drilling 12 2.4.2. Lateral and whirling vibrations in drilling 15 2.5. Conclusions 22 3 . Drill Bit Model 24 3.1 .Twist drill geometry 24 3.1.1. Point thinned drill geometry 25 3.1.2. Grinding errors and tool misalignment 27 3.2.Indexable drill geometry 28 3.3.Dynamic properties of drill bits 30 3.3.1. Beam theory 30 3.3.2. Finite element model of drill bit 32 3.3.3. Experimental modal analysis - stationary drill 33 3.3.4. Experimental modal analysis - rotating drill 35 3.3.5. Experimental modal analysis - indexable drills 38 iv 4. Mechanics of Drilling 39 4.1 .Introduction 39 4.2. Torque and thrust models for drilling 40 4.2.1. Cutting mechanics approach for the prediction of torque and thrust 40 4.2.2. Mechanistic cutting force model 50 4.2.3. Experimental verification of models for torque and thrust in drilling 61 4.2.4. Lateral cutting force prediction using the cutting mechanics approach 75 4.3. A higher order mechanistic model for force prediction of piloted holes 77 4.3.1. Mechanistic model calibration for torque, thrust and lateral forces 78 4.3.2. Comparison experiments and mechanistic model for torque, thrust and lateral forces 79 4.4 .Conclusions 81 5. Numerical Modeling of Drilling Dynamics 82 5.1 .Introduction 82 5.2. Drill geometry 83 5.3. Workpiece model 85 5.3.1 .Bottom hole surface 85 5.3.2.Hole wall surface 86 5.4. Time domain simulation model 87 5.4.1. Geometry of tool motion 88 5.4.2. Dynamic generation of hole profile 90 5.4.3. Cutting force calculation 95 5.4.4. Workpiece surface finish 96 5.4.5.Integration scheme 97 5.5.Static drilling simulation 98 5.5.1.Static torque and thrust 98 5.5.2.Lip height, tip angle grinding errors and runout 99 5.5.3.Imposed whirling vibrations 102 5.6. Dynamic drilling simulation 110 5.6.1. Axial and torsional-axial vibrations in drilling 110 5.6.2. Lateral vibrations in drilling 114 5.6.3. Combined torsional-axial and lateral vibrations in drilling 117 5.7. Dynamic cutting tests 118 5.7.1 .Torsional-axial chatter stability 118 5.7.2. VvTiirling experiments 126 5.8. Conclusions 130 v 6. Frequency Domain Chatter Stability of Drilling 131 6.1. Dynamic chip thickness in three dimensional drilling 131 6.2. Frequency domain solution procedure 139 6.3. Partial chatter stability laws 142 6.3.1 .Lateral chatter stability using a rotating coordinate system 143 6.3.2. Lateral chatter stability of a stationary tool 146 6.3.3. Torsional-axial chatter 147 6.4. Chatter stability lobes for drilling 149 6.4.1. Comparison of proposed frequency domain solution with time domain model 151 6.4.2. Proposed frequency domain solution versus experimental results 155 6.4.3. Proposed frequency domain solution versus Bayly's frequency domain solution 156 6.4.4. Proposed frequency domain solution versus Arvajeh's frequency domain solution 158 6.4.5. Proposed frequency domain solution versus previously presented partial stability laws.... 160 6.5 .Conclusions 162 7. Conclusions and Recommendations for Future Work 164 7.1 .Conclusions 164 7.2.Recommendation for future work 166 Bibliography 168 Appendix A. Stationary Drilling Dynamometer Calibration 174 Appendix B. Decomposition of Three-dimensional Forces in Cutting Mechanics Approach ....178 vi List of Tables Table 3 1 Table 3 2 . Table 3 3 • Table 3 4 Table 3 5 Table 3 6 Table 4 1 Table 4 2 Table 4 3 Table 4 4 Table 4 5 Table 4 6 Table 4.7 : Table 5.1 : Table 5.2 : Table 5.3 : Table 5.4 : Table 5.5 : Table 6.1 : Table 6.2 : Natural frequencies and stiffnesses of a uniform cylindrical beam [69] 31 High speed steel (HSS) material properties 31 Drill bit natural frequencies and stiffnesses when modeled as a cylindrical beam 32 Dynamic properties of Guehring HSS drill bit, geometry #217, TiAIN coating; Length=176mm; Diam-eter=16mm (L/D ratio=ll) 36 Natural frequencies of Guehring HSS drill bits, geometry #217, Diameter=16mm 37 Natural frequencies of the Sandvik indexable U-drill, 21mm diameter. 38 Cutting conditions orthogonal cutting experiments for AL7050-T7451 database 62 AL7050-T7451 cutting database for orthogonal to oblique transformation of cutting lip section; h=[mm], Vc=[m/min] an=[deg] 65 Cutting force coefficients chisel edge region, from orthogonal cutting tests 66 Model coefficients CI , C2, p, q for 7.94mm twist drill in AL7050-T7451 68 Normal and friction force coefficients from 7.94mm in AL7050-T7451 68 Geometrical properties of Kennametal Screw Machine Length twist drills used in experiments to verify cutting mechanics approach and Chandrasekharan's mechanistic model; Types: S110FX, surface treated (7.94-17.46mm), S100FX, surface treated (20.64mm) 70 Geometrical properties of the drill tip of a Guehring 16mm twist drill, geometry #217, TiAIN coating; this drill is used for the mechanistic model in Eq. (4.55) 78 Geometrical properties of two fluted Guehring twist drill, geometry #217; Tool material: HSS (high speed steel), TiAIN coating; 98 Mechanistic cutting coefficients (refer to Eq. 4.57) for a Guehring #217 twist drill (geometry: Table 5.1), cutting AL7050-T7451 and using flood coolant; 98 Geometrical properties of two fluted Guehring twist drill, geometry #217; Tool material: HSS (high speed steel), TiAIN coating 105 Summary of simulation results of imposed whirling motions with 100mm amplitude; Cutting condi-tions: 2400rpm, f=0.30mm/rev, pilot hole diameter=4mm, hole depth=15mm. Tool geometry: Table 5.3; Cutting coefficients: Table 5.2 105 Dynamic properties of Guehring HSS drill bit, geometry #217 (see Table 5.3); Tool length=176mm; Diameter=16mm (L/D ratio=ll) 110 Dynamic properties of drill bit 150 Linear cutting force model parameters for the proposed frequency domain solution; Workpiece mate-rial: AL7050-T7451. Drill geometry: Table 5.3; Cutting conditions: Cutting speed=120m/min, fee-drate=0.30mm/rev, pilot hole diameter=4mm.w 150 vii List of Figures Figure 1.1: Drilling operation on a horizontal machining center; Workpiece on the left is fed to the rotating drill on the right 1 Figure 1.2 : Hole shapes resulting from drilling blind holes in full material; a) Stable cut, no visible vibrations; b) Sunray pattern due to unstable torsional-axial chatter vibrations; c) Trigon caused by whirling vibra-tions; d) Surface resulting from combined torsional-axial chatter and whirling vibrations 2 Figure 2.1 : a) twist drill geometry; b) indexable drill geometry 6 Figure 2.2 : a) 18mm hole profile generated by a twist drill (three sided, tool length 95mm); b) 18mm hole profile generated by a indexable drill (oval shape, tool length 70mm) (after Venkatesh [8]) 6 Figure 2.3 : a) definition of feed fr, feed speed Vf, radial depth of cut b; b) cutting speed Vc; c) thrust force Fz and torque Tc acting on the tool 8 Figure 2.4 : Photographs of different cutting regions for a twist drill, from extrusion at the chisel edge to oblique cutting at the periphery; 2W is the drill web thickness, 2kt is drill tip angle (after Ernst et al. [11]) 8 Figure 2.5 : Experimental and predicted thrust and torque using a mechanistic model (after Chandrasekharan et al. [16]); 1: Air cutting; 2: Lip engagement; 3: Steady state lip cutting; 4: Lips and chisel edge both fully engaged 11 Figure 2.6 : Effect of lip height error on hole oversize; a) position of axis of drill rotation; b) experimental hole oversize as a function of lip height error in steel (after Galloway [14]) 15 Figure 2.7 : Formation of shaped hole profile, after Lee et al. [50]; a) Trigon shaped hole (Nw=l); b) Pentagon shaped hole (Nw=2); c) Heptagon shape hole (Nw=4) 16 Figure 2.8 : Trajectory of stationary drill exhibiting whirling vibration at seven times the rotational frequency of the workpiece; (S designates starting point, E the ending point) (after Ema et al.[28]) 19 Figure 2.9 : Predicted and experimental hole shape in reaming; a) 5-sided hole; b) 7-sided hole (after Bayly et al. [47]) 21 Figure 3.1 : Twist drill geometry specifications 24 Figure 3.2 : Point thinning of the chisel edge to reduce thrust force 25 Figure 3.3 : Geometrical details of point thinned drill geometry. 26 Figure 3.4 : Measurement of drill runout and lip grinding errors 27 Figure 3.5 : Geometry of indexable drill; a) insert configuration, b) balance angle 29 Figure 3.6 : Detail of insert geometry used in indexable drill 29 Figure 3.7 : Dynamic model of drill bit 30 Figure 3.8 : Drill bit model in C A D system, and forces applied to create a pure torque 33 Figure 3.9 : Experimental setup for measurement of torsional-axial, lateral, and axial transfer functions 34 Figure 3.10 : Experimental setup for lateral transfer function measurement 35 Figure 3.11 : Experimental direct lateral transfer functions of Guehring HSS drill bit, geometry #217, TiAIN coat-ing; Length=176mm; Diameter=16mm (L/D ratio=ll) 37 Figure 3.12 : a) Indexable drills examined in this thesis; Sandvik U-drill, 21mm diameter; Drill lengths 4, 3 and 2 times Diameter, mounted in Sandvik solid HSK63A tool holder; b) detail of the tip geometry of an indexable drill, also refer to Fig. 3.5 38 Figure 4.1 : Cross sectional views of drilling cutting process; a) indentation at the center of the chisel edge; b) orthogonal cutting with high negative rake angle on the chisel edge; c) oblique cutting with negative rake angle on the cutting lip; d) oblique cutting with positive rake angle on the cutting lip (after Ernst et al. [11]) 39 Figure 4.2 : Twist drill geometrical analysis (after Wiriyacosol et al. [12]) 43 Figure 4.3 : Feed speed effect strong on chisel edge (a), but weak on cutting lip (b) 45 Figure 4.4 : Typical variation of the cutting angles on a twist drill 46 Figure 4.5 : Element discretization on twist drill cutting lip and chisel edge 47 Figure 4.6 : Definition of normal and friction force [16] 51 viii Figure 4.7 : Figure 4.8 : Figure 4.9 : Figure 4.10 : Figure 4.11 : Figure 4.12 : Figure 4.13 : Figure 4.14 : Figure 4.15 : Figure 4.16 : Figure 4.17 : Figure 4.18 Figure 4.19 Figure 4.20 Figure 4.21 Figure 4.22 Figure 4.23 Figure 4.24 : Figure 5.1 : Figure 5.2 : Figure 5.3 : Figure 5.4 : Figure 5.5 : Figure 5.6 : Figure 5.7 : Figure 5.8 : Figure 5.9 : Figure 5.10 : Figure 5.11 : Figure 5.12 : Figure 5.13 Figure 5.14 Derivation of oblique cutting forces Fcut, Fth and Flat from normal and friction force 51 Trigonometric decomposition of elemental oblique-cutting forces into the drilling thrust, torque and radial directions 54 Indentation process at the drill center [75] 55 Variation of chip load Ac during drilling calibration experiment 58 Experimental thrust and torque 16mm twist drill; Vc=40m/min, fr=0.30mm/rev, 4mm pilot 59 Orthogonal cutting database AL7050-T7451: tangential forces for cutting lip region; experiment and prediction; For each rake angle four cutting speeds are tested at five different feedrates 63 Orthogonal cutting database AL7050-T7451: feed forces for cutting lip region; experiment and predic-tion; For each rake angle four cutting speeds are tested at five different feedrates 64 Orthogonal cutting: tangential forces for chisel edge region; experiment and prediction 66 Orthogonal cutting: feed forces for chisel edge region; experiment and prediction 67 Normal cutting coefficient for Chandrasekharan's mechanistic model; AL7050-T7451; 7.94mm twist drill; experiment and prediction 69 Friction cutting coefficient for Chandrasekharan's mechanistic model; AL7050-T7451; 7.94mm twist drill; experiment and prediction 69 Experimental thrust and torque for 7.94mm twist drill in AL7050-T7451 versus prediction 71 Experimental thrust and torque for 11.11mm twist drill in AL7050-T7451 versus prediction 72 Experimental thrust and torque for 14.29mm twist drill in AL7050-T7451 versus prediction 73 Experimental thrust and torque for 17.46mm twist drill in AL7050-T7451 versus prediction 74 Experimental thrust and torque for 20.64mm twist drill in AL7050-T7451 versus prediction 75 Three-dimensional representation of cutting force components acting on an element on the cutting edge 76 Single flute experimental thrust, torque & lateral forces versus mechanistic model 80 Drill bit geometry and coordinate system 83 Cutting edge discretization; Number of elements along cutting edge m=7 84 Initial workpiece discretization; Number of grid circles m=7; Number of points per grid circle Ng=60 85 Example of hole wall and bottom surfaces: a) start of simulation b) end of simulation 87 Structure of proposed three dimensional time domain simulation model for drilling 88 Intersection points of cutting edges with grid circles and illustration of time varying widths of cut; Number of elements along cutting edge m=7 89 Exact kinematics approach for surface updating; a) tooth in cut, regular cutting b) tooth out of cut, due to excessive vibrations 92 Cutting edge points as moved with respect to workpiece; Illustration of chip height 93 Mechanism of wall surface updating; points on each radial grid circle are pushed outwards 94 a) three dimensional representation of tooth surfaces left by teeth; b) finished workpiece surface con-structed from tooth surfaces. Note: feed is 1.5mm/rev for visualization 96 Comparison between experimental and predicted torque and thrust during engagement into piloted workpiece (AL7050-T7451); pilot hole radius Rp=2mm; Drill geometry: Table 5.1, cutting coeffi-cients AL7050-T7451: Table 5.2; Cutting speed Vc=40m/min, Feed per revolution fr=0.30mm/rev; Number of elements along cutting lip m=30 99 Simulated static cutting forces, fourier spectrum and elemental chip thicknesses for: a) Lip height error; b) Lip angle error; c) Radial runout error; Pilot hole diameter=4mm; Feedrate=0.30mm/rev; Spindle speed=2400rpm; Number of elements per cutting edge, m=50; Tool geometry: Table 5.1; Cut-ting coefficients AL7050-T7451: Table 5.2 101 Physics of whirling vibrations in drilling; Drill seen from shank side 103 Traces of the cutting edges during backward whirling motion at exact integers of the spindle frequency fs at four different tool rotations: a) fwf=2fs; b) fwf=4fs; c) fwf=6fs; Drill diameter = 16mm, web thickness 2W= 2.12mm, fixed whirling amplitude = 500micron (for visualization) 104 ix Figure 5.15: Simulated chip thickness, thrust force, lateral forces and Fourier spectra for a drill backward whirling frequency a) 3.00fs; b) 2.90fs; c) 4.90fs; d) 6.90fs; 2400rpm, f=0.30mm/rev, pilot hole diame-ter=4mm, hole depth=15mm. Tool geometry: Table 5.3; Cutting coefficients: Table 5.2 107 Figure 5.16 : a) Tangential and radial force Ft, Fr in the frame rotating with the drill bit, and Fourier spectrum; b) Lateral forces Fx, Fy acting on the drill bit in the stationary frame, and Fourier spectrum; Refer to Fig. 5.15b for tool geometry and cutting coefficients 108 Figure 5.17 : Simulated cross sections of hole wall for six different whirling frequencies (defined in the rotating frame), at the top and the bottom of hole; a) 3.00fs (refer to Fig. 5.15a); b) 2.90fs (refer to Fig. 5.15b); c)5.00fs; d)4.90fs (refer to Fig. 5.15c); e) 7.00fs; f) 6.90fs (refer to Fig. 5.15d); hole depth= 15mm; 2400rpm, f=0.30mm/rev; Tool geometry: Table 5.3; Cutting coefficients Aluminum AL7050-T7451: Table 5.2 109 Figure 5.18 : Whirling cross sections as a function of whirling amplitude; Whirling frequency is 7.00fs in the rotat-ing frame; a) 1mm; b) 2mm; c) 5mm; d) 10mm; e) 20mm; f) 50mm; hole depth=15mm; 2400rpm, f=0.30mm/rev; Tool geometry: Table 5.3; Cutting coefficients AL7050-T7451: Table 5.2 109 Figure 5.19 : Time domain simulation results - axial vibration only: a) 3D surface of piloted hole; b) Detail of sur-face at pilot hole; c) Simulated tool vibration and Fourier spectrum; d) Side profile of the surface along pilot hole, generated by one tooth; 2400rpm, pilot hole diameter=4mm, feedrate=0.30mm/rev; Tool geometry: Table 5.3; Cutting coefficients AL7050-T7451: Table 5.2; Tool dynamics: Table 5.5 I l l Figure 5.20 : Simulation of combined torsional-axial vibrations: a, b) 3D surface of piloted hole; c) Axial tool vibra-tion and Fourier spectrum; d) Torsional tool vibration and Fourier spectrum; e) Side profile of the sur-face along pilot hole; f) Total tool rotation angle under fully developed chatter vibrations. Tool geometry, cutting coefficients, tool dynamics and cutting conditions: same as Fig. 5.19 113 Figure 5.21 : Simulation for drill with 10mm lip height error on flute 1, and lateral flexibility with a damping of 300%; a) tool deflection and Fourier spectrum; b) tool trajectory; c) chip thickness for each flute, and details; Conditions: 2400rpm, fr=0.30mm/rev, pilot hole diameter=4mm, hole depth=6mm; Tool geometry: Table 5.3; Cutting coefficients AL7050-T7451: Table 5.2; Tool dynamics: Table 5.5 114 Figure 5.22 : Simulation for drill with lateral flexibility and radial runout; a) bottom workpiece surface; b) lateral tool deflection and Fourier spectrum; Cutting conditions: 2400rpm, fr=0.30mm/rev, pilot hole diame-ter=4mm, hole depth=6mm; Tool geometry: Table 5.3; Cutting coefficients AL7050-T7451: Table 5.2; Lateral stiffnesses of the drill bit are 2.0xl07N/m, natural frequency 340Hz and 5% damping (symmet-ric dynamics are considered). The torsional and axial modes are rigid 116 Figure 5.23 : Simulation for drill with flexibility in lateral and torsional-axial directions; a) bottom workpiece sur-face; b) cross-sectional surface profile at pilot hole; c) axial tool deflection and Fourier spectrum; d) lateral tool deflection and Fourier spectrum; Conditions: 2400rpm, fr=0.30mm/rev, pilot hole diame-ter=4mm, hole depth=6mm; Drill geometry: Table 5.3; Tool dynamics: Table 5.5. The dynamics in X and Y directions are equal (symmetric dynamics), with 5% damping ratio and 2.5x107N/m stiffness; Torsional-axial damping is 3%; Cutting coefficients AL7050-T7451: Table 5.2 117 Figure 5.24 : Details of experimental cutting forces: Thrust, FFT Thrust, FFT X Y forces; a) full hole 800rpm; b) 4mm pilot hole 800rpm; c) 12mm pilot hole 800rpm; d) full hole 2400rpm; e) 4mm pilot hole 2400rpm; f) 12mm pilot hole 2400rpm; Drill geometry: Table 5.3; Workpiece material: AL7050-T7451 119 Figure 5.25 : Comparison of (a) experimental and (b) simulated surface, 2400rpm, 4mm pilot hole, feedrate 0.30mm/rev; c) simulated axial tool deflection and fourier spectrum. Tool geometry, cutting coeffi-cients, tool dynamics: same as Fig. 5.20. Workpiece material: AL7050-T7451 120 Figure 5.26 : Experimental torsional-axial chatter stability chart for Guehring 1 lxD twist drill (200-1400rpm); Dark grey regions indicate transition from stable to unstable drilling process; fr=0.30mm/rev; Drill geome-try: Table 5.3; Workpiece material: AL7050-T7451; Flood coolant was used during experiments. Depth of cut=drill radius-pilot hole radius; Pilot hole diameters 12-10-8-6-4-0mm correspond to depths of cut 2-3-4-5-6-8mm in the chart 121 X Figure 5.27 : Experimental torsional-axial chatter stability chart for Guehring l l x D twist drill (1400-2600rpm); Dark grey regions indicate transition from stable to unstable drilling process; fr=0.30mm/rev; Drill geometry: Table 5.3; Workpiece material: AL7050-T7451; Flood coolant was used during experi-ments. Depth of cut=drill radius-pilot hole radius; Pilot hole diameters 12-10-8-6-4-0mm correspond to depths of cut 2-3-4-5-6-8mm in the chart 122 Figure 5.28 : Process damping in drilling 123 Figure 5.29 : Experimental torsional-axial chatter stability charts for Guehring 16mm twist drills; a) Tool length 8xD; b) Tool length 15xD; fr=0.30mm/rev. Drill geometry: Table 5.3; Workpiece material: AL7050-T7451; Flood coolant was used during experiments. Depth of cut=drill radius-pilot hole radius; Pilot hole diameters 12-10-8-6-4-0mm correspond to depths of cut 2-3-4-5-6-8mm in the chart 124 Figure 5.30 : Sound measurement, frequency spectra and experimental surface finish for two cutting experiments with a 21mm indexable drill (Sandvik U-drill, length 4xD, fr=0.10mm/rev. Workpiece material: AL7050-T7451; Flood coolant was used during experiments 125 Figure 5.31 : Details of 4 cutting experiments: Full hole at 400rpm (a) and 2400rpm (b), 4mm piloted at 400rpm (c) and 2400rpm (d); Thrust, lateral force FFT's from 0-5kHz and from 0-1 Ofs; fr=0.30mm/rev. Drill geometry: Table 5.3; Workpiece material: AL7050-T7451; Flood coolant 128 Figure 5.32 : Details of 4 cutting experiments: 8mm piloted at 400rpm (a) and lOOOrpm (b), 12mm piloted at 400rpm (c) and 800rpm (d); Thrust, lateral force FFT's from 0-5kHz and from 0-10fs; fr=0.30mm/rev. Drill geometry: Table 5.3; Workpiece material: AL7050-T7451; Flood coolant 129 Figure 6.1 : Elemental forces acting on the cutting edges of a two-fluted drill bit 132 Figure 6.2 : Chip thickness change due to lateral vibrations while drilling a piloted hole 133 Figure 6.3 : Time varying directional coefficients for drilling 136 Figure 6.4 : Definition of rotating and stationary coordinate systems 143 Figure 6.5 : Limit depth of cut for the proposed frequency domain solution as a function of chatter frequency, for lateral and torsional-axial modes. Drill geometry: Table 5.3; Tool dynamics: Table 6.1; Cutting coeffi-cients AL7050-T7451: Table 6.2; Feedrate=0.30mm/rev 151 Figure 6.6 : Comparison of proposed frequency domain with time domain simulation results. Drill geometry: Table 5.3; Tool dynamics: Table 6.1; Cutting coefficients AL7050-T7451: Table 6.2; Feedrate=0.30mm/rev. 153 Figure 6.7 : Time domain simulation details for three cases in Fig. 6: a) 14000rpm, 5.5mm depth - lateral chatter; b) lOOOOrpm, 5.5mm depth - torsional-axial chatter; c) HOOOrpm, 5.5mm depth - stable cut; Drill geome-try: Table 5.3; Tool dynamics: Table 6.1; Cutting coefficients AL7050-T7451: Table 6.2; 154 Figure 6.8 : Comparison of experiments and proposed frequency domain solution for torsional-axial chatter stabil-ity; Drill geometry: Table 5.3; Tool dynamics: Table 6.1; Workpiece material: AL7050-T7451; fee-drate=0.30mm/rev.; Flood coolant was used during experiments; Depths of cut 2-3-4-5-6-8mm in the chart.correspond to pilot hole diameters 12-10-8-6-4-0mm (the drill diameter is 16mm) 155 Figure 6.9 : Comparison of the proposed frequency domain solution with Bayly's frequency domain solution and experiments [1]; Drill diameter D=9.525mm, length=107mm; feedrate=0.203rnm/rev; Workpiece material: AL7075 156 Figure 6.10 : Verification of frequency domain solution by Bayly [1] using the proposed time domain model; Drill diameter D=9.525mm, length=107mm; feedrate=0.203mm/rev; Number of elements m=30; Work-piece material: AL7075 157 Figure 6.11: Comparison of the proposed frequency domain solution with Arvajeh's frequency domain solution and experiments [68]; Drill diameter D=9.525mm, length=107mm; feedrate=0.10mm/rev; Workpiece material: AL6061-T6 159 Figure 6.12 : Comparison between proposed frequency domain solution and a) Bayly's torsional-axial stability [1]; b) Lateral stability for stationary tool (Bayly [45]) and rotating coordinates method (modified approach of Bayly [45]); Drill geometry: Table 5.3; Tool dynamics: Table 6.1; Cutting coefficients AL7050-T7451: Table 6.2; Feedrate=0.30mm/rev. 161 xi Figure A. 1 : Drilling dynamometer calibration setup with pulleys and weight 174 Figure A.2 : Detail dynamometer calibration setup with two equal, opposite forces 175 Figure A.3 : Torque and thrust signal readings during calibration experiment (torque sensitivity) 176 Figure A.4 : Torque and thrust signal readings during calibration experiment (thrust sensitivity) 177 Figure B. 1 : Three-dimensional representation of elemental cutting forces in the cutting mechanics approach.... 178 Figure B.2 : Force decomposition of radial cutting force into X Y Z coordinate system of drill bit 179 Figure B.3 : Force decomposition of feed force into X Y Z coordinate system of drill bit 180 Figure B.4 : Force decomposition of tangential cutting force into X Y Z coordinate system of drill bit 181 xii Nomenclature A = cross sectional area of cylinder, tool [mm2] Ac = chip area [mm2] AAk = elemental chip area orthogonal to oblique transformation [mm2] ax...a6 = mechanistic cutting force model coefficients (thrust) [div.] bv..b6 = mechanistic cutting force model coefficients (torque) [div.] B> Bred = dynamic drilling coefficient matrix and reduced matrix [-] B\\>B22>B\2-> B 2 \ dynamic drilling coefficient sub matrices [-] = first order approximation of dynamic drilling coefficient matrix [-] b = radial depth of cut [mm] blim = critical depth of cut [mm] blip, 1' blip,2 = total width of cut for individual lips [mm] h h h h °xx> °yy> °xy °yx = time varying directional factors in drilling [-] Ab = elemental width of cut [mm] Abk = elemental width of cut of a cutting lip element [mm] Abj = elemental width of cut of a chisel edge element [mm] Ab23 = (time varying) width of element 3 on cutting edge 2 [mm] bt = back taper of drill (reduction of flute diameter over flute length) [mm] C = damping matrix of drill bit [Ns/m] cx, c2 = power law constants thrust and cutting force 2 [N/mm ] c c = tangential and thrust force coefficients [N/m2] c<, = tangential force per unit width of cut (chisel region) [N/mm] = feed force per unit width of cut (chisel region) [N/mm] c , . . .c 6 = mechanistic cutting force model coefficients (tangential force) [div.] c c xx"1 yy = viscous damping [Ns/m] D = drill diameter [mm] D P = pilot hole diameter [mm] dx...d6 = mechanistic cutting force model coefficients (radial force) [div.] du = tool deflection in the direction of the cutting lips [mm] xiii dx, dy, dz, dQ = regenerative tool displacements [mm] E = Young's modulus [N/m2] e = whirling motion amplitude [mm] E§, £\, &2' E3 = tangential cutting force coefficients chisel edge [-] Ft = impact force during tap test [N] Fn> Ffr = normal and friction force in mechanistic model [N] Ft, Ff, Fr = tangential, feed and radial force acting on cutting edge element or flute [N] Fth' Fcuf Flat = thrust, cutting and lateral forces in mechanistic model [N] P P W' V = forces on drill bit in frame rotating with the tool [N] F • Wl = contact force between hole wall and drill flute [N] r r p x>1y> z = cutting forces acting on tool tip [N] /o'/l'/2'/3 = feed cutting force coefficients chisel edge [-] fc = chatter frequency [Hz] fn = natural frequency [Hz] fr = feedrate [mm/rev] *fr = feed increment per time step in time domain simulation [mm] fs = spindle frequency [Hz] 'w = whirling frequency in fixed coordinate system [Hz] f r = whirling frequency in coordinate system rotating with the tool [Hz] ft = tooth passing frequency [Hz] G = shear modulus [N/m2] G P = frequency response matrix in rotating frame [m/N] h = chip thickness [mm] hind = indentation depth [mm] = chip thickness on a chisel edge element [mm] h = chip thickness on a cutting lip element [mm] hlip = distance from drill tip to start of the flutes (in spindle direction) [mm] K = static chip thickness [mm] hceA(h k) = chip height for cutting edge point i on lip 1 at time step k [mm] hx(i, k) = elemental chip thickness cutting edge element / at time step k on lip 1 [mm] xiv I K K Ktc> Kfc> Krc Kte> Kfe> Kre ,vcp k k k ntc> arc' ^ac k k xx' yy kZFz kQFz k/,i,kf,2 k.. Jcl 4 cross sectional moment of inertia [mm ] : local inclination angle [rad] -• stiffness matrix of drill bit [N/ m ] 2 : cutting stiffness matrix frequency domain solution [N/mm ] 2 : normal and friction coefficient mechanistic model [N/mm ] 2 : tangential, feed and radial cutting coefficient; orthogonal to oblique [N/mm ] transformation = tangential, feed and radial edge force coefficient per unit width; [N/m] orthogonal to oblique transformation : integration time step number [-] 2 : cutting constant empirical torque expression [N/mm ] 2 -• cutting constant empirical thrust expression [N/mm ] -• number of modes included in a vibration direction [-] : modal stiffness [N/m ] = specific radial force [N/mm ] 2 • specific tangential force [N/mm ] 2 : specific thrust force [N/mm ] 2 ; specific torque force [N/mm ] 2 • tangential, radial and axial cutting coefficients frequency domain sohi-[N/mm ] tion = lateral stiffnesses [N/m] -• direct axial stiffness, axial deflection over axial force [N/m] • cross axial stiffness, axial deflection over torque load [Nm/m] • direct torsional stiffness, torsional deflection over torque [Nm/rad] • cross torsional stiffness, torsional deflection due to thrust loading [N/rad] • estimated location of cutting edge 1, 2 with respect to workpiece grid [-] : discrete location of cutting edge 1, 2 with respect to workpiece grid [-] 4 : warping function constant [m /rad] -• drill length [mm] •- flute length (measured from drill tip) [mm] -• cutting lip length [mm] xv T r -'pth ''eel "ceh UP Al t M m m m mQ...m4 N f N. n0...n4 Pp j Pt 1 z * c p Pind q = height of cutting lip section = point thinned length = projected point thinned length = height of point thinned section = chisel edge length = projected chisel edge length = height of chisel edge section = drill dimension used for initialization of workpiece grid = drill dimension used for initialization of workpiece grid = drill dimension used for initialization of workpiece grid = indentation contact length = elemental length along cutting lip = mass matrix of drill bit = number of elements along the cutting edge or chisel edge = lumped mass at drill tip = cutting force coefficients mechanistic (normal force) = number of teeth on drill/reamer = number of grid points on a grid circle = number of revolutions passed in simulation = whirling frequency number = spindle speed = cutting force coefficients mechanistic (friction force) = working reference plane = working cutting edge plane = normal plane (plane normal to cutting edge) = working plane = indentation force per unit length in thrust and cutting torque directions [N/m] = power coefficient of thrust force [-] 2 = indentation pressure [N/mm ] = power coefficient of cutting force [-] [mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm] [kg] [-] [ k g ] [• [• [• [• [• [rev/min] [-] xvi R = drill radius [mm] Rav = average radius of the cutting force [mm] Rc = chisel edge radius [mm] R P = pilot hole radius [mm] = torque arm (torque from tangential and radial forces) [mm] r = radial distance from drill axis [mm] rind = radius of indentation zone on drill chisel edge [mm] rs = sensor arm [mm] rr = radial drill runout (in direction of cutting lips) [mm] rt = tangential drill runout (perpendicular to cutting lips) [mm] rz = axial drill runout (lip height error) [mm] rQ = torsional-axial coupling factor [-] Ar = vector with regenerative displacements [mm] T = tooth period [s] Tc = cutting torque acting on tool tip [Ncm] t = time [s] At = time step [s] •> u = unity vector of size [m+1] [-] vc = cutting speed at drill periphery [m/min] = dynamic cutting speed [m/min] Vf = feed speed in axial direction of drill bit [mm/min] K = local cutting speed [m/min] 2W = drill web thickness [mm] Wx, 1' ^y, 1' ^ z , 1 = workpiece surface coordinate matrices for surface generated by flute 1 [w/w] xc>yc>zc = lateral and axial tool tip deflections [mm] > > > Xe, l,ye, l,Ze, 1 = coordinates of cutting edge points on lip 1 for workpiece initialization [mm] xfi,i>yfi,i = coordinates of intersection point of flute with wall grid circle [mm] = coordinates of intersection points of cutting edge 1 with grid circles [mm] XP, i^p, vzp, i = coordinates of peripheral point flute 1 [mm] > > > *t,\,yt, i,zt, i = coordinates of tool tip points on cutting edge 1 [mm] xvii > > > xwg, i» y-wg, 1? zwg, 1 = coordinates of workpiece grid points for flute 1 for grid angle g [mm] z l i» Z2/ = state space variables (velocity, displacement) [m/s],[m zrf = height of whirling grid layer in global coordinate system [mm] Z- !