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Mechanics and dynamics of milling thin walled structures Budak, Erhan 1994

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MECHANICS AND DYNAMICS OF MILLING THIN WALLED STRUCTURESByErhan BudakBSc. Middle East Technical University, Ankara, Turkey 1987;MSc. Middle East Technical University, Ankara, Turkey 1989A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF MECHANICAL ENGiNEERiNG1994© Erhan Budak, 1994We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAIn presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives, It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.__Department of fri €cI&Cl4 ca I En InrThe University of British ColumbiaVancouver, CanadaDate l39SDE-6 (2/88)AbstractPeripheral milling of flexible components is a commonly used operation in the aerospaceindustry. Aircraft wings, fuselage sections, jet engine compressors, turbine blades anda variety of mechanical components have flexible webs which must be finish machinedusing long slender end mills. Peripheral milling of very flexible plate structures made oftitanium alloys is one of the most complex operations in the aerospace industry and it isinvestigated in this thesis.Flexible plates and cutters deflect statically and dynamically due to periodically varying milling forces and self excited chatter vibrations. Static deflections of the plate andcutter cause dimensional form errors, whereas forced and chatter vibrations result in poorsurface quality and chipping of the cutting edges. In this thesis, a comprehensive model ofthe peripheral milling of very flexible cantilever plates is presented. The plate and cutterstructures are modeled by 8 node finite elements and an elastic beam, respectively. Thecutting forces are shown to be very dependent on the magnitude of the plate and cutterdeformations which are irregular along the helical end mill-plate contact. The interaction between the milling process and cutter-plate structures is modeled, and the millingforces, structural deformations and dimensional form errors left on the finish surface areaccurately predicted by the simulation system developed in this study. A strategy, whichconstrains the maximum dimensional form errors caused by static deformations of plateand cutter by scheduling the feed along the tool path, is developed. The variation of theplate thickness due to machining and the partial disengagement of the plate and cutterdue to excessive static deflections are considered in the model. The simulation systemis proven in numerous peripheral milling experiments with both rigid blocks and veryflexible cantilevered plates.11The self excited vibrations observed during peripheral milling of very flexible structures with multi-degree of freedom dynamics is investigated. A novel analytical model ofmilling stability is developed. The stability model requires structural transfer functionsof plate and cutter, milling force coefficients and helical end mill geometry. Chattervibration free cutting speeds, axial and radial depths of cut, i.e. stability lobes, arepredicted analytically without resorting to computationally expensive time domain simulations. The analytical chatter stability model is verified in various peripheral millingexperiments, including the machining of plates.The cutting force and chatter stability models developed in this thesis can be used toimprove the productivity of peripheral milling of thin webs by enabling simulation andprocess planning prior to production.111Table of ContentsAbstract iiTable of Contents ivList of Tables ixList of Figures xAcknowledgments xvNomenclature xvi1 Introduction 12 Literature Survey 72.1 Overview 72.2 Geometry of Milling 82.3 Milling Force Models 92.4 Stability of Dynamic Milling 112.4.1 Dynamic Cutting 122.4.2 Chatter Stability Models 162.5 Peripheral Milling of Flexible Structures 193 Structural Modeling of the Workpiece and End Mill 213.1 Introduction 21iv3.2 Structural Modeling of Workpiece 223.2.1 Bending of Thin Plates 233.2.2 Series Solutions for Cantilever Plate 283.2.3 The Finite Element Modeling of Plate Defiections 343.2.4 Dynamics of Cantilever Plate 463.2.5 Simulation and Experimental Examples 483.3 Structural Modeling of End Mill 513.3.1 Cantilever Beam Model for End Mill 523.3.2 Simulation and Experimental Examples 543.4 Summary 554 Modeling of Milling Forces 564.1 Introduction 564.2 Mechanistic Modeling of Milling Forces 574.2.1 Exponential Force Coefficient Model 584.2.2 Linear Edge-Force Model 654.3 A Mechanics of Milling Approach for Milling Force Prediction . 684.3.1 Force Coefficient Expressions 694.3.2 Procedure for Milling Force Calculation 814.4 Simulation and Experimental Results 834.4.1 Cutting Conditions in Milling and Orthogonal Tests 854.4.2 Analysis of Orthogonal Data: Identification of Cutting Parameters 864.4.3 Prediction of Milling Force Coefficients 934.4.4 Accuracy of Milling Force Calculation by Predicted Coefficients 984.5 Summary 101v• 5 Effects of Milling Conditions on Cutting Forces and Accuracy 1045.1 Introduction 1045.2 Surface Generation by Statically Flexible End Mill 1045.3 Identification of Cutting Conditions for Minimum Dimensional Surface ErrorlO75.4 Simulation and Experimental Results 1115.4.1 Cutting Force and Surface Finish 1115.4.2 Selection of Optimal Cutting Conditions 1135.5 Summary 1226 Static Structure-Milling Process Interaction 1236.1 Introduction 1236.2 Statically Regenerative Milling Force Model (Variation of Chip ThicknessDue to Defiections) 1246.3 Flexible Milling Force Model (Variation of Radial Depth of Cut Due toDeflections) 1356.4 Summary 1407 Peripheral Milling of Plates 1427.1 Introduction 1427.2 Static Modeling of Plate Milling 1447.2.1 Structural Model of the Plate . . . 1447.2.2 Structural Model of the Tool 1467.2.3 Cutting Force Distribution-Rigid and Flexible Force Models . 1477.3 Simulation of Peripheral Plate Milling. . . 1507.3.1 Plate Surface Generation. 1517.3.2 Control of Accuracy. 1527.4 Simulation and Experimental Results . . . 153vi7.5 Summary 1698 Analysis of Dynamic Cutting and Chatter Stability in Milling 1708.1 Introduction 1708.2 Formulation of Dynamic Milling Forces 1738.2.1 Dynamic-Regenerative Chip Thickness 1758.2.2 Differential Dynamic Milling Forces 1768.2.3 Total Dynamic Milling Forces 1778.2.4 Dynamic Displacements of Cutter and Workpiece 1848.3 Stability Analysis 1878.3.1 Stability Theory of Periodic Systems 1878.3.2 Stability Analysis of Milling Using Periodic System Theory . . 1898.3.3 Milling Stability Analysis Based on the Interpretation of Physicsof Milling Dynamics 1938.3.4 Truncation of the Characteristic Equation of Dynamic Milling 1978.3.5 Summary of the Calculation of Milling Stability for the General Case2O28.3.6 Accuracy of the Chatter Limit Prediction by the Truncated Characteristic Equation 2038.3.7 Solution of the Characteristic Equation to Determine the ChatterStability Limit in Milling 2058.4 Solutions of Milling Stability Equation for Special Cases 2108.4.1 Milling of a Single Degree-of-Freedom Workpiece 2118.4.2 Milling of a Flexible Structure with a Flexible End Mill-Single Axial Element 2188.4.3 Milling of a Flexible Structure with a Rigid End Mill-Varying Dynamics in Axial Direction 224vii8.5 Dynamic Peripheral Milling of Plates 2298.6 Summary 2399 Conclusions 242Bibliography 247Appendices 261A Chip Flow Angle Formulation 262viiiList of Tables3.1 Eigenfunction parameters for clamped-free and free-free beams 314.1 Helical flute engagement limits to be used in cutting force calculations. 644.2 Cutting parameters identified for Ti6A14V from orthogonal cutting tests 924.3 Cutting force coefficients for different rake angles as identified and transformed from orthogonal data by using the linear edge-force model 964.4 Cutting force coefficients for different rake angles as identified from millingtests by using exponential force model 975.1 Cutting conditions for up and down milling tests conducted for surfaceerror verification and identification of optimal milling conditions 1147.1 Cutting conditions for experiments 1 and 2. (Material: Titanium AlloyTi6A14V) 154ixList of Figures2.1 Dynamic cutting process 122.2 Variation of effective clearance angle in dynamic cutting 143.1 Internal forces and moments on a differential plate element 243.2 Displacement of a point P to F’ due to bending of plate 253.3 3D isoparametric solid element 383.4 The 3D solid element in the natural coordinates 393.5 Flow chart of the developed Finite Element Program 453.6 One element example to test the developed FE Program 483.7 Structural model for end mill: Cantilever beam with elastically restrainedend 524.1 Differential milling forces applied on the cutting tooth 594.2 Contact cases of a helical flute with workpiece 634.3 Orthogonal and oblique cutting geometries 704.4 Orthogonal cutting force diagram 714.5 Cutting forces and chip flow geometry on a helical milling cutter 734.6 Detailed view of the oblique cutting geometry 744.7 The oblique cutting force components in the normal plane 754.8 Predicted variations of chip flow angle with rake and friction angles. 804.9 Variation of chip flow angle with cutting ratio for different values of frictionangle 804.10 The effects of the inclination angle and cutting ratio on the chip flow angle. 81x4.11 Generalized milling force prediction algorithm 824.12 The orientation of differential milling force components on a ball end millflute 844.13 Measured cutting and feed forces in orthogonal cutting tests with differentrake angles 874.14 Edge forces as identified from the orthogonal cutting tests 884.15 Variation of the measured cutting ratio r(= h/he) with chip thickness andrake angle 894.16 The identified values of the shear stress at the shear plane from the orthogonal cutting tests 914.17 The friction angle calculated from the orthogonal cutting forces 924.18 Predicted values of the chip flow angle for 300 inclination (helix) angle.. 934.19 Variations of the predicted milling force coefficients with the chip thickness. 944.20 The statistical error analysis of the milling force predictions 994.21 Measured and simulated milling forces (linear-edge force model) 1024.22 Measured and simulated milling forces (linear-edge force model) 1024.23 Measured and simulated milling forces (exponential force model) 1034.24 Measured and simulated milling forces (exponential force model) 1035.1 Statically flexible end mill model 1055.2 Variation of the optimum exit angle (for up-milling) with K, 1115.3 Measured and simulated milling forces for half immersion-up milling. 1165.4 Measured and simulated milling forces for half immersion-down milling 1165.5 Simulated and measured surface profiles for half immersion-up milling 1175.6 Simulated and measured surface profiles for half immersion-down milling 117xi5.7 Variation of the predicted m&ximum dimensional surface error due to tooldeflection with the radial depth of cut and the feed per tooth 1185.8 Variation of the specific-predicted maximum dimensional surface error(SMSE) with the radial depth of cut and the feed per tooth for up milling.1195.9 Variation of measured and simulated maximum normal cutting force Fy,naxwith radial depth of cut for different values of feed per tooth 1205.10 Variation of measured and simulated maximum dimensional surface erroremaa, with radial depth of cut for different values of feed per tooth 1215.11 Variation of measured and simulated specific maximum surface error SMSEwith radial depth of cut for different values of feed per tooth 1216.1 Statically regenerative chip thickness geometry in milling 1256.2 Simulated rigid and statically regenerative milling force in x direction. . 1356.3 Variation of radial depth of cut and immersion angles due to deflectionsin up and down milling 1377.1 (a) Peripheral down milling of flexible plates, (b) Finite element model ofthe plate, (c) Corresponding nodal stations on the tool 1457.2 Experiment #1 - A sample window of simulated and measured forces. 1567.3 Experiment #1- (a) Simulated, (b) Measured surface finish dimensions 1587.4 Experiment #2- (a) Rigid model, (b) Flexible model, (c) Measured surfacefinish dimensions 1597.5 Experiment #2 Predicted and measured surface profiles near the beginning, middle and exit feed stations 1607.6 Experiment #2- (a) Measured and simulated average forces 1617.7 Experiment #2 - Flexible force model predicted variation of q58t in thebeginning, middle and close to exit feed stations 162xii7.8 Scheduled feedrates for Experiment 1 and Experiment 2 for surface errortolerances of 80 m and 250 IIm, respectively 1647.9 Experiment # 1 with scheduled feedrate for 80 m tolerance - a) Simulatedsurface errors b) Measured surface errors 1657.10 Experiment # 2 with scheduled feedrate for 250 m tolerance- a) Simulated (flexible model) surface errors b) Measured surface errors 1667.11 The variation of the machining time with tolerance value in Case # 1 and#2 1677.12 The simulated variation of the machining time with tool radius and radialwidth of cut in Experiment # 2 for tolerance of 250 .tm on the surface. 1688.1 Dynamic milling process 1748.2 Node numbering on cutter and workpiece 1868.3 The effect of number of teeth on peak to peak (AC component) value ofdirectional coefficient 2048.4 Variation of chatter frequency with spindle speed as predicted by the orthogonal cutting chatter theory 2078.5 Single degree-of-freedom milling system model 2128.6 The phase angle and transfer function at the chatter stability limit. . . 2148.7 Variation of the directional milling coefficient a, with the immersion anglein up and down milling 2168.8 Comparison of the predicted chatter limit with the published data for thesingle degree-of-freedom milling system example 2188.9 Milling system model with two degree-of-freedom cutter and workpiece. 2198.10 Experimental and time domain simulated stability limits for end millingtests. (data by Weck, Altintas and Beer, 1993) 222xlii8.11 Analytically predicted stability lobes for the case for which the experimental data and time domain simulations are shown in Figure 8.10 2238.12 Analytical and time domain stability limit predictions for a case analyzedby Smith and Tlusty 2258.13 Stability limit calculation algorithm for flexible workpieces with varyingdynamics in the axial direction 2278.14 Predicted effect of vibration mode shape on the stability limit 2288.15 Stability diagram for the cantilever titanium Ti6A14V plate down milledby 8 flute carbide end mill 2308.16 Cutting force spectrums in x and y directions for n = 6500 rpm anda = 0.4 mm 2328.17 Sound signal and its spectrum at n = 6500 rpm, a = 0.4 mm 2338.18 Cutting force spectrums in the x and y directions for n = 5000 rpm anda 0.4 mm 2348.19 Cutting forces in the y direction for m = 5000 rpm and n 6500 rpm fora = 0.4 mm 2358.20 Cutting forces in x and y directions for n = 6500 rpm and a = 2 mm. . 2368.21 Sound signal and its spectrum for n = 6500 rpm and a = 2 mm 2378.22 Cutting force spectrums for n = 6500 rpm and a = 2 mm 2388.23 Sound spectrums for different spindle speeds showing the effect of processdamping on chatter stability, a = 44 mm 2408.24 Effect of spindle speed and the process damping on the cutting forces.a=44mm 241xivAcknowledgmentsI would like to express my sincere appreciation to my research supervisor Dr. YusufAltinta for his support, guidance and encouragement throughout this work.I greatly appreciate the assistance of Yetvard Hosepyan and Peter Lee in cutting tests,and Dr. Ercan Köse with computer software. I thank the Manufacturing DevelopmentDivision of Pratt £4 Whitney, Montreal for providing the cutters and workpieces used inthis work. I also appreciate the valuable comments of Dr. Ian Yellowley on my research.Finally, I am most thankful to my wife and daughter, Asuman and Ece, for their sacrifice, continuous support and understanding during the course of this study. I dedicatethis work to them.This research has been supported by the Natural Sciences and Engineering ResearchCouncil of Canada and Pratt £4 Whitney Aircraft of Canada.xvNomenclaturea axial depth of cutaltm limiting axial depth of cut for chatter stabilitydirectional dynamic milling coefficients[A(t)] directional dynamic milling coefficient matrixb radial depth of cutbj(z) effective radial depth of cut at axial position z[B] strain-displacement matrix for 3D elastic bodyd0 cutter diameterde effective diameter of the cutterdF, dF, dFa differential tangential, radial and axial cutting forces inmillingD flexural rigiditye( k) surface form error at node kE Young’s modulus of elasticity[EJ elasticity matrixF, Ff cutting and feed force in orthogonal cuttingF3,F3,F3 milling forces in feed x, normal y and axial z directionson flute j[Ge], [Gm] cutter and workpiece transfer functionsI end mill area moment of inertia of end mill based on theequivalent diameterxvih uncut chip thicknessha average chip thicknessh cut chip thickness[J] Jacobian matrix of coordinate transformationICC collet stiffness[K] stiffness matrix of structureK, K, K tangential, radial and axial milling force coefficients (exponential force model)K7,Kac tangential, radial and axial milling-cutting force coefficients (linear-edge force model)Kte, Kre, Kae tangential, radial and axial milling-edge force coefficients (linear-edge force model)m rth modal residual[M] mass matrix of structureN number of teeth on the cutter{Q} load vectorr chip thickness ratio or cutting ratio1? cutter radiusSt feed per tootht, t, uncut and cut plate thicknessT tooth periodV cutting velocityx feeding direction coordinate axisxviiy normal direction coordinate axisw deflection of platez axial direction coordinate axisz axial coordinate of the surface generation pointz3,1z3,2 lower and upper engagement limits for flute ja cutter rake angle in orthogonal cuttinga, a1, normal and velocity rake angles in oblique cutting/3 friction angle in orthogonal cuttingf3 normal friction angle in oblique cutting&(k), 6(k) end mill deflections in x and y directions at node kaxial element thickness,ey, e, linear and shear strainsphase between succesive waves in chatter vibrationsic chip flow anglefriction coefficient on the rake faceLI Poissons’s ratiodamping ratioo, a, o stresses in x—y planeT shear stress at the shear plane{} normalized mode shape of the structurerotation angle of the cutter4i(z) immersion angle of tooth j at axial position zcutter pitch anglexviiishear angle in orthogonal cuttingnormal shear angle in oblique cutting48t, qes start and exit angles of cut in millinghelix anglew tooth passing frequencychatter frequencyspindle speed in (rad/sec)xixTo Asuman and Ece PolenxxChapter 1IntroductionThe peripheral milling of flexible components is a commonly required operation.Aircraft wing structures, fuselage sections, jet engine compressors, turbine blades andprecision instrumentation housings all have flexible webs which must be finish machinedby long slender end mills. In general, the majority of the aerospace components listedhere are machined from aluminum or titanium blocks. While the aluminum alloys have agood machinability rating (due to their low yield stress) the titanium alloys are difficultto machine because of their poor thermal conductivity. Furthermore, most aerospace andinstrumentation components have tight dimensional tolerances which have to be satisfiedduring machining.The peripheral milling of very flexible, cantilevered plate structures made of titaniumalloys is studied in this thesis. The project was originated and supported by a jet enginemanufacturer, who produces rotors (i.e. blisks), impellers, scrolls and turbine bladesusing slender helical carbide end mills. The workpiece material is titanium (Ti6A14V)for all components. Impellers and blisks are milled on five axis CNC machining centers,and the cutting time varies from 7 hours to 40 hours depending on the size of each modelcomponent. The wall thickness and the height from the cantilevered bottom (i.e. hub)of these parts are between 1.0mm to 2.5mm, and 25mm to 75mm, respectively. Therefore, the parts resemble clamped-free-free- free (CFFF) plates and are very flexible. Thecarbide end mills used to machine these parts have a diameter of 10mm to 20mm, and1Chapter 1. Introduction 2a gauge length of 35mm to 100mm from the clamping chuck. Because both the platetype workpiece and the long slender end mill represent very flexible structures, severestatic and dynamic deformations are experienced during peripheral milling of husks andimpellers.The flexibility of the slender end mill becomes dominant during the roughing of platesfrom solid blocks. The end mill removes the material in slotting mode, the resulting forceshave both a strong dc component as well as a dynamic component at the tooth passingfrequency. These forces may then lead to excessive static deformations which may breakthe cutter at the shank, as well as severe chatter vibrations which may chip the cuttingedges during roughing.Flexibilities of both cutter and plate must be considered during semi-finishing andfinishing operations. The end mill is most flexible at its free end which is in contactwith or adjacent to the root, i.e. the most rigid portion of the cantilevered plate. Theplate’s flexibility increases towards the free edge, which is closer to the clamped partof the end mill. As a result of the milling forces, the cutter statically deflects most atthe plate’s bottom end, and the deflection of the plate increases towards its free edge.Due to the cutter helix angle, the distribution of loads are very irregular in three cartesian directions. The amplitudes and direction of the forces change as the cutter rotates.Furthermore, the cutting forces are dependent on the local chip thickness removed fromthe plate, which continuously varies due to static deformations of both plate and toolstructures. The problem is further complicated by the partial disengagement of plate andtool due to excessive static displacements which are irregular along the axial direction.In order to understand the physics of the process and constrain the plate deformationswithin the required tolerances, a comprehensive model of the milling mechanics, theChapter 1. Introduction 3structures involved and the interaction between the metal cutting process and structuraldeformations have been developed. The plate is represented by its finite element modelwith varying thickness, and the cutter is modeled as a continuous elastic beam. Thelocal cutting forces and displacements of both plate and cutter at each elemental zoneare evaluated by predicting the chip thickness and cutter-plate intersection boundaries.Furthermore, the model developed in this work is able to predict the chip loads (i.e.feed rates) along the feed direction so that the static deformations are kept within theprescribed tolerances as the material is removed from the plate. The developed methodshave been experimentally verified.The flexibilities of the cutter and the plate produce severe forced and self excited (i.e.chatter) vibrations during the peripheral milling operations. Forced vibrations can beavoided by selecting spindle speeds whose corresponding tooth passing frequency harmonics do not coincide with the natural modes of the plate. Chatter vibrations are initiatedby transient vibrations, and their stability depends on the axial and radial depths of cut,cutting speed, workpiece material hardness and structural properties of both tool andthe workpiece. Chatter free axial and radial depths of cut, and cutting speeds have beendetermined by time domain simulations of the cutting-structure interaction, explained inthe static case, but including dynamic properties and regeneration of chip thickness atsuccessive tooth periods. The time domain simulations have been found to be time consuming, they do not provide a physical insight, and are prone to errors due to sensitivityof digital differentiation and integration techniques when the chip thicknesses are verysmall and the deformations are very large. A novel analytical technique, which predictsthe chatter vibration free stability lobes for multi-degree freedom flexible cutter and flexible workpieces, has been developed. The method has been experimentally verified whenmilling with flexible end mills and plates.Chapter 1. Introduction 4The peripheral milling of such flexible plates has not been investigated in depth before. The models developed in this thesis for static and dynamic deformations of theplate during machining are quite comprehensive and are believed to contribute to themilling literature.The chapters of the thesis are organized as follows:In Chapter 2, the relevant literature on milling geometry, milling force modeling,dynamic milling and chatter stability, and the milling of flexible workpieces is reviewedin general. Detail reviews are provided when related methods and approaches are usedor introduced in individual chapters.The static and dynamic modeling of slender end mills and clamped-free-free-free platesare presented in Chapter 3. The end mill is modeled as a continuous beam and the plateis represented by a finite element model with 8 node isoparametric elements. Bothstructural models are verified using analytical and experimental techniques.The modeling of milling forces using mechanistic and oblique cutting approaches ispresented in Chapter 4. The influence of edge forces is considered. It is shown that whilemechanistic approaches may provide coefficients to predict the cutting forces and corresponding structural deformations more accurately, the oblique cutting model providesmore insight to the physics of the process yet it still has sufficient accuracy to predictthe cutting forces in milling. The oblique model reduces the amount of tests requiredin mechanistic models as it uses a generalized orthogonal cutting data base to calculatethe milling force coefficients for different milling cutter geometries. Both techniques areexperimentally proven and compared.The influence of milling conditions, such as feed rate, axial and radial depth of cutand cutting coefficients, on milling forces and dimensional form errors produced by theChapter 1. Introduction 5flexible slender end mills are presented in Chapter 5. A method of finding optimal radialdepth of cut to achieve minimum dimensional form errors on finished surfaces is developedand experimentally proven.The static regeneration of chip thickness in milling is modeled in Chapter 6. It isanalytically and numerically proven that for a chatter free stable milling operation theeffect of deflections on the chip thickness diminishes in a few tooth periods. Then, themain mechanism of deflection-milling process interaction is identified as the variationof the cutter-workpiece immersion boundaries or radial depth of cut when milling veryflexible parts. A flexible milling force model, which uses the effective radial depth of cutunder the deflections, is developed. The model is used in the simulation of the peripheralmilling of plates in Chapter 7.The simulation model for peripheral milling of flexible plates with flexible end millsis presented in Chapter 7. The local changes in the radial width of cut, are considered in• calculating the chip thickness, milling forces and displacements using the model developedin Chapter 7. The Finite Element modeling of the plate with varying structural propertiesdue to metal removal, the beam model of the slender end mill, the flexible milling forcemodel which considers the partial disengagement of the structures along the cutter axisand milling mechanics are integrated to a comprehensive simulation model. The modelpredicts the milling force distribution and surface form errors caused by static flexibilitiesof the plate and tool. A feed scheduling technique which constrains the form errors withinthe prescribed tolerances has been developed and integrated with the plate simulationmodel. The model has been experimentally verified in peripheral milling of very flexibletitanium plates.A novel general chatter stability model for multi-degree of freedom systems is introduced in Chapter 8. The variations in the structural dynamic properties along thecutter axis, which is the case in plate milling, are considered. The stability model isChapter 1. Introduction 6analyzed by two different approaches which converge to the same results. The first solution is based on the application of the known periodic system theory to the dynamicmilling model introduced, while the second approach is based on the physics of dynamicmilling formulated. The chatter free axial or radial depth of cuts and cutting speedsare predicted analytically as opposed to being determined using the numerical and timedomain simulation approaches proposed before. The model is verified with experimentaland time domain simulation results for various structures and milling modes includingthe peripheral milling of plates. In addition, the forced and chatter vibrations observedduring peripheral milling of plates are presented in Chapter 8.The thesis concludes with a summary of contributions and suggestions for future workin Chapter 9.Chapter 2Literature SurveyStatics and dynamics of peripheral milling of very flexible webs are highly interdisciplinary. They include theories and methods of metal cutting, milling mechanics,structural mechanics and dynamics, and stability of chatter vibrations in milling. Because of this, the relevant literature is cited and explained in each section throughout thethesis. In this chapter, only a brief review for metal cutting mechanics, milling mechanics and dynamics, dynamic cutting and chatter stability literature is given to provide atheoretical base for the remainder of the work.2.1 OverviewAlthough machining processes have been in use in some form or other since the earlyages, it is only during this century in general and since the mid-forties in particular, thatsystematic attempts have been made to bring this field into a scientific basis. This canbe attributed to several factors which characterize machining processes: unconstrainedflow of material with large strains, high strain rates, high stresses and temperatures,and unusual friction conditions. The comprehensive work by F.W. Taylor [1] on the Artof Cutting Metals published in 1907 was the beginning of serious and systematic studies on the various aspects of metal cutting. However, it was M.E. Merchant’s cuttingprocess model [2] in 1944 that took the remarkable step from the art of metal cuttingto the science of metal cutting. Since then, progress in machining research has beenconsiderable. Many models have been developed towards the understanding of the chip7Chapter 2. Literature Survey 8formation, shearing, plastic and elastic contact, friction and wear mechanisms, the prediction of forces, stresses, strains and temperatures involved in the machining process.Also, extensive research efforts have been spent on the understanding and modeling ofdynamic cutting and cutting stability, in the last four decades. These analyses can befound in several books written on machining and machine tools [3, 4, 5, 6, 7, 8, 9, 10]. Inrecent years, more and more emphasis has been put on the modeling of the machiningprocess because of the increasing demand for untended machining, improved CAD/CAMsystems, advanced process planning, control and monitoring techniques.2.2 Geometry of MillingMilling is a multiple point, interrupted cutting operation. Because of the multipleteeth, each tooth is in contact with the workpiece for a fraction of the total time. Thefinished surface, therefore, consists of a series of elemental surfaces generated by the individual cutting edges of the cutter. Due to the nature of relative contact between theworkpiece and the tool, the chip thickness is not constant but starts with a zero thickness and increases in up-milling and starts with a finite thickness and decreases to zeroin down milling.The early research in milling mechanics [11, 12, 13, 14, 15, 16] dealt with the chipformation mechanism and spindle