(/, A:) = height of cutting edge at intersection points with grid [mm] a = torsional-axial coupling parameter [-] a / r = local reference rake angle (function of the radius r) [degrees] ag = workpiece grid angle [degrees] = local normal rake angle [degrees] andr = local dynamic rake angle [degrees] Po = drill bit helix angle at the drill periphery [degrees] P f l = friction angle [degrees] P„ = normal friction angle [degrees] Pr = local helix angle [degrees] Pxx' Pxj" Pyx' $yy = lateral coefficients of dynamic drilling coefficient matrix [-] $zz> Pz9' Pez' Pee = torsional-axial coefficients of dynamic drilling coefficient matrix [-] Yc Yi. Y2> Y3 = coefficients characteristic equation [N/m] lind = wedge angle [degrees] Jr = local reference angle [degrees] Y w = chisel wedge angle [degrees] e = phase shift between vibration marks of consecutive teeth [rad] = slip line H C' C,-, = modal damping ratio [-] = chip flow angle [degrees] = torsional tool tip deflection [degrees] = twist per unit length [rad/m] K = ratio of imaginary part and real part of eigenvalue [-] = tip angle point thinned section [degrees] 2 K , = drill tip angle [degrees] = tip angle grinding error [degrees] X, = normalized radial coordinate [-] xviii = clearance angle at the drill periphery [degrees] K = local clearance angle [degrees] ^•fnr = local reference normal clearance angle [degrees] A, AR, Aj = eigenvalue of characteristic equation, real and imaginary parts [N/m] V = poisson's ratio [-] p = density [kg/m3] T , = shear stress [N/mm2] = local cutting speed angle [degrees] CP(5) = transfer function [N/m] = cross transfer function [N/m] o x x , o Y Y = direct lateral transfer functions [N/m] ®zz> ^ee = direct axial and torsional transfer functions [N/m] G>ze> %z = cross axial and cross torsional transfer functions [N/m] <t>c = shear angle [degrees] = entry angle of the drill [rad] = exit angle of the drill [rad] = pitch angle of the drill [rad] = chisel edge angle [degrees] = point thinning angle [degrees] = phase shift of eigenvalue [rad] = local reference angle [degrees] Q = angular speed of the tool [rad/ s] (D0 = web angle at drill periphery [degrees] = chatter frequency [rad/s] = natural frequency [rad/s] = local web angle [degrees] xix Acknowledgements I wish to thank my research supervisor Dr. Yusuf Altintas for the opportunity to study at the Man-ufacturing Automation Laboratory, and valuable support and guidance throughout this work. This research was sponsored by NSERC and Pratt & Whitney Canada. Guehring and Sandvik compa-nies provided the cutting tools and tool holders. The Mori Seiki SH403 machining center was donated by Mori Seiki, Japan. Many thanks to the fellow slaves, especially Andrew Woronko, Yuzhong Cao, Dimitri Ostafiev, Arnaud Larue, Will Ferry, Simon Park, Kaan Erkorkmaz and Fuat Atabey, for support and help in the lab as well as making time in and outside of the lab very enjoyable. I want to thank my friends and family from the Netherlands: Michiel van Vliet, Leen Lambregtse, Daniel van der Beek and Marinus Jochemsen, for their patience, support and visits - which have made for good holidays. Participation in the UBC Graduate and Faculty Christian Forum (GFCF) has been a wonderful experience, for which I'd like to express my appreciation to Phil Hill, Olav Slaymaker, Gord Carkner, Paul Stanwood, Bill Reimer, Ed Piers and fellow students from the Graduate Christian Union. The International Group (IVCF) initiated by Ian Elliot has been very good in welcoming me to Vancouver, and has provided me with many friends and good memories. I would like to thank Dick Williams for being a great mentor throughout my time here. Finally, I would like to thank my wife Yoko for her unwavering patience and continuous supply of good food. xx 1 Chapter 1. Introduction Drilling is the most important method used to produce holes in part manufacturing, mainly due to its economy and speed. Tool vibrations during drilling can cause errors in hole size and shape that may be unacceptable. The demand for increased productivity and high quality has stimulated interest in automated drilling equipment for high speed, high precision operations. Progress in improving the speed and precision of drilling operations is hampered by an incomplete under-standing of the mechanisms that cause tool vibrations and the inability to predict the hole shape accurately. Drilling is used to create cylindrical openings in parts. In a drilling operation, a typically flat workpiece is clamped on the machine table and moves towards the cutter, which is held in a rotat-ing spindle with a tool holder, see Fig. 1.1. This feeding motion occurs in spindle direction only and the position of the tool with respect to the workpiece determines the axial engagement. The radial depth of cut is determined by the size of the pre-drilled hole, called pilot hole. Figure 1.1 : Drilling operation on a horizontal machining center; Workpiece on the left is fed to the rotating drill on the right. The mechanics of drilling, vibrations and hole shape formation with slender twist drills are stud-ied in detail in this thesis. From literature it is known that lateral, axial and torsional vibrations Chapter 1. Introduction 2 occur in drilling. Lateral and torsional-axial vibrations have been studied separately, but no com-prehensive model that includes lateral, axial and torsional vibrations has been published in the lit-erature. Figure 1.2 shows four different hole types generated by a regular twist drill. Figure 1.2 : Hole shapes resulting from drilling blind holes in full material; a) Stable cut, no visible vibrations; b) Sunray pattern due to unstable torsional-axial chatter vibrations; c) Trigon caused by whirling vibrations; d) Surface resulting from combined torsional-axial chatter and whirling vibrations. The hole shown in Fig. 1.2a was drilled with a very short drill bit, is perfectly round, and has a smooth surface without drill vibration marks. Fig. 1.2b shows a sunray pattern obtained from an unstable cut (chatter), due to coupled vibrations in axial and torsional directions. Torsional-axial chatter occurs due to a lack of dynamic stiffness [1], or a lack of process damping [2]. The hole is round as in Fig. 1.2a, however, chatter is undesirable because of increased cutting forces, possible damage to tool and spindle, as well as a poor surface finish. Figure 1.2c shows a three-sided poly-gon shape at the bottom and a smooth surface, generated by lateral whirling vibrations [3]. Due to misalignment of the tool with the spindle axis, grinding errors on the tool and workpiece material inhomogenities, the drill deflects from the spindle axis and will contact the wall surface causing the drill centre to trace circles in the direction opposite to the tool rotation. This motion is called a backward whirl, and the most common whirl is one that results in a three sided hole as shown in Fig. 1.2c. Figure 1.2d shows a three-sided hole with sunray pattern at the bottom, left by a drill undergoing torsional-axial chatter as well as lateral whirling vibrations. The holes shown in Figs. 1.2b,c,d were drilled with a slender drill bit. These photographs show that torsional-axial and lat-eral whirling vibrations can occur independently or at the same time. Chapter 1. Introduction 3 This thesis proposes a numerical time domain simulation model for the drilling process which includes torsional, axial and lateral vibrations at the same time. The numerical model enables study of the coupling between structural deformations and the resulting cutting forces acting on the drill. Time domain modeling is also the only method to obtain the surface finish of both the hole bottom (Fig. 1.2) as well as the wall surface. Non-linearities in the system such as the loss of contact between the vibrating tool and stationary workpiece, variation of cutting process parame-ters along the cutting edge and tool geometry are also taken into account. Analytical stability solution of drilling vibrations is also solved in the thesis. The lateral, axial and torsional vibrations are coupled with the drilling process mechanics, and the stability solution is reduced to a fourth order eigenvalue problem. The stability leads to rapid prediction of stable drilling speeds and depths of cut as a function of tool dynamics, geometry and work material properties. The chapters of the thesis are organized as follows: In chapter 2, the relevant literature on drilling geometry, drilling force modeling, dynamic drilling and chatter stability is reviewed in general. The drill geometry of regular twist- and indexable drills is modeled and its dynamic properties are established in chapter 3. A dynamic model of the drill is presented, and the sources of the flexibilities are identified. Model parameters are identi-fied both analytically and by experiment. The modelling of drilling forces using mechanistic and cutting mechanics approaches is presented in chapter 4. In the cutting mechanics approach, the cutting edge geometry of the twist drill is modeled analytically, and the orthogonal to oblique cutting transformation method is used in pre-dicting the force. In the mechanistic approaches, the cutting forces are related directly to cutting Chapter 1. Introduction 4 conditions such as the feedrate and cutting speed, and some drill geometry parameters. Both tech-niques are experimentally verified and compared. Chapter 5 details the drilling time domain model developed for simulation of piloted holes. This model allows for vibrations in lateral, axial and torsional directions, and provides a realistic repre-sentation of the drilling process. Grinding errors on the drill, misalignment due to tool clamping, and nonlinearities such as the tool jumping out of the cut and nonlinear cutting force models are implemented in the model as well. Experimental cutting forces and the sound of the drilling pro-cess are captured to analyze the vibration behavior and compare it with simulations. The frequency domain solution for drilling vibrations is presented in Chapter 6. The dynamic chip thickness is derived as a function of axial, torsional and lateral tool deflections. The dynamic drilling expression leads to an eigenvalue problem which is solved to obtain the chatter stability chart of the drilling process. The chart shows the limiting radial depth of cut as a function of spin-dle speed. The proposed frequency domain solution is compared with the time domain model developed in chapter 5, experiments and previously published stability laws. Chapter 7 concludes this thesis with a summary of contributions and suggestions for future work. 5 Chapter 2. Literature Review Recent advances in cutting tool materials and coatings allow for increased cutting speeds and feeds in drilling, which are employed on high speed machining centers. Although the productivity is increased by higher cutting speeds, vibrations and tool breakage due to a lack of tool, spindle and tool holder stiffness may prohibit increases in productivity, and may lead to deviations from the intended round hole profile and poor hole quality. Improvements in productivity and hole quality can only be obtained if the mechanisms behind cutting forces and drill vibrations are fully understood and modeled. 2.1. Overview In this chapter, the relevant literature on drilling geometry, drilling force modeling, dynamic drill-ing and chatter stability is reviewed in general. Detailed reviews are provided when related meth-ods and approaches are used or introduced in individual chapters. 2.2. Drill geometry The design of the twist drill was patented in 1863 by Morse [4] and is symmetric, having two cut-ting edges and a chisel edge, Fig. 2.1a. The chisel edge in the center works as an indenter. If the drill is ground perfectly and the tool does not deflect, each flute cuts the same chip. Variations on the two-fluted design include three and four fluted models, drills with special tip shapes and mod-ifications to the chisel edge. Modern, symmetric drill designs include solid carbide drills and the brazed drills, which use a (regrindable) solid carbide tip (for the cutting zone) and steel for the drill body. Indexable drills (Fig. 2.1b) also use steel bodies, but accommodate exchangeable, coated inserts with complex geometries. The indexable drill is the only non-symmetrical type in which the peripheral insert creates the hole shape and the center insert cuts the center part of the hole [5,6]. The total number of inserts depends on the drill diameter. Chapter 2. Literature Review 6 Figure 2.1 : a) twist drill geometry; b) indexable drill geometry. The difference in tool geometry leads to different hole profiles [7], as shown in Fig. 2.2. A two-fluted twist drill typically generates a three-sided hole, as shown in Fig. 2.2a. Lateral deforma-tions of the slender drill are caused by uneven distributions of cutting forces, due to grinding errors on the tip, workpiece inhomogenities and misalignment of the drill with the spindle axis, and lead to whirling vibrations. The triangular shape produced by a twist drill is attributed to whirling vibration at odd multiples of the rotational frequency, in the case of a three sided hole whirling at three times the rotational frequency [3]. An indexable drill is always slightly unbal-anced because of the asymmetric design. The force unbalance results in an oval hole shape, due to unequal stiffness in X- and Y-directions of the machine tool-spindle structure [8]. The hole shapes are illustrated in Fig. 2.2b. Huang et al. studied the geometry of Multi Faceted Drills, which have a strongly modified tip geometry with an improved rake and clearance angle distribution [9,10]. Figure 2.2 : a) 18mm hole profile generated by a twist drill (three sided, tool length 95mm); b) 18mm hole profile generated by a indexable drill (oval shape, tool length 70mm) (after Venkatesh [8]). Chapter 2. Literature Review 7 2.3. Mechanics of drilling The mechanics of drilling deals with the prediction of cutting forces acting on the tool. A mechan-ics model leads to the prediction of cutting forces for a drill geometry under a variety of cutting conditions. The following main parameters, illustrated in Fig. 2.3, affect the drilling forces: a) Cutting speed Vc - measured at the periphery of the drill in [m/min], b) Feedrate fr - the dis-tance the tool moves into the workpiece per spindle revolution in [mm/rev], and c) Radial depth of cut b - determined by the pilot hole diameter size Dp and the drilled hole diameter D. A pilot hole is generated with a drill of a smaller diameter. The feed velocity Vj- is the speed with which the tool penetrates into the material in [mm/min]. The uncut chip is shown in Fig. 2.3a, and its area is determined by the radial depth of cut and the feedrate. The twist drill shown has two flutes, and every tooth therefore cuts a chip thickness of half the feed per revolution. The cutting torque Tc and thrust force Fz acting on the tool are deter-mined by the uncut chip area. When the drill is not vibrating, the uncut chip thickness does not change over time, except when the tool enters the workpiece and exits from it. Certain drills have irregular cutting edges, which lead to irregular distributions of the chip load along the cutting edge. Since the geometry changes along the edge, the governing laws of cutting mechanics may vary significantly, which is investigated in this thesis. Chapter 2. Literature Review 8 Figure 2.3 : a) definition offeed fy, feed speed Vp radial depth of cut b; b) cutting speed Vc; c) thrust force Fz and torque Tc acting on the tool. Fig. 2.4 shows the tip-geometry of a twist drill and detailed pictures of the cutting regions along the chisel edge and cutting lip [11]. At the center of the drill, the chisel edge works as an indenter and merely pushes the material sideways. From the edge of the chisel edge towards the periphery of the drill, the chip formation changes from extrusion, with discontinuous chip formation [12], to cutting through shearing of the material. Drilling is characterized by oblique cutting, where rake angle, oblique angle and cutting speed vary over a wide range along the cutting lips. Figure 2.4 : Photographs of different cutting regions for a twist drill, from extrusion at the chisel edge to oblique cutting at the periphery; 2 Wis the drill web thickness, 2tct is the drill tip angle (after Ernst et al. [11]). Chapter 2. Literature Review 9 2.3.1. Mechanistic cutting force models Before the cutting mechanisms at the different regions along the cutting geometry were known, Kronenberg [13] presented simple formulas for torque and thrust, which are functions of drill diameter and feedrate only. Early work was done by Galloway [14], who studied the effects of drill geometry on the cutting forces, tool life and hole geometry errors. Using dimensional analy-sis, Oxford and Shaw [15] also derived these formulas. These formulas assume geometric similar-ity between the different drill diameters, which does not exist in practice. Geometric similarity means that for example the web thickness W (Fig. 2.4) is a constant percentage of the drill diam-eter. The torque Tc and thrust Fz were expressed by the following empirical relationships: Tc-kc*f"*D" F . - k ^ . l T (2-0 where (&C], kcJ were empirical cutting constants, fr the feedrate and D the drill diameter. The empirical cutting constants were identified from cutting tests. This approach in cutting force mod-eling is called mechanistic as coefficients are calibrated from experiments directly. Chan-drasekharan et al. [16,17] predicted the thrust and torque using a mechanistic oblique-cutting force model for the cutting lips (the tertiary cutting edges), continuing on the work of Stephenson et al. [18,19]. The oblique cutting was modeled through normal and friction coefficients, which were a function of the chip thickness, cutting speed and rake angle. The very center of the chisel edge was modeled as an indenter, and a yield stress characterizing this process. This yield stress was assumed to be independent of the cutting speed. The remaining part of the chisel edge, the secondary cutting edge, is modeled as an orthogonal cutting edge. All model coefficients were calibrated from only four drilling experiments (two cutting speeds and two feedrates), in which blind pilot holes were drilled. The contributions to the thrust and torque by the cutting lips and the chisel edge were determined separately, which was necessary for this approach. The normal and friction force coefficients were determined from experimental cutting forces when only the lips were cutting. The yield stress was determined from the thrust measurement of cutting a full hole Chapter 2. Literature Review 10 at the high feedrate by subtracting the thrust contributed by the cutting lips and remaining part of the chisel edge, thus obtaining the thrust contributed by the indentation zone. Although the model was mechanistic, it related the cutting forces to the actual cutting geometry, and could therefore predict the cutting forces for drill geometries with a different diameter, web thickness, point angle, cutting lip and chisel edge shape with reasonable accuracy (within 20%). An example of the torque and thrust for a 9.5mm uncoated carbide drill cutting cast iron at 400rpm and 0.25mm/rev feed is shown in Fig. 2.5. The web thickness is 1.3mm and the blind pilot hole is 2.38mm. The model coefficients were calibrated with a 12.7mm drill cutting 0.05mm/rev and 0.30mm/rev at lOOrpm and lOOOrpm, so the cutting speed used in Fig. 2.5 corre-sponds to 500rpm for the 12.7mm drill. Stage 1 shows air cutting and in stage 2 the cutting lips engage with the material and full engagement of the cutting lips starts at 2.0s (stage 3). At 4.4s, the chisel edge also starts to cut (stage 4), and the thrust increases significantly, while the torque only increases slightly, as the torque arm of the chisel edge is very small. As can be seen from the thrust curve, the chisel edge contributes significantly to the thrust, despite its small size. Inaccu-rate modeling of the chisel edge region therefore easily leads to strong discrepancies between pre-dicted and measured thrust force. Chapter 2. Literature Review 11 Time (sees) Time (sees) Figure 2.5 : Experimental and predicted thrust and torque using a mechanistic model (after Chandrasekharan et al. [16]); 1: Air cutting; 2: Lip engagement; 3: Steady state lip cutting; 4: Lips and chisel edge both fully engaged. 2.3.2. Cutting mechanics force model Armarego et al. [12,20-25] have conducted extensive research in cutting force modeling of machining operations in general and in modeling of the drilling thrust and torque in particular. Armarego used Merchant's thin shear plane theory [26] to predict shear stress, shear angle and average friction angle from two dimensional orthogonal cutting tests. The orthogonal to oblique cutting transformation was used to predict the cutting forces for both the oblique drill and the chisel edge with good accuracy for point thinned general purpose twist drills [12]. Deviations in thrust and torque averaged about -10%. They did not use an indentation model for the center of the drill point. For drills with special point geometries, differences in thrust force between model prediction and experiment were large due to modeling difficulties, thrust deviations averaged -50% [21]. Torque deviations averaged -5%, as the torque was not influenced significantly by the chisel edge. One of the problems in using the orthogonal to oblique transformation lies in the cor-rect geometrical description of the drill, and especially the drill point. Due to different grinding techniques, drills that show only slight differences in geometry, can show large differences in cut-ting forces. Additionally, it is cumbersome to obtain an accurate cutting database for the drill geometry, as the rake angle typically varies from - 3 0 ° until + 3 0 ° degrees on the cutting lips. The Chapter 2. Literature Review 12 chip formation for large negative rake angles, down to - 6 0 ° on the chisel edge, becomes discon-tinuous, and the assumptions used in the orthogonal to oblique transformation do not hold any-more. 2.4. Dynamics of drilling Tool vibrations during drilling can cause errors in hole size and shape that may be unacceptable. Figure 1.2 showed four different hole types generated by a regular twist drill. The demand for increased productivity and consistent, high quality in hole making has stimulated drilling research. Understanding of the mechanisms that cause drill vibrations will allow for improve-ments in speed and precision of drilling operations. Accurate prediction of the cutting force sys-tem, the tool dynamic properties, and the chip generation mechanism are key in numerical simulation of the drilling process. Section 2.4.1 gives a review of torsional-axial chatter vibrations in drilling while section 2.4.2 reviews the literature on lateral and whirling vibrations. 2.4.1. Torsional-axial chatter vibrations in drilling Galloway [14] analyzed chatter vibrations on radial drilling machines, which have flexibility in the axial direction of the bit. He also provided a stability chart with experimental verification. Galloway noted that as the drill enters the workpiece, the cutting forces rapidly increase and the machine is deflected from its unloaded position, causing a reduction in feed. Rapid changes in torque and thrust occur as the drill point breaks through the underside of the workpiece, and there may be a substantial increase in the instantaneous feed as the elastic energy stored in the structure is released and the machine returns to its undeflected position. Breakthrough is often accompa-nied by vibration, which can especially be seen in the torque, showing fluctuations as large as the average torque, after the chisel edge has broken through. Chapter 2. Literature Review 13 Ema et al. [27-29] focused on lateral (whirling) vibrations first, and they studied lateral chatter vibrations of long drills in [30]. The lateral chatter frequency was found to be the natural fre-quency of the drill when supported in a machined hole (clamped-pinned) and loaded with a typi-cal thrust force (without rotating the workpiece). They studied the effect of drill length, cutting conditions, and adding mass to the bit. From experiment they observed whirling vibrations when the drill was engaging with the workpiece, but at a certain depth the vibration frequency steeply increased, from which time chatter was defined to start. The vibration perpendicular to the cutting edge was much larger than the vibration along the cutting edge in this stage. The amplitude of the lateral vibrations decreased as the drill penetrated into the workpiece. Variation of spindle speed, feedrate or pilot hole size hardly affected the chatter frequency. An increase in spindle speed, reduction of the feed or the use of a pilot hole lowered the hole depth from which chatter devel-oped. From experiment they concluded that the chatter frequency was an odd integer of the drill rotation frequency, but they did not provide a model to explain this. In [31] Ema et al. found that cutting conditions could not eliminate chatter, except by choosing a low cutting speed. At higher cutting speeds they used an impact damper to suppress the chatter vibrations. Ema et al. noted that the chatter frequency measured was lower than the clamped-pinned natural frequency of the tool. Using the Finite Element method, Tekinalp and Ulsoy [32-34] studied the bending natural fre-quency as a function of drill geometry, rotation speed and boundary conditions such as pin sup-ports and axial forces applied to the drill bit. Rincon and Ulsoy continued this study by including rotary inertia and gyroscopic moments [35,36]. In [37] Rincon and Ulsoy calculated the influence of induced lateral vibrations on the torque and thrust for piloted holes, using the orthogonal to oblique transformation. Their model underestimated the torque and thrust fluctuations, and strongly over predicted the lateral forces. The studies by Ulsoy et al. did not deal with self-excited (unstable) drill vibrations. Tarng and L i [38] detected axial chatter vibrations in drilling from experimental cutting forces and eliminated the chatter vibrations by adjusting the spindle speed Chapter 2. Literature Review 14 such that the ratio of chatter frequency and tooth passing frequency became an integer. Bayly et al. [1] developed a frequency domain solution for torsional-axial chatter in drilling, using Trusty's stability law for orthogonal cutting [39]. The torque acting on the drill unwinds the drill, while the bit elongates at the same time. This effect has been investigated in pre-twisted beams by Hodges et al. [40], and lead to a chip thickness regeneration mechanism in axial direction. Depending on the radial depth of cut and spindle speed, the oscillatory force will grow and the vibrations may become unstable. A heavy fixture was mounted on the stationary drill bit to measure the tool vibration and change the torsional-axial frequency of the drill-fixture system from 5800Hz to 350Hz in [41]. As a result, the lobes that would have occurred at high cutting speeds, shifted to 17 times lower spindle speeds. The experimental verification of the stability lobes showed a good match for the speed range and radial depths tested. Dilley [42] noted that the sunray patterned sur-face generated by a bending vibration can look the same as one caused by torsional chatter because the fixed-embedded tool has a higher lateral natural frequency than a freely vibrating drill. Dilley [43] proposed to use a spring end condition to represent the effect of the chisel edge on both bending and torsional-axial chatter frequencies in drilling full holes. In [44] Dilley et al. included the influence of the drill margin engagement on the chatter behavior of twist drills, also by adding a spring to the drill tip. From comparison of drilling piloted holes and piloted cylinders, the chatter frequency appeared to shift upwards as the drilled hole depth increased. This phenom-enon was attributed to the drill interacting with the hole wall. Bayly et al. studied lateral chatter vibrations and whirling as well [45-47], which is discussed in the next section. If the vibration amplitudes are large, then part of the tool may lose contact with the material being cut, which is a nonlinearity in the process, as first modeled by Tlusty [48]. Montgomery and Altintas [49] included this phenomenon in their time domain simulation model for milling as well. Until now, only static deflections have been studied in drilling. No time domain model has been presented that allows detailed analysis of dynamic drilling vibrations, such as regenerative tool vibrations, forces and the resulting surface. Chapter 2. Literature Review 15 2.4.2. Lateral and whirling vibrations in drilling Galloway [14] noted that hole oversize strongly depended on the symmetry of the drill point. A lip height error rz caused a hole oversize of rz x tame,, where K , is half the drill tip angle. After some initial skidding, the drill would tend to rotate about a new axis, displaced du - 1/2 x rz x tanK? from the spindle axis, as illustrated in Fig. 2.6. In the deflected position, the chip areas cut by each flute were the same again. Figure 2.6b shows the experimental hole oversize for two different tip angles, as a function of lip height error, denoted by circles. The lines show the predicted hole oversize matched experiments well. Galloway noted that using a pilot hole larger than the chisel edge resulted in increased tool displacements when the drill was engag-ing with the workpiece, and that the trace of the movement of the drill point formed a series of tri-angles centred about a point close to the spindle axis. The general center of motion was closer to the spindle axis when a pilot hole was used, thus improving the hole alignment. The slope of the drilled hole depended on the deflection of the drill when it became fully engaged with the work-piece. Figure 2.6 : Effect of lip height error on hole oversize; a) position of axis of drill rotation; b) experimental hole oversize as a function of lip height error in steel (after Galloway [14]). Chapter 2. Literature Review 16 Lee et al. [50] analyzed skidding and wandering motion of drill bits. They assumed that the tool followed an elliptical or circular path, opposite to the cutting direction, as seen from the spindle, and that each edge cut at the same position after one tooth period. The authors [50] found the rota-tion frequency of the whirling motion to be an integer multiple of the tool rotation speed. The tool center motion was defined by: x c ( 0 = e(0cos(27t / / r ) / / v c ( 0 = e(0sin(27r / w '0 where Q was the spindle frequency, fj the whirling frequency in the global frame, e(t) the amplitude of the drill axis motion, and Nw a positive integer. Polygonal shapes with odd numbers of sides were generated as a result of the elliptical motion, as shown in Fig. 2.7. (2.2) Figure 2.7 : Formation of shaped hole profile, after Lee et al. [50]; a) Trigon shaped hole (Nw=l); b) Pentagon shaped hole (Nw=2); c) Heptagon shape hole (Nw=4). In experiment they observed that the number of sides on the polygon increased as the drill pene-trated the workpiece, which meant the hole being generated became rounder. Whirling motion did persist though. The use of a pilot hole eliminated the skidding that took place when the drill started to touch the workpiece, but whirling still took place, starting with a higher number of sides on the polygons created. The skidding affected the hole positional accuracy. The model by Lee et al. did not predict the amplitude of the whirling vibration, or when and why the number of sides Chapter 2. Literature Review 17 on the polygon changes. Zhou and Lin [51,52] explained the whirling motion during initial pene-tration as a regenerative chatter vibration. Reinhall and Storti [53] presented the drill as a rod impacting a rigid plate and, using a simple model, showed that the number of sides created on the polygon depended on the spindle speed. They noted that softer materials increased the probability of non circular holes. Zelentsov [54] wrote that lobing was always observed in the presence of conventional enlarge-ment of holes, even at the bottom of deep holes. Polygon shaped profiles in different sections along the axis were rotated with respect to the preceding by a certain angle, in one or another direction. A change in drill rotation frequency and feed could only distort the contour of the trans-verse hole section. Lobing in drilling could be reduced by increasing the drill stiffness and reduc-ing the runout. In [36] Rincon and Ulsoy analyzed the effect of the drill geometry, rotary inertia and gyroscopic moments on the fundamental bending frequency of the drill bit, using finite element analysis. Rin-con and Ulsoy [37] examined the effect of forced elliptical motion of the drill bit on cutting torque and thrust. While the torque and thrust varied strongly, their mean values changed little. The anal-ysis was not extended to include the effect of these forces on the drill vibrations. Ema et al. [27] studied whirling behavior of stationary drills cutting holes into rotating workpieces. From experi-ment the number of sides created always appeared to be an odd number for a two fluted drill. They observed that an increase of the chisel edge length delayed the initiation of lateral whirling vibrations, as well as reducing the vibration amplitude, as a result of the increased bending stiff-ness of the bit. The duration of whirling vibrations was also shortened with increased chisel edge size. After the lips had fully engaged, the whirling vibration decreased gradually. Increasing the Chapter 2. Literature Review 18 feedrate delayed the initiation of whirling vibration, but also increased the amplitude and duration of the whirling vibrations. A drill with medium web thickness generated a five-sided shape when drilling a full hole, and a seven-sided shape when using a pilot hole, the whirling vibration amplitude being twice as large when using a pilot hole. A drill with a smaller chisel edge created a three-sided polygon when drilling a full hole. When drilling a piloted hole, the stationary drill moves counterclockwise, trac-ing near ellipses with a tilted axis. The number of ellipses traversed by the drill during one work-piece revolution was just under an odd integer, and this phase increased with pilot hole size and feedrate. In [28] Ema et al. observed from experiment that a whirling vibration of seven cycles per revolu-tion, see Fig. 2.8, evolved into a vibration of five cycles per revolution, as the drill engaged fur-ther into the workpiece. Two out of seven vibration cycles decreased in amplitude while the other five increased. During engagement the number of whirling cycles could start from infinity, then evolving down step wise from 13 to 11, 9, to 7 and finally 5. The shape of the ellipse was hardly affected by feedrate, point angle, relief angle or helix. The authors established a flank collision index to explain that the flank was likely to impact the cut surface close to the chisel edge. Whirl-ing would start early after engagement if a large flank collision index was used, and was more likely to result in five or three sided holes. Ema et al. indicated that spindle speed did not affect the whirling behavior, but did not develop a model to explain the observed phenomena. Chapter 2. Literature Review 19 Figure 2.8 : Trajectory of stationary drill exhibiting whirling vibration at seven times the rotational frequency of the workpiece; (S designates starting point, E the ending point) (after Ema et al.