power estimation. Martelotti [17, 18] showed that thetrue path of the milling cutter tooth is trochoidal, but it can be approximated as circular• if the radius of the cutter is much larger than the feed per tooth. This approximationsimplifies the analysis of the process and, in practice, the necessary condition for the feedper tooth to radius ratio is usually satisfied. Martelotti also derived an expression forChapter 2. Literature Survey 9the amplitude of the tooth marks left on the surface:where Ii is the height of the tooth mark, St is the feed per tooth and R is the cutterradius.2.3 Milling Force ModelsDue to the large number of variables involved in the milling geometry, an abundantamount of data is required for the analysis of milling force and surface finish with empirical techniques [19]. Therefore, the analytical or semi-analytical prediction of millingforces is essential. In early studies, some expressions for the amplitude of the pulsatingcutting force in milling were developed from purely geometrical considerations [14, 13].Salomon [14] based his equation, for the work done with a straight tooth cutter, on theassumption that the specific cutting pressure was an exponential function of the chipthickness. In their analytical milling force expressions, Sabberwal and Koenigsberger[20, 21] used similar exponential specific milling coefficients (both in tangential and radial directions) which are identified experimentally. This approach for the milling forcecoefficients is referred to as the “mechanistic model” which has been adopted by manyresearchers in the analysis of the milling process [22, 23, 24, 25]. In another type of mechanistic modeling, edge milling force coefficients are separated to yield constant millingforce coefficients [26, 27, 28]. In the mechanics of milling models, the milling force coefficients are determined by using an oblique cutting model [29, 27, 30, 31].Milling force and surface generation models can be classified as suggested by Smithand Tlusty [32]. The simplest milling force model is the average rigid force model whichChapter 2. Literature Survey 10assumes that the average power cousumed, torque, tangential cuttiug force aud the dimensional error ou the machined surface are proportional to the material removal rate[33]. This model though cannot provide accurate results as, in general, there is no simple,direct relationship between the material removal rate and the cutting forces and cutterdeflections. For accurate predictions, the cutting forces at the tip of the tooth have tobe considered. In the instantaneous rigid force model, the milling force on the helicalcutting edges is computed. The model of Koenigsberger and Sabberwal [21] was the firstcomplete model in this group. Kline et al. [23, 34, 35] included runout in the millingforce calculations by dividing the end mill into a number of axial elements. Sutherlandet al. [24] and Armarego et al. [36] included the effect of cutter defiections on the chipthickness calculations by using iterative algorithms.The accuracy of the milled surfaces was modeled by Kline et al. [35] by calculatingthe cutter and workpiece deflections at the surface generations points. Montgomery andAltintas [37] nsed a dynamic cutting model to simulate the surface produced by a vibrating end mill. They used true kinematics of milling presented by Martelotti. Ismailet al. [38] included the effect of tool wear and tool dynamics on the surface generation.In addition to the force and surface accuracy predictions, milling force models have beenused extensively in adaptive force control [39, 40, 41, 42, 43] and cutter breakage detection [44, 45] of milling operations.In this thesis, several milling force models are developed to analyze milling forces andsurface accuracy in milling flexible workpieces. The static interaction between the cutterand workpiece deflections and the milling process is modeled in two ways. First, ananalytical milling force model is developed by formulating the chip thickness under theeffect of static cutter and workpiece deflections. It is shown that the effect of defiectionsChapter 2. Literature Survey 11on the chip thickness diminishes very quickly (in a few tooth periods) for static milling.Then, a flexible milling force model, which considers the effect of static deflections onthe cutter-workpiece immersion boundaries, is developed and used for peripheral millingof plates. The model accurately predicts the milling forces and surface errors. It isshown that if the deflections are not used to update the immersion boundaries (as doneby Kline et al. [35]), the predictions are not accurate, especially in peripheral millingof very flexible cantilever plates. In order to generalize the milling force prediction fordifferent cutter geometries, an improved mechanics of milling method is developed. Theaccuracy of the model predictions is found to be satisfactory. The model can be used inmilling cutter design, process planning and CAD/CAM systems.2.4 Stability of Dynamic MillingBoth forced and self-excited vibrations arise in milling operations. Periodic millingforces excite the cutter and workpiece, and may cause resonance. Self-excited chattervibrations occur due to dynamic interactions of the cutting process and structure. Forcedvibrations and chatter stability are particularly important in the peripheral milling ofthin-walled components due to very flexible workpiece and slender end mill. Generally,there are two approaches used in the analysis of chatter in machining: since chatter isundesirable, researchers establish stability limits and consider the cutting process as ablack box; the other approach is to understand the mechanics of the cutting processunder dynamic conditions. In the following sections, a brief review of the literature ondynamic cutting and chatter stability is given.Chapter 2. Literature Survey 12hFigure 2.1: Dynamic cutting process.2.4.1 Dynamic CuttingDynamic orthogonal cutting process is shown in Figure 2.1. The tool is removingchip from an undulated surface which was generated during the previous pass when thetool’s vibration amplitude was z0 (outer modulation or wave removing). Simultaneously,the tool is vibrating with amplitude z (inner modulation or wave generation). The process can be visualized as a superposition of these two distinct mechanisms. There hasbeen extensive research efforts towards the understanding and modeling of the dynamiccutting process in 60s and 70s [46, 47, 48, 49, 50, 51, 52, 53]. An excellent review of therelated literature is given by Ilusty [54].Different mechanisms have been proposed for the dynamic cutting process. Knight[51] observed that the shear angle oscillates during dynamic cutting. Albrecht [46] considered the oscillations in the shear angle to be a part of the chip segmentation (or cycliczoTOOLx WORKPI ECEChapter 2. Literature Survey 13chip formation) mechanism. The shear angle oscillation was considered to be the mostsignificant source of chatter in [55, 56, 7], as it results in oscillations in the cutting forcesand vice versa. The shear angle oscillation is attributed to the variations of the surfaceslope (wave removing) and cutting velocity direction (wave cutting). The variation ofthe rake angle due to a vibrating tool and changing cutting velocity direction can also beconsidered to have negative damping effects on the dynamic cutting system [54]. Also,the variation of the friction coefficient between the chip and the rake face of the tool withthe continuously varying cutting velocity in dynamic cutting may have a small effect onthe shear angle variation [49]. The effects of different parameters on the shear angle oscillation were investigated by Nigm and Sadek [57]. Their results show that the magnitudeof shear angle oscillation decreases as cutting speed, feed and rake angle increase. Thevibration amplitude does not have a significant effect on the shear angle oscillation whichslightly increases with the frequency of the chip thickness modulation [57]. In both wavegeneration and wave removing processes, the clearance angle does not seem to have aneffect on the shear angle oscillation. However, the clearance angle has a strong effecton the stability of the chatter vibrations. Sisson and Kegg [58] formulated the effectiveclearance angle in the dynamic cutting by considering the cutting speed and vibrationvelocity normal to the workpiece. The normal and horizontal flank force componentswere calculated by assuming simple elastic contact between the flank face and the workpiece. These forces were treated as damping forces (process damping), and the dampingcoefficient derived is inversely proportional to the cutting velocity. Sisson and Kegg [58]could explain the high cutting stability at low speeds, but, by this model, they could notexplain the high speed stability. The mechanism behind the process damping is explainedby Tiusty [54]. In Figure 2.2, for a vibrating tool, the variation of the clearance anglebetween the flank face and the cut surface is shown. In the middle of the downward slopethe clearance angle is at minimum, 7mm, and in the middle of the upward slope it is atChapter 2. Literature Survey 14Figure 2.2: Variation of effective clearance angle in dynamic cutting.maximum, 7mar• It has been shown that the decrease of clearance leads to an increaseof the thrust component of the cutting force. Therefore, during the half cycle from (A)to (C) the normal force is greater than that during the upward motion from (C) to (D).This variation of the thrust force is 900 out of phase with displacement, thus it representsa positive damping in the cutting process. The damping coefficient is larger for shortwaves as the slope is steeper. The wave length A of the undulations produced on the cutsurface is A = -, v is the cutting speed and f is the vibration frequency. The processdamping coefficients are usually determined empirically, however, there have been a fewattempts to formulate them analytically [58, 59]. In these methods, the definition ofDynamic Cutting Force Coefficients (DCFC) are used mainly to determine how muchdamping arises in the chip formation process. The effects of outer and inner modulations(wavy surface and vibrating tool) can be superposed by DCFC and the correspondingtransfer function of the dynamic cutting process can be written as follows:= a(Kdz + Ad0Z)= a(Kz+K0z)Az‘minChapter 2. Literature Survey 15where F and F are the normal and tangential cutting forces, a is the width of cut, zis the amplitude of the vibration normal to the cut surface, z0 is the amplitude of themodulations on the surface (which is equal to the tool vibration in the previous pass)and Kd, are direct and cross-inner modulation DCFCs and Kd0,K0 are direct andcross DCFCs for outer modulation. Unlike the static cutting process, in dynamic cuttingthe cutting force coefficients are complex numbers. The real components represent thecutting stiffness whereas the imaginary components are due to the process damping generated in the cutting process. Tremendous effort has been spent in the determination ofthe effects of cutting parameters on DCFCs by using complicated test rigs, some of theseworks are cited here [3, 47, 60, 61, 62, 63, 64, 52, 65]. In most of these works, DCFCswere identified from the controlled dynamic cutting tests as the tool vibration amplitudeand frequency; cutting speed and the tool geometry were varied. The results of thesetests [54] show that the imaginary component of Kd is the largest damping term andhas a special effect on the dynamic cutting process. This also indicates that the processdamping is generated due to the flank contact resulting from vibrations of the tool.is strongly affected by the cutting speed and wearland on the flank face.Periodic cutting forces can cause forced vibrations in milling systems. The Fourieranalysis of the milling forces was done by Gygax [66] and Yellowley [67]. Doolan et al.[68] designed the optimal pitch angles between the milling cutter teeth to minimize forcedvibrations. Tlusty [69] and Smith [70] used process damping in the modeling of dynamicmilling forces. Montgomery and Altintas [37] developed a comprehensive dynamic millingmodel by considering the contact between cutter and workpiece in different zones.Chapter 2. Literature Survey 162.4.2 Chatter Stability ModelsIn his classical paper, On the Art of Cutting Metals (1907) [1], F.W. Taylor statesthe following opinion which is based on the experimental observations:“Chatter is the most obscure and delicate of all problems facing the machinist, and inthe case of castings and forgings of miscellaneous shapes probably no rules or formulaecan be devised which will accurately guide the machinist in taking the maximum cuts andspeeds possible without producing chatter.”Taylor was partly right in that the first comprehensive treatise on the mechanism ofcutting tool vibration by Arnold [71] appeared four decades later than his paper. Arnoldexplained a theory of self- induced vibration based on the decrease in the cutting forcewith cutting speed. If the force-speed curve exhibits a negative slope, this implies a negative damping coefficient in the equation of motion and may lead to instability. Hahn [72],however, pointed out that in general the slope of the force-speed curve is not sufficientlysteep to explain for the self-induced vibration.In the early stage of the machining chatter research, the existence of negative dampingwas considered a necessary condition, and the only source, for chatter to occur. However,it was later recognized that the most powerful sources of self-excitation, regenerationand mode coupling, are associated with the structural dynamics of the machine tooland the feedback between the subsequent cuts. Tlusty and Polacek [73] showed theimportance of the structural dynamics by modeling the machine tool system as a multidegree-of-freedom structure with positional mode coupling. They analyzed the stability ofmode coupling and regenerative chatter mechanisms and obtained the following classicalChapter 2. Literature Survey 17equation for the regenerative chatter stability1aiim= 2KsR[Gjminwhere aiim is the limit width of cut for the chatter stability, K is the specific cutting forcecoefficient and Re[G]mjn is the minimum value of the real part of the structure’s transferfunction, oriented with respect to the cutting force and to the direction of the normalto the cut surface. Later Tlusty [5] improved this formula to include the lobing effectby considering the effect of the spindle speed on the chatter frequency. Tobias and Fish-wick [74] combined the two aspects of the chatter vibrations: the process damping anddynamics of the machine tool structure. They developed a comprehensive mathematicaltheory of chatter of lathe tools by taking into consideration the instantaneous variationin chip thickness, the penetration effect, and the slope of the cutting force-cutting speedcurve. Merrit used the feedback control theory to develop stability lobe diagrams forregenerative chatter. Similar to Tiusty, he also neglected the dynamics of the cuttingprocess. The effect of cutting dynamics on the chatter stability is strong in the slowcutting speed range where the process damping is high.Due to the rotating cutter with multiple teeth and the periodically varying directional coefficients, the dynamics and the stability of milling are more complicated thanthe orthogonal cutting case. That is why, in the beginning, the stability of chatter inmilling was analyzed using the orthogonal cutting-chatter theory [5, 3]. Later Tlusty etal. [75, 76, 77] concluded that the time domain simulation of the milling chatter is thebest method to obtain the stability diagrams. Sridhar et al. [78, 79, 80] formulated themilling dynamics for the straight tooth cutter, and used a numerical algorithm to analyzethe stability of dynamic milling. Minis et al. [81, 82] used Nyquist criterion to analyzethe stability of the milling process and followed a numerical procedure to obtain theChapter 2. Literature Survey 18stability limits. There is no analytical method of chatter stability prediction in millingavailable in the literature.Various active and passive chatter suppression methods have been developed. In order to prevent the full development of chatter, the phase between the inner and outermodulation can be disturbed by using variable pitch milling cutters [83, 84] or variablespindle speeds [85, 86, 87]. Smith [88] used the chatter sound spectrum for the on-lineselection of the spindle speed to utilize the high stability pockets in the stability lobes.Nachtigal and Srinivasan [89, 90], Shiraishi and Kume [91] and Liu [92] developed feedback controllers for the control of chatter in turning. Vibration dampers have also beenused in chatter suppression [3, 93].In this thesis, a comprehensive dynamic milling model is developed by consideringthe dynamic displacements of the workpiece and cutter. Unlike the point contact modelsconsidered in the previous milling chatter research [3, 5, 78, 94, 81], the dynamic interaction between the cutter and workpiece is modeled along the axial direction by consideringthe variations in the dynamics of structures in this direction. A novel stability analysis,which is based on the physics of dynamic milling, is given. The resulting stability equations are solved analytically by obtaining a relationship between chatter frequency andspindle speed. The general theory is applied to several common milling cases such as theperipheral milling of flexible workpieces. Analytical stability conditions are derived foreach case. The analytical solutions are verified by the time domain simulation results.Chapter 2. Literature Survey 192.5 Peripheral Milling of Flexible StructuresIn end milling operations, a flexible structure may be defined as a workpiece whoseflexibility is significant as compared to the cutter and machine tool. The peripheralmilling of flexible components is a commonly practiced machining operation, especiallyin the aerospace industry [95, 96], used in: machining of thin webs, jet engine components(such as impeller or blisk blades), instrument housings, microwave guides etc.Kline [35] considered milling of a clamped-clamped-clamped-free (CCCF) plate witha flexible end mill. He used the Finite Element Method to model the plate. The interaction between the milling forces and the static and dynamic structural displacementswere neglected in his study. Therefore, the model can only be used for the static millingof relatively rigid workpieces. Montgomery [97] and Altintas et al. [98] modeled thedynamic peripheral milling of a plate by employing the true kinematics of the millingwhich is trochoidal. The end mill was assumed to be rigid and the plate was modeled bythe Finite Element (FE) method. The dynamic milling forces and the detailed surfacefinish were obtained by considering the dynamic regeneration in the chip thickness. Dueto the fact that the plate displacements were obtained by a FE package, off-line withthe developed computer program, and long computer run times, the process is simulatedonly at the middle of the plate, along the feed direction. Sagherian et al. [99] improvedKline’s model by including the dynamic milling forces and the regeneration mechanisms.However, they did not consider the effect of tool and workpiece deflections on the cuttinggeometry, i.e. the radial depth of cut. They also used a numerical force algorithm and theFE method to simulate cantilever plate displacements. Anjanappa et al. [100] showedthe imprints of high frequency vibrations on thin rib milling.Chapter 2. Literature Survey 20As a summary, the peripheral milling of plates has been modeled by numerical algorithms. The dynamic regeneration mechanism has been included in some of thesemodels, however, the complete static and dynamic interaction of the milling process andstructural displacements have not been investigated. Also, the stability of milling veryflexible workpieces has not been studied.In this thesis, the interaction between the flexible plate and cutter is accurately modeled. The variation of the immersion boundaries under the deflections is considered.Finish dimensions of the plate are accurately predicted by an integrated Finite Elementscutting process model. In order to achieve prescribed tolerances, the peripheral millingof plates is planned by scheduling feedrates. The simulation system is experimentallyverified. Also, a novel stability model of milling multi degree-of-freedom systems is developed. The model is applied to the peripheral milling of plates and is verified.Chapter 3Structural Modeling of the Workpiece and End Mill3.1 IntroductionFlexible cutter and workpiece structures deflect and vibrate under the milling forces.If not controlled, static deflections cause dimensional surface errors which may violate thetolerance requirements on the machined surfaces. The cutter and workpiece vibrations,on the other hand, result in poor surface quality, chipping of the cutting edges, microcracks on the finished surface and may even damage the machine tool if the self excitedchatter vibrations become excessive. The effect of structural deformations on the cuttingprocess needs to be investigated for the prediction of cutting conditions which result inthe required dimensional accuracy in milling flexible structures. Here, structural models of the flexible workpiece and tool are studied to analyze the milling of very flexibleworkpieces.The flexible workpiece is modeled as a cantilever plate since it represents the mostextreme case for the family of very flexible aerospace components, such as the impellerand rotor blades. Also, due to its very high flexibility, the cantilever plate is a very goodchoice for analyzing the interaction betweell structural deformations and the cutting process. The thickness and the structural properties of the plate continuously vary due to theremoval of metal during machinillg. Because of the continuous variation of the structural21Chapter 3. Structural Modeling of the Workpiece and End Mill 22properties, and the interaction between the milling process and the structural deformations, the structural and milling force calculations have to be performed together, in thesame computer program. Previously, Kline et al. [35] used a commercial Finite Elements(FE) software to calculate the workpiece (clamped-clamped- clamped-free, CCCF, plate)defiections under the milling forces by neglecting the interactions between the deflectionsand the milling forces. Altintas et al. [98] used a commercial FE package too in studyingthe dynamic milling of a cantilevered plate at the middle section only, so that they didnot have to consider the variation in the plate dynamics. However, they considered thetrue kinematics of milling to predict dynamic deformation marks left on the plate surface.Kline et al. [35] and Sutherland et al. [24] used a beam model for the tool deflectionsby defining an equivalent length to account for the clamping stiffness at the collet. Inthis study, the plate and tool deformations and the milling forces are calculated in theintegrated algorithms developed for plate milling. A beam model with linear springs atthe fixed end (to account for the collet stiffness) is used for the deflection analysis of theend mill. This model closely represents the real structure and gives satisfactory results.In the following, dynamic and static models of the plate and tool are given. These modelsare used in Chapters 5,6,7 and 8 to analyze the static and dynamic interactions betweenthe milling process and the structures, to predict and control dimensional surface erroron the plates, and to study the stability of self excited chatter vibrations.3.2 Structural Modeling of WorkpieceThe workpiece considered in this study is a cantilever (clamped-free-free-free CFFF)plate. The thickness of the plate is reduced continuously during machining. Thereare a number of methods in the literature for the deflection analysis of plates [101,102]. The accuracies of different methods in the analysis of plate problems depend onChapter 3. Structural Modeling of the Workpiece and End Mill 23the type of boundary conditions, loading, homogeneity of the thickness and magnitudesof the deflections considered. There exist exact analytical solutions for some specialboundary conditions and loading types. Unfortunately, there is no closed form solutionfor cantilever plate problems. In the following, Plate Theory will be briefly reviewed.Series solutions for the cantilever plate deflection are formulated. Due to the steppedthickness and nonuniform loading during milling of the plates, the series methods donot provide accurate results. Thus, the Finite Elements Method (FEM) is used in thedeflection analysis of the stepped plate. For the constant thickness plate, however, seriessolutions may give acceptable results. This applies to the negligible step size which is thecase in some finishing operations. For these cases, the series solution may be preferred, asit is faster than the FEM. On the other hand, the FE can be used for different boundaryconditions and workpiece geometries and provide accurate results for different types ofloading. In the following sections, the FE formulation and the developed algorithms areexplained. The dynamic analysis of the plate structure is performed by the FEM as well.The developed models are experimentally verified before being used for milling processsimulation.3.2.1 Bending of Thin PlatesThe basic theory of thin plate bending, so-called Kirchhoff/Love Theory, will bereviewed. In this theory, it is assumed that the points on the mid surface of the platemove only in the z direction as the plate deforms in bending, and a line that is straightand normal to the mid surface before loading is assumed to remain straight and normalto the mid surface after loading. The theory is applicable to the cases where the platedeflections (w) are less than half-plate thickness (t/2).Consider the differential plate element shown in Figure 3.1. In order to be consistentChapter 3. Structural Modeling of the Workpiece and End Mill 24Mx+MxxdXdy-/ M÷Mxy,xdxPrFigure 3.1: Internal forces and moments on a differential plate element.with the notation used in the plate literature, the z axis will be taken along the thicknessof the plate. The moments, Mr, M and the shear forces, q and qy, their differentialvariations and the load distribution in z direction, p(x, y), are shown on the element. Ifthe plate deformations are small (w < t/2) and if there are no loads at the boundariesin the plane of the plate (x—y plane), then the in-plane forces can be neglected. Thetransverse shear deformation is assumed to he zero. The first step in the solution is themoment and force equilibrium equations,OM 81VI+ pgPM 8M— (3 1q= —p(x,y)M Mx,uI Ip(x,y)s÷y,vMdyChapter 3. Structural Modeling of the Workpiece and End Mill 25Undeflected plate. XDeflected plateFigure 3.2: Displacement of a point P to F’ due to bending of plate.from which the following can be obtained:82MX 321tJ 82M+ 2 DxDy + 02 = —p(x, y) (3.2)Figure 3.2 shows the displacement of a point F, which is not on the middle plane dueto the bending of the plate. u0(x, y) , v0(x, y) and w(x, y) are the displacements of themiddle surface in x, y and z directions, respectively. Assuming that the straight linesnormal to the middle surface of the plate before bending remain straight and normal afterbending, (u(x, y) and v(x, y)), the displacements of the point F in x and y directions,can be obtained as followsU = U0—ZW(3.3)V = Vo—ZWywhere w = and w = 8?t. From the equations of the linear elasticity,9v lOu Ovw(x,y)uo-z AX(3.4)Chapter 3. Structural Modeling of the Workpiece and End Mill 26assuming that the mid-plane does not stretch,- and - terms drop out and thefollowing is obtained:= U3: = — ZtIisxEy = Vy = ZWyy (3.5)——r V3:) — —ZW3:The stress-strain equations reduce to the following form for plane stress (o = 0) andisotropic, homogeneous and linear elastic material,= E + vo= e,,E + vo (3.6)E—where v is the Poisson’s ratio and E is Young’s Modulus of Elasticity. If the strains givenby equation (3.5) are substituted in the stress-strain relations given by equation (3.6) thefollowing is obtained:0•3:o.y (3.7)o.z,yThe moment expressions can be obtained by integrations of the stresses along thethickness of the plate,t/2M3: JaxzdzM= £/2uzdz (3.8)t/2M3: I uzdzJ—t/2from which the following is obtained:M3: w + vwEt3‘‘ 12(1—i,) (3.9)IvI (1—Chapter 3. Structural Modeling of the Workpiece and End Mill 27where t is the thickness of the plate. The following governing equation of plate bendingis obtained if the moments given by equation (3.8) are substituted in the equilibriumequation (3.2),84w 94w ö4w—+2 +-—=- (3.10)8x4 0x28y 9y4 Dor= (3.11)where .cç7” is the biharmonic operator and D is the flexural rigidity and given byD= 12(1_v2) (3.12)The solution of equation (3.10) gives the deflection of the plate, w, under the specifiedload, p. The boundary conditions are listed below:1. Fixed (clamped) edge : Displacement and slope are zero at the edge, i.e. w = 0and = 0 on boundary C, where n is normal to C.2. Simple supported edge: Displacement and moment are zero, i.e. w = 0 andM = —D(w + vw33) = 0, where s is tangent to boundary C.3. Free edge: Moment and combined-shear are zero, i.e.M = 0 and q = —D + (2 — v)ww33 = 0.The exact solution of equation (3.10) exists for a few boundary conditions andloading cases. As an example, consider a simple supported plate with dimensions (axb)and sinusoidally varying loading p = Po sin sin A solution of the form w =w0 sin sin -, which satisfies all the boundary conditions, is assumed and substitutedin equation (3.10) to determine the unknown coefficientPowo— 1 1Dir4(-+)2Chapter 3. Structural Modeling of the Workpiece and End Mill 28This result can be used to find a solution for different loads on a simple supportedrectangular plate. The method is called the Navier solution in which the load is expressedby a double Fourier sine series, i.e.p(x,y) = pmnsinmSi11n (3.13)m=1n=1 awhere4 a b . irx. ryPmn= —J J p(x,y)sinm—sinn----dxdy (3.14)ab x=O y=O a bThen, the solution is given by00 00 Pmn . 