[28]). In [29] Ema et al. further investigated whirling vibrations by numerical simulation. The drill bit was given a small initial displacement to model eccentricity between drill and workpiece axes. Mass, stiffness and damping of the drill bit were obtained experimentally. In simulation, increas-ing the feed reduced the number of sides being generated, while the whirling frequency remained below the first bending frequency. At low feedrates, an increase in spindle speed reduced the number of sides generated, while increasing the pilot hole diameter resulted in a higher number of sides. During engagement of the drill into a piloted workpiece, the whirling frequency decreased with cutting depth. The simulated vibration pattern for a seven-sided hole agreed well with exper-iment, where vibration in the direction of the cutting edge was almost sinusoidal. The tool vibra-tion perpendicular to the cutting edge was more jagged, and increased in amplitude with cutting speed. The simulation showed that the number of sides generated should decrease with increased spindle speed, but this was not confirmed by experiment. Lin et al. [55] found in experiments that the tool whirling frequency was slightly smaller than an odd multiple of the rotation speed, dismissing the assumption made by Lee et al. [50]. The poly-gon would rotate as the drill penetrated into the workpiece, forming spiral lines on the hole wall. Chapter 2. Literature Review 20 They concluded that the wandering motion was a self-excited regenerative vibration. Wijeyewick-rema et al. [56] used a finite element model that included the effects of transverse shear, rotary inertia and gyroscopic moments to study the drill response to sinusoidal lateral force patterns. They also used smoothed experimental forces measured from whirling vibrations as input to their model. The phase shift between cutting force and tool displacement was much larger in experi-ment than their model suggested. Gong et al. [57] used the same finite element model to deter-mine the influence of drill geometry on critical speeds and buckling loads for micro-drills. Deng et al. [58] proposed that whirling in BTA (boring and trepanning association) drilling was similar to chatter in milling. They suggested that the number of waves generated around the hole profile was the same as the number of waves generated between two consecutive teeth in milling. They showed this principle in simulation, but did not verify it by experiment. Bayly et al. [45] developed chatter stability models for drilling holes with large pilot hole sizes and reaming operations, which were restricted to bending vibrations of the tool. Their simulation model showed three- and five-sided holes resulting from lateral chatter vibrations in drilling. Sta-bility lobe predictions were made based on measured static stiffness, assumed damping of 2% and a natural frequency of 250Hz. The spindle speed at which the lobe was penetrated determined whether the whirling would be forward or backward. The model was not verified experimentally. In [46,47] Bayly and coworkers developed a quasi-static model of reaming in which they explained the formation of reamed holes with (Ny+ 1) or (Jvy- 1) sides, where the reamer has Nj- teeth. Eigenvalues were found by substituting characteristic solutions in the equations of equi-librium, and solving the resulting characteristic equations. A six fluted reamer generated five and seven sided holes in experiment, depending on the feedrate, see Fig. 2.9. The model predicted these shapes by changing the rubbing coefficient. Chapter 2. Literature Review 21 simulation experiment simulation experiment Figure 2.9 : Predicted and experimental hole shape in reaming; a) 5-sided hole; b) 7-sided hole (after Bayly et al. [47]). Bayly et al. [3] also analyzed the formation of lobed holes in drilling using a quasi-static model that includes regenerative cutting and rubbing forces. Unstable retrograde whirling modes were found in terms of eigenvalues and eigenvectors of a discrete state-transition matrix. These unsta-ble modes corresponded closely to behavior seen in practice. Dilley et al. [42-44] studied the effects of the chisel edge and margins on the drill natural frequen-cies. Pirtini and Lazoglu [59,60] predicted oval hole shapes from a static drilling model, but experiments showed round hole shapes with small distortions. An unbalance force acted on the drill due to grinding errors. The oval shape is generated by their simulation as they fix the unequal drill stiffnesses to the machine frame instead of rotating them with the tool, which would have resulted in a round, but oversized hole. Gupta et al. [61] developed a cutting force model that took drill alignment errors into account. The authors used beam theory to verify the resulting tool deflection and radial forces as a result of the drill misalignment. Gupta et al. [62] included the drill wandering motion of Lee et al. [50] to generate three-sided hole shapes, the amplitude of the wandering motion being dependent on the drill deflection due to unbalance forces. When the cutting lips were fully engaged with the work-piece, the wandering motion was switched off. The hole gradually became round and of smaller diameter due to interaction with the wall surface. The margin forces due to wall contact were Chapter 2. Literature Review 22 modeled by Jalisi et al. [63]. The prediction of the average diameter along the hole by Gupta et al. [62] agreed well with experiment. Gong et al. [64,65] studied the skidding and wandering motion of micro-drills. Recently, Arvajeh and Ismail [66,67] modeled bending stability in drilling by assuming the drill tip is pin supported in the hole being cut, which was proposed by Ema et al. [30]. Tool rotation at the tip provides for a chip thickness regeneration mechanism that explains bending instability with vibration frequencies three to four times higher than the lateral bending frequency. A long drill (length to diameter ratio, L/D=20) would vibrate in torsional-axial mode at certain speeds, but in bending at other speeds. The bending frequency, as identified from X , Y cutting forces, would be the pin-supported bending frequency minus the spindle frequency. The thrust force would show a peak at twice this frequency. The bending frequency did not match well for a drill with L/D=10. Arvajeh and Ismail modeled the variation in cutting and geometrical parameters along the drill lip to improve the accuracy of their frequency domain torsional-axial chatter stabil-ity prediction in [68]. 2.5. Conclusions Self excited, chatter vibrations can occur in drilling, leading to a wavy surface finish and increased cutting forces. The chatter frequency is typically higher than the whirling frequencies. The mechanism for chatter in drilling is attributed to lateral, torsional and axial vibrations which generate a time varying chip thickness distribution along the cutting edges. There is a chain rela-tionship between the vibrations, chip thickness, cutting forces and vibrations, making the system self excited. Although individual vibrational modes have been investigated by previous research-ers, a model that includes all vibrational modes has not been developed yet, except in this thesis. Also, the dynamic models that have been developed use simple, linear cutting force models. Chapter 2. Literature Review 23 This thesis presents a time domain model that will: - simulate combined torsional, axial and lateral vibrations in drilling - include coupling between lateral, axial and torsional vibrations - predict the vibrations during penetration of the drill into the workpiece with a pilot hole - predict unstable, regenerative chatter vibrations - accommodate non-linear force models Based on the drill tip geometry and dynamic properties, the time domain simulation will provide the history of cutting forces, tool vibrations, surface finish, and the chatter stability of drilling operations. The combination of accurate force modeling and distribution of chip thickness along the cutting edge based on the true kinematics of drilling, which takes all vibrational modes into account, will lead to the most accurate drilling process model developed. 24 Chapter 3. Drill Bit Model This chapter describes the geometry of drill bits commonly used in industry, and their dynamic properties. Two main drill types can be distinguished: twist and indexable. The twist drill is a solid body made of high speed steel (HSS), solid carbide, or a combination of both (brazed). The geometry is obtained by grinding a cylindrical blank. In a brazed drill, the tip material is solid car-bide, while the body is made from steel. Typically, a coating is applied to the tool to improve the tool life and wear resistance. The indexable drill consists of a machined steel body on which coated inserts made from solid carbide or other high quality tool materials are mounted. Indexable drills are more economic, as the inserts can easily be replaced. Twist drills can be reground sev-eral times, but each time a new coating needs to be applied as well. 3.1. Twist drill geometry A basic twist drill geometry with cutting diameter D is shown in Fig. 3.1. Figure 3.1 : Twist drill geometry specifications. The chisel edge is located at the tip of the drill. The two helical cutting lips with helix angle P 0 meet with the flutes at the drill periphery. The drill tip angle is specified as 2K, . The flutes do not Chapter 3. Drill Bit Model 25 cut, but are used to evacuate the chips from the drilled hole. The chisel edge has a width of 2 W, called web thickness, and radius Rc. The chisel edge angle cp orients the chisel edge with respect to the cutting lips. The flute geometry is defined by the helix angle P 0 at the drill periphery and the drill cross section. It should be noted that the diameter of the flutes is reduced slightly towards the shank, to provide clearance as the drill penetrates into the workpiece. 3.1.1. Point thinned drill geometry Although the chisel edge is small, it contributes significantly to the thrust necessary to feed the rotating drill into a workpiece. In order to reduce this effect, several tip modifications can be made. One of the most common methods is point-thinning, in which the length of the chisel edge is reduced. Fig. 3.2 shows the details of a point-thinned chisel edge. The cross-hatched area is cre-ated by an additional grinding operation. As a result of this grinding step, the drill has a small chisel edge and an additional set of cutting edges. i Figure 3.2 : Point thinning of the chisel edge to reduce thrust force. The geometrical details that define the complete cutting edge geometry are shown in Fig. 3.3. A lip consists of three sections: a) chisel edge, b) point thinned chisel edge section, and c) main cut-ting lip. Plane A contains the cutting lip and is parallel to the drill axis. Two details of the top view (view from spindle) show the dimensions of the chisel edge and point thinned section. Chapter 3. Drill Bit Model 26 chisel edge section (top) ; point thinned section (top) 1 W - L c e h j point thin tip angle in plane C: LC|Cos(coQ) R Lptcos((ppt-co0) T , plane B k i . Z . Lcecos((p-C00) Figure 3.3 : Geometrical details of point thinned drill geometry. Plane C contains the point thinned section of the chisel edge, is parallel to the drill axis and shows the point thin tip angle K ^ , . Plane B shows the projection of the cutting edge geometry on a plane through the drill axis. Chapter 3. Drill Bit Model 27 3.1.2. Grinding errors and tool misalignment The drill geometry described above is the perfect geometry. Theorethically, as a twist drill is sym-metric, no lateral forces should act on the tool. In practice a drill may not be ground properly, or there may be misalignment of the tool in the tool holder. Both grinding errors and misalignments will cause force unbalances, which affect the hole geometry. It is difficult to measure the tilt angle of the drill because of the flutes, or the angles of the cutting lip with respect to the drill axis. A more practical way is to measure the radial location of the flutes while the tool is mounted in the tool holder on the machine, to determine the radial runout. The axial locations of two points on each cutting lip are measured to establish the axial runout. Figure 3.4 : Measurement of drill runout and lip grinding errors. The radial runout rr in the direction of the cutting lips causes equals deflections on the flutes, but in opposite directions. Therefore, the measured runout equals rr } + rr 2 ~ 2rr, as can be seen in Fig. 3.4. Tangential runout rt is measured perpendicular to the cutting edges and is established by measuring the flutes as well. Chapter 3. Drill Bit Model 28 Secondly the tip angle for each individual flute is determined. For one flute, the lip height differ-ence between radius rx and r2 determines the tip angle for that lip. If the lips have the same tip angle, but 8/zL and 8h2 are not zero, then the lips are not ground at the same height, and there is an lip height error rz, also called axial runout. Note that a radial drill axis runout rr causes addi-tional opposite height errors on the lip, equal to r r/tanK,. 3.2. Indexable drill geometry The indexable drill geometry is shown in Fig. 3.5. The indexable drill is different from regular twist drills in that it does not cut in the center of the hole. Instead, it leaves a very small cylinder that breaks while the hole is being generated. Because there is no chisel edge, there is no indenta-tion at the center, resulting in lower thrust forces than regular twist drills. When the indexable drill starts drilling a hole, the peripheral insert is the first to contact the workpiece. The peripheral insert creates the hole geometry (wall) and this insert is tilted with respect to the drill axis, so the wall will not be recut by the top part of the insert. The drill body has a reduced diameter as well. While the hole is being machined, the bottom of the hole has a special shape, shown in Fig. 3.5, due to the arrangement of the inserts. When drilling a through-hole, a disc is ejected from the hole, when the drill reaches the bottom of the workpiece. In Fig. 3.5b, the peripheral insert is aligned with the r-axis. The center insert is mounted under a small angle (balance angle) with respect to the peripheral insert, to create a zero force balance in r -direction. If there is a force unbalance in r -direction, the generated hole would either be smaller or larger than intended. Chapter 3. Drill Bit Model 29 clearance angle top part of insert ejected disc Figure 3.5 : Geometry of indexable drill; a) insert configuration, b) balance angle. Fig. 3.6 shows an insert in detail. The insert has a tip angle 2K,and a corner radius. Along the edge there is a chip breaker, that facilitates the chip breaking process by forcing the chip to deform with a smaller curvature. The edge itself is chamfered, which means it's not sharp, but has a small radius. This is necessary as the insert is made from solid carbide, which is very brittle and chips easily. The peripheral and center insert typically have different coatings and carbide grades, as the cutting conditions change from periphery to center. chip breaker groove chamfered cutting edge corner radius cross sectional view chamfered cutting edge\ Figure 3.6 : Detail of insert geometry used in indexable drill. Chapter 3. Drill Bit Model 30 3.3. Dynamic properties of drill bits The general equations of motion for the dynamic drilling system can be formulated in the station-ary frame as follows: ' * c ( 0 " xc(t) ' xc(t) ' Fx(t) ' •+ Tel- •+ Ik]-. yc(0 Fy(t) z c ( 0 Fz{t) - e c (0 - . e c(o . . e c ( 0 . . TCU) . (3.1) where (xc, yc) denote positive lateral, (zc) the axial and (0C) the torsional deflections of the drill center, as defined in Fig. 3.7. The matrices [M], [C] and [K] contain mass, damping and stiff-ness characteristics reflected at the drill tip, respectively. The external loads acting on the drill tip are two lateral forces (Fx, F ), thrust force (Fz), and torque (Tc). Due to the twisted shape of the twist drill, torsional and axial deflections are coupled. Figure 3.7 : Dynamic model of drill bit. 3.3.1. Beam theory Beam theory explains how lateral, axial and torsional natural frequencies are related to a cylinder with modulus of elasticity E, density p, length L and cross-sectional properties A, I. The shear modulus G is defined as: Chapter 3. Drill Bit Model 31 G " 2 ( 1 ^ ) <32> The natural frequencies and its respective stiffnesses of a uniform, cylindrical beam are given in Table [69]: Table 3.1 : Natural frequencies and stiffnesses of a uniform cylindrical beam [69]. Lateral Axial Torsional Natural frequency 0.55959 [FJrrr n Stiffness [N/m] L — [N/m] The axial and torsional natural frequencies are inversely proportional to the length of the cylinder, and do not depend on its diameter. The lateral natural frequency is inversely proportional to the second power of the length and proportional to the square root of the ratio moment of inertia/ cross-section. The lateral stiffness is inversely proportional to the third power of the length, while axial and torsional stiffnesses are inversely proportional to length. The material properties of the high speed steel (HSS) drill bits used in this thesis are provided in Table 3.2. Table 3.2 : High speed steel (HSS) material properties. Property Symbol Value Young's modulus E 2.17 x lOUN/m2 Density P 8.1 x 103kg/m Poisson's ratio v 0.28 Shear modulus G 8.477 x l o V / w 2 If we model a drill bit which has a diameter of 16mm and a stick out length of 176mm (L/D=ll) as a cylindrical beam with an effective diameter of 12mm (75% of the diameter), the natural fre-quencies and stiffnesses are as listed in Table 3.3. A reduced diameter is used to account for the material ground away to create flutes. Chapter 3. Drill Bit Model 32 Table 3.3 : Drill bit natural frequencies and stiffnesses when modeled as a cylindrical beam. Lateral Axial Torsional Natural frequency 280.5I Hz 7352.2i7z 4595.1 Hz Stiffness 1.2155 x \05N/m 1.3944 x 10SN/m 9.80 x l02Nm/rad Due to the pretwisted shape of the bit, the cross-section is not constant but varies along the tool length. As a result, these formulas can only be used as an approximation and the lateral natural frequencies in X and Y directions may be different. Also, a cylindrical beam has distinct torsional and axial natural frequencies, while a twist drill only has a single torsional-axial frequency, due to strong coupling in the bit geometry. Coupling between the lateral directions X and Y can also exist. Hodges [40] modeled the coupling between axial and torsional deformation for pretwisted bars. He derived an equation for the torsional deflection of a pretwisted beam due to an applied static axial load. Later work by Hodges and coworkers [70,71] has extended this theory. By the principle of reciprocity, the axial deflection at the end of a beam zc due to an inequivalent tor-sional load Tc is identical. The relationship is: z = - ^ - T (3.3) GIA c K } where 0, is the twist per unit length, L the length, A the cross-sectional area, / the moment of inertia and kw a warping function constant. In order to use this expression, the warping function for the drill needs to be obtained. The warping function is different for each drill bit, and therefore this approach is not employed in this thesis. 3.3.2. Finite element model of drill bit A Finite Element (FE) model of the drill bit has been developed to study the coupling between the lateral, torsional and axial modes numerically. The twist drill consists of a revolved extrusion of the cross-section and a cylindrical part. The tip of the drill is also modeled with a simple boolean Chapter 3. Drill Bit Model 33 operation. The clamping of the tool into the tool holder is assumed rigid, because the drill is slen-der and flexible relative to the spindle and tool holder. The model is shown in Fig. 3.8. The natural frequencies of the drill bit are found from modal analysis in the C A D software, which has a built in Finite Element program. The stiffness in lateral, axial, torsional and the cross-stiffnesses are determined by applying concentrated loads, and determining the corresponding static deflections, see Table 3.4. To apply a pure torque, two equal forces with opposite directions are applied at the peripheral points of the cutting edges. Figure 3.8 : Drill bit model in CAD system, and forces applied to create a pure torque. 3.3.3. Experimental modal analysis - stationary drill For experiments with a stationary drill bit, the tool is mounted in a tool holder on the table of the machine tool and the workpiece is rotated in the spindle. In order to measure the dynamic proper-ties of the drill in this setup, four optical displacement sensors were used. A lightweight, alumi-num fixture is mounted on the tip of the drill with setscrews to provide surfaces for both excitation of the drill and measurement of the resulting vibrations. A cap is used as there is no face on the tip of the drill that is perpendicular to the drill axis. Two sensors (SI and S2, at a dis-tance rs from the drill axis) are used to capture the torsional deflection as well as the lateral Chapter 3. Drill Bit Model 34 deflection in the 7-direction. The third sensor (S3) measures the Z-deflection. The transfer func-tion in X-direction is determined in a separate measurement setup. The measurement setup for the torsional-axial transfer function is shown in Fig. 3.9a. The torsional transfer function is obtained by impacting the fixture along the working axis of sensor SI with an instrumented hammer, and summing displacement sensor signals dS^ and dS2, cancelling out the deflections in 7-direction. The drill will also deflect in Z-direction due to the applied torque, and sensor S3 reads this axial vibration, yielding transfer function: Figure 3 . 9 : Experimental setup for measurement of torsional-axial, lateral, and axial transfer functions. Secondly, an excitation force Fi is applied in Z-direction (Fig 3.9b) to obtain the direct axial transfer function and the torsional response due to axial force: Chapter 3. Drill Bit Model 35 ^zz=7F[N/m] dS. (3.5) ez (dSl + dS2)/r, [N/rad] Finally, the direct lateral transfer functions are obtained using the setups shown in Fig. 3.10: (3.6) (a) lateral excitation (XX) ;?;*g (b) lateral excitation (YY) Figure 3.10 : Experimental setup for lateral transfer function measurement. 3.3.4. Experimental modal analysis - rotating drill For experiments with a rotating drill, the tool is clamped with a collet chuck mounted in the spin-dle. It is difficult to find correct stiffness through impact testing due to the high flexibility of the drill tip. Therefore values from the Finite Element model of the drill bit were used for the stiff-ness, and the experimental measurements are used for the damping ratios and natural frequencies. Chapter 3. Drill Bit Model 36 Experimental impact modal tests on the drill while mounted on the spindle have been performed with an instrumented hammer (PCB model 086D80). The resulting response is captured with an accelerometer, (Kistler model 8778A500, 0.29gram weight). The mass of the accelerometer is neglected. The results from the experimental test (Experiment) and F E model are summarized in Table 3.4. The frequency of the torsional-axial mode is found by hitting the drill in axial direction on one flute, while the accelerometer is mounted on the other flute in radial direction. The damp-ing ratios of the vibration modes can be determined only from the experimental modal tests applied to the drill mounted in the spindle. The natural frequencies determined from the modal tests and those predicted from the FE model are in good agreement. The slight deviation may be due to approximation of the drill shape in the FE model. Due to the twisted shape of the drill bit, the natural frequencies and stiffnesses of XX and YY directions (experimental transfer functions shown in Fig. 3.5) are not the same, which is seen in both the FE model and experiment. Table 3.4 : Dynamic properties of Guehring HSS drill bit, geometry #217, TiAIN coating; Length=176mm; Diameter=16mm (L/D ratio=ll). Mode Symbol Property F E Model Experiment X X Frequency [Hz] 340 362.7 Kx Stiffness [N/m] 1.187xl05 1.405xl05 Damping [%] - 0.2575 Y Y Frequency [Hz] 390 338.6 kyy Stiffness [N/m] 1.053xl05 1.300xl05 Damping [%] - 0.2533 zz,ee Frequency [Hz] 2955 3357.5 ^ZF, Direct axial stiffness [N/m] 5.277xl07 -kzrr Cross axial stiffness [Nm/m] 2.186x105 kQTr Direct torsional stiffness [Nm/rad] 3.889xl02 -kQF7 Cross torsional stiffness [N/rad] 2.462x105 Damping [%] - 0.5024 Chapter 3. Drill Bit Model 37 x10"' 600 800 Frequency [Hz] 1000 200 400 600 800 Frequency [Hz] 1000 Figure 3.11 : Experimental direct lateral transfer functions of Guehring HSS drill bit, geometry #217, TiAIN coating; Length=176mm; Diameter=16mm (L/D ratio=ll). Due to the twisted shape of the drill bit, a torque loading produces axial deflection (extension of the bit). As well, a thrust loading produces torsional deflection (twist). As the natural frequency for torsional and axial vibration is the same, the direct axial stiffness (kZF , axial deflection over axial force) and the cross stiffness (kZT , axial deflection over torque load) are obtained from the FE model, and shown in Table 3.4. The direct torsional stiffness (kQT , torsional deflection over torque) and the cross stiffness (kQF , torsional deflection due to thrust loading) are used to predict the torsional deflection. In a normal load case, the thrust causes the drill to be compressed, while the torque extends the drill. Table 3.5 provides the natural frequencies for four different tool lengths of the Guehring #217 geometry with 16mm diameter. Table 3.5 : Natural frequencies of Guehring HSS drill bits, geometry #217, Diameter-16mm. Length Lateral [Hz] Torsional-axial [Hz] 19xD 102 1700 15xD 185 2340 l lxD 360 3350 8xD 602 4160 Chapter 3. Drill Bit Model 3 8 3.3.5. Experimental modal analysis - indexable drills For the three indexable drills shown in Fig. 3.11, the natural frequencies were determined, and provided in Table 3.7. Figure 3.12 : a) Indexable drills examined in this thesis; Sandvik U-drill, 21mm diameter; Drill lengths 4, 3 and 2 times Diameter, mounted in Sandvik solid HSK63A tool holder; b) detail of the tip geometry of an indexable drill, also refer to Fig. 3.5. Table 3.6 : Natural frequencies of the Sandvik indexable U-drill, 21mm diameter. Length Lateral [Hz] Torsional-axial [Hz] 4xD 1000 4600 3xD 1500 6100 2xD 2600 8700 39 Chapter 4. Mechanics of Drilling 4.1. Introduction This chapter presents the prediction of cutting forces generated by the drilling operation, the mechanics of drilling. The cutting forces are predicted as a function of drill geometry, cutting conditions and workpiece material properties. The twist drill is one of the most complex cutting tools, and the geometry of the cutting process changes dramatically along both cutting lip and chisel edge. In Fig. 4.1a, the drill center acts as a rotating indenter extruding the material on both sides of the chisel edge. Figure 4.1b shows that a little away from the drill center, the chisel edge forms irregular chips due to the strongly negative rake angle and low cutting speed. In Fig. 4.1c,d the variation in rake angles along the cutting edge is shown, and oblique cutting occurs in these regions. Close to the chisel edge the rake angle is negative and irregular chips are formed, but at the drill periphery a positive rake angle and a higher cutting speed result in regular chip formation. Figure 4.1 : Cross sectional views of drilling cutting process; a) indentation at the center of the chisel edge; b) orthogonal cutting with high negative rake angle on the chisel edge; c) oblique cutting with negative rake angle on the cutting lip; d) oblique cutting with positive rake angle on the cutting lip (after Ernst et al. [11]). Chapter 4. Mechanics of Drilling 40 4.2. Torque and thrust models for drilling Previous research on cutting force prediction in drilling focused mostly on torque and thrust, as these are limited by the machine tool. This section deals with the prediction of torque and thrust. As drill bits are symmetric, little research has been done on lateral forces in drilling caused by drill geometry errors and asymmetry. Prediction of the lateral forces in drilling is necessary to study lateral drill vibrations, and is covered as part of Section 4.3. 4.2.1. Cutting mechanics approach for the prediction of torque and thrust In order to use the cutting mechanics approach for drilling force prediction, it is assumed that the material is cut by the cutting lips and sheared away in a thin shear plane. The cut chips are evacu-ated by the flutes. First, an orthogonal cutting database is obtained from two-dimensional cutting tests, yielding shear angle, friction angle and shear stress as functions of the cutting conditions and the drill cutting edge geometry. Secondly, the helix angle, normal rake and oblique angles along the cutting geometry are identified. The cutting lips and chisel edge are discretized into small elements for which the cutting angles are assumed to be constant. Finally the cutting forces for each element are predicted through the orthogonal to oblique cutting transformation, and summed to obtain the total cutting forces. Armarego [12,25] has done significant work in this area, and this approach is adopted here. Cutting data for the cutting lips and the chisel edge is obtained and used for that region exclusively. R e g i o n 1 - c u t t i n g l i p s The rake face curvature away from the cutting edge is ignored, so it is assumed that the rake sur-face is flat. The tool geometry at each element is considered to be a satisfactory approximation of the varying geometry along the rake face, as was shown by previous workers. This assumption implies that the feed velocity is much smaller than the workpiece velocity. The cutting angles at an arbitrary point Q will be determined and will be a function of radius r, denoted by subscript. Chapter 4. Mechanics of Drilling 41 The local web angle co,. in Fig. 4.2a is defined as: - l GO„ = sin (3 Helix angle: Cross-sectional view A-A in Fig. 4.2b is the plane that contains the feed velocity and the cutting speed. This plane is perpendicular to the radius r. The helix angle P r is the angle between the line tangential to the flute and the working reference plane Pre: -V2rtanBn\ Br = tan ( — ( 4 . 2 ) The clearance angle XR is the angle between the tangent of the flank and the cutting speed, which lies in plane Pr. XR = tan l Q (4.3) Oblique angle: The projection of the cutting speed Vr onto the cutting edge is FrsinoursinKp see Fig. 4.2c. The oblique angle ir is the angle between the cutting speed Vr and the normal to the cutting edge: ~X(Vrsinco s i n K A _j ir = sin I J = sin ( s i n m ^ s i n K , ) (4.4) r Normal rake angle: Cross-sectional view B-B (Fig. 4.2d) is the normal plane Pn and it is per-pendicular to the cutting lip. VrcosG>r and P^sinco^cosK, are the components of the cutting speed projected in the normal plane. The normal rake angle anr is the angle between the tangent to the flute in the normal plane at point Q and the normal to the projection of the resultant cutting speed Vr/Jsin rarcos K,+ cos oo r which also lies in plane Pn. Points a and b lie in plane Pj-e (Fig. 4.2a) on the tangents to flute and flank faces respectively. By projecting points Q, a and b in the Chapter 4. Mechanics of Drilling 42 different views it is possible to establish the mathematical relationship between the drill geometry and the normal rake angle. By projecting points Q and a onto the normal plane Pn reference rake angle a^r is found: xcoscor tana f r = — r-11 (4.5) J r vsinK/-xsincorcosK/ From view A - A (Fig. 4.2b) the following expression for the helix angle can be found: tanpr = * (4.6) and if we substitute this into the expression for oy r, we find: tanBr x cosoo,. tana,, = ^ — : - (4.7) J r sinK -^tanp,. x sincor x COSK, Reference angle \\fr is found by projecting Vr in the normal plane Pn as: V x sinco x COSK, tan vi/, = — - tanco„ x COSK, (4.8) r Vrx cosco,. r 1 The normal rake angle a n r we are looking for can be expressed as a n r = a.yr - \ j / r : tanPr x coscor t a n a „ r = : tan ox x COSK, (4.9) " r sinK,-tanPr x sino),. x COSK, R ' Chapter 4. Mechanics of Drilling 43 B - B r S i n O ) r C O S K t normal plane P n ;j cutting edge plane P s e | (normal rake angle) jj V r sina>rsinKt (oblique angle) | 1 1 v ^ v r " ' * ' i yV r y sin 2 oo rcos 2 K t+cos 2 (B r .-'11" cutting edgeY projection of component V rsinco r into normal plane P n V r sincor V r sincDrsinKt j V rsinco rcosKt 0 Pr V Iworking plane P f e a V ] ^ - ^ ^ |(helix angle) \ ^re Figure 4.2 : Twist drill geometrical analysis (after Wiriyacosol et al. [12]). Chapter 4. Mechanics of Drilling 44 The helix angle fir can be expressed as: tanpr = ^tanp 0 (4.10) which allows the normal rake angle to be expressed as a function of r only: tanpn(r sin K.) — R Wsinic.cosK, t a n a „ r = — ^ f = = '• (4.11) Region 2 - chisel edge The cutting action of the chisel edge will be mostly one of orthogonal cutting with negative rake angles and small Vr due to the small r as evident from the region 0 < r < Rc, where Rc is the chisel edge radius (see Fig. 3.1). In the region 0 <r<rirld, where rind bounds the indentation zone, the chisel edge acts as a rotating indenter. The combination of cutting conditions is unfavor-able for continuous chip formation and the application of the well known thin shear zone analysis fails. However, due to the lack of an adequate deformation model for force prediction of indenta-tion, force data from orthogonal cutting tests is used. In the chisel edge region, the local cutting speed Vr becomes small when compared to the feed velocity Vj. Therefore the resulting cutting speed angle u r can no longer assumed to be small, and the dynamic cutting angles must be considered. The oblique angle is zero for all points on the chisel edge so that orthogonal cutting may be considered to occur. The static normal rake angle on the chisel edge is negative and numerically equal to the wedge angle an = -yw. The wedge angle yw is evaluated as: tanyw = tanK/sin(7i: - (p) (4.12) The cutting speed angle is the ratio between feed speed and local cutting speed: Chapter 4. Mechanics of Drilling 45 Vf "fr fr tanu r = -f = = ^ - (4.13) r Vr nznr 2nr where n is the spindle speed in [rpm]. Vd is the dynamic cutting speed. The dynamic rake angle is found from geometry: andr = U r - Y w ( 4 - 1 4 ) Fig. 4.3 shows that the cutting speed angle has a large value on the chisel edge (Fig. 4.1b). At the drill periphery (Fig. 4. I'd) the cutting speed angle vr is very small and is neglected (taken to be zero). Figure 4 . 