7rx . Kyw(x,y) = 2 2 sin rn—sinn--- (3.15)m=1n=1DW4(+’) aAnother special case is the plate with two opposite sides simple supported and anycombination of boundary conditions on the other sides. For this case, the Levy methodis used for the solution. In this method, the homogeneous and the particular solutions ofequation (3.10) are determined separately. The unknown coefficients in the homogeneoussolution are determined from the boundary conditions. These methods cannot be usedin the analysis of the cantilever plate. The numerical solutions for the cantilever plateare explained next.3.2.2 Series Solutions for Cantilever PlateThe numerical methods for the plate deflection analysis discussed in this section arebased on a series type solution. Their accuracy depend on boundary conditions, typeof loading and the approximating functions used in the series. In the following, the• Galerkin and Ritz formulations are given for the cantilever plate. Although it is shownthat accurate results cannot be obtained for the stepped plate by the series solutions,the accuracy is acceptable for the constant thickness plate. This may be an acceptableChapter 3. Structural Modeling of the Workpiece and End Mill 29approximation for some finish milling operations where the radial depth of cut is verysmall.The Galerkin Formulation for the Cantilever PlateThe general form of the approximate solution is in the form ofN=c1X(x)Y(y) (3.16)where N is the total number of terms considered in the series, X(x) and Y(y) are admissible functions for plate defiections. Q sign indicates that the solution is an approximation.The Galerkin formulation requires that at least the kinematic boundary conditions besatisfied by the approximation functions [103]. The order of the kinematic boundaryconditions, m, is determined by the order of the original differential equation (n), i.e.m = n/2 —1. In the case of the plate problem, n = 4 thus m = 1. Therefore, the approximation functions should satisfy the displacement and the slope boundary conditions. Inthe Galerkin weighted residual statement, the unknown coefficients, c, are determinedsuch that the total domain and boundary weighted error introduced by the approximatesolution is minimum. This is stated by the following weighted residual statement for aplate with dimensions (axb):jajbRWdXdY+JRW0 j=1,2,...,N. (3.17)where RD and RB are the domain and boundary residuals, Wj and W3 are the domainand boundary weighting functions.The displacement and the slope should be equal tozero at the clamped edgeth =0(3.18)Chapter 3. Structural Modeling of the Workpiece and End Mill 30The following zero moment and zero equivalent-shear stress boundary conditions apply to the free edges(w + = 0(3.19)D(w + (2— = 0x=O, aAssume that the loading due to the cutting force in the z axis1 is represented by aline force applied at the tool position, x =p(x, y) = F(y)6(x — x1) (3.20)where 6(x) is the Kronecker delta function which specifies the location of the line forceand the F(y) is the normal cutting force distribution along the y direction of the plate.The domain residual becomesRD = D 4 t — F(y)6(x — x1) (3.21)and the boundary residuals evaluated at the boundaries:RBr —D + (2—+ +x=O,a (3.22)— D (ti + v) + (z + (2 — v)th)y=bThe residuals can be substituted in equation (3.17) to determine the coefficients, c.This formulation was followed and the resultant set of equations were solved by computer.Free-free and cantilever beam eigenfunctions (vibration mode shapes) were used for the1n order to be consistent with the plate formulation, the normal axis was chosen to be z. In fact,the cutting force in this direction is F as defined in the cutting force model.Chapter 3. Structural Modeling of the Workpiece and End Mill 31Table 3.1: Eigenfunction parameters for clamped-free and free-free beams.rn am m m1 - 0.734- 1.8752- 1.019- 4.6943 0.982 1.0 4.73 7.8564 1.0 1.0 7.853 11.05 1.0 1.0 11.0 14.1376 1.0 1.0 14.137 (2m — 1)K/2approximation functions, X and Y, respectively [104]:Xi =1X2() = 1—2wXm() = cosh EmX + cos mX (3 23)— am(sinh + sin + sin m) (m = 3,4, ...)Ym(V) = coshm—cosm—(m=1,2....)where = and = . The constants in equation (3.23) are given in Table 3.1 for thefirst six modes.In the Galerkin method, the weighting functions are chosen to be the approximationfunctions themselves, i.e. Wj = However, the resulting weighted error expressionsshould be consistent from the energy point of view. The boundary residuals in equation(3.21) include both shear force and moment residuals. Therefore, the moment shouldbe multiplied by the rotational displacements or slopes, w and w. When it is doneso, the formulation becomes exactly the same as the Ritz energy method. The case ofthe stepped plate will be discussed before the Ritz formulation is given. One possiblesolution for the stepped plate is to consider it as a combination of two plates with differentChapter 3. Structural Modeling of the Workpiece and End Mill 32thicknesses. Thus, two different domains with two different solution functions, say w1and w2, have to be considered in the formulation. These two solution functions shouldsatisfy the continuity equations at x = x1. The first two simple continuity requirementsare the continuity of displacement and slope:= w2(x=x1)(3.24)wi(x = x1) = = x1)These are kinematic conditions for the plate deflection formulation, so they have to besatisfied by the approximating functions before a residual statement can be written.However, it is not a straightforward task, if it is not impossible, to determine the typeof the functions that would satisfy these conditions. Also, the solution procedure withadditional continuity residuals becomes very lengthy. Thus, the solution of the steppedplate problem with the Galerkin formulation is not practical and it is dealt with theFinite Element method.Ritz Energy MethodThe strain energy stored during the bending of a plate is given by [101]:Dabu = j j [(w + w)2 — 2(1 — v)(ww — w)] dxdy (3.25)The work done on the plate by the external force, p, is= jafp(xy)wdxdy (3.26)The potential energy of a system, H, is defined as the total internal and externalwork done in changing the configuration from a reference state to the displaced state.Then, it is given byll,=U+V (3.27)Chapter 3. Structural Modeling of the Workpiece and End Mill 33The approximate solution given by equation (3.16) is to be used in the energy statements. Then, the coefficients, c, are determined by using the principle of stationarypotential energy which states that [105]:“Among all admissible configurations of a conservative system, those thatsatisfy the equations of equilibrium make the potential energy stationary withrespect to small admissible variations of displacement.”Therefore, each approximation function, X and , must be admissible; that is, eachmust satisfy compatibility conditions and essential boundary conditions. Then, accordingto the principle of stationary potential energy, the equilibrium configuration is definedby the N algebraic equations9II 9U 8V=—+=0 (n=1,2 ,N) (3.28)aSubstituting the series solution form given by equation (3.16) and the line force appliedat x = x1 defined by equation (3.20) into equations (3.25-3.26) the following is obtained:= D ja [ + xiYi”xjYj,,+ 2vX”YX1” + 2(1 — v)XY’X’] dxdy(3.29)=_j0:1cixi= —j F(y)cX(xi)1dyj=1where (F) indicates differentiation with respect to x or y. The following matrix equationis obtained when U and V are substituted in equation (3.28):[k] {c}= {f} (3.30)Chapter 3. Structural Modeling of the Workpiece and End Mill 34where the elements of the stiffness matrix, [k] and the force vector f are defined as— Db f j X1”X7YiYj + r”1”XX3+ 2r(1 — v)X i’’XY’ + vr (x’YxY” + xi”x”) Jf = X(1)jF(y)Yjdwhere ra = b/a is the aspect ratio of plate, and = x/a,g = y/a and = xi/a. Inorder to make the integrals in equation (3.31) independent of the plate dimensions, thedifferentiations indicated by (i) are performed with respect to and . The cantilever andfree-free beam vibration modes given by equation (3.23) are used for the approximatingfunctions X and Y. In both directions, the first six modes are included in the series. Afterthe integrals are evaluated numerically, the matrix equation (3.30) is solved to obtain thecoefficient vector {c}. The coefficients and the approximating functions are then used inequation (3.16) to determine the displacements of the plate at different locations. Thisis programmed on computer and the program listing is given in the internal report [106].Alternative extended Simpson’s rule [107] (page 115) is used to evaluate the numericalintegrals. The force F(y) should be known at the integration points along the y axis todetermine the force vector. However, if the force is a point load or uniform loading, thenF(y) moves outside of the integration.3.2.3 The Finite Element Modeling of Plate DeflectionsThe Finite Element method (FEM) is superior to the other presented numerical platedeflection solutions due to several reasons. First of all, different workpiece geometriescan be modeled by the FEM. In addition, the variable workpiece geometries, such as astepped plate, can easily be handled by the FEM. In this section, the FEM formulationsare given for the 3 dimensional (3D)-isoparametric solid element and the structure of theChapter 3. Structural Modeling of the Workpiece and End Mill 35developed computer program is explained.Structural Formulations for 3D Elastic SolidBecause of the stepped thickness the workpiece is modeled as a 3D elastic structure.The stiffness matrix and the force vector for the 3D structure are given by [105]:[K]=(3.32){Q} = j[N]TFdV+{F}where the structure is assumed to be free of prescribed surface tractions and[K] stiffness matrix of the structure[B] strain-displacement matrix for 3D elastic body{ u} 3D stress vector{ Q} consistent load vector[N] matrix of shape functions{F} body forces{ P} concentrated forces on the nodes[E] elasticity matrix(1—i’) 1’ 0 0 0xi (1—v) v 0 0 0xi ii (1—v) 0 0 0[E] = E’ (3.33)0 0 0 1—2v 0 00 0 0 0 1—2xi 00 0 0 0 0 1—2vChapter 3. Structural Modeling of the Workpiece and End Mill 36where= (1 + v)(l — 2v) (3.34)E and v are the modulus of elasticity and the Poisson’s ratio. The strain-displacementmatrix [B] relates the strains, {e}, to the displacements of the structure, {z.}{} = [B]{} (3.35)wherea 0 0n 0U000[B]= a a (3.36)7xya a7yz 0 — y—a a7zx ‘T U 7Taz axU{z} = (3.37)zwhere u, v and w are the displacements in x, y and z directions. As [B] includes onlyfirst order derivatives, the shape functions should have C° continuity2.This requirementis satisfied by the linear shape functions. The convergent rate is determined from thefollowing expression [105]:CR 1/2(P+l_m/) (3.38)where p is the order of the polynomial used for shape functions, m is the order of the integrand in the definition of the stiffness matrix given by equation (3.32) (in = 2 for elasticsolid) and n is the number of finite elements. For linear shape functions the convergence2Jj general, cr continuity means that the shape functions have continuous r1 order derivativesChapter 3. Structural Modeling of the Workpiece and End Mill 37rate is 1/n2. In FEM, the shape functions N are used to express the displacements(u, v, w) in terms of the nodal displacements, {S}. The linear 3D solid element has 8nodes, thus 8 shape functions. The same shape functions are used in three directions:ENjuiN1 0 0 N2 ... N8 0 0{z}= = 0 N1 0 0 N2 ... N8 0 {6} (3.39)8 ri A A A 7T> U U LV1 U U 1V2 •.. tV8N1wwhere {5T} = (u1,v1,w1 , U8,v8, wg). From equations (3.35, 3.36) and (3.39) thefollowing expression is derived for the matrix [B]:ON1,-ON ON8V V ...o ON1 0 0 ON2 ON8 0--ON1 ON ON8ri— 0 0-—0 040— ON1 ON1 ON8 ON8---- V ...----ON1 ON1 0 0 ON8 ON8---- ...----Element FormulationThe 3D isoparametric element (8 node-brick) is shown in Figure 3.3. Unlike the3D cubic element, the isoparametric element can have different edge lengths and anglesbetween the edges. This is, in general, necessary to model three dimensional geometries such asa stepped plate, a variable thickness or a variable length plate, withoutincreasing the number of elements significantly. The same element can also be usedfor curved geometries like impeller blades. As it was discussed in the previous section,an 8 node-3D solid element (also referred to as trilinear isoparametric element or 3DChapter 3. Structural Modeling of the Workpiece and End Mill41Figure 3.3: 3D isoparametric solid element.x38isoparametric-linear tetrahedron) is the simplest 3D element which satisfies the continuity requirements. It is simpler to formulate and faster to compute compared to higherorder elements like a quadratic solid element. The quadratic solid element has 20 nodes(corner nodes and a node on each edge), and a stiffness matrix of (60x60) whereas alinear-8 node solid element has a (24x24) elemental stiffness matrix. On the other hand,the convergence rate with the quadratic element is 1/n4, which is obtained at thecost of longer elemental computations (and formulations). In the developed computerprogram, the 8 node-isoparametric solid element was used to model the plate deflections.In the following, the formulations and the program algorithm will be given for this typeof element. However, the formulation of quadratic or higher elements are quite similar,and can easily be implemented in the computer program.y376z5Mathematically, it is very difficult to deal with the irregular element shape shown inChapter 3. Structural Modeling of the Workpiece and End Mill 3941Figure 3.4: The 3D solid element in the natural coordillates.Figure 3.3. Thus, another coordinate system is introduced, natural coordinates (, i, ),in which the element looks like a cube, as shown in Figure 3.4. Two coordinate systemsare related to each other by mapping. In isoparametric mapping the global coordinates(x, y, z) are related to the natural coordinates as follows:x =xN1(,C) , y = yN(,C) , z = zN(,C) (3.41)where (xi, y, z) are the nodal coordinates in the global coordinate system. The shapefunctions (Ni) are used to define both the coordinates aild the displacements inside theelement, that is why this element is called isoparametric. The serendipity shape functions[105] are used in the natural coordinates where (—1 > , , < 1):1137CChapter 3. Structural Modeling of the Workpiece and End Mill 40N1 =N2 =N3N4 (1-e)(1+)(1+C) (342)N5 =N6 =N7 =N8 =A shape function N has a unity value oniy at the node i, and zero at the other nodesof the element. The stiffness matrix defined by equation (3.32) should be calculated inthe natural coordinates as follows:[Ke] = Je[B1T[E ]dxdydz= I L1 f[B]T[Ej[B}det[J]dddC (3.43)where [Ke] is the elemental stiffness matrix and [J] is the Jacobian of the transformationwhich will be defined in the following formulation. The determinant of [J] can be regardedas a scale factor that yields volume in the above integral from dxdydz to ddiid [105].The derivatives of the shape functions with respect to the global coordinates,etc.)are required to calculate the element stiffness matrix, i.e. in the strain- displacementmatrix [B] (equation 3.40). The derivatives in the natural coordinates can be written asfollows by using the calculus chain rule:a aa_[i a (344a aChapter 3. Structural Modeling of the Workpiece and End Mill 41ôx ô?;rjl_ öx ôy OzLiOx 8?;! OzThe derivatives of the global coordinates with respect to the natural ones can beobtained from equation (3.41):88 8N 88 8N8 8N 8 0N8 8N 8 8N88—1L8 8where [L] = [J]’. Then, the derivatives of the shape functions in the global coordinates8N--=[L]8N--It should be noted that the derivatives of the shape functions in the natural coordinates can be obtained analytically,ON1= -(1 - )(1 + C); = -(1 - )(1 + ()where [J] is the Jacobian matrix of the transformation and is given as:(3.45)8 IiT.Oz —Equation (3.44) can be inverted to obtain the following:(3.46)(3.47)are determined as follows:8N--8N--(3.48)Chapter 3. Structural Modeling of the Workpiece and End Mill 42and the others are similar. The Jacobian [J], however, has to be inverted numerically.Finally, the elemental stiffness matrix can be obtained by integrating equation (3.43)numerically using the Gauss quadrature:,1 r1 tlj J J [B]TE][B]det[J]dd?]d(—1 —1 —1(3.49)m m m= YEHiHjHk ([B]T[E][B]det[J])ki=lj=lk=1 ‘where m is the number of sampling points, H are the weighting for the i point and(i, ‘ii C) is the coordinate of the sampling point. In the above expression, det[J] and[B] have to be calculated at the sampling points. The Gauss quadrature is used in theabove integration, as a polynomial of degree (2m— 1) is integrated exactly by rn-pointGauss quadrature. In FEM, usually the degree of det[J] is considered in the selectionof the number of sampling points for the Gauss quadrature. This is the minimum orderrequirement for convergence. The idea is that as the elements become very small, theyonly have to represent constant stress, so [B]T[E] [B] goes outside of the integration.det[J] has quadratic terms in the natural coordinates; therefore, 2 point integration ineach direction is necessary. In three dimensions, 2x2x2 integration is performed withintegration points F(±1/v”, +1/’, +1/i/) and weighting of H = 1.0. The procedureto obtain the elemental stiffness matrix can be summarized as follows:• Calculate the Jacobian [J] given by equation (3.45) at the Gauss integration pointsPi.• Invert [Jj numerically to obtain [L].• Calculate the strain-displacement matrix [B] (equation 3.40) at P.• Calculate the elemental stiffness matrix:Chapter 3. Structural Modeling of the Workpiece and End Mill 43[Ke]=( [BIT[E] [Bidet [J] )This element has been observed to be too stiff in bending. This is due to the parasiticshear [105] which is created because of the linear shape functions. This can be improvedby adding the missing modes as internal freedoms [105]:+ (1 —2)a + (1— 2)a + (1—v = Nv+ (1 —2)a4+(1 —2)a5+(1 —2)a6 (3.50)w = ENw + (1—2)a7+ (1—2)as+ (1—where the a, are the nodeless degree of freedom. The following elemental equation isobtained with the inclusion of internal freedoms:[Kel(24,) [1(eal(24,9) = {Q}(24) (3.51)[Kae](9,24) [Ka](gg) I {a}9 J I {0} Jwhere {a} is the nodeless degrees of freedom, and the matrices [Ka], [Keel and [Keelcontain the quadratic terms. The size of the new strain-displacement matrix with thequadratic terms is (6x33). The nodeless degree of freedom, {a} can be eliminated fromthe above equation by static condensation [105]. The resultant matrix equation is asfollows:[K]{S} = {Q} (3.52)where the new elemental stiffness matrix is given by[K] = [[Ke] — [Keal[Ka]’[Kaei] (3.53)Chapter 3. Structural Modeling of the Workpiece and End Mill 44The new element with the quadratic terms gives more accurate results.Finite Element ProgramThe formulation of the elemental stiffness matrix has been given. The stiffness matrix is the most important part of the FEM, however there are many other functionsin the Finite Element program developed. In this section, description of the differentsubroutines in the developed program will be given.The program flow chart is shown in Figure 3.5. The input data is read from a filewhich contains the plate geometry, number of elements in each direction, material constants, location of the cutter along the feeding direction and the cutting conditions. Eachsubroutine will be briefly described in the following.MESH: Generates the finite element grid data, i.e. coordinates of the nodes and thenode numbers for elements. It was specifically written for milling of cantilever plates. Itreads the plate dimensions, the number of elements ill each direction, the location of thecutter and the radial depth of cut from the input file. An uniform mesh is generated inx and y directions, i.e. the element size is constant which is equal to the length of theplate in this direction divided by the number of elements. In the z direction, however,the thickness of the elements are reduced in the cut portion of the plate, which is determined from the location of the cutter. The variable plate thickness can be handled. Thissubroutine can be modified to model plates with different boundary conditions.LAYOUT:Reads all finite element grid data.STIFF: Generates the stiffness matrix for a 3D isoparametric-linear tetrahedron.Chapter 3. Structural Modeling of the Workpiece and End Mill 45Figure 3.5: Flow chart of the developed Finite Element Program.Chapter 3. Structural Modeling of the Workpiece and End Mill 46Calls a number of subroutines to evaluate the derivatives of the shape functions, the Jacobian matrix and its inverse and the stress-displacement matrix. The improved stiffnessmatrix given by equation (3.53) is returned to the main program.SETUP: Places elemental stiffness matrix in the correct position in the total globalstiffness vector. Due to the large size and symmetric-banded nature of the global stiffness matrix, only the terms inside the half-band are placed in a vector which makes thestorage possible.DISCRD: Eliminates the elements from the global stiffness vector for homogeneousboundary conditions.DFBAND: Solves the system of linear equations to obtain the displacements. Theforce vector contains the lumped forces on the relevant nodes in the cutting zone whichis explained in detail in Chapter 7.EXPAND: Expands the solution vector back to gross size by placing zeros wherevariables were eliminated by DISCRD.3.2.4 Dynamics of Cantilever PlateThe dynamic response characteristics of the plate are necessary in the analysis offorced and self excited chatter vibrations. For that, natural modes of the plate should bedetermined. Although for relatively rigid workpieces the dynamic modes of the machinetool have to be studied (usually by experimental modal analysis) for the chatter prediction, due to the very high flexibility of the plate, the milling machine table on which theplate is mounted can be assumed to be rigid. Cantilever plate vibration modes can beChapter 3. Structural Modeling of the Workpiece and End Mill 47determined by series methods [104]. The vibration modes of stepped plates have beenconsidered in many studies [108, 109, 110, 111, 112]. However, as the FE formulation isalready used for the static deformations of the plate, dynamic modes can be calculatedby the same method as well. Also, vibration modes of plates with arbitrary dimensions,e.g. varying thickness and length, milled only certain portion of the length (due to chatter limitations) etc., can easily be determined by the 3D element based FE analysis.Therefore, the mass matrix by FE formulation is given next.Finite Element Model for Plate DynamicsThe mass matrix for an elastic structure is given by [105]:[MJ= Jp[N]T[N]dV (3.54)where [N] is the shape function matrix and p is the mass density. Similar to the stiffness matrix formulations, the elemental mass matrix can be integrated in the naturalcoordinates by using the Gauss quadrature:[Me]= ffj p[N]T[N]det[J]dddC = (p[N]T[N]det[J]) (3.55)where P are the Gauss points. The procedure in the program is the same as for thestiffness matrix. Again, a vector is used to store the global mass matrix which is banded.A library subroutine which uses Power’s iteration method is used to determine the eigenvectors (mode shapes) [q] and eigenvalues {w2} of the structure. The mode shapes canbe normalized to obtain a unity modal mass which is useful in modal analysis:{qr} = {r}/mr(3.56)mr = {r}T[M]{qr}Chapter 3. Structural Modeling of the Workpiece and End Mill 48iN1 2Figure 3.6: One element example to test the developed FE Program.where {q.} is the rth mode shape. The modes of the plate must be updated as the millingcutter advances in the feeding direction and the thickness of the plate is reduced.3.2.5 Simulation and Experimental ExamplesSeveral simulation and experimental results are presented for static and dynamicplate displacements. The developed FE program has been tested for different cases. Forthat, the results obtained from the program were compared with the ones obtained fromANSYS. A one-element example is given here. Consider the cubic geometry shown inFigure 3.6. The material is steel with E = 207 GPa, ii = 0.3. Two (iN) nodal forces areapplied at nodes 3 and 4 as shown in the figure. The displacements of the nodes in the zdirection as obtained from the developed FE program (by using the trilinear stiff elementand the improved 3D element with internal-nodeless degrees of freedom) and ANSYS aregiven below in (m):Chapter 3. Structural Modeling of the Workpiece and End Mill 49Node FE (trilinear) FE (improved) ANSYS1 0.581395D-09 0.72096311-09 0.72096251D-092 0.581395D-09 0.72096311-09 0.72096251D-093 0.78249311-09 0.949195D-09 0.94919485D-094 0.78249311-09 0.949195D-09 0.94919485D-09The stiffness of a cantilever steel plate has been measured at the middle of the freeend. For this, the plate was mounted on a dynamometer on the milling machine tablewhich was given small displacements while the tip of a bar clamped in the collet wastouching the plate at the middle of the free end. Therefore, the point force appliedon the plate was measured as a function of the displacement at the same point. Theplate dimensions were (63.5x63.5x3.81 mm). The displacement range of (0-100 tm) wascovered at 5 steps. The stiffness was calculated by linear regression askexp = 610N/mmThe stiffness at the same point was calculated by FE and Ritz methods by applying a• point load at the middle of the free end. (16x16) elements were used in the FE solutionand the first six functions were taken in the series in the Ritz solution. The followingstiffness values were obtained from two methods:kFE = 714 N/mm ; k2 = 800 N/mmTherefore, the FE solution is closer to the measurements. The difference of about 15%(which is more than 30% for the Ritz solution) between the experimental value andthe FE result can be attributed to the measurement error, material uncertainties andunmodeled surface residual stresses left form the grinding of the plate. The differencebetween the FE and Ritz solutions becomes larger for a stepped plate. Consider the casein which the thickness of the plate is reduced from 3.81 mm to 1.905 mm in the 3/4 ofChapter 3. Structural Modeling of the Workpiece and End Mill 50the plate (between x = 0 and x = 48 mm). In the Ritz solution, an average thicknesscan be used. In order to be consistent with the energy approach , the average thicknesscan be defined asta = /(3 * 1.905 + 3.81)/4 = 2.67mmas the strain energy is proportional to the thickness of the plate. The stiffness at x =48, y = 63.5 mm (on the free edge, at the step location). The FE and Ritz solutions givekFE = 245 N/mm ; k2 = 281 N/mmIf an arithmetical average of the thicknesses is used in the solution, then kRt = 200N/mm is obtained. This shows that the variation of the plate’s stiffness during themachining is quite significant and has to be considered in the analysis. Comparing theresults obtained from the FE and Ritz methods it can be concluded that the FE givesmore accurate results than the Ritz solution.Natural modes of a cantilever plate are considered next. The plate has the dimensionsof 63.5x44x3.81 mm. The plate material is Ti6A14V titanium alloy with volumetricdensity of p = 4350 kg/rn3 and E = 110 GPa. The following equation is given in [104]for cantilever plate natural frequencies:= a/7 (3.57)where y = pt is the mass per unit area of the plate, t is the plate thickness, D is thefiexural rigidity, and a is the length (in y direction) of the plate. ) depends on the aspectratio a/b of the plate , where b is the width of the plate. For a/b = 0.7, ). for the firstthree modes are : 3.48, 6.63, 15.48. By substituting the values in the frequency equation(3.57), the first three natural frequencies are obtained. The natural frequencies obtainedfrom equation (3.57), the FE program and the impact tests are given below in (Hz):Chapter 3. Structural Modeling of the Workpiece and End Miii 51Mode Equation 3.57 FE Measurement1 1658 1657 15252 3159 3046 28003 7377 7110 6725The small difference between the predicted and the measured natural frequencies ispartly due to the structural damping. Assume that half of the plate thickness is removedby milling. The variation of the first 3 natural frequencies of the plate obtained from theFE program along the feeding direction of the tool is shown below:Feed position (mm) fi (Hz) f2 f0 1657 3046 711016 1571 2790 600032 1361 2192 488148 1119 1887 423563.5 839 1552 3617Comparing the frequencies of the uncut and the completely machined plate, it is seenthat the natural frequencies linearly vary with the thickness which is also predicted byequation (3.57).3.3 Structural Modeling of End MillAlthough the plate is usually very flexible compared to the end mill, long slender endmill deflections are important especially at the most flexible part (the free end) whichis in contact with the most rigid part of the plate, i.e. the cantilevered end. Therefore,end mill deflections should be modeled for accurate surface error predictions. However,dynamic modeling of the cutter may not be necessary in plate milling as the chatterChapter 3. Structural Modeling of the Workpiece and End Mill 52Figure 3.7: Structural model for end mill: Cantilever beam with elastically restrainedend.vibrations are expected to occur at the plate modes due to the very high flexibility ofthe plates. Also, due to the strong dynamic coupling between the end mill, tool holderand spindle, it is not possible to model the cutter dynamics by analytical or numericalmeans without experimental identification. For that, modal analysis techniques are usedin modeling end mill dynamics [77]. Therefore, only a static model will be given for theend mill; however, some dynamic testing results are discussed.3.3.1 Cantilever Beam Model for End MillStatic deflection tests performed on a vertical milling machine showed that for amoderate size end mill, the tool deflections are much higher than the deflections of thespindle and tool holder. The tests were performed by loading the cutter at its free andfixed ends. The deflections at the fixed and free ends of the cutter and tool holder, andon the spindle just above the tool holder were measured. As a result, the end mill wasmodeled as a cantilever beam with linear springs at the fixed end to account for thestiffness at the collet (see Figure 3.7). This is because the deflection measurements atkc/2Chapter 3. Structural Modeling of the Workpiece and End Mill 53the fixed end of the tool were almost the same when it was loaded at the free and fixedends separately. This implies that the stiffness at the collet k can be modeled by alinear spring. Then, the static deflection of the end mill under the cutting forces can becalculated from the beam theory [113]. The deflection in x or y direction at axial positionzk caused by the force applied at Zm is given by the cantilever beam formulation as;:i 2____F6EI(3im)+7’ ,O<lIk<lJmS(k, m) = (3.58)/Fy,ml17136E1 — Vm) + V <Vkwhere E is the Young Modulus, I is the area moment of inertia of the tool, k is theexperimentally measured tool clamping stiffness in the collet and vk = 1—ZJ, I beingthe gauge length of the cutter. The area moment of the tool is calculated by using anequivalent tool radius of Re = sR, where s ( 0.8) is the scale factor due to flutes [114].In the previous studies by Kline et al. [34] and Sutherland et al. [24], the collet stiffnesswas taken into consideration by defining an equivalent tool length which is calibrated atthe free end of the tool by using the cantilever beam formula. This does not representthe real physics and is not practical as each tool, even the same tool with differentclamped-gauge lengths, should be calibrated to determine the equivalent length. Thecollet stiffness k, on the other hand, was determined to be the characteristic of the toolholder and the collet diameter. End mill deflection has been determined using lumpedmilling forces by Kline et al. [35], Sutherland et al. [24] and Altintas et al. [98]. Althoughthis approximation may lead to reasonably accurate results for short end mills, for longcutters the distribution of the milling forces on the cutter has to be considered for accuratepredictions. In this study, the end mill deflections are calculated by using distributedmilling forces as explained in Chapter 5. The dimensional surface error predictions, byusing the tool deflection model given in this section, are quite accurate as shown inChapter 3. Structural Modeling of the Workpiece and End Mill 54Chapters 5 and 7.3.3.2 Simulation and Experimental ExamplesStiffness measurements for two different end mills will be given and the beam modelfor the end mill deflections will be verified. The first cutter has 4 flutes which are 300helical, 25.4 mm diameter, 115 mm gauge length and is made of HSS (High Speed Steel).The end mill was loaded by using weights (at the free and fixed ends separately) and thedeflections at the loading points were measured by a dial gauge. From linear regressionanalysis, the stiffness of the end mill at its free end, kT, and the collet stiffness, k, arefound askT = 3580 N/mm ; k = 19600 N/mmThe stiffness of the tool at its free end can be calculated from the beam deflection formulagiven by (3.58):— 3Mwhere E = 210 GPa for steel, 1 is the gauge length and the area moment of inertia,I is calculated by using the effective diameter which is identified as de = 0.85dfor 25.4 mm diameter tools. Then, the stiffness at the free end is found by determiningthe equivalent stiffness of the collet and the tool which are connected in series1.