3 : Feed speed effect strong on chisel edge (a), but weak on cutting lip (b). Variation of cutting angles on cutting lip and chisel edge Using the above derived formulas, the typical distribution of cutting angles over the two regions can be determined and is shown in Fig. 4.4. The rake angle varies over a very wide range, from + 30° on the cutting lip at the periphery to over - 50° on the chisel edge. The cutting speed varies from its maximum value at the periphery to zero at the drill center. Obtaining the basic cutting data for the above analysis can be a major task. Orthogonal cutting tests need to be performed that cover both the rake angle and cutting speed ranges. The uncut chip thickness h should also be varied and amply tested if adequate estimates of the edge forces are to be obtained (10 to 15 h values is suggested). Chapter 4. Mechanics of Drilling 46 Element discretization The cutting lip is divided into m equal elements. The web angle at the outer corner (Fig. 4.5) is defined through: c o 0 = sin (j^J (4.15) The elemental cutting edge length is determined as: Dcoscon - 2 ^ / tan (7r - (p) A / / / n = — : 1 — (4-16) "P 2msinK, v J The radial distance rk between the drill axis and the mid-point of the cutting lip element k is: J^ c o s c o 0 - ^ - ^ A / / ( p s i n K r ) 2 + W2 (4.17) Figure 4.5 illustrates the element discretization. The elemental chip thickness hk and the elemen-tal chip width Abk are determined as: Chapter 4. Mechanics of Drilling 47 / r sin K, cosy, (4.18) Figure 4.5 : Element discretization on twist drill cutting lip and chisel edge. The chisel edge is also divided into m equal elements. The chisel edge element width Abj is determined as: Ab; = W /wsincp and the radial distance r. between the drill axis and the chisel edge element mid-point is: (4.19) W (. n W J sincp The elemental chip thickness hj is determined as: wsincp (4.20) (4.21) Cutting forces on cutting lips The cutting forces on the cutting lip are predicted through the orthogonal to oblique cutting trans-formation, in which the orthogonal cutting database and the cutting angles of section 4.2.1 are used. The cutting forces in tangential, feed and radial directions can be expressed as [12]: Chapter 4. Mechanics of Drilling 48 F t = K t c A A k + K t e A b k FfrKfckAAk+KfekAbk (4-22) Frk = K r c A A k + KreAbk (4.23) The cutting coefficients Kte, Kj-e are edge force coefficients determined in the orthogonal cutting experiments, Kre is assumed to be zero. The cutting coefficients Ktc, Kj-e and Kre are determined through the orthogonal to oblique cutting transformation: x5 _ c o s ( B a - a „ t ) + tan^tanri A s inp„ t t C k sin())c / 2 2 2 * VC°S + K ~ + t a n ^ksm P„t K ^ .. si"(P.-%) •fo sine)) cosz'j. / 2 2 2 A / C 0 S ( * c t + P«t - % ) + t a n ^ A * 5 " 1 P«, _ cos(Pa- ank)tonik- tanrusinp^ K :—:— x , r C k Sin(()c / 2 2 2 " V C 0 S (<t>ct + P„t" % ) + t a n T ^ " 1 P„t in which the shear stress xs, friction angle Pa and shear angle <j)c are determined from experi-ments, as functions of the rake angle, cutting speed and chip thickness used in the orthogonal cut-ting tests. The normal friction angle Pw is determined through: P„ = tan_1(tanpacosz) (4.24) and Stabler's rule [72] is used for the chip flow angle yielding T , t = ik (4.25) Cutting forces on chisel edge Armarego used a mechanistic method to model the cutting forces at the chisel edge [12], and the same method is used here (see Table 4.3). The tangential and feed force for element /', along and normal to the resultant cutting speed are expressed as: Chapter 4. Mechanics of Drilling 49 F,=C,x Ab, = e° x hei xf2x (90 + a„f3 x Ab, h h J J (4.26) Ffj = Cf x Abj = / ° x / ' x / 2 x (90 + a „ / 3 x Abj where Ct = Ct(hj, Vj, anj) is the tangential force per unit width of cut, and it is an exponential function of the local rake angle a . and chip thickness hj. The cutting constant Ct is multiplied by the element width Abj to obtain the elemental force. The constants e0, e l 5 e2, e3 and / 0 , / j , / 2 , / 3 are determined from a least squares fit to the experimental data. Resulting cutting torque and thrust force For cutting lip element k, the torque and thrust contributions are determined as: Tc HPk ~ Ftk x rk FZi ^ = F ^ c o s v i / j . s i n K , - F ^ C O S / J . C O S K , + sin/^sinxi/^sinK,) (4.27) Important to note here is that in the thrust force, the elemental feed- and the elemental radial force contributions act in opposite directions. The torque and thrust contributions at the chisel edge are expressed as: Tr rh = (Ft cosv ;-F f sino,) x r, c,chj v tj j Jj j' j (4.28) Fz,chJ = F t J s i n » j + FfJC0™j Chapter 4. Mechanics of Drilling 50 4.2.2. Mechanistic cutting force model Chandrasekharan [16] developed a mechanistic model in which the three dimensional cutting forces in drilling are predicted. In a mechanistic approach, the model coefficients are determined from a curve fit to experimental data. The drill geometry is divided into three zones: oblique cut-ting on the cutting lips, indentation by the chisel edge in the very center of the drill, and orthogo-nal cutting on the remaining part of the chisel edge. Oblique cutting at the cutting lips Merchant [26] showed by using conservation of energy methods that if the chip is held in equilib-rium by the resultant machining force, the cutting forces acting on the tool in the equivalent two-dimensional rake-face or machine tool coordinate system are proportional to the uncut chip area or chip load. The chip load is the projected area of the shear plane measured in a plane normal to the cutting-velocity vector. In the force modeling, this rake face coordinate system is used. The forces in the cutting-tool coordinate system are determined from the forces in the rake-face coor-dinate system by a rotational transformation. Figure 4.6 illustrates the oblique cutting force sys-tem, where a friction force i y r is assumed to act along the flow of the chip and the normal force Fn is defined as the force normal to the rake face of the tool. The chip flow direction is character-ized by the chip flow angle r\c, which is measured from the normal to the cutting edge that lies on the rake face of the tool. The two forces and the chip flow angle fully characterize the force sys-tem on the rake face of a tool. The magnitude of the three-dimensional oblique-cutting forces act-ing on an element on the cutting lip, thrust Fth, cutting Fcu( and lateral force Flat can be written in terms of the magnitude of the normal and friction forces as: \Fth\ = |^ 7r| cosicosaB - |i^„| sina„ \Fcut\ - {sinr)csinz + cosr|ccos/'sinan} + |F M | cos/cosa, \Flat\ = \Ffr\ i cosr| csimsina n - sinr|ccos/} + |F n |sin/cosa w (4.29) Chapter 4. Mechanics of Drilling 51 The derivation of Eq. 4.29 is shown graphically in Fig. 4.7. rake angle a n normal force O topview T| c chip f low angle rake face cutting edge i inclination angle Fth thrust friction force lateral F | a t Figure 4.6 : Definition of normal andfriction force [16]. 7^ F c u t cutting sideview Ffr c o s r l c cutting force F c u t F n cosa n cos i topview F c u t Ff r cos r | c s ina n cos i + Ff r s inr| c sini lateral force F | a t Ff r cosr j c s ina n s in i - Ff r s inr | c cosi F n cosa n s in i Figure 4.7 : Derivation of oblique cutting forces Fcut, Fih and Fiatfrom normal and friction force. Chapter 4. Mechanics of Drilling 52 Kn and Kj-r are defined as the specific normal and friction force, and they are computed from empirical equations in which the chip thickness h, cutting speed Vc and rake angle an are taken into account. The magnitude of the normal and friction force are assumed to be proportional to the chip area Ac: Fn = KnAc Ffr = KfrAc Kn = Kn(h,Vc,an) Kfr = Kfr(h,Vc,an) (4.30) The chip area Ac is defined to be in the plane whose normal coincides with the direction of the cutting speed. The dependency of the normal and friction force on the chip thickness and cutting speed can be accurately modeled through the power law. The dependency on the rake angle is also modeled through the power law, but as the normal rake angle can be negative on a drill geometry, the term (1 - sinctw) is used instead of an as was done previously by Stephenson and Bandyo-hadhyay [19]. The term (1 - sinaw) is always positive and this form fits experimental data best for a large number of data points in a wide range of rake angles. The empirical equations relating the specific normal and friction forces to chip thickness, cutting speed and the normal rake angle The model coefficients mry..m4 and nQ...n4 can be determined from a regression given the experimental specific forces for at least two levels of each variable. A cutting lip is divided into m equal elements. For each element the chip thickness, cutting speed and the normal rake angle are determined. The chip thickness is defined to be normal to the cutting edge and in the plane of the chip load. For a regular twist drill the chip thickness is: are: lnATrt = mQ + mllnhc + m1\x\Vc + m 3 ln( l - s ina„) + m4lnhc\nVc InKfr = n0 + nl\nhc + n2inVc + « 3 l n ( l - sina n) + n4lnhclnVc (4.31) h = /,-sinK, 2 (4.32) Chapter 4. Mechanics of Drilling 53 The cutting speed is proportional to the distance between the midpoint of the element and the drill axis. The normal rake angle is given by Armarego [25] (Eq. 4.11): tanpn(r sin K,) — RFFsinK,cosK, t a n a „ r = — ^ f = = (4.33) and depends on the helix angle at the periphery p 0 , the radial distance from the drill center r, the web thickness 2 W, the tip angle 2K, and drill radius R. The chip flow angle at each element is assumed to be equal to the inclination angle of the element (Stabler's rule, x\c = i). The local inclination angle is given by: ir = sin 1(sincorsinK/) (4.34) The local web angle cor is given by: cor = s i n " ^ (4.35) The trigonometric decomposition of the elemental oblique-cutting forces into drilling thrust, torque and radial forces is done by a vector analysis. The elemental thrust, cutting and lateral force components are represented as vectors and by taking the dot product with the unit vectors in the drilling thrust, torque and radial force directions, and summing the products, the total three-dimensional drilling forces can be determined. A coordinate system is attached to the point of the drill as shown in Fig. 4.8. The Z-axis is aligned with the drill axis and the y-axis is aligned with one of the cutting lips. A unit vector in the direction of the cutting lip element ( Z ) and the cutting speed (V) at the element are represented by: Z = 0ex + sinK,^ + cosK ,e z . -ifm (4-36) V= coscoe .^- svL\Giey + 0ez co = sin J Chapter 4. Mechanics of Drilling 54 topview Figure 4 . 8 : Trigonometric decomposition of elemental oblique-cutting forces into the drilling thrust, torque and radial directions. The elemental cutting force (dFcut) is in the direction of the cutting speed and is given by: dFcut = rFcut\{~C0S(£>ex+ s m G ) e y - c o s c o e z } (4.37) The elemental oblique-cutting force thrust force (dFthr) is normal to the plane that contains the cutting speed vector and the cutting edge. The direction is given by the cross product of these two f ^ c x Z 1 vectors, \ — — — \ , and is equal to: VcxL dFthr = \dF,hr\ - s i n co C O S K , C O S CO C O S K . c o s co s i n K . 2 , . 2 2 * 2 , . 2 2 y 2 , . 2 2 * cos Kt+sm K , C O S co c o s K ( + s i n K ( C O S co c o s K , + s i n K ( C O S co (4.38) The elemental lateral force (dFlat) is given by: d F l a , = \ d F l a \ - s i n co c o s co s i n K . c o s CO s i n K , - e „ + • 2 , . 2 2 ~x 2 , . 2 2 ~y 2 , . 2 2 « c o s K ( + s i n K ( C O S co c o s K , + s i n K ^ C O S CO c o s K , + s i n K ^ C O S CO (4.39) The magnitude of the elemental drilling thrust force (\dFz\) and radial forces (\dFz\) and (\dFz\) are the Z - , X- and 7-components of the vector representation of the three-dimensional oblique-cutting forces at each element. The elemental torque (\dTc\) is the product of the radial coordinate of the element and the cutting force [16]. Chapter 4. Mechanics of Drilling 55 dFz = dFthr cos co sin K 2 , . 2 2 COS K, + sin K,COS 00 \dF, COSK lal\ 2 2 2 cos K, + sin K,COS co dTc =\r\\dFi cut\ dFx = \dF. -sincocosK thr\ 2 . 2 2 COS K, + sm K,COS CO dFcut cosco- J F j sin co cos co sin K 2 2 2 cos K, + sin KjCos co \dFy\ = \dFthr -COSCO COSK 2 2 2 cos K , + sin K,COS co + |c/FcJsinco-|c/F cos co sin Kt lat\ 2 , . 2 2 cos K , + sin K,COS co 2 2 2 . Chandrasekharan replaced the term cos K,+ sin K,COS CO with cos/. (4.40) Indentation zone at the chisel edge In the region around the center of the drill, material removal occurs by extrusion. This region is called the indentation zone. In this zone, the drill acts as a rigid wedge indenting a plastic material extruding on both sides of the wedge. The size of the indentation zone r i n d is determined using the geometrical analysis for flank interference by Mauch and Lauderbaugh [74] as: rind fr 2tan(7t/2-K,) (4.41) The indentation process is illustrated in Fig. 4.9. indentation zone chisel edge plastic region sliplines Figure 4.9 : Indentation process at the drill center [75]. Kachanov's [75] slip line field solution is used to determine the normal force acting on the wedge. The included angle of the wedge 2yind is equal to twice the static normal rake angle at the chisel Chapter 4. Mechanics of Drilling 56 edge an and the height of indentation h i n d is equal to half the feed per revolution fr. The normal rake angle at the chisel edge can be computed from the drill tip angle and chisel edge angle, and is equal to: tany l W = tanK,sin(7t - cp) (4.42) The solution for slip-lines es, the contact length lc, and the pressure pind to the wedge surface is given by: 2Y,w=s+cos j t a n^-^1, /= 1/1 - = 2 t (T + e ) <4"43) where xs is the shear yield stress of the material, which is assumed to be a constant. The force per unit length in the thrust direction and the cutting torque direction is then given by: PF = 2pindlcsm.an and PT = 2pjndlccosan. The contribution to the total drilling thrust and torque due to the indentation zone is given by the expressions: p _ 4 / r T , ( l + e , ) r f W s i n a w ^ _ 2 / r T J ( l + £ , ) r ^ c o s a „ ( 4 . 4 4 ) z,wd c o s a „ - s i n ( a w - s j c > m d c o s a „ - s in(a w -& s ) Orthogonal cutting at the chisel edge The forces on the non-indenting part of the chisel edge, called the secondary cutting edge, are determined using the same mechanistic model as that for the cutting lips of the drill. The chisel edge is divided into elements and the normal and friction forces are computed using Eq. 4.31. In the chisel edge region the feed velocity can no longer be ignored, and the dynamic angles must be considered. The inclination angle is zero for all points on the chisel edge so that orthogonal cut-ting may be considered to occur. The cutting speed angle u r is found from tanu r = - — and Chapter 4. Mechanics of Drilling 57 depends on the radial distance r to the drill axis and the feedrate. The dynamic rake angle andr is found from geometry as andr = u w - yw. Model calibration The specific normal and friction forces for the oblique cutting can be computed from the chip thickness, cutting speed and the normal rake angle of the cutting edge using Eq. 4.31. The model coefficients m0...m4 and n0...n4 must first be determined from force data collected during drill-ing into a flat workpiece with blind piloted holes. Fig. 4.10 illustrates how the chip load during entry varies and Fig. 4.11 shows a typical thrust and torque profile while drilling a blind piloted hole. Since the power law has been used to model the specific normal and friction forces in terms of the model variables, the elemental drilling thrust | dFz\ and cutting force | d F c u / | are assumed to have a similar form along the radial distance of the cutting lip: \dFz\ = ClXpdAc \d¥^\ = C2kqdAc (4-45) The cutting torque is found from \dT^ = | r e / | • | ^ C M / | , where is the radial distance between r the element and the drill axis. C, and C , are constants, X = — is the normalized radial coordi-1 Z ft nate and dAc is the elemental chip load. The thrust and torque at any instant can be determined by integrating the elemental forces between the limits of the pilot hole (lim^) and the farthest point on the cutting lip that is in cut (lim2), which is a function of time: lim2(t) lim2(t) lim2(t) lim2(t) \Wz\(t)= \ 2-\dFz\= \ C{kpdAc |M Z |(0= J 2 . | 5 f J = J C2XqdAc (4-46) / i iw j limx limx lim\ The limits of integration are shown in the Fig. 4.10 and are equal to: Chapter 4. Mechanics of Drilling 58 limx = lim2(t) = ^ W 2 + [fr{t-™) tunic, + J i ^ W 2 ) +^W,COSK,]~2 (4-47) Given that y = rcosco and co = sin W \\rel\J , it can be determined that dy_ _ 1 dr cos co limit 1 W—i- limit 2 Figure 4.10 : Variation of chip loadAc during drilling calibration experiment 1 W2 1 W^sin 2^ Approximating 77777 s 1 + — — - and 777-; ^  1 7-7-, substituting in Eq. 4.46 and simpli-cosco 2R2X2 cos/ 2R2X2 tying yields: W ( 0 W ( 0 = c 1/ rx o ? + i ) | c/ycos2^-0 c/y.mV(p-3)-p+1 2t f (p- l ) 4 i ? > - 3 ) lim2(t) limx C / ^ ( g + 2 ) | C/Vcos2^ C^ysin\^{q 2 > q + 2 2q 4R\q-2) lim2(t) lim. (4.48) Chapter 4. Mechanics of Drilling 59 1800 1500 1200 z • in 3 900 600 300 0 1500 1200 1" [Nc 900 a> 3 D" k_ rt 600 w 1-300 0 i <r 1 t i i 4 2 > V ^ Ifc ^ P ^ P A P -i i i -Time [s] Figure 4.11 : Experimental thrust and torque 16mm twist drill; Vc=40m/min, fr=0.30mm/rev, 4mmpilot. Equations 4.48 are fitted to the thrust and torque force data at entry and the parameters C , , C2,p and q can be estimated for each test condition. Substituting Eq. 4.29 into Eq. 4.40 yields expres-sions of the thrust and torque at an element in terms of the normal and friction forces at an ele-ment: \dFz\ = I^Fy^lcosa^cososinK,-!- sin/sinK,- sina ncosK,sin/} C sin a n cos co sin K , ] - dF„ \ 1- cos a „ COSK, tan/ \ " cost " 1 . 2 2. . dTc = rel [ dFj-r {sin /+ cos /sinan} + dFn {cos/cosaw}] (4.49) Chapter 4. Mechanics of Drilling 60 Equating Eq. 4.49 for the thrust and torque at an element with Eq. 4.45 and factoring out the chip load dAc the following equations are obtained: C{kp = .ryr{cosawcoscosinK, + sim'sinK,- s ina n cos s in i } fsina„coscosinK, 1 ( 4 5 Q ) -KJ : r cosa„cos ic / tam > ^ • J « ; C2Xq = A^.{sin2/+ cos2/'sinaw} + ATw{cos/cosaw} from which Kn and Kjr can be determined for each element along the cutting lip. Cutting lip element discretization The cutting lip is divided into m equal elements. The web angle at the outer corner is defined through co0 = sin • The elemental cutting edge length is determined as: Z)coscoft - 2 W/tan(n - (p) Ahio = : " — (4-51) l'P 2/wsinK, The radial distance rk between the drill axis and the mid-point of the cutting lip element is: j^R cos Q>Q-(k-^J Allip sinK^ 2 + W2 (4.52) The model coefficients for one tool-workpiece combination are determined as follows: 1) Collect the cutting forces (thrust and torque) drilling four piloted holes at two different feed rates and two different cutting speeds. 2) From Eq. 4.31 ,the constants Cx, C2,p and q are determined for each cutting test, using the thrust and torque of the cutting lips only. The drill web thickness W, radius R and tip angle K, are used in this step. The term Cj x Xpk is the specific thrust at element k. Chapter 4. Mechanics of Drilling 61 3) The cutting lip is discretized, and for each of the M elements the normalized radial coordinate X, rake angle an, web angle co and inclination angle i are computed. For each of the four combi-nations of cutting speed and feed rate, the specific normal and friction forces Kn and Kjr at an element can be evaluated from Eq. 4.31. The drill tip angle K, , web thickness W, radius R, chisel edge angle cp and drill helix angle p 0 are used in this step. 4) Equations 4.31 relating the specific normal and friction force to the chip thickness, velocity and normal rake angle are linear in log space. Two levels of spindle speed and feed rate therefore suf-fice. The experimental specific normal and friction forces for all elements are known from step 3 and through a multi-variable regression the model coefficients /w0 to m4 and » 0 to n4 can then be determined. 5) Finally, the thrust force on the cutting lips and the orthogonal cutting chisel edge are deter-mined using coefficients mQ...m4 and nQ...n4. This force is subtracted from the experimental thrust force to determine the indentation force contributed by the very center of the drill. Using the indentation mechanics described earlier, the shear stress xs is determined. The shear stress is only determined from the high feed rate conditions, using the average indentation force of the two different cutting speeds, as the shear stress is assumed to be a constant. Only the high feed rate conditions are used, as the indentation force is only significant at high feedrate. 4.2.3. Experimental verification of models for torque and thrust in drilling In order to determine the accuracy of the approaches described, experiments have been carried out. This section details the orthogonal database developed for using the cutting mechanics approach and describes how coefficients were obtained for Chandrasekharan's approach, provid-ing details of the normal and friction force curves found. Finally, the experimental verification of these 2 models is presented. Chapter 4. Mechanics of Drilling 62 Cutting database for cutting mechanics approach In order to determine the orthogonal cutting database for the cutting mechanics approach, special experiments have been conducted with tools of varying rake angle. Five different feedrates have been used. Combinations of cutting speed and rake angle were chosen to be similar to the combi-nations occurring in the drilling process. The cutting conditions are summarized in Table 4.1. Tubular workpieces of aluminum AL7050-T7451 with a wall thickness of 2.5mm have been machined for these cutting tests. Flood coolant was used during the experiments, and the forces were measured using a Kistler model 9121 three component dynamometer. Table 4.1 : Cutting conditions orthogonal cutting experiments for AL7050-T7451 database. Rake angle Cutting speeds Feedrate [degrees] [m/min] [mm/rev] +30 80.0, 120.0, 200.0 0.04, 0.08, 0.12, 0.16, 0.20 +20 14.3, 57.3, 100.2, 143.2 0.04, 0.08, 0.12, 0.16, 0.20 +10 10.2, 40.8,71.4, 102.0 0.04, 0.08, 0.12, 0.16, 0.20 0 7.3, 29.0, 50.8, 72.6 0.04, 0.08, 0.12, 0.16, 0.20 -10 6.3,22.0,38.5,55.0 0.04, 0.08, 0.12, 0.16, 0.20 -20 6.3, 17.3,30.2, 43.2 0.04, 0.08, 0.12, 0.16, 0.20 -30 6.3, 15.7, 27.4, 39.2 0.04, 0.08, 0.12, 0.16, 0.20 -40 6.3, 11.8, 20.6, 29.4 0.04, 0.08, 0.12, 0.16, 0.20 -50 11.8, 20.6, 29.4 0.04, 0.08, 0.12, 0.16, 0.20 -60 15.7, 27.4, 39.2 0.04, 0.08, 0.12, 0.16, 0.20 Fig. 4.12 and 4.13 show the experimental tangential and feed forces that are used to create the database for the cutting lip section. The database parameters are determined from cutting tests with rake angles 0, +10, +20 and +30 degrees, as the chip ratio for negative rake angles cannot be determined due to discontinuous chip formation. The plot shows experimental forces for the range of - 3 0 ° . . . + 30° degrees rake angle (circles, dashed lines). The predicted tangential and feed forces are shown in solid lines. The experimental and predicted forces match well. Chapter 4. Mechanics of Drilling 63 1200 0 20 40 60 80 100 120 Cutting test # [-] Figure 4.12 : Orthogonal cutting database AL7050-T7451: tangential forces for cutting lip region; experiment and prediction; For each rake angle four cutting speeds are tested at five different feedrates. Chapter 4. Mechanics of Drilling 64 1200 0 20 40 60 80 100 120 Cutting test # [-] Figure 4.13 : Orthogonal cutting database AL7050-T7451: feedforces for cutting lip region; experiment and prediction; For each rake angle four cutting speeds are tested at five different feedrates. Chapter 4. Mechanics of Drilling 65 Table 4.2 provides the equations for the shear stress TS , shear angle cj)c, friction angle Pc and tan-gential and feed force edge coefficients K(e, Kj-e, which are all functions of chip thickness, cutting speed and rake angle. Table 4.2 : AL7050-T7451 cutting database for orthogonal to oblique transformation of cutting lip section; h=[mm], Vc=[m/min] ocn=[deg]. 2 Shear stress [N/mm ] xs = 266.8 + 174.1/2-0.0437^+ 0.8961a„ Shear angle [rad] tyc = 19.4 + 42.0/2 + 0.0200 F c + 0 .3842a„ Friction angle [rad] p c = 25.9- 1.2837/2 - 0.0076FC + 0 .1818a„ Tangential edge force coefficient [N/mm] Kie = 20.94-0.0152 F c -0 .3588a w Feed edge force coefficient [N/mm] Kfe = 18.13 +0.0049 F c - 0 . 4 4 1 5 a „ For the chisel edge region, mechanistic equations are fitted to the experimental data. The tangen-tial and feed force, along and normal to the resultant cutting speed are expressed as (Eq. 4.26): F, = C, x Ab, = e* x h£] xf2x (90 + or)*3 x Ab, Ffj = Cf x Abj = efo x hfx x / 2 x (90 + a „ / 3 x Abj where Ct = C((hj, Vc j, an]) is the tangential force per unit width of cut, and it is an exponential function of the local rake angle anj and chip thickness h-. The constants e0... e 3 and / 0 . . .f3 are determined by a least squares fit to the experimental data. The experimental data of the cutting tests with rake angles range - 6 0 ° . . . 0° degrees is used. The experimental data and model fits are shown in Fig. 4.14 and 4.15 for tangential and feed force respectively. The model coefficients are provided in Table 4.3. The cutting speed appears to have little influence, as seen from the magni-tude of the coefficients e2,f2- The database prediction matches well with the experimental results, showing that the model is suited for the experimental data. Chapter 4. Mechanics of Drilling 66 Table 4.3 : Cutting force coefficients chisel edge region, from orthogonal cutting tests Component eO'/o [N/mm] e i » / i (chip thickness [mm]) (cutting speed [m/min]) E 3 ' / 3 (rake angle [deg]) Tangential(a) 10.3231 0.8056 -0.0489 -0.7757 Feed(b) 12.9901 0.7720 0.0539 -1.5695 It should be noted that for rake angles 0° and negative, discontinuous chip formation occurred, as well as sidespreading of the material. This means that the experimental force is too low. When these coefficients are applied in drilling mechanics, they will lead to under prediction of the cut-ting forces. 1600 1400 1200 S 1000 o £ D) c 5 3 O c a> o> c ro I-800 600 400 h 200 '0 20 40 60 80 100 110 Cutting test # [-] Figure 4.14 : Orthogonal cutting: tangential forces for chisel edge region; experiment and prediction. Chapter 4. Mechanics of Drilling 67 3000 " 0 20 40 60 80 100 110 Cutting test # [-] Figure 4.15 : Orthogonal cutting: feedforces for chisel edge region; experiment and prediction. Chapter 4. Mechanics of Drilling 68 Model coefficients for mechanistic approach (Chandrasekharan, [16]) Chandrasekharan's mechanistic force model requires only four torque/thrust measurements of piloted holes. A 7.94mm diameter twist drill is used, at cutting speeds 16m/min and 80m/min. The feedrates chosen are 0.08mm/rev. and 0.16mm/rev. First, curves are fit to the experimental torque and thrust for the piloted section of the hole. Table 4.4 provides the coefficients. Table 4.4 : Model coefficients Ch C2, p, qfor 7.94mm twist drill in AL7050-T7451. Condition Thrust: Cx Thrust: p Torque: C 2 Torque: q 640rpm, 0.08mm/rev 535.65 -0.5632 1710.00 -0.4310 640rpm, 0.16mm/rev 216.78 -1.1040 1250.00 -0.4709 3200rpm, 0.08mm/rev 389.51 -0.8311 1455.00 -0.4177 3200rpm, 0.16mm/rev 155.29 -1.4083 980.00 -0.6880 Using the geometrical properties of the drill, the coefficients for the normal and friction force can be determined, and these are provided in Table 4.5. The cutting edge is divided into 50 elements for this step, and the experimental and predicted coefficients are shown in Figs. 4.16 and 4.17. The match for both normal and friction force is quite good. Table 4.5 : Normal andfriction force coefficients from 7.94mm in AL7050-T7451 Term Normal Friction Constant mQ 7.2795 "o 6.5008 Chip thickness m l -0.2126 ni -0.2794 Cutting speed m2 -0.3117 n2 -0.4373 Rake angle m3 1.1096 n2 0.2543 Chip thickness/Cutting speed m4 -0.0632 n3 -0.1093 Finally, using the coefficients of Table 4.5, the indentation stress for the center of the chisel edge is determined from (16m/min-0.16mm/rev) as 1447MPa, from (80m/min-0.16mm/rev) as 1873MPa. The average of these two values is used in the model, yielding a 1660MPa indentation stress. Chapter 4. Mechanics of Drilling 69 4500 4000 3500 E E z g 3000 o E g 2500 u O) • | 2000 3 u 1500 1000 • K n experiment • K n predicted 20 40 60 80 100 120 Element number [-] 140 160 180 200 Figure 4.16 : Normal cutting coefficient for Chandrasekharan's mechanistic model; AL7050-T7451; 7.94mm twist drill; experiment and prediction. 1800 600 - Kfr experiment - Kfr predicted 20 40 60 80 100 120 Element number [-] 140 160 180 200 Figure 4.17 : Friction cutting coefficient for Chandrasekharan's mechanistic model; AL7050-T7451; 7.94mm twist drill; experiment and prediction. Chapter 4. Mechanics of Drilling 70 Experimental verification cutting mechanics approach and mechanistic model Piloted holes have been drilled with five drills of different diameter. The pilot hole diameter was slightly larger than the chisel edge in order to identify the contributions of the chisel edge and cut-ting lips separately. All drills were used at one cutting speed and five feedrates. Each.cutting con-dition has been tested three times, and the average is taken as the experimental result. The torque measurements are most consistent with deviations from the average being within 7%, typically only 1 or 2% different from the average value. The thrust results show more variation, for three drills the deviations are within 5%, for one within 10% and the last drill within 22%. AL7050-T7451 has been used as workpiece material, and minimal lubrication has been applied manually. Table 4.6 provides the geometrical details of the drills used to verify the cutting force prediction methods. They are short, general purpose HSS drills. Short drills were chosen to avoid vibrations and large tool deflections. Table 4 . 6 : Geometrical properties ofKennametal Screw Machine Length twist drills used in experiments to verify cutting mechanics approach and Chandrasekharan's mechanistic model; Types: S110FX, surface treated (7.94-17.46mm), S100FX, surface treated (20.64mm). Diameter [mm] Web thick-ness [mm] Web ratio [-] Chisel edge angle [deg] Point angle [deg] Helix angle [deg] Pilot hole diameter [mm] 7.94 1.42 0.179 125 110 28 1.8 11.11 1.79 0.161 125 118 33 2.2 14.29 2.24 0.157 125 118 30 2.8 17.46 2.72 0.156 125 116 31 3.3 20.64 3.06 0.148 125 118 30 3.8 In Figs. 4.18 through 4.22 the experimental results for the five different drill diameters are shown using circles. The torque and thrust are split in contributions of the cutting lips and the chisel edge. Predicted values are shown for the cutting mechanics approach in squares, the mechanistic prediction is shown in diamonds. Chapter 4. Mechanics of Drilling 71 E o 3 200 160 120 80 40 0 1200 1000 =•800 600 400 200 0 Cutting lips Chisel edge Full drill O o • oo 5 o • (A 6 9 0.08 0.10 0.12 0.14 0.16 O o ° & & 6 6 6 o O o O o O 8 • 0.08 0.10 0.12 0.14 0.16 Feedrate [mm/rev] O O • ° A 9 0 O H O experimental result • cutting mechanics O chandrasekharan O o O o O o O o O ° „ • ° 0.08 0.10 0.12 0.14 0.16 Figure 4.18 : Experimental thrust and torque for 7.94mm twist drill in AL7050-T7451 versus prediction. Results for the cutting lips The cutting lip torque is consistently under predicted (-12% on average) by the cutting mechanics approach, but the trend is followed well. The mechanistic approach by Chandrasekharan mostly under predicts (-25%) as well, but the torque does not increase as strongly with feedrate as exper-imental data shows. The cutting mechanics approach under predicts the lip thrust as well, but less consistently and with larger error (-30%). The thrust prediction error for the mechanistic approach increases with feedrate, as was the case for the torque. Chapter 4. Mechanics of Drilling 72 500 400 o z 300 V 3 CT k_ O 200 1-100 0 2000 Cutting lips o O • ° • g 9 o 0 • Chisel edge Full drill o o o • o 0 • I - S 0 . O experimental result • cutting mechanics 0 chandrasekharan 1600 _ l J1200 l 800 400 0 a 9 e • 8 • o o O o • O o • 0 0 0 o o o 0 • • i—i n L J • 0.09 0.12 0.15 0.18 0.21 0.09 0.12 0.15 0.18 0.21 Feedrate [mm/rev] 0.09 0.12 0.15 0.18 0.21 Figure 4.19 : Experimental thrust and torque for 11.11mm twist drill in AL7050-T7451 versus prediction. Results for the chisel edge The two prediction methods are different for the chisel edge region in that the mechanistic approach uses an indentation model for the center of the drill. The indentation adds mostly to thrust, and little to the torque. Both methods predict the chisel edge torque with poor accuracy, the cutting mechanics approach with -51% and the mechanistic with -44% on average. The error is smaller for larger drill diameters. The chisel edge thrust force is strongly under predicted by the cutting mechanics approach, were errors range from -30% to -60%, average -44%. The errors are smaller for larger diameter and higher feedrates. The mechanistic approach predicts the thrust well for small feedrates, but the indentation contribution becomes very large for high feedrates, Chapter 4. Mechanics of Drilling 73 leading to large over prediction errors. Since indentation is calibrated directly from drilling tests in the mechanistic approach, better prediction was expected. 1000 800 E u 2,. 600 0) 3 O 400 J— 200 0 3000 2400 —1800 •4-* (A 3 £1200 Cutting lips Chisel edge 6 9 o o • o • O O 8 § o O 600 o ° O 9 0.12 0.16 0.20 0.24 0.28 0 o CO o o o CO O • • • I—1 • 1 1 0.12 0.16 0.20 0.24 0.28 Full drill O O • O • o n / s • 8 9 0 * O experimental result • cutting mechanics O chandrasekharan o o O o o O 8 0.12 0.16 0.20 0.24 0.28 Feedrate [mm/rev] Figure 4.20 : Experimental thrust and torque for 14.29mm twist drill in AL7050-T7451 versus prediction. Results for the full drill Finally the cutting torque for the full drill, mostly determined by the cutting lip torque, is followed well by the cutting mechanics approach, both in trend and magnitude, with an average error of -19%. The prediction error in the mechanistic approach increases with feedrate yielding a -29% average. The thrust for a full hole, mostly determined by the chisel edge thrust, is under predicted by -30%...-50% in the cutting mechanics approach, the mechanistic approach shows errors rang-ing from -17%...