— cb — 0 mmCI 6which is very close to the measured stiffness. The second example is a 4 flute, 30° helical,19.05 mm diameter carbide end mill with a gauge length of 55.6 mm. k = 19800 N/mmand kT = 10180 N/mm were measured on the tool. The predicted value is kT = 10450N/mm (E = 620 GPa for carbide tool).Chapter 3. Structural Modeling of the Workpiece and End Mill 55The effect of support flexibility on the natural modes of beams has been investigatedin many studies [115, 116, 117, 118, 119, 120, 121]. One may think that the mode shapesand the natural frequencies of end mills can be determined by using the beam modelconsidered in the static deflection analysis. However, the frequency predictions do notmatch with the measured data. For example, for a 4 flute, 00 helix, 19.05 mm diametercarbide end mill, the first 5 modal frequencies were measured as 113, 625, 2113, 2800and 3500 Hz in an impact test. The maximum amplitude of the transfer function occursat the third mode, 2113 Hz. Another impact test performed on the tool holder revealedthat the first two modes are not related to the tool. The frequencies calculated by usingthe equations given in [121] for the natural frequencies of elastically supported beams(k = 19800 N/mm) are : 875, 1770, 15770 Hz. This suggests that the tool dynamicscannot be modeled by the simple beam model. A complete analysis of the spindle, toolholder and end mill assembly is necessary for modeling and prediction purposes. However,this is not a simple task, and modal testing is used to obtain the modal characteristicsat the free end of end mills [77, 122, 123, 124].3.4 SummaryThe cantilever plate and the end mill structures have been modeled. The staticdeformations and the dynamic modes of the plate are calculated by using the FiniteElements method. The end mill is modeled as a cantilever beam with elastic restraintsat the fixed end. Both methods are verified experimentally.Chapter 4Modeling of Milling Forces4.1 IntroductionThe prediction of cutting forces in machining is of fundamental importance as theyare the key factors in determining dimensional surface accuracy, machining power and therequired strength for workpiece holding mechanisms and the cutting tool. Practical andaccurate cutting force prediction methods are also required to optimize process planningin CAD/CAM environments. In this chapter, different approaches and models are givenfor the prediction of milling forces. In peripheral milling operations, the mechanisticapproach is usually used where the cutting forces are related to average chip thickness bycutting force coefficients calibrated experimentally for a particular workpiece material-tool pair [35, 32]. Then, the cutting forces produced by the same cutter are predictedanalytically by using the force coefficients as shown by Sabberwal et al. [20, 21], Armarego et al. [30], Tlusty et al. [22] and Altintas et al. [25]. The mechanistic modelgives accurate milling force predictions, however, milling forces have to be calibrated forevery tool geometry and cutting conditions and thus it has little use in tool design andprocess planning. Armarego et al. [27, 30, 36, 125] used the oblique cutting model formilling force predictions which is called the mechanics of milling approach. This methoduses an orthogonal cutting database in determining the milling force coefficients withoutcalibration tests, and therefore it is very practical for optimal tool design and processplanning.56Chapter 4. Modeling of Milling Forces 574.2 Mechanistic Modeling of Milling ForcesIn the mechanistic approach, the cutting force coefficients are directly identified fromthe milling tests unlike the mechanics of milling approach which is discussed in the nextsection. Two different mechanistic models are considered in this section, the exponentialforce coefficient model and the linear edge force model. In the exponential model, the cutting forces are proportional to the chip thickness. Although the linear relation betweenthe chip thickness and the cutting forces in orthogonal cutting was modeled by Merchantin 1944 [2], Koenigsberger and Sabberwal [21, 20] reported the proportional relationshipbetween the uncut chip thickness and the tangential milling force component in 1961.This model has been extended by Tiusty and MacNeil [22] and Kline et al. [23, 34] toinclude a radial force component and has been used widely in the milling force analysis [24, 126, 25]. The experimental results show that as the chip thickness approacheszero, cutting forces converge to some values different than zero. Because of this, in theexponential force model, the cutting force coefficients increase indefinitely as the chipthickness approaches zero. This is due to the finite sharpness of the tool edge whichresults in the ploughing of some material under the tool nose, and the flank contact whichfollows it. Flank contact has special importance in the dynamic cutting process as it generates process damping in cutting which is discussed in the chatter stability analysis, inChapter 8. Masuko [127], Albrecht [128] and Zorev [4], independently, proposed cuttingmodels which consider the edge forces. The complicated physics of the ploughing andthe flank contact led the researchers to identify the edge forces by experimental methods.Zorev [4] proposes different ways of identifying the edge forces, however they are usuallyfound by extrapolating the cutting forces to zero chip thickness. The linear edge forcemodel was used by Armarego and Epp [129] in formulating the milling forces for zerohelix cutters and by Yellowley [28] for analytical mean force and torque formulations inChapter 4. Modeling of Milling Forces 58peripheral milling operations.Exponential and linear edge force models give satisfactory force predictions as long asthe cutting force coefficients are calibrated accurately. The linear edge force model hasthe advantage of having linear force coefficients and a better physical interpretation ofthe cutting process. However, there is no acceptable model for the prediction of the edgeforces, so they have to be determined experimentally. Also, it may not be completelycorrect to assume that the edge forces (ploughing and flank forces) are independent of thechip thickness as suggested by the linear edge force model, although this is an acceptablefirst order approximation [29]. Therefore, in this study, the exponential force coefficientmodel is used when the accuracy is of primary importance, such as in surface errorpredictions, and the linear edge-force model is used when the interpretation of cuttingdata and identification of some cutting characteristics, such as shear stress and friction• between chip and the tool rake face, are necessary.4.2.1 Exponential Force Coefficient ModelThe elemental tangential ( dF), radial (dFr) and axial (dFa) cutting forces actingon flute j of a rigid end mill are shown in Figure 4.1, and given bydFt3(,z) = Kth(q,z)dz ; dF(qS,z) = KdFt3(q,z) (41)dFaj(5,z) = KadFtj(b,Z)where is the immersion angle measured from the positive y axis. The uncut chipthickness, z), can be approximated ash(ç,z) = St sin çj(z) (4.2)Chapter 4. Modeling of Milling Forces 59Figure 4.1: Differential milling forces applied on a milling cutter tooth. Total millingforces are calculated by integrating the differential forces within the engagements limits.Directions of the differential milling forces change as the cutter rotate and also, alongthe axial direction due to helical flutes.where s is the feed per tooth and (z) is the immersion angle for flute j at axial depthof cut z. Due to the helix, the immersion for flute j changes along the axial direction z,= — k1pz (4.3)where t/’ is the helix angle , = tan /R and q5,, = 2K/N is the cutter pitch angle. Rand N are the radius and number of flutes of the cutter. Cutting parameters I(t, Kr andKa are expressed as exponential functions of the average chip thickness per flute period,ha, as follows= KTha ; Kr = Kh iç = KAh8 (4.4)where constants KT, K, KA, p, q and s are determined experimentally for a tool-workpiecematerial pair as explained later in this section. The average chip thickness, ha, is definedyzChapter 4. Modeling of Milling Forces 60as the ratio of the chip volume removed to the exposed chip area:I’ exa] stsingdgIi =aa(ex— st) (4.5)— cos st — cos cex— 5t , ,Yer Ystwhere a is the axial depth of cut and q and are the start and exit angles, respectively. According to the geometry shown in Figure 4.1, for up milling q 0 whereas= K for down milling.In this section only rigid cutting forces, which will shortly be referred to as cuttingforces, are considered, i.e. the effects of the deflections on the cutting geometry are neglected. The elemental cutting forces can be resolved in feed, x, and normal, y, directionsasdF(,z) =(4.6)dF,(q5,z) dFt,(b,z) sin qj(z) — dFr,(q5,z)coscj(z)Substituting equations (4.1) and (4.2) in (4.6), cutting force intensities for flute j areobtained asdF(,z)= —Ks [cos(z) + Kr sin(z)] sin(z)z)= Ks [sin(z) — Kr cos (z)] sin(z) (4.7)dF2(q,z)= 1(tKastsinq5j(z)The cutting force intensities for flute j are normalized by the axial depth, but dependenton feed rate s and cutting force coefficients. The total cutting force contributed by flutej can be found by integrating the intensities along the in cut portion of the flute,Chapter 4. Modeling of Milling Forces 61F,() = { - cos2(z) + Kr(2z) - sin 2(z))] 3,2Ks {2(z) — sin 2(z) + ITcos2j(z)]32 (4.8)4k zj,i(q)KK z,2()F() = — aS [cosi(z]Z.()where z,1() and zj,2(q) are the lower and upper limits of the in cut portion of the flutej. Total cutting forces on the cutter are found by summing forces contributed by allflutes which are in cut.N—i N—i N—i= F3() ; F(q5) = F(4) ;F(qf) = F,(q) (4.9)j=O j=O j=OIt should be noted that for a zero helix case (b 0), the cutting force intensities given byequation (4.7) are constant along the axial direction as the local immersion angle j, andthus the chip thickness does not vary in the axial direction z (see equations 4.2 and 4.3).Therefore, for zero helix cases the integration of the force intensities is not necessary todetermine the cutting forces as they can directly be obtained by multiplying the forceintensities by the axial depth of cut a.The cutting force intensities are normalized by the total cutting forces to obtainpurely geometric unit cutting force intensity functions for a cutter whose reference flute(j = 0) is at immersion q,df(q,z) — dFr(q,z)/dz df,(,z) — dF(çb,z)/dzdz — F(q5) ‘ dz —(4.10)df(q,z) — dF(çf,z)/dzdz—Chapter 4. Modeling of Milling Forces 62For example, the unit cutting force intensity in x direction isj=N-14kg, 0.5sin2cbj(z)+Krsinqj( )df(ç,z)—____________________________________________dz — jN-1— cos2q(z) + Kr (2q5z) — sin2Ai(z))jz,i()Equation (4.10) shows that the unit cutting force intensity depends on the tool geometry(, ,), the limits of engagement zj,2 and zj,1, and cutting constant Kr. The cutting forceintensities in x and y directions are particularly important in deflection analysis. For amultiple flute helical milling case the force intensities vary periodically along the tool axisdue to the engagement of flutes at different axial locations. At some points there maynot be any cutter-workpiece contact resulting in zero intensities. As the cutter rotates,the force intensities move along the axial direction due to the helix effect.Limits of EngagementThe limits of engagement are explained in Figure 4.2 which shows the unrolled formof the cylindrical part surface [25]. The part surface is now bounded by the two verticallines qf ç and q = ex, and the two horizontal lines z = 0 and z = a, where a isthe axial depth of cut. Four different possible intersections of the cutter flutes with theworkpiece are shown. As it can be seen from the figure, the engagement limits depend onthe axial depth of cut (a), start and exit angles (qst) and (qex), and the lag angle of theflute, (ak). The lag angle is the angle between the tip of the flute (z = 0) and the highestpoint in the cut (z = a). Equation (4.3) can be inverted as z(qj)= —[ + jq, — 4j]to obtain the intersections of a flute with the boundaries when necessary. Engagementlimits for the intersections shown in Figure 4.2 are given in Table 4.1.Chapter 4. Modeling of Milling Forces 63aFigure 4.2: Contact cases of a helical flute with workpiece. The contact geometry dependson the axial depth of cut, the immersion angle and the helix angle and it determines thelimits of integration for the milling force calculation which are given in Table 4.1Average Forces and Identification of Milling Force CoefficientsThe cutting force coefficients can be identified from the average cutting forces, i,and,as they are assumed to be constant over the full rotation of the cutter. Theaverage cutting forces are independent of the helix angle as the total chip removed inone rotation of the cutter does not depend on the helix angle. Whitfield [130] integratedthe cutting forces over a rotation of the tool for the different contact cases shown inFigure 4.2 and obtained the same average force expression for each case. Therefore, thecutting force intensities given by equation (4.7) are first multiplied by the axial depth ofcut to obtain the cutting forces for the zero helix case, then they are integrated over onerotation of the cutter and divided by 2r to obtain the average forces as:1This is equivalent to integrating the forces over one tooth period and dividing by 2ir/N as the flutesare equally spaced.Chapter 4. Modeling of Milling Forces 64Table 4.1: Helical flute engagement limits to be used in cutting force calculations. Thenumbers shown in the parenthesis correspond to the contact cases in Figure 4.2.gj(O) zj,1st < j(°) < çtex 0 (1,2) (4j(O) — ak) < qstZj,2 = (q5j(0)- ) (1)(qj(O) — ak) > cst= z,2 = a (2)qS(O) > ex(0) (q3O) — ka) < ex (j(O) — ak) < gst= -(q(0)— ex) (3,4) = q(0)— g5st) (3)t < ((O) — akü) <= zj,2 = a (4)= Kt(PKrQ)Fy = —K(Q + P1(r) (4.11)= StKaKtTwhereaNP = —Fcos2l2ir LaN [ exQ = — 2ç — sin 2I (4.12)JstaN rT = —lcosqSl27r LChapter 4. Modeling of Milling Forces 65ct and qer are the start and exit angles of the cut, a is the axial depth of cut and N isthe number of teeth. Then, the cutting force coefficients can be obtained as follows:K — QF+PF—K — F (4.13)— t(P+QKr)17 —_________1a— sKTAfter the cutting coefficients are obtained for different feedrates, they are expressed asexponential functions of the average chip thickness as shown in equation (4.4). Due toedge components, the cutting forces converge to a finite value as the feedrate approacheszero. As a result, the cutting force coefficients are very high at small feeds, and theydecrease exponentially as the feedrate increases.4.2.2 Linear Edge-Force ModelIn the linear edge-force model, the cutting forces are modeled as having two fundamental components: the edge (due to ploughing and flank contact) and cutting components [28]. The cutting constants (Kte,Kre,Kae) and (Ktc,Krc,K,) relate the totalcutting forces to ploughing and rubbing, and actual chip cutting mechanisms, respectively. The force coefficients are determined experimentally for a certain tool-workpiecematerial pair when the mechanistic approach is used.The elemental tangential, dF, radial, dFr, and axial dFa cutting forces acting on fluteChapter 4. Modeling of Milling Forces 66j of an end mill are shown in Figure 4.1, and given bydFt(,z) = [Kte+Ktchj(q!,z)]dzdFrj(75, z) = [Kre + Krchj(q’?, z)]dz (4.14)dFaj(q,z) = [Kae+Kachj(q,z)]dzwhere h(q, z) = St sin (z) is the uncut chip thickness and st is the feed rate per tooth.4 is the immersion angle measured clockwise from the positive y axis to a reference flutej = 0, which has immersion at its tip z = 0. The elemental forces are resolved intofeed (x) and normal (y) directions and integrated analytically along the in cut portionof the flute j to obtain the total cutting force produced by the flute.R [Kte sin (z)— Kre cos (z)tan+ [Krc(2(Z) - sin 2(z)) - cos 2(z)]]:= ta [ Kresin(z) — Ktecos(z) (4.15)+ [Kt(2(z) - sin 2(z)) + Krc cos 2(z))1]R Z,2()tan — stKaccos(z)]Z.