+54%, as the high feedrate values give large over predictions from the indenta-tion model. Chapter 4. Mechanics of Drilling 74 1600 •^ •1200 u 3 800 cr o I-400 0 4000 3000 3 21000 Cutting lips Chisel edge O 9 o o • • O O o o o o o o 6 O 9 3 0.15 0.20 0.25 0.30 0.35 8 8 8 8 o o 8 ° ° _ • D o • 0.15 0.20 0.25 0.30 0.35 Feedrate [mm/rev] Full drill o o • o • O • 0 o 0 O experimental result' • cutting mechanics 0 chandrasekharan o 8 o 0.15 0.20 0.25 0.30 0.35 Figure 4.21 : Experimental thrust and torque for 17.46mm twist drill in AL7050-T7451 versus prediction. Chapter 4. Mechanics of Drilling 75 2500 2000 u 2.1500 0) 3 XT O 1000 500 0 6000 4800 Cutting lips Chisel edge Full drill OT 3 '3600 2400 1200 Ol o • O o • O o • O o • o • O o o o o o o 6 O 9 9 0.18 0.24 0.30 0.36 0.42 @ 3 @ 8 8 0 0 0 0 o o o o • • D • • 0.18 0.24 0.30 0.36 0.42 o O • o • • C • 0 O , o 0 O experimental result • cutting mechanics 0 chandrasekharan 0 O 0 o o 0 o o 0 • • • • • 0.18 0.24 0.30 0.36 0.42 Feedrate [mm/rev] Figure 4.22 : Experimental thrust and torque for 20.64mm twist drill in AL7050-T7451 versus prediction. 4.2.4. Lateral cutting force prediction using the cutting mechanics approach In the case of lateral vibrations, the chips cut by each flute will be different resulting in a force unbalance in lateral direction. This force unbalance needs to be predicted in order to study the resulting vibrations. In case the drill is misaligned, or ground improperly, there will be a force unbalance as well. The forces on a single flute are measured in experiment, and modeled mecha-nistically in section 4.4.3. The cutting mechanics approach can also be used to predict the lateral cutting forces, as it allows prediction of different drill geometries. Figure 4.23 shows the force components acting on an element on the cutting edge, tangential Ft, radial Fr and feed Fj-. For Chapter 4. Mechanics of Drilling 76 clarity, a coordinate system is attached to the tool, with the Z-axis coinciding with the drill axis, and the X-axis parallel to the cutting edge. The thrust force on the tool, in Z-direction, depends on the radial and feed force, as given in Eq. 4.27. Details of the full decomposition of tangential, radial and feed forces is provided in Appendix C. F r s in i r s in l | f r Figure 4.23 : Three-dimensional representation of cutting force components acting on an element on the cutting edge. In matrix form the force decomposition is give by: Ft ' Fy > = [A]- Ff • { FZJ I F r \ (4.54) [A] = - cos^cos^ - s i n ^ , . - s i n i r c o s ¥ r cos/,, sin ^ C O S K ^ sin / , .s inK, - cos C O S K , sin/ r sin ^ r cos K , - c o s / r s i n K , - c o s / r s i n T r s i n K , + s i n / r c o s K , c o s x F r s i n K , - s i n / ^ s i n ^ s i n K , - c o s / r c o s K , Chapter 4. Mechanics of Drilling 11 4.3. A higher order mechanistic model for force prediction of piloted holes Accurate cutting force prediction for the cutting lip region is required to investigate the stability of and vibrations in the drilling process when drilling piloted holes. The radial depth of cut in drilling b, in [mm] is defined as the difference between the drill radius R and pilot hole radius, b = R-r . The depth of cut is a gain that can easily be varied in cutting experiments. A mecha-nistic approach is proposed that allows the cutting coefficients to change as a function of radial distance to the drill center and chip thickness. The web thickness and tip angle of the cutting edge geometry are used in this approach. For a specific drill geometry, the torque and thrust are mea-sured for several pilot hole sizes and feedrates. For each feedrate, torque and lateral forces are modeled to have second order dependency on the pilot hole size, and thrust has third order depen-dency. The cutting lip is then discretized in small elements and from the curve fits the specific torque and thrust for each element is calculated. The specific thrust (z), torque (p), radial (r) 2 and tangential (t) forces (in [N/mm ]) are modeled to have second order dependency on chip thickness and third order dependency on the radial distance of the element midpoint to the drill axis: 2 2 3 kcz(r, h) = tfjO + a2h + a3h )(1 + a4r + a5r + a6r ) kcp(r, h) = bl(l+b2h + b3h2)(l + b5r + b5r2 + b / ) ( 4 . 5 5 ) 2 2 3 kct(r, h) = C j ( l + c2h + c3h )(1 + c4r + c5r + c6r ) kcr{r, h) = dl(l+d2h + d3h2)(\ + d4r + d5r2 + d6r) The cutting constants al...a6, bl...b6, cx...c6 and dx...d6 are estimated from a linear least squares fit to the calculated specific thrust, torque and lateral force data. Equation 4.55 is used to predict the total thrust Fz, torque Tc, radial force Fr and tangential force Ft for flute /' by sum-ming the contributions of each element: Chapter 4. Mechanics of Drilling 78 X Z kcz(r> h) x A b x h Tc= £ £ kcp(r, h) x Ab x h x r flutes elements Fr= £ kcr(r, h) x Ab x h elements flutes elements Ftj = ^ Jfcc,(r, h)x Abxh elements (4.56) 4.3.1. Mechanistic model calibration for torque, thrust and lateral forces For calibration of the mechanistic model, the torque and thrust have been measured using a rotat-ing dynamometer at six feedrates and five pilot hole sizes. The lateral forces have been measured for the same cutting conditions with a special drill with only one flute (one flute is ground away). The experimental cutting forces are measured in a coordinate system fixed to the drill with a three-axis dynamometer on which the drill is mounted. The workpiece rotating in the spindle is fed to the drill bit. The X-direction of the dynamometer is aligned with the cutting lip (R), the Y-direction is perpendicular to the cutting lip (T) and the Z direction is the thrust direction. The twist drill specifications are provided in Table 4.7. Note that the chisel edge is not modeled in this mechanistic approach. Table 4.7 : Geometrical properties of the drill tip of a Guehring 16mm twist drill, geometry #217, TiAIN coating; this drill is usedfor the mechanistic model in Eq. (4.55). Diameter [mm] Web thickness [mm] Web ratio [-] Chisel edge angle [deg] Tip angle [deg] Helix angle [deg] 16.00 2.12 0.133 125 118 30 First, curves are fitted to the experimental torque, thrust and lateral forces. The torque and lateral forces are assumed to have second order dependency on the pilot hole size, the thrust third order. From these curve fits, the model coefficients are determined through a least squares fit as: Chapter 4. Mechanics of Drilling 79 kcz = 2028 x (1 - 8.211 A + 21.77/z2)(l -0.1656r- 4.662 x 10"V + 1.684 x 10~V) kcp = 5228 x(l-7.235/z + 1 7 . 7 8 £ 2 ) ( 1 - 0 . 2 7 0 0 r + 3.506 x 10~V-1.669 x 10~V) kct = 1449 x (1 -2.772/z + 3.854/z2)(l + 0.2278r-4.782 x 10~V + 2.629 x 10~V) kcr = 523.1 x (1 - 14.89/2 + 37.66/*2)( 1 + 0.2424r- 4.861 x 10~V + 2.666 x 10 _ 3 r 3 ) 4.3.2. Comparison experiments and mechanistic model for torque, thrust and lateral forces Fig. 4.24 shows the predicted thrust, torque, radial and tangential forces acting on one lip, for two different feedrates. The match between the experiments and the mechanistic model is good, as it is a higher order curve fit model. Both thrust and torque are nonlinearly dependent on the pilot hole size. The cutting mechanics model results detailed in sections 4.2.1 and 4.2.4 are included for comparison. Using the cutting mechanics approach, the torque and thrust are under predicted, as found in earlier experiments (Section 4.2.3). The radial force is predicted poorly, but the predicted tangential force follows experiments well in both magnitude and trend. Chapter 4. Mechanics of Drilling 80 o experiment f=0.1 mm/rev O experiment f=0.6mm/rev — mechanistic model cutting mechanics model 6 8 10 Pilot hole diameter [mm] Figure 4.24 : Single flute experimental thrust, torque & lateral forces versus mechanistic model. Chapter 4. Mechanics of Drilling 8 1 4.4. Conclusions The cutting mechanics approach provides good insight into the prediction of drilling forces. This modeling technique follows experimental trends well, except for the radial force acting on the flutes. However, due to several reasons the cutting mechanics approach is not very accurate. For the cutting lip region it should be noted that only one chip is formed, although the rake angle var-ies tremendously. Drilling is therefore strongly oblique, and the assumption that the cutting edge can be divided into small elements for which the orthogonal to oblique transformation can be applied, loes its accuracy. The forces associated with the oblique cutting are higher than pre-dicted. When drilling holes with a fixed pilot hole size, it cannot be determined from which zone on the cutting lip the error actually arises. Cutting coefficients from orthogonal tests are used, although the material is deformed in a more complex way. For the chisel edge region, where the chip formation resembles extrusion rather than cutting, both the torque and thrust are under pre-dicted as the cutting coefficients determined for this region are too low, due to sidespreading in the orthogonal experiments. For accurate force prediction for the cutting lip region, a high order polynomial is fitted to the measurements to identify mechanistic cutting coefficients as a function of chip thickness, cutting speed and radial location along the lip. Thrust and torque for piloted holes are measured using a regular drill and three orthogonal forces are measured while drilling piloted holes with a one fluted drill. These experiments show that the cutting forces are very non-linear. The nonlinearities are captured well by the proposed model. The main drawback of mechanistic modeling is that experiments are required for each drill geometry, and the method cannot be used for drill design and general analysis. 82 Chapter 5. Numerical Modeling of Drilling Dynamics 5.1. Introduction In order to investigate the details of the drilling process, a numerical time domain simulation model has been developed. This model uses an exact kinematics strategy, tracking the exact posi-tions of both tool and workpiece, and allowing for nonlinearities such as the tool jumping out of cut. The cutting mechanics and mechanistic force models developed in chapter 4 are integrated allowing cutting coefficients to be a function of cutting edge geometry, cutting speed and chip thickness. In this time domain simulation, the surface finish left by the tool, time history of cutting forces and tool vibrations are obtained. Sections 5.2-5.4 cover the geometrical and numerical treatment of the exact kinematics modeling. Static cutting force calculation with tool imperfec-tions and simulation of forced whirling motion, including their effect on wall surface generation are presented in section 5.5. Dynamic axial, torsional-axial, and lateral regenerative vibration simulation results are provided in section 5.6 and compared with experiments in section 5.7. In the model, the intersections between the cutting edges and circular workpiece grids are found. These grids represent the surface generated by the drill on the hole bottom. The chip thickness distribution along each cutting edge is calculated from the surface left by the previous tooth and the location of the cutting edge at the current time step. The model is developed for drilling piloted holes initially, and can later be extended to drilling full holes by considering the chisel edge region as presented in Chapter 4. The model also predicts the hole wall formation, by keep-ing track of the hole shape errors (deviations from the circular shape) along the hole depth, using an additional wall grid. With the best knowledge of the author, the numerical model of integrated mechanics, kinematics and dynamics of drilling is the first reported in the literature. Chapter 5. Numerical Modeling of Drilling Dynamics 83 5.2. Drill geometry The drill geometry is defined by the tip angle 2 K, , the chisel edge angle cp, the diameter D and the web thickness 2 W, as shown in'Figs. 5.1a and 5.1b. A two-fluted drill is considered here. A three-dimensional representation with coordinate system is provided in Fig. 5.1c and a cross-sec-tional view of the drill geometry in Fig. 5. Id shows the direction of rotation of the tool, as seen from the tool holder side. The rotation angle of the tool equals Q.t in [rad]. Figure 5.1 : Drill bit geometry and coordinate system. The initial workpiece geometry model is derived from the tool diameter and pilot hole diameter Dp by discretizing each cutting lip into m elements, yielding m + 1 points, as illustrated in Fig. 5.2. The number of grid elements affects the accuracy and smoothness in the cutting force predic-tion. Dimension Lx is the distance between the y-axis and the peripheral point of the cutting edge, when the cutting lips are aligned with the X-axis, and depends on the web thickness: The length of the cutting lip equals: Chapter 5. Numerical Modeling of Drilling Dynamics 84 Dimension L2 depends on the radius of the pilot hole Rp = -Dp and half the web thickness (W), L3 depends on the web thickness and chisel edge angle cp: H1^ w tancp (5.3) w / e -^ i 1 x 3 — x 4 — < x 5 — x 6 i — x 7 i — » \ Figure 5.2 : Cutting edge discretization; Number of elements along cutting edge m=7. When the cutting lip is parallel to the positive X-axis, the coordinates of the cutting edge points on cutting lip 1 are given by: * e , i ( 0 (L2 + (i- 1) x (Ij -L2)/m) -W •, / = 1, 2, m + 1 - z e , l ( 0 , {(L2-L3) + (i- 1) x (Lx - I 2 ) / w } / t a n K , The vectors (x g ; i,ye,i) and (ze> i) of size (m + 1) define the points on a cutting edge 1, and are used to initialize the workpiece grid. Chapter 5. Numerical Modeling of Drilling Dynamics 85 5.3. Workpiece model The main material removal takes place at the tip of the drill, where the cutting lips generate the hole. Due to lateral vibrations, small amounts may also be taken away from the wall. First, the modeling of the bottom hole surface will be described, followed by the hole wall model. 5.3.1. Bottom hole surface Each flute generates a tooth surface, in this thesis the drill has two flutes, hence two surfaces are used in the simulation model. Each tooth surface is modeled by points arranged in a number of circles. The number of points in a grid circle is an even integer N , and is chosen such that the selected time increment At is at least 15 times smaller than the period of the highest vibration mode of the system. An initial grid is generated for each tooth by projecting a repeated rotation about the Z-axis of the cutting edge points onto the z = 0 plane, as shown in Fig. 5.3, distribut-ing the points evenly. The grid depends on the drill- and pilot hole diameters. 8 o cutting edge points on twist drill • workpiece points • hole periphery -8 -10 -8 -6 -4 -2 0 2 4 6 8 10 X-axis [mm] Figure 5.3 : Initial workpiece discretization; Number of grid circles m=7; Number of points per grid circle N=60. Chapter 5. Numerical Modeling of Drilling Dynamics 86 Using the center of the hole and the flat upper surface of the workpiece as a reference, the initial coordinates of the points on the surface or flute 1 are expressed using the grid angle ag in Eq. (5.5). The workpiece surface coordinates for flute 1 are stored in three matrices, WY , , W , and Wz^ j , each of size Ng by m + 1. At the end of a simulation, the final bottom hole finished work-piece is created from the tooth surfaces. a = ^ x ( g - l ) g = 1,2,...,/V +1 c o s ( a g ) sin(ag) 0 r > i xe,l (5.5) • ywg,\(g) > — sin(cxg) c o s ( a p 0 • > ye,i > . >zwg,\{g) 0 0 1 > l ze i J 5.3.2. Hole wall surface The hole wall surface is modeled by a number of circles with an evenly spread, fixed number of points. These circles provide cross sections of the machined hole at axial increments, that can be compared with measurements from a hole measurement machine. The points are initialized at a radius smaller than the drill, so that their radial coordinate can be updated to account for the cut-ting action of the drill flutes. The circles form a cylindrical surface. Figure 5.4 provides a three-dimensional view of the finished bottom workpiece and hole wall surfaces, at the beginning of the simulation (a), and at the end of the simulation (b). Chapter 5. Numerical Modeling of Drilling Dynamics 87 Figure 5.4 : Example of hole wall and bottom surfaces: a) start of simulation b) end of simulation. 5.4. Time domain simulation model The chip geometry in drilling is constant when the drill bit does not experience any vibrations or static deflections. When engaging into and exiting from the workpiece, the chip geometry experi-ences transient changes. As the tool vibrates due to structural flexibility, the chip thickness may change with time at any stage of the drilling process. The change in chip thickness depends on the tool geometry, feed, vibration amplitudes generated at the present moment, as well as vibration marks left on the surface during previous revolutions. Montgomery and Altintas [49] used exact kinematics of milling in predicting the chatter vibra-tions in time domain by allowing the tool to vibrate in the lateral directions. The chip thickness was evaluated by subtracting the present tooth trajectory from the previously generated cut sur-face. Lazoglu et al. used a one-dimensional approach in analyzing boring chatter [76], allowing the tool to vibrate radially. Roukema et al. used a similar, one-dimensional fixed grid approach to study axial chatter vibrations in drilling [77]. A three dimensional model is proposed in this thesis Chapter 5. Numerical Modeling of Drilling Dynamics 88 to simulate lateral, axial and torsional vibrations in dynamic drilling, by employing a flexible grid. Lateral motion of the tool is assumed to be small relative to its length, and to not influence the axial tool position. When the wall is cut by the flutes, contact and friction forces may act on the tool, that can be significant. These contact forces are not modeled in this thesis. The tool is allowed to vibrate due to its natural modes while the workpiece is rigidly fed towards the tool. The dynamic system is simulated at discrete time intervals, and the chip geometry removed at each time instant is evaluated by identifying the dynamic tool and workpiece engage-ment. The chip thickness is evaluated by calculating the distance between the surface generated by the current tooth and the surface generated by the previous tooth. The chip thickness is used to predict dynamic cutting forces. Figure 5.5 shows the structure of the time domain simulation model and provides references to the sections detailing the modules developed for the model. cutting speed feedrate drill diameter pilot hole size cutting coefficients r WORKPIECE (5.3) meshing of workpiece geometrical | i model TOOL GEOMETRY (52) kinematics & tool position TOOL MOTION (5.4.1) WORKPIECE MOTION (542) 1 dynamic chip thickness I I structural dynamics cutting force calculation .INTEGRATION (54.4).: CUTTING FORCE (5A3) K cutting forces tool vibrations surface finish Figure 5.5 : Structure ofproposed three dimensional time domain simulation model for drilling. 5.4.1. Geometry of tool motion In the simulation, the time increment Ar is coupled with the workpiece discretization through: At = (5.6) n x Ng where n is the spindle speed in [rev/min] and Ng is the number of points on each grid circle. The rotational angle of the drill's rigid body motion can be written as: Chapter 5. Numerical Modeling of Drilling Dynamics 89 n t = f 2 n x ^ x k x A t - ( 2 n x ^ x k x ( - ^ - ) = ^ (5.7) V 60/ \ 60J \nxNgJ Ng K J where integer k is the time step counter. At each time step, the intersection points x-,t \ (i, k), yh \ (i, k) of the cutting edges with the grid circles are determined as illustrated in Fig. 5.6. Due to lateral deflections of the drill bit, the distance between any two adjacent intersection points on a cutting edge varies over time. Hence, the fixed discretization as presented in Eq. 5.4 cannot be used for expressing the elemental width of cut. The figure shows the width of inner (Ab{ j, Ab2 j) and outer elements (Abx 7 , Ab2 7 ) for a large tool deflection. The time varying elemental width of cut is calculated from the distances between intersection points. Due to the lat-eral vibrations, one cutting edge will move outside of the largest grid circle, while one moves inside of the largest circle or an even smaller one. The cutting edge peripheral point's coordinates are compared with the intersection points to determine where the peripheral point is. For any lat-eral vibration, one cutting edge will have a wider peripheral element, while the other edge has a peripheral element with a reduced width, and may have some elements with zero width. Using this approach, the total width of cut for each cutting edge is continuously variable. Hence, the cut-ting force fluctuations due to lateral deflections are smooth. Figure 5.6 : Intersection points of cutting edges with grid circles and illustration of time varying widths of cut; Number of elements along cutting edge m=7. Chapter 5. Numerical Modeling of Drilling Dynamics 90 For each of the intersection points, the height of the cutting edge at that location, z,-; k), is cal-culated based on the tool geometry. The calculated cutting edge height at the location of the grid circle is used for the chip thickness calculation and surface grid updating. The location of the peripheral point of the cutting edges, including runouts rr, rt and rz, can be expressed as a function of time and tool vibrations: xP,\ik) xp, 2 W I zpt2(k) j (Z, + r r )cos(^ + ec(/)) -{W-rr)sinffi + 0c(r)) +xc(t) g g -(£, + rt)sin[^ + 9 C(0) -(W-r^cosf^ + 6 c(o) +yc(t) g g z(m+\) + z(t) - ( L j - O c o s ^ + O ^ ) ) + ( » r + r r ) 5 i « ( ^ + ec(0) +xc(t) g g ( L , - rt)sin{^- + 9 C(0) +(W+ r,)cos(^ + 9 c(o) +yc(t) g g z(m+\) + r+z(t) (5.8) 2kn The total rotation of the cutter, —— + 0 c(r), also provides the direction of the cutting edge, which g is used to find the intersection points with the grid. 5.4.2. Dynamic generation of hole profile The bottom hole is flat in the beginning (Fig. 5.4a). The drill starts rotating while the workpiece surface is fed to the drill. The drill will start removing material when the workpiece has been fed over a distance equal to zx, shown in Fig. 5.2. Distance zx depends on the pilot hole size. The hole diameter gradually increases until it becomes steady state and equal to the hole diameter, after the workpiece has traveled a distance in Z-direction equal to z 8 . For each integration time Chapter 5. Numerical Modeling of Drilling Dynamics 91 step (kAt), two modifications are made to update the bottom surface. First, the feeding motion in positive Z-direction is accounted for by adding a feed increment to every entry of matrices Wz x and Wz 2 , which contain the Z-coordinates of the workpiece surfaces. The feed increment equals: A / r - §- (5.9) g where fr is the feedrate in [mm/ rev]. Secondly, the cutting action of the drill is accounted for on matrices Wx x , Wy x , Wz x and Wx 2 , W 2» Wz 2 . The X, 7-surface points are confined to the circles defined in the initialization of the surface. Chip thickness calculation The simulation finds the points on the previously generated surface (Wx 2 , Wy 2 and Wz 2 for flute 1) that are close to the intersection point of the cutting edge with the grid: one point ahead of the cutting edge, one behind. The index of these points (Fig. 5.3) in the surface matrix is estimated based on the time step k: k'f,i=k-NrxNg ( f l u t e 1) ( 5- 1 0) where Nr is the integer number of revolutions that has passed in the simulation. The surface matrix index for interpolation, (Ay j), is identified through iteration. The height of the workpiece at the location of the cutting edge is then calculated by linear interpolation of the height of the two workpiece points, as illustrated in Fig. 5.7. For each intersection point of the cutting edge z ( ) i(/, k), the Z-coordinate is compared with the corresponding interpolated Z-coordinate of the workpiece surface point, Wz2'(i,kjX), at the exact location of the cutting edge point. If z? ; i(z', k) < Wz> 2(i, k^ x), tooth 1 is in cut. A new point is generated on workpiece surface Wz x using the current coordinates of the cutting edge point xi j(z, k), yi x(i, k), zf- x(i, k) and the chip height hce x(i, k) is determined: Chapter 5. Numerical Modeling of Drilling Dynamics 92 Wx,i(i>kf,\) = xi,\(i> k) wy,i(i>kf,0 = yi,i(i>k) Wz, \(j>kf,l) = z i , l ( i > k ) hce, k) = Wz, 2'0> kf,\)-Zi,\i^ k) The workpiece updating procedure is shown schematically in Fig. 5.7. Figure 5.7 : Exact kinematics approach for surface updating; a) tooth in cut, regular cutting b) tooth out of cut, due to excessive vibrations. If z ( ; \ (i, k) > Wz 2'(i,kj-\), the tooth is out of cut, due to excessive vibrations or because it has not engaged with the workpiece yet. The Z-coordinate of the new workpiece point is obtained from the interpolated height of the previous surface, and the chip height is set to zero: wx, kf,\) = xi,\(i> k) wy,i(i>kf,\)=yi,\(i>k) (5.12) WzA(i,kfA)=W^(i,kfA) This case is illustrated in Fig. 5.7b. The same procedure is followed for flute 2. The two modifica-tions to the surface vector (feed motion, cutting action) completely define any change to the bot-tom workpiece surface. The chip thickness is evaluated at the m + 1 intersection points, for which the chip height hce j(z, A;) is determined. The chip thickness of element /' is then determined by the average of the chip heights on the sides of the element. Chapter 5. Numerical Modeling of Drilling Dynamics 93 hi<J,k)= {hcel(j,k) + hceJ(j+lk)}/2 <j=\...m) (5.13) Fig. 5.8 shows different positions of the cutting lips with respect to the workpiece. Figure 5.8 : Cutting edge points as moved with respect to workpiece; Illustration of chip height. Surface grid re-ordering As the drill can move forward and backward due to lateral vibrations, points are not necessarily generated in orderly fashion as in Fig. 5.3. This occurs mainly near the chisel edge region, where the grid is very dense. In Montgomery's milling simulation [49], each tooth surface is completely reordered after the tooth has left the workpiece, which occurs once per revolution. The proposed drilling simulation algorithm checks at each time step whether the cutter moved forward or back-ward for each point generated. If the cutter has moved backward, the surface is reordered when the currently generated point lies behind one or more previously generated points. This reordering ensures that the angular locations of all points in a grid circle increase monotonously. In drilling, the number of points on a grid circle is much larger than in a milling array, hence a full reordering would seriously compromise the simulation speed/efficiency. Chapter 5. Numerical Modeling of Drilling Dynamics 94 Wall surface updating The wall surface is generated by the flutes. The cylindrical wall surface is fed towards the drill simultaneously with the bottom hole surface grid. At each time step, the location of the flute on each layer of the whirling grid is determined. If the radius of this point is greater than the work-piece radius at that location, the layer is updated (flute in cut), otherwise it is left unchanged. Fig-ure 5.9 illustrates this process. The input for calculating these intersection points is the height of the whirling grid layer, zd, which is measured from z = 0. The coordinates of the intersection point depend on the helix angle (50, back taper, tool runout and tool vibrations. hlip is the distance from z = 0 to the start of the flutes. The drill has a back taper which reduces the fluted diameter by an amount bt (in [mm]) along flute length Lj, to provide clearance as the drill penetrates the workpiece. The back taper is included as a linear diameter reduction. The tool displacements xc(t),yc(t) are included by assuming the tool deflects as a simply supported beam, clamped in the tool holder. Finally, the drill runout rr, rt is included, which is assumed to affect the coordi-nates of the intersection points linearly, which means that the runout at the tool holder is zero. Figure 5.9 : Mechanism of wall surface updating; points on each radial grid circle are pushed outwards. Chapter 5. Numerical Modeling of Drilling Dynamics 95 The intersection point of flute 1 with the grid circle at height zd is found from: xi, i(zd) = \{D ~btx Z d Lh''P) c o s ^ p 5 x (Zd-hlip)-o)(t) - c o 0 ) (L-ZjfilL + Zj) L-zd + xc(t) x *L + — x { r co sco ( 0 + r , s i n c f l ( f ) } 2U L~niip ( \ lfn u zd~hlip\ • r 2 t a n P o , u ^ t.\ y-x, \(zd) = 2{D~b'x Lf J S 1 \ D x ( Z r f ~ h n p ) ~ ^ t ) (5.14) c o 0 (L-zd)\2L + zd) , L - z ^ . + ^ c ( 0 x : + 7—:— x {-rrsinco(r) + /^cosm^)} 2L L ~ nliP where the influence of zc(t) is neglected. Axial runout rz is considered only to influence the Z-coordinates of the cutting edges, and not the flute geometry. As the flute intersection points do not necessarily coincide with the fixed wall grid, the radial location of the flute intersection point is interpolated between the intersection point of the flute in the current time step and the intersection point of the flute in the previous time step. The intersection point for flute 2 is found by replacing angle -co(r) - co0 by -co(r) - co0 + n in Eq. 5.14. 5.4.3. Cutting force calculation The elemental chip thickness distribution is used to calculate the cutting forces using the mecha-nistic model (Eq. 4.57). In chapter 4 the specific torque, thrust, and lateral forces were determined as a function of radial distance from the drill axis. Once the chip thickness for each element is cal-culated, the torque, thrust, and lateral force contribution for that element is simply calculated by multiplying element width, average chip thickness (yielding area of cut) and specific force. The force is multiplied with the radial distance of the element to the drill axis to obtain the torque. The resulting forces are obtained by summing the contributions of all elements along the lips: Chapter 5. Numerical Modeling of Drilling Dynamics 9 6 Fz= 2 X kcz(r> h) x A b x h Tc= £ kcp(r, h) x Ab x h x r flutes elements flutes eiements (5.15) elements Ft= £ kct(r,h)xAbxh elements e ateral forces Z*^ , Z^ in the stationary global coordinate system are obtained by projecting the resultants of the radial and tangential forces Fr, Ft: Fy{t) sin(Qr) cos(Q/) cos(Q/) -sin(Qr) F -F -Fr +Fr (5.16) 5.4.4. Workpiece surface finish At the end of the simulation, the finished hole surface is established from the two surfaces gener-ated by the flutes, see Fig. 5.10a, by finding the points that have the smallest Z-coordinate. This method can easily be extended to handle three- and four-fluted drills. (intersection points of cutting edge with grid circles) Figure 5.10 : a) three dimensional representation of tooth surfaces left by teeth; b) finished workpiece surface constructedfrom tooth surfaces. Note: feed is 1.5mm/rev for visualization. Figure 5.10b shows the three-dimensional view of finished workpiece surface (without the wall surface) and the intersection points of the cutting edges with the grid circles. A high feedrate and course discretization are used for visualization purposes. Chapter 5. Numerical Modeling of Drilling Dynamics 97 5.4.5. Integration scheme The dynamic deflections of the drill bit are determined from the structural model that is excited by the cutting forces. The general transfer function of the drill structure in the axial direction (Z) at the tip of the drill is obtained in the Laplace domain as: (5.17) where (km) is the total number of modes in that direction of the system. (oon;), (kt) and (Ct) are the natural frequency, modal stiffness and modal damping, respectively. The modal properties of the drill bit are given in Table 3.4. The transfer function is converted into observable state-space canonical form [78] for each mode as: 1 *2i(0 J z At) = z2(t) o - 4 <°2ni/ki .1 -2C,<v 1 * 2 / (0 J 0 {Fz(t)} (5.18) The total axial displacement is evaluated by summing the vibrations contributed by all natural modes of the drill bit in axial direction: zc(t) = £ A z c . ( 0 /= 1 (5.19) The Runge-Kutta fourth order formula is by far the most common and most preferred numerical integration algorithm and is used to calculate Eq. 5.18. Chapter 5. Numerical Modeling of Drilling Dynamics 98 5.5. Static drilling simulation This section shows the chip thickness distributions, cutting forces and surface finish in case the drill bit is rigid and deflections are zero. The cutting forces arising from the material removal are evaluated, including tools with grinding errors and misalignment. Imposed whirling vibrations are also simulated, and the resulting cutting force spectra and hole wall shape are investigated. The tool geometry is provided in Table 5.1, the cutting coefficients in Table 5.2: Table 5.1 : Geometrical properties of two fluted Guehring twist drill, geometry #217; Tool material: HSS (high speed steel), TiAIN coating; Diameter [mm] Web thickness [mm] Chisel edge angle [deg] Tip angle [deg] Helix angle [deg] Drill length [mm] 16.00 2.12 125 118 30 40 Table 5.2 : Mechanistic cutting coefficients (refer to Eq. 4.57) for a Guehring #217 twist drill (geometry: Table 5.1), cutting AL7050-T7451 and using flood coolant; Kz = 2028 x ( l -- 8.21 l/2 + 21.77/?2)(l-0.1656r-4.662 x 10" V + 1.684 x 10"V) KP = 5228 x ( l -7.2356+ 17.78/z2)(l-0.2700r + 3.506 x 10" "V-1.669x 10"V) Kt = 1449 x ( l -- 2.772/z + 3.854/22)( 1 + 0.2278r- 4.782 x 10" V + 2.629 x 10"V) kc r ~ 523.1 x ( l - 14.89/2 + 37.66/22)( 1 +0.2424r-4.861 x 10 "V + 2.666 x 10"V) 5.5.1. Static torque and thrust Figure 5.11 compares the time domain cutting force prediction for the static case with the experi-mental torque and thrust, showing good agreement. The time domain simulation model predicts how the cutting forces change over time when the drill engages with the piloted workpiece. The mechanistic model (Table 5.2) is used, which is based on curve fitting experimental data. The torque and thrust were measured using a Kistler 9125 rotating dynamometer. Chapter 5. Numerical Modeling of Drilling Dynamics 99 500 5- 400 i * 300 jC 200 100 0 1200 •g-1000 Z 800 » 600 | 400 *" 200 "0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Time [s] Figure 5.11 : Comparison between experimental and predicted torque and thrust during engagement into piloted workpiece (AL7050-T7451); pilot hole radius Rp=2mm; Drill geometry: Table 5.1, cutting coefficients AL7050-T7451: Table 5.2; Cutting speed Vc=40m/min, Feed per revolution fr=0.30mm/rev; Number of elements along cutting lip m=30. In stage 1 air cutting takes place, and the torque and thrust are zero. In stage 2 the drill starts to engage with the workpiece, and the torque and thrust increase until full engagement at 1.1 sec-onds. Stage 3 shows the steady state cutting with constant thrust and torque. A short drill (L/D ratio=2) was used, reducing tool deflections to a minimum. A small periodical fluctuation in the measured cutting force profile can be seen, caused by drill runout. The forces in X and Y direction are assumed to be zero, as the drill is assumed to be perfect and no vibrations are present. 5.5.2. Lip height, tip angle grinding errors and runout Figure 5.12 provides simulation results to show the effects of drill grinding errors and misalign-ment. The tool geometry and cutting coefficients provided in Table 5.1 are used in the simulation. If there is a lip height error rz = 10JJ./W (a), the chip thickness on one lip will be increased by this amount, and on the other lip the chip thickness will be reduced by this amount. In this case the intended chip thickness is 150u,m, but each element on lip 1 one cuts 160u,/w and the elements on lip 2 cut 140u./w. As a result of the unequal chip thickness, an unbalance force acts on the tool. - © —1 1 1 1 I —1 1 1 Chapter 5. Numerical Modeling of Drilling Dynamics 100 The unbalance force grows during engagement and remains constant after full engagement. The force rotates with the tool, and can be identified from the single peak in the fourier spectrum at the spindle frequency fs. A tip angle grinding error is similar to a lip height error, but the lip height error varies linearly along the cutting edge. As a result, elements from both lips close to the chisel edge cut almost the same chip thickness, but the elements close to the drill periphery show a larger difference in chip thickness. The resulting force balance develops more parabolically dur-ing engagement, but stays constant after full engagement as well. Misalignment of the tool with the spindle axis occurs due to inaccuracy of the tool holder-spindle interface, tool holder and mounting of the tool in the tool holder. Figure 5.12c shows the forces for a radial runout rr = 10u,w (in the direction of the cutting lip). In this case the lips will cut different chip thick-nesses, but the lips will also cut different width of cut. The result is similar to the tool grinding errors, with an unbalance force rotating with the tool, yielding a peak at the spindle frequency in the fourier spectrum. Chapter 5. Numerical Modeling of Drilling Dynamics 101 150 100 50 0 -50 -1001 -15oJ 6 5 4t 3J 2 1 0 (a)( Lipheight error=0,010mm ] (b)( Lipangle error=0.100 deg. ] (c)( Radial runout error=0.010mm) 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 x 10 T i m e [ s ] x 10 T i m e [ s ] x 10 T i m e [ s ] 0 0.18 0.15 0.12 2f s 4f s 6f s F r e q u e n c y 2f s 4f s 6f s F r e q u e n c y s 4f s 6f s F r e q u e n c y in in a> c a. 5 o 0.18 0.15 0.12 V pilot 1 \ -[element peripheral-) . element J E E, <M a in in a> c u Q. o 0 0.1 0.2 0.3 0.4 0.5 0.6 T i m e [ s ] pilot [element peripheral] element 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 T i m e [ s ] T i m e [ s ] Figure 5.12 : Simulated static cutting forces, fourier spectrum and elemental chip thicknesses for: a) Lip height error; b) Lip angle error; c) Radial runout error; Pilot hole diameter-4mm; Feedrate=0.30mm/rev; Spindle speed=2400rpm; Number of elements per cutting edge, m=50; Tool geometry: Table 5.1; Cutting coefficients ALT050-T7451: Table 5.2. Chapter 5. Numerical Modeling of Drilling Dynamics 102 5 . 5 . 3 . Imposed whirling vibrations It has been experimentally observed by several authors [3,50,53,54] that a drill whirls at odd inte-ger multiples of the drill spindle frequency fs, as measured in the frame rotating with the drill. This whirling motion consists of the deflected drill center tracing a circular or elliptical path in the opposite direction of the drill rotation. The whirling frequency in the fixed frame fj can be obtained from fj = fwr —fs where fwr is the whirling frequency in the rotating frame, and the minus sign applies to backward (regressive) whirling modes, as established by Bayly et al. [47]. These whirling vibrations do not depend on the spindle speed, so they are different from whirling vibrations found in rotating shafts. The physics of whirling vibrations in drilling can be explained as follows. A tool deflection in the negative X direction results in an increased cutting force on flute 2, F2 + AF, and a decreased cutting force on flute 1, Fx - AF, see Fig. 5.13a. These two forces can be replaced by a single unbalance force through the drill center, 2AF and the torque Tc + 2RAF, Fig. 5.13b. Den Hartog showed that the torque would merely retard the rotational motion Q of the tool [79, pp. 291/Fig. 212], but the unbalance force pushes the tool down, in the direction opposite of the tool rotation. This force excites a counterclockwise motion - backward whirling. When the drill bit contacts the wall of the hole being cut, a contact force Fwi will arise, Fig. 5.13c, which results in a backward whirling motion as well. Unlike the example by Den Har-tog of a flexibly supported rotating disk that can experience contact forces on both side of the disk, drilling deflections cannot result in precessional (forward) whirling vibrations. In drilling, contact forces only occur on one side, the drill tip. Chapter 5. Numerical Modeling of Drilling Dynamics 103 F 2 + A F Figure 5.13 : Physics of whirling vibrations in drilling; Drill seen from shank side. If we assume the drill is executing a circular whirling motion in the fixed frame, the tool cen-ter position is defined by: xc(t) = esin(-//0 / yc(t) = ecos(-//0 >fw 2^wfs>^w 1> 2, ... (5.20) where e is the amplitude of the whirling motion, fs is the spindle frequency, and Nw is a positive integer. Due to the minus sign, the whirl is backwards - counterclockwise. A backward whirling motion as in Eq. 5.20 can be seen as a deflected drill motion restrained by the round shape of the hole. Figure 5.14 shows the cutting edge traces for whirling at two, four and six times the spindle fre-quency. Three, five and seven sided hole shapes can clearly be seen. The intended hole has a diameter of \6mm, and a whirling amplitude e = 500u,/w is used for visualization. After half a tool revolution ( 1 8 0 ° ) , the cutting edge peripheries coincide with the profile generated by the previous flute - the tool center is at the same location as it was at 0° tool rotation. The location of the drill peripheral points is given by: Chapter 5. Numerical Modeling of Drilling Dynamics 104 x p j(0 = L{cos(£lt)-Wsm(Clt) + esm(-fjt) XP , 2(0 = ~Li cos (Q0 + JFsin(Q0 + es in ( - / /o yp, i(0 = -^iSin(Q0-^cos(Q0 + eco&(-fJt) >^,2(0 = z i s i n ( ^ 0 + Wcos(Clt) + ecos(-fwJt) (5.21) and depends on drill dimension L, (Eq. 5.1) and the drill web thickness 2 W. (c) intended hole profile C=-6f. tool axis (1083) trajectory Figure 5.14 : Traces of the cutting edges during backward whirling motion at exact integers of the spindle frequency fs at four different tool rotations: a) fj=2fs; b) fj—4fs; c) fj=6fs; Drill diameter = 16mm, web thickness 2W= 2.12mm, fixed whirling amplitude = 500micron (for visualization). Cutting forces during whirling vibrations Commonly observed whirling motions were imposed on a slender drill bit to investigate the chip thickness, thrust force and lateral forces as a function of time. The drill geometry is provided in Table 5.3, and the cutting coefficients from Table 5.2 are used in the simulation. The simulation results are summarized in Table 5.4, and shown in Fig. 5.15. Chapter 5. Numerical Modeling of Drilling Dynamics 105 Table 5.3 : Geometrical properties of two fluted Guehring twist drill, geometry #217; Tool material: HSS (high speed steel), TiAIN coating. Diameter Web Chisel Tip angle Helix Drill Back taper Flute [mm] thickness edge angle [deg] angle length [mm] length [mm] [deg] [deg] [mm] [mm] 16.00 2.12 125 118 30 176 0.052 149 Table 5.4 : Summary of simulation results of imposed whirling motions with 100pm amplitude; Cutting conditions: 2400rpm,f=0.30mm/rev, pilot hole diameter=4mm, hole depth=15mm. Tool geometry: Table 5.3; Cutting coefficients: Table 5.2. Imposed whirling frequency in rotating frame (/, =spindle frequency) 3.00/, 2.90/, 4.90/, 6.90/, Details in Figure: 5.14a, 5.15a 5.15b, 5.16 5.15c 5.15d Peaks in chip thickness spectrum (hx) : none 2.90/, 4.90/, 6.90/, Peaks in width of spectrum ( b i i p \,blip 2): 3.00/, 2.90/, 4.90/, 6.90/, Peaks in thrust force spectrum (Fz): none 5.80/, 9.80/, 13.80/, Peaks in lateral force spectrum (Fx, F ) : 2.00, 4.00 1.90,3.90 3.90, 5.90 5.90, 7.90 A backward whirling motion at fwr = 3.00/, (superscript r denotes the frequency is measured in the rotating frame) is achieved by imposing a circular tool vibration (Eq. 5.20) in the fixed frame using 7v"w = 1 and amplitude e = 100u,m. In Fig. 5.15a, the elemental chip thickness fluctuates during one drill revolution, but remains constant in the following as the wave left on the surface is in phase with the tool motion. This phenomenon only occurs when the whirling frequency in the rotating frame is an odd integer of the spindle frequency fs. The width of cut on each flute ( b l i p i , b l i p 2 ) varies from the very beginning at 3.00/,, and as a result, the tangential and radial forces in the frame rotating with the tool vary with the same frequency, 3.00/,. As shown in Fig. 5.14a the resulting hole shape is three sided. The spectrum of the lateral forces in the stationary frame shows peaks at 2fs and at 4fs. The occurrence of two peaks, rather than one, can be explained using the common trigonometric product-to-sum identity: Chapter 5. Numerical Modeling of Drilling Dynamics 106 (5.22) where s\n(Js x 2rcr) represents term sin(Q/) in Eq. 5.16 (2nfs = Q , the speed of the frame rotating with the tool), and sin (3/; x 2nt) is the force measured in the frame rotating with the tool (at 3fs). The force functions on the right hand side represent the forces in the stationary frame. Figure 5.16a shows the tangential and radial forces in the frame rotating with the drill for the case of Fig. 5.15b (backward whirling frequency fwr = 2.90fs). The Fourier spectrum of the tangential and radial forces shows peaks only at 2.90fs. In Fig. 5.16b (and Fig. 5.15b) the lateral forces in the stationary frame are shown, and the spectrum shows two peaks. Using Eq. 5.22 the first peak occurs at 2.90fs-fs = 1.90/; and the second at 2.90fs+fs = 3.90/;. The chip thick-ness in Fig. 5.15b varies throughout the cut at 2.90/;. The lateral forces fluctuate strongly com-pared to Fig. 5.15a, and grow until the drill has fully engaged with the workpiece. The thrust force fluctuates with a small amplitude at 5.80/; because the drill has two flutes. The coupling between lateral vibrations and torque and thrust is very weak. In Fig. 5.15b, lateral forces in the order of 200N are generated for a 100u,w whirling vibration while the thrust force only fluctuates by less than 0.1% of its 465N steady state average. Figures 5.15c and 5.15d show the simulation results for fwr = 4.90/; and fwr = 6.90/; respectively. Similar to Fig. 5.15b, the chip thickness spectra show a peak at fwr, the thrust spectra show a peak at 2fwr, and the lateral force spectra show peaks at fj -fs and fj +fs. Chapter 5. Numerical Modeling of Drilling Dynamics 107 0 2fg 4fs 6fs 8fs F r e q u e n c y 0 2f, 4f, 6fs 8fs F r e q u e n c y 0 2fs 4f, 6fs 8f8 F r e q u e n c y 0 2f, 4fs ef8 Bf8 F r e q u e n c y 6.10 •g"250 0 9 6 T i m e r s ] 1 0 0 0 9 6 T i m e r s ] 1 00 0.96 T i m e [ s ] 1.00 0.96 T j m e [ s ] 1.00 CM JO 3.0f! 0 2 f s _4f s 6fs 8f8 2.9f« 0 2f8 4f8 6f, 8f8 F r e q u e n c y 4.9f« 0 2f8 4f8 8f8 8f8 F r e q u e n c y 6.9f, 0 2fs 4f8 6f, 8fs F r e q u e n c y / 466.0 / 464.5 • f 1 1 / 0.42 0.46 0 4f. 8fc 12fc 16f, ^ 2 0 0 z LL 0 K LL -200. z LL* >< LL h-LL LL ' S _ " ' S ' " S ' " ' S F r e q u e n c y NUefg r i x10 3 T i m e t s ] 1 0 ° x 1 0 3 T i m e [ s ] 3r \ 1 \ 1 \ 1 \ 1 i \ i >^ i j 9 8fs| 1 i 1 ! ! F r e q u e n c y 0 4fs 8f8 12f8 16f8 F r e q u e n c y 0 4f8 8f8 12f, 16fs F r e q u e n c y x10« T i m e [ s ] (a) 2.0fs 4.0fs| 0 2l . 41 „ 6f. 8f. 1.0 0 x10 4 T i m e [ s ] 10 5 0 (b) 1.9f, 3.9f, 0 2fc 4fc 6fc 8f, ° X 1 0 4 T i m e r s ] 1-0 0 ~ T i m e [ s ] s u , s u , s 10 5 0 (C) 3.9fJ -4 A 5.9t 0 2fc 4fc 6fc 8f, s u , s "'s 10 5 0. (d) 5.9f, 0 2fc 4fc 6fc 8 7.9f„ F r e q u e n c y ( D ) F r e q u e n c y ( c ) " F r e q u e n c y " W " F r e q u e n c y Figure 5.15 : Simulated chip thickness, thrust force, lateral forces and Fourier spectra for a drill backward whirling frequency fj = a) 3.00fs; b) 2.90fs; c) 4.90fs; d) 6.90fs; 2400rpm,f=0.30mm/rev, pilot hole diameter=4mm, hole depth=15mm. Tool geometry: Table 5.3; Cutting coefficients: Table 5.2. Chapter 5. Numerical Modeling of Drilling Dynamics 108 Rotating frame Stationary frame Frequency Frequency Figure 5.16 : a) Tangential and radial force Fp Fr in the frame rotating with the drill bit, and Fourier spectrum; b) Lateral forces Fx, Fy acting on the drill bit in the stationary frame, and Fourier spectrum; Refer to Fig. 5.15b for tool geometry and cutting coefficients. Hole wall formation Figure 5.14 showed the trace of the cutting edges during whirling vibrations. In reality, the flutes may cut the previously generated surface again, depending on tool geometry and tool deflections. This process is simulated in the proposed time domain model. Figure 5.17 shows the simulated cross sections of the hole wall at the top and bottom of the hole, including the average diameter of the hole cross section. It can be seen that if the whirling frequency in the rotating frame is an odd integer of the spindle frequency (see Eqns. 5.20, 5.21), the odd sided hole shape is maintained at the top of the hole, despite being cut again by parts of the flute away from the tip. The shape of the cross section resulting from repeated cutting action by the flutes, depends on the drill back taper, the whirling frequency and the amplitude of the whirling vibration. Figure 5.18 shows cross sec-tions for whirling at fwr = lfs for the amplitudes 1, 2, 5, 10, 20 and 50LIW . For amplitudes up to 5\xm (Figs 5.18a,bc,d), the seven sided shape is very visible at every level in the hole. For ampli-tudes 10, 20 and 50p.m the profile at the top of the hole becomes more round, followed by the middle section and finally the bottom of the hole. Chapter 5. Numerical Modeling of Drilling Dynamics 109 • intended hole profile (=tool diameter) simulated hole profile Figure 5.17 : Simulated cross sections of hole wall for six different whirling frequencies (defined in the rotating frame), at top and bottom of hole; a) 3.00fs (refer to Fig. 5.15a); b) 2.90fs (refer to Fig. 5.15b); c) 5.00fs; d) 4.90fs (refer to Fig. 5.15c); e) 7.00fs;f) 6.90fs (refer to Fig. 5.15d); hole depth=15mm; 2400rpm,f=0.30mm/rev; Tool geometry: Table 5.3; Cutting coefficients AL7050-T7451: Table 5.2. { 1um ) ( 2um ) ( 5um ) (lOum) (20um ) (50um) •• intended hole profile (=tool diameter) simulated hole profile Figure 5.18 : Whirling cross sections as a function of whirling amplitude; Whirling frequency is 7.00fs in the rotating frame; a) lpm; b) 2pm; c) 5 pm; d) 10pm; e) 20pm; f) 50pm; hole depth=15mm; 2400rpm, f=0.30mm/rev; Tool geometry: Table 5.3; Cutting coefficients AL7050-T7451: Table 5.2. Chapter 5. Numerical Modeling of Drilling Dynamics 110 5.6. Dynamic drilling simulation This section details simulation results for cases where the drill is considered flexible. The dynamic properties of the drill bit are provided in Table 5.5. First, axial and torsional-axial vibra-tions and chatter stability are studied. Secondly, lip height errors and lateral regenerative chatter vibrations are simulated, and finally lateral and torsional-axial vibrations are combined. Table 5.5 : Dynamic properties of Guehring HSS drill bit, geometry #217 (see Table 5.3); Tool length=176mm; Diameter=16mm (L/D ratio=ll). Mode Symbol Property FE Model Experiment X X Frequency [Hz] 340 362.7 kxx Stiffness [N/m] 1.187xl05 1.405x105 Damping [%] - 0.2575 Y Y Frequency [Hz] 390 338.6 kyy Stiffness [N/m] 1.053xl05 1.300xl05 Damping [%] - 0.2533 ZZ,00 Frequency [Hz] 2955 3357.5 Direct axial stiffness [N/m] 5.277xl07 -kzTr Cross axial stiffness [Nm/m] 2.186xl05 Direct torsional stiffness [Nm/rad] 3.889xl02 -kQF, Cross torsional stiffness [N/rad] 2.462x10s Damping [%] - 0.5024 5.6.1. Axial and torsional-axial vibrations in drilling The proposed time domain simulation model can handle combined torsional-axial vibrations. The inclusion of torsional vibration allows for chip thickness modulation due to torsional deflections. It is possible for the cutter to rotate back and recut just generated surface marks. Due to the. rota-tional flexibility, the waves generated on the surface are not restricted to a purely sinusoidal form. Figure 5.19 shows the simulation results when only axial vibrations are presented by Roukema et al. [77], and using a damping ratio of 3.0% in the axial mode. Sinusoidal waves are left on the sur-Chapter 5. Numerical Modeling of Drilling Dynamics 111 face, which quickly grow in amplitude. The tool vibration in Fig. 5.19c shows the rapid growth of the tool deflection, in the Fourier spectrum characterized by a single peak. Figure 5.19d shows the waves left at the pilot hole periphery between consecutive teeth. There is a light beating effect vis-ible in the waves left on the surface, which can also be seen in the additional small peak in the vibration spectrum left of the large peak. Figure 5.19 : Time domain simulation results - axial vibration only: a) 3D surface of piloted hole; b) Detail of surface at pilot hole; c) Simulated tool vibration and Fourier spectrum; d) Side profile of the surface along pilot hole, generated by one tooth; 2400rpm, pilot hole diameter=4mm, feedrate=0.30mm/ rev; Tool geometry: Table 5.3; Cutting coefficients ALT050-T7451: Table 5.2; Tool dynamics: Table 5.5. Chapter 5. Numerical Modeling of Drilling Dynamics 112 The simulation was stopped when the tool deflection exceeded 200 micron. Figure 5.20 shows the results for combined torsional axial vibrations (the damping ratio of the torsional-axial mode is also set to 3.0%). All simulation parameters are exactly the same as in the axial vibration case (Fig. 5.19), except that torsional vibrations are included. Figure 5.20a shows that the waves look sinusoidal when the vibration amplitude is small, but the wave shape becomes different when the vibration amplitude grows. Figures 5.20c and 5.20d show that the steady state tool deflection would be about 40 micron extension and 1.4 degrees untwist. Under fully developed chatter vibrations, the axial vibration amplitude is approximately 100% of the steady state amplitude, and the torsional vibration amplitude about 70% of the steady state amplitude. In the Fourier spectra, the same torsional frequencies are dominant, the largest peaks occurring at 3125, 3198 and 3270Hz. In this case the beating effect is more pronounced, as can be seen in the tool deflection graphs, their Fourier spectra, and the surface profile. It should be noted that the chatter vibrations in the torsional-axial model do not grow indefinitely, but are limited to a certain amplitude. This can be seen in Fig. 5.20e, which shows the profile along the pilot hole periphery, where the amplitude of the waves left on the surface varies. The waves are smaller than those in Fig. 5.19d. Finally, when inspecting Fig. 5.20f, which shows the total tool rotation as seen at the drill tip under fully developed vibration, the drill bit does not rotate back in this case. The tool rotation does almost plateau, indicating the drill experiences cut-ting speeds varying from zero to cutting speeds higher than set on the machine due to counter tor-sional vibration speeds. Chapter 5. Numerical Modeling of Drilling Dynamics 113 (b) „ 1 6 0 c" § 1 2 0 I £ 8 0 <u = 40 * o ! ? 6 0 ? 40 •o « S 20 1 1 1 1 1 1— I. . . ^ »l«>«^fclAli>.»t»r^lvdl 1 , , : 0.20 0.30 T i m e [ s ] 0.50 F r e q u e n c y [ k H z ] (e) 3 4 D i s t a n c e a l o n g p i l o t p e r i p h e r y [ m m ] •5T7160r-E | 7120 o | 7100f-(0 T i m e [ s ] Figure 5.20 : Simulation of combined torsional-axial vibrations: a, b) 3D surface ofpiloted hole; c) Axial tool vibration and Fourier spectrum; d) Torsional tool vibration and Fourier spectrum; e) Side profile of the surface along pilot hole.f) Total tool rotation angle under fully developed chatter vibrations. Tool geometry, cutting coefficients, tool dynamics and cutting conditions: same as Fig. 5.19. 0.4950 0.4960 0.4970 0.4980 0.4990 0.5000 Chapter 5. Numerical Modeling of Drilling Dynamics 114 5.6.2. Lateral vibrations in drilling In this section, the tool is considered to be flexible in lateral directions only and the tool deflection due to lip height errors and regenerative, lateral chatter vibrations is considered. Lip height error For simulating the effect of a lip height error, the drill geometry in Table 5.3 is used with a lip height error rz = IOLI /W . The dynamic properties of the drill are provided in Table 5.5, but in X and Y directions a damping ratio of 300% is used in order to obtain a quasi static response. The torsional and axial modes are kept rigid. The cutting coefficients are provided in Table 5.2. — Deflection: in X-direction) Deflection in Y-direction 160 240 Frequency [Hz] 400 (flute 1) -e f f v . (flute 2) .1 0.1510 0.1505 0.1500 0.1495 0.1490 0.1510 •0.1505 0.1500 0.1495 element at [periphery J (element at 1 pilot hole d 0.1490 element at [pilot hole J element af| periphery 0.1 0.2 0.3 0.4 Time [s] 0.5 0.45 Time [s] 0.50 Figure 5.21 : Simulation for drill with 10pm lip height error on flute 1, and lateral flexibility with a damping of300%; a) tool deflection and Fourier spectrum; b) tool trajectory; c) chip thickness for each flute, and details; Conditions: 2400rpm, f=0.30mm/rev, pilot hole diameter=4mm, hole depth=6mm; Tool geometry: Table 5.3; Cutting coefficients AL?'05 0-T7451: Table 5.2; Tool dynamics: Table 5.5. Chapter 5. Numerical Modeling of Drilling Dynamics 115 The results in Fig. 5.21 show that as the tool engages, the drill quickly deflects and the chip areas on each flute become almost identical, resulting in an lateral unbalance force of only IN (the forces are not shown in Fig. 5.21, but have a pattern similar to the deflection). This force is much smaller than the 92N for a rigid tool shown in Fig. 5.12, because the drill can now deflect to create a force balance between the two flutes. The steady state tool deflection is about 8 yxm, because the average lateral tool stiffness is approximately 1.12 x 105 [N/m]. The simulation predicts a 14\xm hole oversize. The detail graphs of the chip thickness show that during two revolutions of the tool the chip thickness varies four times - twice per spindle revolution, due to alternating excitation of the X and Y directions, which have different stiffnesses. Galloway established an empirical rela-tionship for the hole oversize and deflected axis position due to lip height error rz. Half the tip angle is K, = 59° for the drill bit. According to Galloway [14] the tool deflection should be 0.5 r z tame, = 8.3u,/n and the hole oversize rztanK, = 16.6\xm. These numbers agree favorably with the simulation presented here. Interestingly, the tool deflection and hole oversize depend on the tool grinding error, and not on the tool stiffness. Regenerative, lateral chatter vibrations 7 Fig. 5.22 presents the simulation result for a drill with lateral stiffnesses of 2.0 x 10 N/m, natu-ral frequency 340Hz arid 5% damping (symmetric dynamics are considered) and a radial runout of 3 \im. The torsional and axial modes are assumed to be rigid. The drill geometry is provided in Table 5.3. A high stiffness and damping are used to consider the wall contact stiffness and friction during actual drilling as well as preventing the numerical instability in digitizing the hole surface. In an actual drilling process, the drill will bump into the hole wall, where interference with the just cut wall will create process damping, which is not modeled here or in the literature. As a result, the drill will show a stiffer and more damped vibration than a freely vibrating drill with the structural stiffness and damping. The drill is given a small runout, because in simulation the lat-eral forces on a perfect drill would be zero, and it would not deflect. Lateral chatter occurs at Chapter 5. Numerical Modeling of Drilling Dynamics 116 362Hz in the lateral deflection spectrum (Fig. 5.22a), leaving 9 waves on the surface (see Fig. 5.22c). There is a small peak at spindle frequency (fs = 40//z) due to the radial runout. The time domain model does not predict lateral whirling motions (circular motion of the tool center in the direction opposite to tool rotation), possibly due to the absence of hole wall contact forces, which are difficult to model. These can be included in the model at a later stage. Figure 5.22 : Simulation for drill with lateral flexibility and radial runout; a) bottom workpiece surface; b) lateral tool deflection and Fourier spectrum; Cutting conditions: 2400rpm,fr=0.30mm/rev, pilot hole diameter=4mm, hole depth=6mm; Tool geometry: Table 5.3; Cutting coefficients AL7050-T7451: Table 5.2; Lateral stiffnesses of the drill bit are 2.0xl07N/m, natural frequency 340Hz and 5% damping (symmetric dynamics are considered). The torsional and axial modes are rigid. Chapter 5. Numerical Modeling of Drilling Dynamics 117 5.6.3. Combined torsional-axial and lateral vibrations in drilling Figure 5.23 shows the simulated tool vibrations for a drill with 3u.w radial runout and which is flexible in torsional, axial and lateral directions. The drill has identical dynamics in X , Y direc-tions (340Hz) with a 5% damping ratio and 2.5 x 10 N/m stiffness. In torsional-axial directions the stiffnesses from Table 5.5 are used with 3% damping. The drill geometry is provided in Table 5.3. Torsional-axial chatter at 2983Hz and lateral chatter at 362Hz develop quickly after full engagement at 0.30s (Figs. 5.23c,d). In the axial frequency spectrum (Fig. 5.23c) there is a clear peak at 3kHz - which leaves about 75 waves on the surface (Fig. 5.23a,b). The lateral chatter at 360Hz (Fig. 21d) leaves about 9 waves on the surface, as is very clear from Fig. 5.23b, but not from Fig. 5.23a. The lateral spectrum also contains a peak at spindle frequency (fs = 40Hz), due to the radial runout This result clearly shows the capability of the model to handle combined lat-eral, torsional and axial vibrations. Cross-sectional surface profile at pilot hole I 1 1 0 1 I 1 1 0 I 0 0.1 0.2 0.3 Time [s] C 1 2 3 4 5 Frequency [kHz] —H 1 1 I 25" 0 0.1 0.2 0.3 Time [s] 1 2 3 4 Frequency [kHz] Figure 5.23 : Simulation for drill with flexibility in lateral and torsional-axial directions; a) bottom workpiece surface; b) cross-sectional surface profile at pilot hole; c) axial tool deflection and Fourier spectrum; d) lateral tool deflection and Fourier spectrum; Conditions: 2400rpm, fr=0.30mm/rev, pilot hole diameter=4mm, hole depth=6mm; Drill geometry: Table 5.3; Tool dynamics: Table 5.5. The dynamics in X and Y directions are equal (symmetric dynamics), with 5% damping ratio and 2.5x107N/m stiffness; Torsional-axial damping is 3%; Cutting coefficients AL7050-T7451: Table 5.2. Chapter 5. Numerical Modeling of Drilling Dynamics 118 5.7. Dynamic cutting tests This section details experimental results used to verify the time domain model predictions. First, cutting tests illustrating the torsional-axial chatter behavior in twist drills of different lengths and indexable drills are presented. Secondly, similar cutting tests are shown to illustrate whirling vibrations in drilling. 5.7.1. Torsional-axial chatter stability To verify the stability prediction with the time domain model, cutting tests have been conducted with the drill with the geometry given in Table 5.3 and tool dynamics shown in Table 5.5. It is a 16mm Guehring drill bit with a 176mm tool length (1 lxD). The cutting tests were conducted on a Mori Seiki SH403 horizontal machining center, using a Sandvik B100 collet chuck (HSK63A interface). Fig. 5.24 shows experiment details for two cutting speeds, 800rpm (40m/min) and 2400rpm (120m/min), and three depths of cut (DOC), full hole (DOC=8mm), 4mm pilot hole diameter (DOC=6mm) and 12mm pilot hole diameter (DOC=2mm). Drilling a full hole at 800rpm gives a stable result, shown by empty frequency spectra in thrust and X Y forces (Fig. 5.24a). The lateral force spectrum is shown to prove that torsional-axial vibrations are dominant in the cutting tests conducted. When the spindle speed is increased to 2400rpm (Fig. 5.24d), clear chatter develops after full engagement, at 1.1s. The dominant frequency is 3323Hz, which leaves 3323/40 = 83 waves on the surface. The X Y force spectrum shows peaks at 3323Hz and 3244Hz, tooth passing frequency lower. When a 4mm pilot hole is used, the cut is stable at 800rpm (Fig. 5.24b), but the thrust force signal is not as smooth. The thrust spectrum shows a peak at 178Hz, which is transient vibration that occurs just before full engagement. At 2400rpm (Fig. 5.24e), chatter occurs at 3238Hz, visible in both thrust and X Y force spectra. The 12mm pilot hole cuts (Fig. 5.24c,f) are unstable at both 800 and 2400rpm. At 800rpm, the chatter devel-ops after 2.3s at 3250Hz, and the thrust force fluctuates from 0 to 300N. Chapter 5. Numerical Modeling of Drilling Dynamics 119 At 2400rpm, the chatter is much more severe, and the recorded thrust varies between -300 and 700N. The chatter occurs at 3230 and 3312Hz. Z 2 0 0 0 = 1 0 0 0 ( 800 RPM ) ( FULL HOLE ) 5" 15 » 10 1.0 .x105 2.0 3.0 Time [s] 4.0 1000 500 0 ( 800 RPM ) ( 0 4mm PILOT HOLE } 2.0 3.0 4.0 Time [s] x10 1 x10 0 1 2 3 4 5 Frequency [kHz] 0 1 2 3 4 5 Frequency [kHz] x10 8 1 0 i 5 x" L 0JL. LL 0 (a) 1 2 3 4 Frequency [kHz] 2 °0 (b) 1 2 3 4 Frequency [kHz] ( C ) 1 2 3 4 Frequency [kHz] Z 2 0 0 0 0.8 1.0 1.2 1.4 1.6 1.8 Time [s] x106 LL Z <U o > X" 1b" 10 5 nil 1.0 1.2 1.4 1.6 1.8 2.0 Time [s] x10 0.8 1.0 1.2 1.4 1.6 1.8 x10= Time [s] 1 2 3 4 Frequency [kHz] 1 2 3 4 Frequency [kHz] 1 2 3 4 Frequency [kHz] (d) 1 2 3 4 Frequency [kHz] (e) 1 2 3 4 Frequency [kHz] (f) 1 2 3 4 Frequency [kHz] Figure 5.24 : Details of experimental cutting forces: Thrust, FFT Thrust, FFT XY forces; a) full hole 800rpm; b) 4mm pilot hole 800rpm; c) 12mm pilot hole 800rpm; d) full hole 2400rpm; e) 4mm pilot hole 2400rpm;j) 12mm pilot hole 2400rpm; Drill geometry: Table 5.3; Workpiece material: AL7050-T7451. Chapter 5. Numerical Modeling of Drilling Dynamics 120 Figure 5.25a shows the experimental and simulated workpiece surface for the cut at 2400rpm with a 4mm pilot hole. The simulated axial deflection history and Fourier spectrum are included in Fig. 5.25. In the simulation, the time step is taken 40 times smaller than the vibration period of the 3357Hz mode, resulting in a workpiece surface that consists of 170,000 points. Simulations of the other piloted hole cutting conditions presented in Fig. 5.24 all yield unstable results, although 800rpm/4mm pilot hole is stable in experiment. Figure 5.25 : Comparison of (a) experimental and (b) simulated surface, 2400rpm, 4mm pilot hole, feedrate 0.30mm/rev; c) simulated axial tool deflection andfourier spectrum. Tool geometry, cutting coefficients, tool dynamics: same as Fig. 5.20. Workpiece material: AL7050-T7451. Figures 5.26 and 5.27 provide the full experimental stability chart in photos: Six depths of cut -from 2mm (12mm pilot hole diameter) to 8mm depth of cut (full hole) by 13 spindle speeds (200-2600rpm). The grey regions in the chart are the transitions from stable to unstable cutting - unsta-ble cuts can be identified from the sunray pattern left on the surface. At 2mm depth of cut and 800rpm, the number of waves is about fc/fs = 3250//z/13.3//z = 244, as can be seen from the narrowly packed waves. The full chart clearly shows that the drilling process is more stable at higher depths of cut, not at smaller ones, and that chatter develops slowly as the spindle speed is Chapter 5. Numerical Modeling of Drilling Dynamics 121 increased. Full holes have also been cut at spindle speeds up to 6000rpm, but chatter remains and becomes very severe. The influence of the radial depth of cut on drilling stability is opposite from its effect on stability in milling, where increasing the depth of cut leads to instability. Both the effect of spindle speed and the effect of depth of cut can be attributed to process damping, which is explained in the following. Spindle speed [rpm] Figure 5.26 : Experimental torsional-axial chatter stability chart for Guehring llxD twist drill (200-1400rpm); Dark grey regions indicate transition from stable to unstable drilling process; fr=0.30mm/rev; Drill geometry: Table 5.3; Workpiece material: AL7050-T7451; Flood coolant was used during experiments. Depth of cut=drill radius-pilot hole radius; Pilot hole diameters 12-10-8-6-4-0mm correspond to depths of cut 2-3-4-5-6-8mm in the chart. Chapter 5. Numerical Modeling of Drilling Dynamics 122 Process damping - effect of depth of cut on torsional-axial chatter stability If the spindle speed is fixed at lOOOrpm, see Fig. 5.26, and the depth of cut is 2, 3 or 4mm, the drilling process is unstable. When the depth of cut is increased to 5, 6 or 8mm depth of cut, the drilling process is stable. At the periphery of the drill bit (Fig. 5.28a), the cutting speed is high, but at the center of the drill bit it becomes zero (Fig. 5.28b). Spindle speed [rpm] Figure 5.27 : Experimental torsional-axial chatter stability chart for Guehring UxD twist drill (1400-2600rpm); Dark grey regions indicate transition from stable to unstable drilling process; fr=0.30mm/rev; Drill geometry: Table 5.3; Workpiece material: AL7050-T7451; Flood coolant was used during experiments. Depth of cut=drill radius-pilot hole radius; Pilot hole diameters 12-10-8-6-4-0mm correspond to depths of cut 2-3-4-5-6-8mm in the chart. Chapter 5. Numerical Modeling of Drilling Dynamics 123 Figure 5.28 : Process damping in drilling. As a result the vibrations waves left on the surface close to the center of the drill are tightly packed and stretched out waves at the periphery. Close to the drill center, the relief face will inter-fere with the just-cut surface as shown in Fig. 5.28b, but at the periphery it may not. This interfer-ence dampens out the vibration of the whole tool, and is therefore called process damping. The effect of process damping is therefore strong close to the drill center, but weak at the periphery. When a large pilot hole is used, the drill will chatter, because the zone where process damping is strong, is absent. Increasing the depth of cut (using a smaller pilot hole) leads to stronger process damping and can stabilize the process although more material is being removed. Process damping - effect of cutting speed on torsional-axial chatter stability The cutting speed determines the length of the vibration waves in Fig. 5.28, both at the periphery and close to the drill center. Lowering the spindle speed results in shorter waves both at the drill periphery and close to the drill center, hence process damping increases along the whole cutting edge. This explains that reducing the spindle speed stabilizes the drilling process, as was observed by Ema et al. [31]. Process damping - effect of feed rate on torsional-axial chatter stability The workpiece moves toward the drill with feed velocity Vp see Fig. 5.28c, and as a result the surface has a small slope. This slope is determined by the feedrate (fr [mm/rev]) and the circular Chapter 5. Numerical Modeling of Drilling Dynamics 124 distance between two consecutive teeth (nRl), which depends on the radial location on the cut-ting edge. The slope can be calculated from o(r) = tan . Close to the drill center and V nr J depending on the clearance angle of the drill geometry, the relief face of the drill will contact the workpiece more easily if the feedrate is high - and increase process damping. Influence of drill length Several cutting tests were conducted with two twist drills with the same tip geometry, but differ-ent tool length. The experimental stability charts for these drills are presented in Fig. 5.29. The drill with length 8xD shows a very similar result as the l l xD drill, but is stable up to higher speeds due to its higher stiffness and natural frequency. The 15xD drill proved relatively unstable drilling full holes, and is only stable at very low spindle speeds. 9 8 7 E E 6 5 5 o o 4 £ 3 ° 2 (Full hole) * * Y 0 0 c • h 0 0 0 0 stable (experiments) 0 torsional-axial chatter (a) 0 400 800 1200 1600 2000 2400 2800 3200 3600 4000 Spindle speed [rpm] 8 7 ¥ E, ¥ 5] o "S 4 ° 2 1 0J (Full hole) • • • ! * -• • 4 o -• • 1 * -• t • I 0 (b) 200 400 600 800 1000 1200 Spindle speed [rpm] Figure 5.29 : Experimental torsional-axial chatter stability charts for Guehring 16mm twist drills; a) Tool length 8xD; b) Tool length 15xD; fr=0.30mm/rev. Drill geometry: Table 5.3; Workpiece material: AL7050-T7451; Flood coolant was used during experiments. Depth of cut=drill radius-pilot hole radius; Pilot hole diameters 12-10-8-6-4-0mm correspond to depths of cut 2-3-4-5-6-8mm in the chart. Chapter 5. Numerical Modeling of Drilling Dynamics 125 Chatter stability of indexable drills Cutting tests were also conducted with the indexable drills presented in Fig. 3.12 and Table 3.6. As these indexable drills are not symmetric, pilot holes cannot be used, and we could only test cutting full holes. The longest drill (4xD) is stable up to 1200rpm, the 3xD is stable up to 1500rpm and the short (2xD) is stable up to 4500rpm. The feedrate used was fr = OAOmm/rev. Figure 5.30 shows the experimental sound, frequency spectra and surface finish for two spindle speeds for the longest drill, which has a torsional-axial natural frequency of 4600Hz. At 1500rpm torsional-axial chatter occurs at 4514 and 4565Hz (below the natural frequency), with smaller peaks at 4489, 4540, and 4590Hz, spindle frequency apart. The observed chatter behavior is simi-lar to torsional-axial chatter in drilling. If the spindle speed is increased to 4500rpm, the chatter occurs at 4492, 4568 and 4644Hz, also spindle frequency apart. Indexable drills show chatter peaks spindle frequency apart, because each flute cuts a different part of the hole. A twist drill recuts the surface with consecutive flute passes, hence chatter peaks occur tooth passing fre-quency apart. Time [s] Frequency [Hz] Figure 5.30 : Sound measurement, frequency spectra and experimental surface finish for two cutting experiments with a 21mm indexable drill (Sandvik U-drill, length 4xD, fr=0.1 Omm/rev. Workpiece material: AL7050-T7451; Flood coolant was used during experiments. Chapter 5. Numerical Modeling of Drilling Dynamics 126 5.7.2. Whirling experiments Cutting tests were conducted on a Mori Seiki SH403 horizontal machining center, using a Sand-vik B100 collet chuck (HSK63A interface) and a Guehring 16mm HSS drill with the geometry in Table 5.3 and tool dynamics in Table 5.5. Figures 5.31 and 5.32 show time domain measurement data of thrust Fz and lateral forces Fx, Fy and two Fourier spectra of each force. The first Fourier spectrum shows frequency components up to 5000Hz, the second spectrum shows the frequency content up to ten times the spindle frequency fs. Drilling tests were conducted at four depths of cut: full hole, 4mm pilot hole, 8mm pilot hole and 12mm pilot hole. For each depth of cut, a low and high spindle speed were used. The feedrate was 0.30mm/rev in all cases. All lateral force spectra contain energy at spindle frequency fs, which can be attributed to runout, lip height errors, or lip grinding errors (see Fig. 5.12). The cutting tests at 400rpm (Figs. 5.31a,c and 5.32a,c) are chatter free due to process damping. Since the tooth passing frequency is 250 times lower than the torsional-axial frequency (3357Hz), 125 waves are left between consecutive teeth, and process damping ensures stability. The full hole, 8 and 12mm pilot hole results (Fig. 5.3 la,b and Fig. 5.32) show a strong peak at 2fs and a smaller one at 4fs in the lateral force spectra, at both low and high spindle speeds, indicat-ing the drill whirls at 3fs in the rotating frame (see Fig. 5.15a), leaving a three sided hole as can be seen in the photographs (Fig. 5.3 la,b). The peaks in the thrust force spectrum close to 6fs indi-cate that the whirling frequency maybe slightly lower, as in Fig. 5.15b. For a full hole at 400rpm, the lateral forces slowly develop after full engagement and are sustained. At 2400rpm, the lateral forces quickly become much larger as the torsional-axial chatter (at 3300Hz) develops from 1.0s, which can be seen from the thrust time domain data (similar to Fig. 5.31a,b). When a 4mm pilot hole is used at 400rpm (Fig. 5.31c), strong peaks occur at 6fs and Sfs but also at 2fs and 4fs in the lateral force spectrum, indicating that there are whirling contributions in the rotating frame at 3fs,lfs and possibly 5 / . Chapter 5. Numerical Modeling of Drilling Dynamics 127 At 2400rpm (Fig. 5.31d), the peaks at 2fs and 4fs are relatively stronger. Due to the pilot hole, no polygon can be distinguished in the chisel edge region. The lateral force spectra also shows small peaks around 1430Hz, however, no clear lateral chatter could be observed - the surface does not show marks that indicate this frequency, hence the lateral vibrations must have a small amplitude. At the higher spindle speeds, both thrust and lateral force spectra show strong peaks near the tor-sional-axial frequency, around 3300Hz. These peaks signify torsional-axial chatter, leaving a sun-ray pattern at the bottom of the hole, as can be seen in the photographs of the experiments, and was accurately predicted before [2]. Chapter 5. Numerical Modeling of Drilling Dynamics 1 2 8 (full hole, 400rpm) 200Q (full hole, 2400rpm) (04mm pilot hole, 400rpm) (04mm pilot hole, 2400rpm) 2 4 6 8 0.8 1.0 1.2 1.4 1.6 1.8 Time [s] Time [s] x10 6 x i o ! x10 5 Time [s] 5. 5 N O U L . x10a Time [s] 10 f 0 Frequency [kHz] 5 0 Frequency [kHz] 5 0 Frequency [kHz] 5 0 Frequency [kHz] 5 20 40 60 0 Frequency [Hz] 200 400 "0 20 40 60 "0 Frequency [Hz] Frequency [Hz] 200 400 Frequency [Hz] Z 200 u_ 0-LL-200 2 4 6 8 0.8 1.0 1.2 1.4 1.6 1.8 Time [s] Time [s] liUJ,^ .kii.lLllLltHHiil 5" 2 x10° LL X 1 x10' 2 f c 10 1 I 5 0 L . J IL , n J x10° 2 4 6 8 1.0 Time [s] x10 s 1.5 2.0 Time [s] 0 Frequency [kHz] 5 0 Frequency [kHz] 5 0 Frequency [kHz] 5 0 Frequency [kHz] 5 „ x10c Z . 2|— f -x10° 2f„ 4f„ L L 6f„ 8f.. .x10 0 20 40 60 0 Frequency [Hz] oUJ 2f. 4f. 6f. - i - - - I • -8fr x10" 200 400 0 20 40 60 0 Frequency [Hz] Frequency [Hz] 200 400 Frequency [Hz] (a) Figure 5.31 : Details of 4 cutting experiments: Full hole at 400rpm (a) and 2400rpm (b), 4mm piloted at 400rpm (c) and 2400rpm (d); Thrust, lateral force FFT's from OSkHz andfrom 0-10fs; fr=0.30mm/rev. Drill geometry: Table 5.3; Workpiece material: AL7050-T7451; Flood coolant. Chapter 5. Numerical Modeling of Drilling Dynamics 129 (08mm pilot hole, 400rpm) (08mm pilot hole, IQOOrpm) (012mm pilot hole, 400rpm) (012mm pilot hole, 800rpm) ^ ^ ^ ^ ^ ^ Time [s] Time [s] Time [s] x10 3 u. 0 2 z £ 1 = 0 0 x10 6 x10° x10° Olli Time [s x10 5 f c Frequency [kHz] 5 0 Frequency [kHz] 5 0 Frequency [kHz] 5 0 Frequency [kHz] 5 2f_ 4f 6f. 8f. • xlO" 'II 2 f s i 4 fs MJJ X103 8f. 2f, 4f c :6fc |8f F s S | 5 | s VI s U si™ x10° 2 f s f s f . £0 40. . . . 60 0 40 ... . 60 0 requency [Hz] Frequency [Hz] -requency [Hz] Frequency [Hz] 2 4 6 Time [s] x10 5 biuftn^yijiui 2 3 4 Time [s] • x10° 0 Frequency [kHz] ° 0 Frequency [kHz] 5 0 Frequency [kHz] 5 0 Frequency [kHz] 5 „ x10 3 i f £ x u. £ 0 LA f, |2f, s! s 4f. 6f. • - - i i 8f. x10° f s i 2 fs 4 f „ i i A 6f 8f. x10 3 . A . 21 4f. i . i ; 6 f „ 8f. ,x10 D ,f, |2f. 1 s ! s oL 4f. 6f. 1 . 1 • ] 8f. 0 20 40 60 0 50 100 150 0 20 40 60 0 50 100 Frequency [Hz] Frequency [Hz] Frequency [Hz] Frequency [Hz] Figure 5.32 : Details of 4 cutting experiments: 8mm piloted at 400rpm (a) and lOOOrpm (b), 12mm piloted at 400rpm (c) and 800rpm (d); Thrust, lateral force FFT's from OSkHz andfrom 0-1 Of; fr=0.30mm/rev. Drill geometry: Table 5.3; Workpiece material: AL7050-T7451; Flood coolant. Chapter 5. Numerical Modeling of Drilling Dynamics 130 5 . 8 . Conclusions An exact kinematics time domain model has been developed for simulation of combined axial, lateral and torsional vibrations when drilling piloted holes. Both the piloted hole bottom surface and the hole wall are generated by stepping through time while the drill deflects due to the vibra-tion dependent cutting forces. The model uses a mechanistic cutting force model for accurate pre-diction of cutting torque, thrust and lateral forces. First, static prediction of torque and thrust, and the effect of tool grinding and misalignment errors are presented. Secondly, whirling motions are imposed on the drill bit and the resulting variation in chip thickness, width of cut and cutting forces are analyzed. The formation of odd-sided shapes on the hole are studied in detail by vary-ing whirling frequencies and whirling motion amplitudes. Thirdly, dynamic simulation results for separate axial, torsional-axial and lateral vibrations are presented. A dynamic simulation that includes both torsional-axial and lateral vibrations is also included. Finally, the simulation results are compared against extensive experimental results. The torsional-axial chatter frequency and development of vibrations can be predicted, as is shown by comparison of the experimental and simulated surface finish and tool vibration spectra. The torsional-axial chatter stability - what vibration free depth of cut can be used for a given spindle speed - cannot be predicted, as process damping is not included in the time domain simulation. The torsional-axial chatter stability is experimentally investigated for different drill lengths of both twist and indexable drills. Although the model does not predict whirling vibrations, the frequency content of simulated imposed whirl-ing vibrations shows good agreement with experimentally observed whirling motion. All experi-ments also indicate drill grinding/alignment errors and light lateral chatter vibrations, which are also predicted by the model. 131 Chapter 6. Frequency Domain Chatter Stability of Drilling This chapter presents a linear frequency domain stability analysis for drilling that takes flexibili-ties in lateral, torsional and axial directions into account. Section 6.1 introduces the dynamic chip thickness, derived from the drill geometry and drill deflections. Section 6.2 proposes a global sta-bility solution for dynamic drilling. Section 6.3 describes alternative, partial stability laws. Sec-tion 6.4 compares the proposed and past stability laws using time domain simulations and experiments as a base. The drill is a slender, pretwisted beam clamped by a tool holder in the spindle and in contact with the metal that is being cut. The lateral, axial and torsional vibrations of the drill cause an irregular distribution of the chip thickness, which depends on the drill edge geometry, as well as the present position and the past history of the cutting edge location, which are governed by the rigid body motion and vibrations of the drill bit. Chapter 4 covered the modeling of cutting forces for typical drill geometries and chapter 5 describes the exact kinematics model that handles all nonlinearities under vibratory conditions. Time domain simulation carries high computational costs. For quick analysis, an efficient frequency domain solution is advantageous to identify chatter free speeds and depths of cut in drilling. 6.1. Dynamic chip thickness in three dimensional drilling In a frequency domain solution, the objective is to study conditions where unstable vibrations start to develop. The variation of cutting forces needs to be expressed as a function of tool deflec-tions. A twist drill is a slender, pretwisted beam with a symmetric cutting geometry. Tool grinding errors and transient vibrations cause force fluctuations, resulting in force unbalances in lateral directions and variation of torque and thrust, that can lead to unstable chatter vibrations. In the following linear frequency domain analysis, a perfect drill geometry is assumed. Chapter 6. Frequency Domain Chatter Stability of Drilling 132 The (positive) force components for each flute are tangential (i), radial (r) and axial (a), as illustrated in Fig. 6.1, are defined as: ktcbhx krcFt, k„„F, ac t. Ft2 = ktcbh2 Fr2 = KcFt2 Fa2 = KcFt2 (6.1) where hx is the uncut chip thickness for flute 1, b is the radial depth of cut, defined by the differ-ence of tool radius and pilot hole radius: b = R-Rp. The radial and axial forces are expressed to be proportional to the tangential force. In the stationary frame, the total cutting forces acting in X, Y and Z directions at the tool tip are (see Fig. 6.1): Fx(t) = (Fti-Ft2)sm(nt)-(FrrFr2)cos(nt) Fy(t) = (FtrFt2)cos(nt) + (FrrFr2)sm(Qt) Fz(0 = F a i + Fa2 Tc = R t ( F t + F t - ( F r + F J ) (6.2) where Rt is the torque arm for calculating the cutting torque Tc from tangential and radial forces. Figure 6.1 : Elemental forces acting on the cutting edges of a two-fluted drill bit. For stability analysis, the cutting force variation due to the tool vibrations needs to be found. The dynamic chip thickness is influenced by vibrations in three orthogonal directions and one tor-Chapter 6. Frequency Domain Chatter Stability of Drilling 133 sional direction. The static chip thickness equals the feed per revolution divided by the number of flutes, which is two in this case (A -^ = 2): The change in chip thickness due to regenerative displacements dx, dy on each flute are: (6.3) dhx dx x cosQt-dy x sinQf tame dh. _ -(dx x cosQt-dy x sinQf) (6.4) tame, where 2K, is the tip angle of the drill, see Fig. 6.2. Qt is the tool rotation angle indicated in Fig. 6.1. The chip thickness change is illustrated in Fig. 6.2, where du = dxx. cosQ.t-dy x sinQ/ is the tool deflection in the direction of the cutting lips. Figure 6.2 : Chip thickness change due to lateral vibrations while drilling a piloted hole. The regenerative displacements are: {Ar} dx xc(t)-xc(t-T) ' . dy y(t)-yc(t-T) dz zc(t)-zc(t-T) dQ . Qc(t)-Qc(t-T) , (6.5) Chapter 6. Frequency Domain Chatter Stability of Drilling 134 2% where T = —— is the tooth period. The dynamic chip thickness due to torsional vibrations is i V L expressed as follows: 1 dh, = dh-y = -dQ x — x f 1 * 2n r (6.6) and depends on the feed per revolution fr. Finally, axial vibrations influence the chip thickness directly: dh\ = dh2 - —dz (6.7) The total change in chip thickness becomes: dhx dhn 1 fr (dxcosClt - dysinClt) -dz- —dQ fame, ' ' 2K -1 fr (dxcosClt - dysinQt)- dz - —dQ tanK, 2TC (6.8) The static chip thickness (hs = fr/2) is neglected since it does not contribute to the stability. The dynamic forces depend on the dynamic chip thicknesses dhx and dh2 '• • ktcb(dhx-dh2) sinQt-k^k^bidh^dh^ cos Qr klcb(dhx-dh2) cosfif + krcktcb(dhx-dh2) sinfif ktckacb(dhx+dh2) d {ktcb(dhx+dh2)}(l-krc)Rt (6.9) where the sum of dynamic chip thicknesses (dhx + dh2) and difference (dhx-dh2) are: dhx + dh2 dhx-dh2 tame fr -2dz--dQ 71" (dx cos Q t - dy sin Q t) (6.10) When the dynamic chip thickness is substituted in Eq. 6.9, the dynamic forces acting on the tool are found to be: Chapter 6. Frequency Domain Chatter Stability of Drilling 135 dx Fy > = bktc[B{t)\ dy . dz dQ (6.11) where the dynamic drilling coefficient matrix [B(t)] depends on time t, spindle speed Q , cutting coefficients krc and kac, tool tip angle K, and feed per revolution fr: [B(t)] = [Bn] [Bl2\ [B2X] [B22] ,[B2l] 0 0 0 0 [Bn] 0 0 0 0 1 tame {2smD.tcosQ.t-2krccos2Q.t} {-2sin 2Qr + 2krcsinQ.tcosQ.t} 2 2 {2cos Qt + 2 k sinOrcosQ/} {-2sinQ/cosQ/-2A: r csin fit} tame. { s m 2 Q . t - k r c - k r c c o s 2 Q t } { cos2Qr-l + £, . c s in2Qr} {cos2Qr+ 1 + £ r c s i n 2 Q f } {-sm2Q.t-krc + krccos2Q.t} bxx bxy byx byy_ [B22] {-2*ac> {-2(1 -*«)*,} ~kaJr rc -(\-krc)fX (6.12) using the double angle formulae to rewrite [Bxx]: 2smQtcosClt = sin2Qr 2 2sin fit = 1 - cos2Q/ 2 2cos Clt - 1 + cos2Q/ (6.13) Only submatrix [Bx x ] depends on time, and its coefficients are shown as a function of cutter rota-tion angle in Fig. 6.3. Because the forces in X and Y directions depend on the chip thickness dif-Chapter 6. Frequency Domain Chatter Stability of Drilling 136 ference (dhl-dh2), axial and torsional deflections do not affect the lateral cutting forces (Fx, Fy)d. Similarly, as the torque and thrust depend on the chip thickness sum {dhx + dh2), lat-eral tool deflections do not affect the dynamic thrust force Fz d or torque Tc d . 90 180 270 Cutter rotation angle [degrees] 360 Figure 6 . 3 : Time varying directional coefficients for drilling. The dynamic cutting forces can be summarized as: {F{t)} = bktc[B(t)]{Ar} (6.14) where {Ar} contains the regenerative displacements from Eq. 6.5. The time dependent matrix [B(t)] is periodic at tooth passing frequency NjD. or tooth period T = , and can be expanded into Fourier series: + oo [B(t)] = £ [ B r ] e l m ' [Br] = \\[B{t)]e-iratdt (6.15) The most simplistic approximation is taking the mean value of the periodic directional coeffi-cients and ignore the higher order Fourier terms as was done for milling by Altintas et al. [73]: [B0] = \\[B(t)]dt (6.16) Chapter 6. Frequency Domain Chatter Stability of Drilling 137 [BQ] is valid only between entry (fyst) and exit ( § e x ) angles of the drill, which are 0 and TC , as 2n the drill cuts throughout the drill pitch angle fy = — Ye* [B0] = ±- l[B(fy)]dfy Nf P** K 0 0 0 0 0 0 Pzz Pze 0 0 Pez Pee The integration only needs to be done for sub-matrix [Bx j ]: o =(2^t^I4 c o s 2^^4C s i n 2 ( t ,]|o =® ^ P*y = (Srarb;I(~ 1 + ^s2Qt + krcsin2Qt)dfy o . _ ! _ r_ t + 1 * 2 ^ 0 - 2 • ! * = (Z\ -=-\2%) t a n K , L 2 Y 2 J V 2 T T V t a m e , o . ( ¥ ) ^ - h + I s f a 2+-^cos2* l " - fa-"-V2Tcy t a n K , L 2 Y 2 J V2rc/ t a m e , 71 P » = tan7, I ( - S i n 2 Q / " krc + Kc^s2Clt)dfy (6.17) (6.18) Chapter 6. Frequency Domain Chatter Stability of Drilling 138 The coefficients of sub-matrix [B22\ are rewritten as: Nf Pe. = ^ { - 2 K ( l - * « ) * , } Nf Pee = ^ { - ( l - ^ c ) / A } (6.19) Nf Factoring out term , we obtain matrix [Bred]: [Bred\ -nkrc -71 tame, tame, 71 -nkrc tame, tanK, 0 0 0 0 0 0 -2nk„ 0 0 (6.20) The dynamic forces are expressed by: { F ( t ) } = ^bk^Bred]{^) (6.21) Chapter 6. Frequency Domain Chatter Stability of Drilling 139 6.2. Frequency domain solution procedure The frequency response function matrix 0(/co) at the tool tip is identified as: 0(i(o) 0 0 0 0 0 0 0 0 <B„(ico) O 2 e(ieo) 0 O 9 z(i(o) %e(iG>) (6.22) assuming there is no coupling between the lateral directions individually, and no cross talk from lateral directions into axial and torsional directions either. Oaa(s) is the direct frequency response function and ^>ab(s) is the cross frequency response function, which is defined as follows: = AaJs} = v <» nh/kh (6.23) h = 1 S + 2Ch(dnhS + (O nh where ab (a or b is x,y, z or 0) denotes displacement of cutter tip coordinate a when a cutting force Fb is applied in direction b. km is the total number of modes in the system, h represents each of these modes and (anh, kh, and Ch are the natural frequency, modal stiffness and damping ratio, respectively. The drilling system is critically stable when the harmonic, regenerative dis-placements {Ar} occur at the chatter frequency o o c with a constant amplitude: {Ar(ia> cO} = {r(i«DcO}-{r(ia)c(/-r))} = (1 - e~iaJ)[^(mc)] {F}e'mtf (6.24) where tacT is the regenerative phase delay between the vibrations at successive tooth periods T. Substituting {Ar(i(oct)} into the dynamic drilling equation 6.21 gives: {F}eaJ = ^bkJBMl-e~iaJ)[O(i(0c)]{F}e /CO-/ 2n (6.25) which has a nontrivial solution if the determinant is zero: Chapter 6. Frequency Domain Chatter Stability of Drilling 140 N, det[[I]-^bktc(l-e~mc )[Bred][0(mc)]) = 0 (6.26) This is the characteristic equation of the closed loop dynamic drilling system. By defining the ori-ented transfer function matrix as: O 0 ( i a > c ) = [5 r e r f][0(ico c)] = 0 0 0 0 0 « * » + P*e*ez P»*xe + P*e*ee P e z ^ z z + P e e ° e z Pez^ze + P e e ° e e rXX XX B <& r'xy yy B O ryx XX Yyy yy 0 0 0 0 (6.27) and the eigenvalue A of the characteristic equation as: Nr A = AR + iAj = ~2^bktc( \ -e c) the characteristic equation becomes: det([I] + A[®0(mc)]) = 0 (6.28) (6.29) The eigenvalues of Eq. 6.29 can be found by scanning through possible chatter frequencies (O c around the natural modes. The directional matrix is evaluated once from Eq. 6.20. The oriented transfer function is found from Eq. 6.27 for each frequency. Substituting e l(°CT = COSG)CT- zsina>cr into Eq. 6.28 gives the critical depth of cut at chatter frequency coc: 2n(AR + iAj) n(A.R + /A 7 ) ( l - cosa^TWsina^T') blim ~ Njktc( \ - coscocr+ /sincocr) jy ,^c(i-cosa>cr) K{Ar(\ - cos(£>cT) + A 7 s inoo c r} ^ TT{A7(1 - cos(ocT) + A^s inco c r} (6.30) Njktc( \ - c o s c o c T ) i V / / c ( l - c o s c o c r ) Chapter 6. Frequency Domain Chatter Stability of Drilling 141 The critical depth of cut b l i m is a physical quantity and must be a real number, therefore the imag-A 7 sin(£>cT inary part should vanish. By substituting K = — = in Eq. 6.30, the critical depth of A ^ 1 — coscoci cut becomes: T C A 'Um Nfi ^(1+K 2 ) (6.31) tc In order to find the spindle speed corresponding to the chatter frequency oi c , write K as: cos(ra„ T/2) K ' t M V " s i n ( m > / 2 ) " t a n [ ' t / 2 " ° - T / 2 l ( 6 3 2 ) The phase shift of the eigenvalue is v|/ = tan ! K and 8 = n - 2v|/ is the phase shift between the present and previous vibration marks. If k is the full number of waves left between two consecu-tive teeth, then: <ocT = s + 2kn (6.33) The spindle speed n can then be calculated from the tooth passing period: T=—(e + 2kn) -> n - Trk coc NfT (6.34) Because matrix (Eq. 6.27) has decoupled lateral and torsional-axial terms, the characteristic equa-tion (Eq. 6.29) can be simplified to: ( m + ABV R<P AB r v 0> AB <D AB O rvx xx r » v v f 1 - A 0 (6.35) Chapter 6. Frequency Domain Chatter Stability of Drilling 142 The roots are then found by solving: (lateral) x (torsional — axial) = 0 ( Y o A 2 + Y l A + l ) ( Y 2 A 2 + Y3A+l) = 0 Yo = (PxxPyy®xx%y - PxyPyx^xx^yy) (6.36) Yl=(Pxx®xx+Pyy%y) Yi = 0 „ P e z * « * r e + PzzPee^zz^ee + PxePex^e^ + P z e P e e ^ e e - PezPzz^zz^e - P e A e ^ e e " P e e P ^ e A e ~ P e e P z e ^ e e ) Y 3 = (K®zZ + Pze^ez + Pez^ze + Pee^ee) resulting in the following four roots: Z l l l M ^ , A - J l z E l ^ ( l a t e r a l ) 1 2 Y o 2 2 Y c A Y3 + A / Y 3 - 4 Y 2 A - Y 3 - A / Y 3 - 4 Y 2 „ . , . A A 3 = , A 4 = (torsional — axial) 2y2 (6.37) 6.3. Partial chatter stability laws Bayly presented a lateral stability solution using rotating coordinates in [45], which was valid for very small radial depth of cut, and depended on feedrate. We here present a lateral chatter stability which uses the same rotating coordinate system, but is valid for any depth of cut and does not depend on feedrate. Section 6.3.2 presents the lateral chatter stability of a stationary tool which was published by Bayly [45] and is based on Tlusty's stability law for orthogonal turning [39]. Section 6.3.3 describes Bayly's method for solving torsion-axial chatter stability [1]. Chapter 6. Frequency Domain Chatter Stability of Drilling 143 6.3.1. Lateral chatter stability using a rotating coordinate system In a fixed coordinate system (X, Y), the equations of motion of the drill are expressed as: mxx 0 U U Kx o 0 myy 1 I _o kyy_ X y (6.38) where viscous, proportional damping is assumed. We can also attach a rotating frame of reference ([/, V) to the tool, where the [/-direction is aligned with the cutting lips, as illustrated in Fig. 6.4. The frame rotates with a constant speed Q. [rad/s]. Figure 6.4 : Definition of rotating and stationary coordinate systems. The conversion between these coordinate systems is done by the following transformation: x y cosQr sinQz* -sinQf cosQf u V cosQf smClt -sinQr cosQr (6.39) where the instantaneous rotation angle is 0 = Qt. The time derivatives of (x, y) are: x y cosQf sinQ/ -sinQr cosQf cosQf sinQ/ -sinQf cosQf u V + Q -sinQr cosQr -cosQr - s inQ/ u v + 2Q -sinQf cosQf -cosCit -sinOr u V - Q cosCit sinQ/ -sinfif cosQf (6.40) u V Chapter 6. Frequency Domain Chatter Stability of Drilling 144 Substituting Eqns. 6.39-6.40 into Eq. 6.38 yields the equations of motion in the rotating frame of reference: mxx 0 0 m yy. + cxx lmxx®--2myyQ. cyy j u V + ^xx mxx^ c Q xx ~cyya kyy-myy£l\ u V (6.41) The off-diagonal terms in the damping matrix are the coriolis forces, or gyroscopic forces. Cen-trifugal forces arise in the diagonal terms of the stiffness matrix due to rotation and structural damping forces are added to the off-diagonal terms in the stiffness matrix. The signs of the off-diagonal terms depend on the direction of rotation in Fig. 6.4. The advantage of using rotating coordinates is that time-invariant matrix equations for the cutting forces are obtained. In chatter stability we need to look at the time-varying portion of the cutting force. The variation of the cut-ting force on a tooth depends on the current vibration u(t) and the vibration u(t- x) one tooth period earlier, and are expressed in matrix form as: F„ -2k.„k 6/tame, 0 tc r c t -2ktcb/tanKt 0 u(t) + u(t-x) \ = b \ K \ \ u(t) + u(t-V ( 0 + V ( / - T ) J L d[ V ( / ) + v ( f -•x ) x) (6.42) Due to the rotation of the frame, the current vibration and the vibration one tooth period earlier are summed, as the displacements of opposing teeth have opposite signs. It is assumed that vibrations in the F"-direction do not affect the chip thickness, hence the zero column in the cutting stiffness matrix [Kc]. Torsional and axial vibrations are also not taken into account. To analyze the chatter behavior, periodic solutions can be assumed: u v U 1 I iat T-t iat le = Ue (6.43) Chapter 6. Frequency Domain Chatter Stability of Drilling 145 which can be found at the boundary between stable and unstable regimes. Substituting this into the equations of motion formulated in the rotating coordinate system, the condition for existence of such a solution becomes: [/+6G p (co;Q)[Zy(l +e-iaT)]U = 0 (6.44) 2 -1 where Gp(co;Q) = [-co M+ iooC(Q) + K(Cl)] is the frequency response matrix of the sys-tem in the rotating frame: GP = k x x - m x X n 2 - m x X ( ° 2 + ( 0 c x x i cyyCl-2myyQ(Di - cxxCl + 2mxxCl(oi ^ ~ myyQ2 - myyV>2 + (OCyyi Gvu Gvv (6.45) The term (1 + e~'mT) in the characteristic equation arises instead of (1 - e mT) found in classical turning stability, because in the rotating frame, the displacements of opposing teeth have opposite signs. Although the equation for chatter stability solution appears to be a matrix equation, it is in fact a scalar equation due to the zero column in the Kc matrix: -/(Br, l + 2 i y G > ; f l ) t r e + G > ; n ) } ( l + c , u") = 0 (6.46) For a given speed Q and corresponding time delay T = n/Q, the chatter frequency coc and lim-iting radial depth of cut blim are sought. As the depth of cut is a real positive number, values of oo need to be found for which the imaginary part of the characteristic equation vanishes: - i a > , 7 \ Im({Guu(coc;Q)k + G t t > c ;Q)} {1 + e < }) = 0 (6.47) The critical radial depth of cut is then found from: -tame 'Urn WtcRe{(Guukrc+Guv)} (6.48) which resembles Tlusty's one dimensional chatter stability law [39]. Chapter 6. Frequency Domain Chatter Stability of Drilling 146 6.3.2. Lateral chatter stability of a stationary tool When the tool is not rotating, the chip thickness is only affected by vibration along the cutting edge (X-direction). In order to determine the chatter stability, Bayly [45] used Tlusty's theory for orthogonal chatter stability [39] and applied it as follows. The dynamic part of the cutting force in ^-direction equals: F * = " ? t a n V X W ) - * c ( ' - r ) > (6-49) and depends on the current tool vibration xc(t) and one tooth period earlier xc(t— T). A deflec-tion of the drill in positive X-direction creates a force in the opposite direction. The static force on the tool is zero as the forces on the cutting lips cancel each other out. The dynamic force excites the drilling structure in X-direction, and the equation of motion can be expressed as: -2k.rkrrb V + V + *«* = tonK/ * {xc(t)-xJt-T)} (6.50) To find the boundary between stable and unstable solutions, we assume a periodic solution x(t) = X(co c )e' C ° c ' and substituting this into Eq. 6.49 yields: -2k^k„bQ>((Qj) - JO, T 1 = ( 1 - e ) (6.51) t a n K , v ' v 2 -1 where O(ooc) = (-coc mxx + i(occxx + kxx) is the transfer function for lateral vibration in the X-direction. The critical depth of cut is found as: -tanK. 2 * / c * „ M > ( a > c ) ( l - e c ) The radial depth should be real and positive. Using O(coc) = G + Hi we find: O((o c ) ( l -e~ , a > e 7 ) = [G(l-cosra c70-//sinm c71 + /[Gsincocr+/7(l-cosa)cr)] (6.53) Chapter 6. Frequency Domain Chatter Stability of Drilling 147 of which we need to equate the imaginary part to zero to obtain a real solution for b l i m . The phase shift \\f of the tool structure's transfer function is: sinco.r cos(co„772) tanvj/ = ± — = . , ' = tan{a) cr/2-(37i)/2} (6.54) cosoo cr-1 -sin(ooc7V2) 0 The phase shift s, tooth period T and spindle speed n are then obtained as: E = 3TC + 2\|/ r = n = TT^ (6.55) where A: denotes the number of full waves between two subsequent teeth. Ay- is the number of teeth on the drill. The critical depth of cut is found to be: -tame, *"» = 4JfcJfc^e{0(a>c)} ( 6 - 5 6 ) which is used to obtain the stability lobes. Bayly's adaptation of rotating coordinates (Eq. 6.48) depends on both spindle speed and the tool dynamics in two directions, whereas application of Tlusty's one dimensional stability law (Eq. 6.56) depends on spindle speed independent tool dynamics in only one direction. 6.3.3. Torsional-axial chatter Bayly et. al. established a frequency domain solution [3] for torsional-axial chatter. They sug-gested the equation governing the axial vibration as: ™PT\P + Cp^P + k P % = & = FzN + QNpTc„ (6-57) where fyp = ^wNp QNp^ = [l 0 ^ is the torsional-axial vibration mode shape, normalized with respect to the axial vibration wNp at the tip. The deformation is a scaled version of the mode shape: x = <t>pTi • mp> c p and k are the mass, damping and stiffness as measured at the tip of Chapter 6. Frequency Domain Chatter Stability of Drilling 148 the drill. The torque and thrust at the tool tip are defined by TCn = -kcbhRav and F Z n = -kcbh respectively, where Rav is the average radius of the cutting force, b the radial depth of cut and h the chip thickness. kc^ and kc^ are experimentally determined specific cutting pressures. The dynamic chip thickness equals: h = hs + wN(t) + wN(t-T) = hav + r)p(t) + r)p(t-T) (6.58) where hs is the static (intended) chip thickness, and T is the tooth period. The axial deflection at the tool tip is the same as the vibration r\ . Using the above expressions and defining K a = -~ 4- (J)^ - R a v , the time-varying component of the modal equation becomes: * c , p mpr\p + cpi\p + kpT\p = -akcb[r\p(t)-r\p(t-T)] (6.59) The constant a includes contributions from both cutting torque and thrust, as well as the tor-sional-axial coupling parameter § N . The physical meaning of the coupling parameter is that torque causes axial deflection and thus affects the chip load. To find the boundary between stable and unstable solutions, a periodic solution r\(t) = Y(<o)eJ(0t is assumed and substituted into the modal equation, yielding: 7(co)[l + akcGk((o)b(l -Z"7)] = 0 (6.60) 2 -1 where Gk((o) = (-co mk+ ja>ck + kk) is the transfer function for the k t n mode. Solving for the critical depth of cut blim, we find: him = — (6.61) a £ c G , ( c o ) ( l - e y ( a 7 ) The radial depth must be a real, positive number, and as a result GA.(co)(l - e j a T ) must be posi-tive and real, which is only true when Re[Gk((o)(l -eJ(oT)] = -Re[Gk((o)], and Im[Gk(d))e~J(oT] = Im[Gk(ca)]. In this case the expression for blim in Bayly's model leads to Tlusty's one dimensional chatter stability law [39]: Chapter 6. Frequency Domain Chatter Stability of Drilling 149 b -1 (6.62) lint 2akcRe[Gk((*)] This equation is identical to the expression for limiting depth of cut in orthogonal cutting, except for the difference in sign, due to the experimentally determined constant a , which is negative. The consequence is that unlike classical chatter which occurs at frequencies where the real part is negative, chatter in this torsional-axial mode will only occur where the real part of the transfer function is positive. This results in chatter frequencies below the natural frequency of the corre-sponding mode, instead of above as in classical turning chatter literature. In this solution, the torque and thrust are combined into one axial loading, effectively untwisting and extending the drill. The cutting constant used in solving the stability of the resulting differential equation is & C j . 6.4. Chatter stability lobes for drilling The chatter stability of drilling is evaluated by comparing the proposed method, which considers flexibilities in all directions, with time domain simulations, experiments, and previously pre-sented partial stability laws (lateral chatter stability using a rotating coordinate system, modified version of Bayly's [45], lateral stability of a stationary tool, Bayly [45], and torsional-axial chatter by Bayly [1]). The natural frequencies and damping ratios presented in Table 6.1 are used. The cutting coefficients are determined from the steady state forces in drilling a 16mm hole which has been predilled with a 4mm drill bit. The feedrate is 0.15mm/tooth. The tangential force Ft and radial force Fr are measured using a single fluted drill (one flute has been ground away). The thrust force Fz and torque Tc are measured using a regular drill with two flutes. F, = 1111.7JV F = 341.1 N (6.63) = 509.9JV = 10.199JV)?! Chapter 6. Frequency Domain Chatter Stability of Drilling 150 Table 6.1 : Dynamic properties of drill bit. Figure 6.6, vs. Time Domain Figure 6.8, vs. Experiments Mode Frequency Stiffness Unit Damping Stiffness Unit Damping Sign [Hz] [%] [%] X X 362.7 1.60xl0 7 [N/m] 2.00 N . A . N . A . N . A . N . A . Y Y 338.6 1.60xl0 7 [N/m] 2.00 N . A . N . A . N . A . N . A . kZFz 3357.5 1.055xl08 [N/m] 2.00 5.277xl0 7 [N/m] 0.5024 positive kQTc 3357.5 7.778xl02 [Nm/rad] 2.00 3.889xl0 2 [Nm/rad] 0.5024 positive kZTc 3357.5 4.336xl05 [Nm/m] 2.00 2.168xl0 5 [Nm/m] 0.5024 negative kQFz 3357.5 4.924xl05 [N/rad] 2.00 2.462xl0 5 [N/rad] 0.5024 negative The cutting coefficients are determined in Eq. 6.64, and provided in Table 6.2. t c b x h 0.006 x 0.00015/w2 1111.7/V m c o ^^^ 6^ ,T/  2 = 1235.2 x 10 N/m , = F I = 341.7N r c Ft \\\\.1N 0.3070 = b z = 509.9^ a c 2xFt 2 x l l l l . 7 J V = 0.2293 10.1997^ 2x(Ft-Fr) 2 x (1111.7-341.3)^ 0.0066193/w (6.64) Table 6.2 : Linear cutting force model parameters for the proposedfrequency domain solution; Workpiece material: AL7050-T7451. Drill geometry: Table 5.3; Cutting conditions: Cutting speed=120m/ min, feedrate=0.30mm/rev, pilot hole diameter=4mm. Parameter Value K 1235.2 x l06N/m2 Kc 0.3070 [-] Kc 0.2293 [-] *, 0.0066193 [m] Chapter 6. Frequency Domain Chatter Stability of Drilling 151 6.4.1. Comparison of proposed frequency domain solution with time domain model Figure 6.5 shows the limit depths of cut for the lateral and torsional-axial parts from Eq. 6.31 for the proposed method, tool dynamics from Table 6.1 and cutting coefficients from Table 6.2. The lateral part of the solution is determined by the smallest of roots Aj and A 2 (Eq. 6.37), which is A j . For the torsional-axial part, A 3 provides the lowest depth of cut. 1000 2000 3000 Chatter frequency 0 ) c [Hz] 4000 1000 2000 3000 4000 Chatter frequency C0 C [Hz] Figure 6.5 : Limit depth of cut for the proposedfrequency domain solution as a function of chatter frequency, for lateral and torsional-axial modes. Drill geometry: Table 5.3; Tool dynamics: Table 6.1; Cutting coefficients AL7050-T7451: Table 6.2; Feedrate=0.30mm/rev. Figure 6.6 shows the comparison between the proposed frequency domain solution and time domain simulation results at 252 cutting conditions from the model developed by Roukema et al. [80] (detailed in Chapter 5 of this thesis). The time domain model uses nonlinear cutting coeffi-cients, but is based on the same workpiece material and drill geometry. The tool is given a three micrometer radial runout to create lateral force unbalance that can result in lateral chatter (otherwise the forces on the two flutes would cancel each other out, and lateral deflections would not occur). From individual inspection of the lateral, axial and torsional tool deflection histories, the time domain results are classified as stable when tool deflections are tran-sient/reach a steady state value and do not grow indefinitely. The classification lateral chatter is Chapter 6. Frequency Domain Chatter Stability of Drilling 152 given when only lateral chatter occurs, torsional-axial chatter when only torsional-axial chatter occurs, and unstable when both lateral and torsional-axial chatter develop at the same time. The lateral chatter stability curve predicts a minimum critical depth of cut of 0.80mm. In time domain however, the process is still stable at 2.0mm depth of cut over the whole speed range (which is clearly above the stability border). While the frequency domain solution is linear, the time domain model considers the nonlinearities such as cutting edge and chip load dependent cutting force coefficients and tool jumping out of cut due to excessive vibrations. Hence, the frequency domain solution gives more conservative stability estimate than the time domain model. The torsional-axial stability curve predicts a minimum critical depth of cut of 2.30mm, which is more close to the time domain solution. Figure 6.7 provides details of three time domain simulation cases denoted by a, b and c in Fig. 6.6. Each column shows the three-dimensional workpiece finish, the cross-section of the surface at the pilot hole, lateral deflection and frequency spectrum, and axial and torsional deflections with their spectra. Case a shows lateral chatter developing very quickly, as can be seen by the lat-eral deflections and the side profile of the surface. The number of waves on the surface is small as the chatter frequency is about 360Hz. Case b shows torsional-axial chatter at about 3122Hz, leav-ing about 19 waves on the surface. The lateral deflection stabilizes at about 2\xm after full engagement due to the runout on the drill, at spindle frequency, fs = \61Hz [80]. Case c is stable and results in a smooth surface, while small lateral deflections at fs = 183Hz occur, again due to the drill runout. Chapter 6. Frequency Domain Chatter Stability of Drilling 153 J 1 1 1 1 1 1 1 1 1 i i i i i i i i i i i 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Spindle speed [krpm] Figure 6.6 : Comparison of proposedfrequency domain with time domain simulation results. Drill geometry: Table 5.3; Tool dynamics: Table 6.1; Cutting coefficients AL7050-T7451: Table 6.2; Feedrate=0.30mm/rev. Chapter 6. Frequency Domain Chatter Stability of Drilling 154 Cross-sectional profile at pilot hole Cross-sectional profile at pilot hole Cross-sectional profile at pilot hole 1 2 3 4 Frequency [kHz] 1 2 3 4 Frequency [kHz] 1 2 3 4 Frequency [kHz] 0.04 0.08 Time [s] 2 3 4 Frequency [kHz] 0.6 0 fc=3122Hz J I0.12 0.12 0.12 0 1 2 3 4 Frequency [kHz] 1 2 3 4 Frequency [kHz] Figure 6.7 : Time domain simulation details for three cases in Fig. 6: a) 14000rpm, 5.5mm depth -lateral chatter; b) lOOOOrpm, 5.5mm depth - torsional-axial chatter; c) HOOOrpm, 5.5mm depth - stable cut; Drill geometry: Table 5.3; Tool dynamics: Table 6.1; Cutting coefficients AL7050-T7451: Table 6.2; Chapter 6. Frequency Domain Chatter Stability of Drilling 155 6.4.2. Proposed frequency domain solution versus experimental results The experiments presented in Chapter 5 (Figs. 5.26, 5.27) indicate that torsional-axial chatter is dominant, and not lateral chatter. Fig. 6.8 compares the predicted torsional-axial stability lobes (proposed solution, Eq. 6.31) with experimental results. The tool dynamics are provided in Table 6.1 (these are different tool dynamics than used for Fig. 6.6, the speed range is also different here). The predicted stability pockets are small and narrowly packed, and cannot be used to select vibration free cutting conditions. Drilling a full hole becomes stable when the spindle speed is reduced to 1800rpm. If chatter occurred at that speed, the number of waves between consecutive teeth would be 56, resulting in significant process damping (see Fig. 5.26). If the speed is reduced further, the number of waves increases, and the process damping effect becomes even stronger. 7r-„ 6 E E ¥ 5 o o JZ 4 Q. CJ> Q 3 0 full hole) ^ chisel edge does not cut • 9 9 • • 9 9 stable (experiments) ^ torsional-axial chatter 0 ^ ^ 0 proposed frequency domain torsional-axial chatter lobes "0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 Spindle speed [rpm] Figure 6.8 : Comparison of experiments and proposedfrequency domain solution for torsional-axial chatter stability; Drill geometry; Table 5.3; Tool dynamics: Table 6.1; Workpiece material: AL7050-T7451; feedrate=0.30mm/rev.; Flood coolant was used during experiments; Depths of cut 2-3-4-5-6-8mm in the chart.correspond to pilot hole diameters 12-10-8-6-4-0mm (the drill diameter is 16mm). Chapter-6. Frequency Domain Chatter Stability of Drilling 156 6.4.3. Proposed frequency domain solution versus Bayly's frequency domain solution Fig. 6.9 compares the proposed frequency domain solution (the torsional-axial part) with Bayly's frequency domain solution (Eq. 6.62 in Section 6.3.3) and experimental results. In the proposed g method, the modal parameters as given in Bayly's work [1], k = 4.37x10 [N/m], fn = 350.2//z and C, = 0.50%, are used to construct <DZZ, which is multiplied with the torsional-axial coupling factor a = -2.7. d>6z, ® z 0 and <Dee are zero. Cutting coefficients k"< = i~ = l a n d k'c - * C 8.0 x 10 N/m are used to calculate the stability lobes (Eq. 6.31). The proposed method predicts a slightly lower depth of cut, but the pocket location and shape is the same. Bayly conducted the experiments shown, which were deemed unstable if the RMS value of the displacement sensor reading exceeded 0.02mm. proposed frequency domain torsional-axial chatter lobes Bayly • stable (experiments) O torsional-axial chatter 900 1000 1100 1200 1300 Spindle speed [rpm] 1400 1500 Figure 6.9 : Comparison of the proposedfrequency domain solution with Bayly's frequency domain solution and experiments [1]; Drill diameter D=9.525mm, length=107mm; feedrate=0.203mm/rev; Workpiece material: AL7075. Chapter 6. Frequency Domain Chatter Stability of Drilling 157 The time domain model presented in Chapter 5 is also compared with the frequency domain solu-tion (Eq. 6.62) established by Bayly et. al. [1]. Figure 6.10 shows the stability chart and the chat-ter frequency diagram from Bayly's work, and details of four time domain simulations using the model presented in Chapter 5. Figure 6.10 : Verification of frequency domain solution by Bayly [1] using the proposed time domain model; Drill diameter D=9.525mm, length=107mm;feedrate=0.203mm/rev; Number of elements m=30; Workpiece material: AL7075. Chapter 6. Frequency Domain Chatter Stability of Drilling 158 A steel fixture was mounted close to the tip of the drill for measurement purposes and to lower the natural frequency of the drill bit. The lower natural frequency of fn = 350.2Hz shifts the big lobe pockets to lower spindle speeds (without the fixture, the torsional-axial frequency is 5800Hz). The damping ratio C, = 0.50% and the stiffness is 4.37 x 10 [N/m]. The graphs show the tool deflection history, fourier spectrum and surface finish for four cutting conditions that have been simulated with the proposed time domain model. Note that the steady state tool deflection for the stable cases B,C are very small due to the apparent high static stiffness of the tool, g 4.37 x 10 [N/m]. The drill bit acts as a torsional-axial spring with a concentrated mass close to the tip - the fixture. Both depth of cut and chatter frequency predicted by the proposed time domain model are in good agreement with Bayly's frequency domain solution [1]. 6.4.4. Proposed frequency domain solution versus Arvajeh's frequency domain solution Arvajeh et al. initially used Bayly's frequency domain solution [1] to calculate the torsional-axial chatter stability [67], but did not find a good match between experiment and prediction, as they deemed a cut unstable if the thrust force was larger than 100N. By taking the variation in geomet-rical and cutting parameters along the cutting edge into account while calculating stability lobes, Arvajeh et al. found an improved match with experiments [68], where chatter was established based on the surface finish. Figure 6.11 presents their experimental results and the stability lobes [68]. Using the same variation of cutting parameters and drill geometry in the proposed frequency domain solution, very similar lobes were obtained and also shown in Fig. 6.11. In the proposed method, the modal parameters as given in Arvajeh's work [68], k = 44 x \06[N/m], fn = 5600Z/z and £ = 0.30%, are used to construct <DZZ, which is multiplied with - 1 . ® 9 z , O z 6 and O 0 e are zero. Ct and Cz are the tangential and thrust force coefficients which depend on the location on the cutting edge, and rQ is the torsional coupling factor which depends on the depth of cut. Chapter 6. Frequency Domain Chatter Stability of Drilling 159 In the proposed method (Eq. 6.31), cutting coefficients kac = — and ktc = rQCt-Cz become dependent on the depth of cut, and are used to calculate the stability lobes. The proposed method predicts a slightly lower depth of cut, but the pocket location and shape is the same. Cz, Ct (in 2 [N/m ]) and rd (r is [mm]) are given by Arvajeh [68]: Ct(r) = 6.96 x 10 V - 1.08 x 108r + 4.97 x 108 Cz(r) = 1.84 x l o V - 1.63 x 108r + 3.93 x 10* rQ(r) = 0.19r (6.65) Arvajeh proposed frequency domain torsional-axial chatter lobes • stable (experiments) ^ torsional-axial chatter 6800 7000 7200 7400 7600 7800 Spindle speed [rpm] 8000 8200 8400 Figure 6.11 : Comparison of the proposed frequency domain solution with Arvajeh's frequency domain solution and experiments [68]; Drill diameter D=9.525mm, length=107mm; feedrate=0.10mm/ rev; Workpiece material: AL6061-T6. Chapter 6. Frequency Domain Chatter Stability of Drilling 160 6.4.5. Proposed frequency domain solution versus previously presented partial stability laws Finally, the proposed frequency domain method is compared with previously published stability laws. For clarity, the stability solution of the proposed method is split into a torsional-axial stabil-ity and a lateral stability part. The torsional-axial stability is compared with Bayly's work [1] in Fig. 6.12a, whereas the lateral stability is compared with stationary tool stability [Bayly, 45] and the rotating coordinate frame approach [modified approach of Bayly, 45, Section 6.3.1] in Fig. 6.12b. In order to use Bayly's solution for torsional-axial chatter [1] the torsional-axial coupling factor needs to be calculated. The static axial deflection for the drill bit (Table 6.) due to thrust and torque would be: 509 27V 10 \99Nm d z = su^ziv lu.iy^vw = 4 8 2 7 _ 23.52u7n = -18.69u./w (6.66) 1.055 x 10 N/m 4.336 x 10 N/m Hence, the torsional-axial coupling factor the drill bit is determined as: a = -}l^m = -3.87 (6.67) 4.827U7W v ' which shows that axial extension due to torque overwhelms the compression due to thrust. The thrust cutting constant is determined as, 509.27V = 5 6 5 g x l Q 6 N / m 2 ( 6 6 8 ) 0.006 x 0.00015m a and kc^ are used to calculate the torsional-axial chatter stability in Eq. 6.62. Bayly's torsional-axial chatter stability law predicts a more conservative depth of cut. The lobe shape is slightly dif-ferent from the proposed method, but the lobe locations line up well. The shape of the lobes pre-dicted by the proposed method are different, as it takes four different transfer functions into account, rather than lumping them into one. For the lateral stability chart comparison, three solu-tions are shown. The lateral chatter stability predicted using a rotating coordinates approach (Eq. 6.48) closely matches the proposed solution (Eq. 6.31). Chapter 6. Frequency Domain Chatter Stability of Drilling 161 1r-torsional-axial stabiliy border (Bayly) torsional-axial stability border (proposed) 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Spindle speed [krpm] Figure 6.12 : Comparison between proposedfrequency domain solution and a) Bayly's torsional-axial stability [1J; b) Lateral stability for stationary tool (Bayly [45]) and rotating coordinates method (modified approach of Bayly [45]); Drill geometry: Table 5.3; Tool dynamics: Table 6.1; Cutting coefficients AL7050-T7451: Table 6.2; Feedrate=0.30mm/rev. Chapter 6. Frequency Domain Chatter Stability of Drilling 162 The rotating coordinates approach uses time invariant cutting coefficients, while the proposed method takes the average of the time varying coefficients. They result in a very similar stability border, which justifies the averaging applied in Eq. 6.18. Altintas used a zero order solution in milling [81], which was shown to work if no closely spaced modes exist in the system. In this case there is only a single mode in each direction. The dynamic cutting force harmonics away from the chatter frequency are then low pass filtered, because the transfer function does not have any strength away from the natural mode (to which the chatter frequency is very close). The chatter stability for a stationary tool - workpiece rotating in the spindle (Eq. 6.52) predicts a higher stable depth of cut, while its lobe locations are determined by the natural frequency in X-direction only. Rotation of the tool makes the drilling process dynamically weak. The first pocket occurs at 363Hz x 60/Iflutes « \0900rpm, the second at half that spindle speed. The proposed lateral chatter stability border (Eq. 6.31) has stability pockets at the same spindle speeds as the other solutions, but predicts a more conservative depth of cut than time domain simulation. Although the frequency domain solution predicts low depths of cut dominated by the lateral modes (X, Y), the time domain solution considers the true dynamics and leads to lobes dominated by the torsional-axial mode. 6.5. Conclusions A new method to calculate the stability lobes for drilling vibrations in lateral, torsional and axial directions is presented and compared with the time domain model and experiments presented in Chapter 5 of this thesis. The new method is also compared with existing, limited methods that only take lateral or torsional-axial vibrations into account. The time domain simulation and pro-posed torsional-axial chatter stability lobes are in good agreement. Bayly's method for torsion-axial chatter does not match as well and provides a slightly conservative estimate for the depth of cut. Chapter 6. Frequency Domain Chatter Stability of Drilling 163 The proposed lateral chatter stability lobes are very similar to the rotating coordinate approach (Bayly's method [45], modified to allow for large depth of cut), indicating that gyroscopics do not play a big role for the cases considered and that the averaging of the time varying directional coef-ficients does not affect the accuracy of the solution. The depth of cut predicted by time domain simulation is less conservative than the proposed frequency domain solution. Experiments did not show any clear cases of lateral chatter. It should be noted that lateral chatter stability in drilling is very different from milling, in the sense that during stable cutting, the resultant lateral force in drilling is zero. Hence, it was neces-sary to include a tool imperfection to excite chatter vibrations in simulation. Torsional-axial chat-ter does have a steady state torque and thrust. In turning, boring and milling chatter the resultant lateral force is always nonzero as well. 164 Chapter 7. Conclusions and Recommendations for Future Work 7.1. Conclusions This thesis presents a comprehensive modeling of drilling mechanics, dynamics and the hole for-mation process. The model allows simulation of the drilling process by considering the drill geometry, drill grinding errors, workpiece material properties, lateral, torsional and axial vibra-tions of the drill. While the proposed numerical model predicts the cutting forces, torque, vibra-tions and resulting hole geometry in time domain, the frequency domain model predicts the chatter free spindle speeds and depths of cut. The thesis explains the fundamental problem in drilling dynamics, which is the unmodeled flank-finish surface contact mechanics and dynamics or the process damping that hinders the accuracy of drilling dynamics modeling. The cutting forces in drilling (mechanics) have been predicted using two methods: a) a cutting transformation in conjunction with a previously established database (cutting mechanics approach); b) a mechanistic approach, which is calibrated directly from experiments. The cutting mechanics approach provides good insight into the prediction of drilling forces, following experi-mental trends well, except for the radial force acting on the flutes. Due to several reasons the cut-ting mechanics approach under predicts the cutting forces. For accurate force prediction for the cutting lip region, a mechanistic model was calibrated from experimental thrust and torque for piloted holes using a regular drill, and from three orthogonal forces while drilling piloted holes with a one fluted drill. The mechanistic model predicts the cutting forces accurately at the expense of experimentation. An exact kinematics time domain model has been developed for simulation of combined lateral, axial and torsional vibrations when drilling piloted holes. Both the piloted hole bottom surface and the hole wall are generated by stepping through time while the drill deflects due to the vibra-Chapter 7. Conclusions and Recommendations for Future Work 165 tion dependent cutting forces. The model uses the cutting force models developed in this thesis, and the static prediction of transient and steady state torque and thrust matches well with experi-ment. The presented kinematic model of the drill is the first reported in the literature. The time domain model is used to predict axial, torsional-axial and lateral chatter vibrations and their effect on cutting forces, torque and the resulting surface finish, which were verified experi-mentally. The torsional-axial chatter frequency and development of vibrations can be predicted, and has been verified by comparing the experimental and simulated surface finish and tool vibra-tion spectra. The torsional-axial chatter stability - what vibration free depth of cut can be used for a given spindle speed - cannot be predicted, as process damping is not included in the time domain simulation. The torsional-axial chatter stability was investigated for different drill lengths of both twist and indexable drills. The effect of tool grinding and misalignment errors are simu-lated and show favorable agreement with published empirical relationships. Although the model does not predict whirling vibrations, the frequency content of simulated imposed whirling vibra-tions shows good agreement with experimentally observed whirling motion. The formation of odd-sided shapes on the hole wall are studied in detail by varying whirling frequencies and whirl-ing motion amplitudes. The mathematical model of whirling mechanism is also the first reported in the literature. A new method to calculate the stability lobes for drilling vibrations in lateral, torsional and axial directions is presented and compared with the time domain model presented in Chapter 5 of this thesis. Time domain simulation results match well with the proposed torsional-axial chatter stabil-ity lobes, which takes both torsional and axial flexibility into account. The proposed lateral chat-ter stability lobes match well with the rotating coordinate approach, suggesting that gyroscopics do not play a big role for the cases considered. From time domain simulation, the lateral stability Chapter 7. Conclusions and Recommendations for Future Work 166 is higher than predicted in frequency domain. Experiments did not show any clear cases of lateral chatter. 7.2. Recommendation for future work The main objectives of this research were to predict chatter and whirling vibrations in drilling. Although the chatter frequency and surface are predicted well, the model is not able to predict how the stability depends on spindle speed and pilot hole size. Process damping due to the tool circular movement needs to be implemented, which poses two difficulties. First, there is no accu-rate model of process damping available, not even for orthogonal cutting. Secondly, as the grid point distribution along the circles is not even, the process damping model needs to provide cor-rect results regardless of the number of grid points in the relief zone of the drill. Lateral chatter is not prevalent in the experiments conducted for this thesis, hence there is also a great amount of process damping in lateral directions. The grid distribution in radial direction is even and fixed, hence implementation of lateral process damping may be more straight forward. It should be noted that prediction of lateral chatter stability needs additional attention, as time domain and frequency domain modeling only agree in trend, but not depth of cut. It should be noted that lateral chatter stability in drilling is very different from milling, in the sense that during stable cutting, the resultant lateral force in drilling is zero. Torsional-axial chatter does have a steady state torque and thrust - time domain and frequency domain modeling for torsional-axial chatter match very well in the proposed models. The contact forces between the hole wall and drill flutes due to lateral deflections have not been modeled yet. Inclusion of these forces may enable the time domain simulation to predict the com-monly observed whirling motion. Hole wall interaction mainly consists of rubbing and friction, Chapter 7. Conclusions and Recommendations for Future Work 167 and little cutting action. Contact forces are also difficult to model as they are very non-linear and change drastically depending on lubrication conditions. The dynamic properties of the drill bit are currently defined in the stationary frame, as was done in the milling simulation model by Montgomery [49]. As was clear from the research by Pirtini and Lazoglu [60], the dynamics of the drill are dominant compared to the spindle, and should be rotated with the tool. This requires deriving coupled equations for the Runge-Kutta scheme. Chatter suppression could possibly be achieved by using variable pitch four fluted drills. 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[76] Lazoglu, I., Atabey, F., Altintas, Y , 2002, "Dynamics of Boring Processes: Part III - Time Domain Modeling," Int. J. Mach. Tools. Manuf. 42(14), pp. 1567-1576. [77] Roukema, J.C., Altintas, Y , "Kinematic Model of Dynamic Drilling Process," IMECE2004-59340, ASME International Mechanical Engineering Congress 2004, Ana-heim, California. [78] Ogata, K., 1997, "Modern Control Engineering," Prentice-Hall, Inc., London. [79] Den Hartog, J.P., Mechanical Vibrations, McGraw-Hill, 1934. [80] Roukema, J.C., Altintas, Y , 2006, "Generalized Modeling of Drilling Vibrations, Part I: Time Domain Model of Drilling Kinematics, Dynamics and Hole Formation," Int. J. Mach. Tools. Manuf. (accepted). [81] Altintas, Y. - Manufacturing Automation, Cambridge University Press, 2000. 174 Appendix A. Stationary Drilling Dynamometer Calibration A drilling dynamometer was built in the Manufacturing Automation Lab, using a Kistler two channel force sensor, model 9065. The dynamometer is capable of measuring torque up to 200Nm and thrust up to 20kN. The torque sensitivity has been determined using a pulley setup with a cable and calibrated weights, shown in Fig. A. 1. Figure A . l : Drilling dynamometer calibration setup with pulleys and weight. Appendix A. Stationary Drilling Dynamometer Calibration 175 Two equal forces are created by running a cable over a pulley which carries the weights. The two cable ends run over two pulleys into a horizontal plane, and the direction of one of the forces is flipped by running it over another pulley. The two cable ends are attached to the dynamometer via screws in order to apply a pure torque, see the detail in Fig. A.l. The calibration is performed as follows: The pulley structure is loaded with the weights. A data acquisition system records the reading from the dynamometer in Volts. The weights are lifted slowly, fully unloading the dyna-mometer, and then the weights are slowly dropped again. The torque applied to the dynamometer can be determined from the weights and the moment arm. The torque loading due to the cable-pulley system remains while unloading the weights, and is assumed to remain constant. Figure A.2 : Detail dynamometer calibration setup with two equal, opposite forces. The change in torque applied to the dynamometer results in deformation of the piezo-electric crystal, generating a charge, which is converted into a voltage signal by the charge-amplifier. The sensitivity of the dynamometer is then determined from the torque loading and the change in volt-age reading from the charge amplifier. The thrust reading shows a similar signal pattern due to Appendix A. Stationary Drilling Dynamometer Calibration 176 cross-talk, but the magnitude is very small. Fig. A.3 shows a typical reading from the drilling dynamometer from which the torque sensitivity is determined. The procedure to determine the thrust sensitivity is as follows: A number of calibrated weights is placed on top of the dynamometer. Data-acquisition is started recording the torque and thrust sig-nals. The weights are carefully lifted from the dynamometer. From the weights and the change in voltage reading from the thrust signal the thrust sensitivity is determined. Fig. A.4 shows a typical reading from the drilling dynamometer from which the thrust sensitivity is determined. The torque reading also shows a very small cross-talk. Time [s] Figure A.3 : Torque and thrust signal readings during calibration experiment (torque sensitivity). Appendix A. Stationary Drilling Dynamometer Calibration 111 0 . 0 4 -20-(/> 3 - 4 0 -JC H - 6 0 -- 8 0 --100L 1 0 T- r i 1 5 Time [s] Figure A.4 : Torque and thrust signal readings during calibration experiment (thrust sensitivity). The torque and thrust sensitivities are determined as 1.3045pC/Ncm and 1.780pC/N respectively. 178 Appendix B. Decomposition of Three-dimensional Forces in Cutting Mechanics Approach The three force components acting on an element on the cutting edge are in tangential (Ft), feed (Fy) and radial (Fr) directions, as shown in Fig. B . l . The magnitude of these force components is determined using the cutting mechanics approach, as detailed in Chapter 4. The carthesian coordinate system shown has the Z-axis aligned with the tool axis, and the X-axis is parallel to the cutting edge. In order to find the forces in the carthesian coordinate system attached to the drill bit, the forces are decomposed graphically. In Fig. B . l , the feed force (FJ) is decomposed into two components, and the tangential and radial forces (Ft, Fr) in three. Figure B . l : Three-dimensional representation of the elemental cutting forces in the cutting mechanics approach. The full decomposition of the radial force is given in Fig. B.2. Appendix B. Decomposition of Three-dimensional Forces in Cutting Mechanics Approach 179 (Fr) plane through cutting edge and F, and F r X' along cutting edge a) plane perpendicular to cutting edge, F rsini r F r s i n i r c o s i | / r Z ' b)z +Y F r sini rsin\|/ rcosK t • j 1 © V F r sini rsini}/ rsinK t, \ \ F r sini rsin\|/ r c) component along cutting edge, F r cosi r F r cosi r cosK t | F r cosi r F r cos i r s inK t (J) Figure B.2 : Force decomposition of radial cutting force into XYZ coordinate system of drill bit. In matrix form the force decomposition is given by: F. --s in/ r cos , F r sin/rsinvF /.cosK, — cos / r s inK , - sin/ r s in^ r s inK , - cos / r cos K, IF, (B.l) The full decomposition of the feed force is given in Fig. B.3. Appendix B. Decomposition of Three-dimensional Forces in Cutting Mechanics Approach 180 Figure B .3 : Force decomposition offeedforce into XYZ coordinate system of drill bit. which in matrix form is expressed by: Ff = -COS^COSK, cos^sinK, (B.2) Finally, the tangential force is decomposed in Fig. B.3, and the components in the carthesian coordinate system attached to the drill are given by: F t ' - cos^cos^ cos zr sin 4^ COSK, + s i n / r s i n K , - cos^sin^sinK, + sinz' rcosK, (B.3) Appendix B. Decomposition of Three-dimensional Forces in Cutting Mechanics Approach 181 (Ft) plane through cutting edge and F t and F r along cutting edge 2) F t s i n i r -a) component perpendicular to cutting edge, F tcosi r 2 ' F t cosi rcos\j/ r b) F t cosi rsin\|/ rsinK t F t cosirsin\j/rcosK,(+Y) F t cosirsin\}/r 2) component along cutting edge, Ftsini, (+z) F t sini rcosK t Figure B.4 : Force decomposition of tangential cutting force into XYZ coordinate system of drill bit. Appendix B. Decomposition of Three-dimensional Forces in Cutting Mechanics Approach 182 The total forces acting on each cutting lip are given by: -cos*' cos*?.. -shW -sin/' c o s ^ F , Ff cos/' rsinCOSK, + sin/^sinK, -cos vF rcosK / s i n / ^ s i n ^ c o s K , - cos/'rsinK, The X, Y and Z components are orthogonal to each other. Using these expressions, the lateral forces on the drill can easily be determined. Through a simple transformation they be converted into the global coordinate system of the time domain simulation model. 

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