()• where zj,i(q) and zj,2(q) are the lower and upper limits of the engagement of the flutej. The cutting forces contributed by all flutes are calculated and summed to obtain thetotal instantaneous forces on the cutter. The determination of the engagement limits isexplained in the previous section.Chapter 4. Modeling of Milling Forces 67Average Forces and Identification of Cutting Force CoefficientsThe average milling forces per tooth period are F, i, and , and can be found byintegrating equation (4.3) over one full rotation of the cutter,KteS+KreT(KtcF+KrcQ)= KteT— KreS + (KCQ + KrcP) (4.16)= Kae(cexcbst)+stKacTwhere F, Q and T are given by equation (4.12) andaNS=— sinq (4.17)2irc5t and ç are the start and exit angles of the cut, a is the axial depth of cut and N isthe number of teeth. After cutting forces are measured and their averages are found atdifferent feedrates, they are put into the following form by linear regression= 1qe + .StFqc (q = x, y, z) (4.18)where Fqe and Fqc are the edge and cutting components of the forces. Finally, the cuttingforce coefficients are identified from equations (4.15) and (4.18) as follows:K — eS+yeT K —_______te—— S2 + T2 , tc — p2 + Q2K — KteS + . — KP — 4F (4 19)re—4rc—2ir F€.- ae = — 11ac =Yex —The cutting force coefficients are directly estimated from the milling data by curvefitting, hence new milling tests are required for the calibration of each new cutter geometry.Chapter 4. Modeling of Milling Forces 684.3 A Mechanics of Milling Approach for Milling Force PredictionThe end mill geometry is complex, having a number of variables such as helix andrake angles which have to be selected properly in order to improve the machining performance. In addition, the cutting edge angles and diameter may vary along the flutesof some special cutters such as tapered-helical ball end mills. Therefore, a vast amountof cutting data is necessary to predict the cutting forces for different cutter geometrieswhich is costly and not practical. Furthermore, the data cannot be generalized as thereis no explicit relationship between the tool geometry, cutting conditions and the cuttingforce coefficients in mechanistic millillg force models. Also, an identification procedurethat is based on experiments entirely does not give physical insight, i.e. the relation ofthe milling force coefficients to the fundamental machining characteristics of the work-piece material and tool geometry, such as friction, shear stress, rake and helix angles.Therefore, although mechanistic cutting force models usually provide accurate predictions, they are not practical for tool design, process planning and analysis of cutters withcomplex geometries.The fundamental metal cutting research has mainly concentrated on simple tool geometries, and cannot directly be applied to practical machining operations like peripheralmilling. In this section, a mechanics of milling approach is presented for the milling forceanalysis. In this method, an oblique cutting model is used to relate the cutting forcecoefficients to the tool geometry, shearing and friction mechanisms. The required cuttingforce parameters are identified from orthogonal cutting tests and used in the obliquecutting model together with the predicted values of the chip flow direction. The samedatabase can be used in the analysis of cutting forces for different tool geometries whichreduces the number of cutting tests considerably. The use of orthogonal cutting data inChapter 4. Modeling of Milling Forces 69force predictions of drilling and milling operations was first introduced by Armarego andWhitfield [27]. They used Stabler’s rule for the chip flow angle. In this study, a moreaccurate model for the estimation of the chip flow direction is used. The method in therevised form predicts the cutting force coefficients with more than 85 % accuracy in theend milling of the titanium alloy Ti6A14V, thus providing an alternative and practicalway of simulating the performance of milling dlltter designs prior to their manufactureand experimental testing.4.3.1 Force Coefficient ExpressionsThe shear stress, shear and friction angles can be identified from orthogonal cutting• tests for different rake angles and cutting velocities [2]. However, in helical end millingthe edges of the cutter flutes are not orthogonal to the cutting velocity but inclined withan angle equal to the helix angle as shown in Figure 4.3. Therefore, an oblique cuttingmodel should be used for the analysis of end milling. The cutting data, however, will beobtained from the orthogonal cutting tests and used in the oblique cutting model. Forthis reason, first the orthogonal cutting model will be briefly discussed and the relatedequations for the identification of shear angle, shear stress and friction on the rake facewill be derived.Consider the force diagram shown for the orthogonal cutting geometry in Figure 4.4.In thin shear zone models, metal is assumed to deform and form chips in the shear plane.The angle between the shear plane and the cutting velocity direction is the shear angle,ç. Shear angle is perhaps the most studied parameter in the metal cutting research, as itis required in cutting force analysis. Although many models have been proposed for theshear angle prediction [2, 131, 132, 133], accuracy of the predictions is usually low anddepends on the cutting conditions and the material. This is due to the complex nature ofChapter 4. Modeling of Milling Forces 701’ if’11+1 1‘7,Orthogonal cutting.Oblique cuttingFigure 4.3: Orthogonal and oblique cutting geometries. In oblique cutting, the edge ofthe tool is not perpendicular to the cutting velocity which results in a three dimensionalcutting geometry and a chip flow direction which is not parallel to the cutting velocity.the machining process which involves plastic deformation of metal under extreme frictioneffects with high temperature gradients. Despite the extensive research efforts in the lasthundred years, the relations between cutting conditions, tool geometry and the shear andfriction characteristics are still not completely modeled. For this reason, the fundamentalparameters of metal cutting, namely the shear angle, the shear stress at the shear planeand the friction on the rake face, are identified from the cutting experiments in thisstudy. The orthogonal cutting model proposed by Merchant [2] is used in the analysis.In Figure 4.4, the length of the shear plane, AE ish = h/cosa—htana (4.20)sm qf cosorr cos atanq8=. (4.21)1 — r sin alic iiwhereChapter 4. Modeling of Milling Forces 71yFigure 4.4: Orthogonal cutting force diagram. The force diagram is used to relate thefrictional and normal forces applied on the rake face to the shear and normal forces onthe shear plane. This force diagram is the basis of the Merchant’s orthogonal cuttingforce and shear angle prediction model.a rake angleh uncut chip thicknessh chip thicknessshear angler = h/he, chip thickness or cutting ratioTherefore, the shear angle can be identified from the cutting ratio r. The averageshear stress at the shear plane, (r), and the average friction angle on the rake face, (9),are obtained from the force analysis. The resultant cutting forces on shear plane andrake face are equal to each other, due to the static equilibrium 2 of the chip. On theforce diagram shown in Figure 4.4, P’ and F,-, are the shear and normal forces on the2The conservation of momentum is not considered in cutting force analysis due to small velocity andnegligible mass of the chip. However, this effect is important and usually taken into consideration in theanalysis of high speed cutting processes.xChapter 4. Modeling of Milling Forces 72shear plane whereas N and F are the normal and frictional forces on the rake face. Fand Ff are the cutting and the feed forces which are normal to each other. The frictioncoefficient on the rake face can be identified from the ratio of the friction force, F, tonormal force, N which can be expressed as followsF = Fsino+Ficosa(4.22)N = FcosQ—FsincThen, the friction ratio () and the friction angle (3) are given by the ratioF Fj+Ftanc[Lf=tan8=—= (4.23)N FC—rftanoThe shear stress at the shear plane (r) can be identified by dividing the shear force F8to the area of the shear plane (bh/ sin ),F cos — Ff sin q= bh sinqf. (4.24)where b and h are the width of cut and the uncut chip thickness, respectively.Both exponential and linear edge cutting force models are used in the analysis of theorthogonal cutting forces and both methods yield accurate force predictions. For theanalysis and identification of shear stress and friction angle, however, the edge forcesshould be separated from the cutting forces as they are not related to the shearing andrake face friction. Therefore, when the linear edge model is used, the cutting componentsof the total cutting forces should be substituted in equations (4.23) and (4.24) . The edgeforces can be identified by extrapolating the cutting forces to zero chip thickness as inthe case of milling force analysis by the linear edge model.Chapter 4. Modeling of Milling Forces 73Figure 4.5: Cutting forces and chip flow geometry on a helical milling cutter.In helical milling, the tooth edge is not orthogonal to the cutting velocity directionbut makes an angle which is equal to the helix angle. This causes the shear plane andthe chip flow direction to be oriented with respect to the edge of the tooth. Therefore,helical milling force analysis requires the use of an oblique cutting model.A simple view of a peripheral milling cutter edge geometry is shown in Figure 4.5.A detailed view of the cutting zone showing the oriented shear plane is given in Figure4.6. A plane which is normal to the cutting edge is considered for force and velocityequilibrium equations. The velocity rake angle, c, and the normal rake angle, a, areshown in different views of the rake face and related asYVd Fad FtTrue view of rake faced FrNormal PlaneV:cutting velocityVn: component ofcutting velocity innormal planetan o = tan a cos (4.25)Chapter 4. Modeling of Milling Forces 74angleiP,Figure 4.6: Detailed view of the oblique cutting geometry showing the oriented shearplane and the chip flow on the rake face. The oblique cutting force analysis are performedon the normal plane which is perpendicular to the cutting edge._4utti velocityChapter 4. Modeling of Milling Forces 75Figure 4.7: The oblique cutting force components in the normal plane.Figure 4.7 shows the cutting force components in the normal plane at the tip of theflute j. dN and dF are the differential normal and friction forces on the rake face of thetool. The components of the differential tangential, radial, dFrn, and shear, dF8,forces in the normal plane are related to the milling forces and geometry as followsdF =dFrn = dFrj (4.26)dF — stsinqj(z) d—Zwhere ç’8 is the shear angle measured in the normal plane and gj(z) is the immersionangle of the flute j at axial depth z. The normal friction angle, is defined as followsVntan/3 = tan/3cos77 (4.27)Chapter 4. Modeling of Milling Forces 76where /3 is the friction angle at the rake face and is the chip flow angle as shown inFigure 4.5 (angle between a perpendicular to cutting edge and the direction of the chipflow over the rake face, as measured in the plane of tool face). The normal shear angle,q, is obtained from the cutting ratio as in the orthogonal cuttingrt cos a,-,tan =. (4.28)1— rt sin a,-,• . . . . . .. cosiwhere the chip thickness ratio in oblique cutting, rt, is related to r by rt = r cos Thisrelation is obtained from the mass continuity equation of the chip before and after the cut.It has been shown [27, 6, 29, 134, 130] that the cutting force coefficients Krc andKac can be determined from the oblique cutting analysis with a satisfactory degree ofaccuracy. In order to derive the cutting force coefficients, the elemental chip is consideredto be in equilibrium under the action of stresses in the shear plane and at the rake faceof the tool. Based on the oblique cutting model, the cutting coefficients which relate thecutting forces to tool geometry are expressed by [6]K — r— SflK — r sin(3 — a) (4 29— sin q’ cos k— r cos(/3n — a)tanL — tanac— sin5 kwherek = \/cos2(q8fl+ /3 — a) + tan2 7?c sin2 [3Therefore, for a given tool geometry, the cutting force coefficients can be calculated fromequation (4.29) if the shear stress, (r), friction angle, (/3), cutting ratio, (r), the chip flowangle, (), and the edge force coefficients, Kje,K,-e and Kae, are known. There is noChapter 4. Modeling of Milling Forces 77acceptable theoretical model for the edge component of the cutting forces, they can onlybe found from cutting tests. Therefore, prediction of the cutting force coefficients requiresthat the shear stress (r), friction angle (/3), cutting ratio (r), chip flow angle () andthe edge force coefficients Kte, Kre and Kae be known. As these parameters depend oncutting conditions and tool geometry, they should be obtained from the cutting tests inwhich chip thickness, rake angle, cutting velocity and the angle of inclination (helix angle)are varied. Based on the experimental results obtained at the University of Melbourne[135, 136], Armarego et al. [27, 130] concluded that r, r, /3 and Kte, Kre, Kae are all largelyindependent of the tool inclination angle /‘ so that these parameters can be obtained fromsimpler orthogonal cutting tests as explained before. This results in an order of magnitudereduction in the amount of needed machining tests. The prediction of chip flow angle iscrucial for the accurate estimation of cutting forces as explained in the following section.Armarego et al. [30] and Whitfield [130] used Stabler’s chip flow rule which assumes that= i/’. This is a crude approximation and may result in significant errors in the cuttingforce predictions.Prediction of Chip Flow AngleThe oblique tool geometry was first rigorously analyzed by Stabler [137]. In thesame study, Stabler also stated the widely accepted Stabler’s rule or chip flow law whichassumes that the chip flow angle is equal to the angle of obliquity. In other words, Stabler’s rule assumes that the chip moves parallel to the cutting velocity vector, withoutbending after it is cut. The rule does not consider the effect of shear angle, friction andtool geometry, i.e. rake angle. Shaw et al. [138] experimentally showed that the chipflow angle varies with the normal rake angle and friction. Therefore, Stabler’s rule mayintroduce significant errors in the prediction of the cutting force coefficients dependingon the cutting conditions. Later, Stabler [139] modified the chip flow law to i = k/’Chapter 4. Modeling of Milling Forces 78where 0.9 < k < 1.0 varies with the work material and cutting conditions. However,the modified chip flow law does not change the results considerably. The effects of cutting conditions and tool geometry on the chip flow angle were experimentally studied inseveral works. Russel and Brown [140] proposed the equation i = tan ‘b cos ce,-, whichindicated dependence of on the rake angle. Zorev [4] considered the effect of cuttingspeed on and suggested i = i1’/V°°8 where V is the cutting speed in m/min. Usui etal. [141] used the minimum energy principle to determine the chip flow angle. Colwell[142], Zorev [4] and Stabler [139] proposed methods for prediction of which considerthe cutting action of the end cutting edge. The experimental and predicted end cutting• edge and tool nose radius effects on the chip flow angle are discussed in detail by Oxley[10]. Whitfield [130] gives a rigorous formulation to obtain the chip flow angle by usingorthogonal cutting data which will be reviewed in the following.In an early study of metal cutting, Merchant [2] indicates that the direction of shearis not perpendicular to the cutting edge in the case of oblique cutting, but makes anangle 6, with the perpendicular. He derived the following equation for 6 from velocityconsiderations.tan cos(qs — a) — tan sintan 6 = (4.30)cos a,-,Stabler [137] later formulated the angle, j, that the shear force makes with the cuttingedge normal as followssin /3 sintanf =. . (4.31)cos /3 cos(—a) — cos /‘ sin /3 s1n(qsn an)In general, it is fairly reasonable to assume that the shear force and shear velocity directions are equal. The following expression is obtained when equations (4.30) and (4.31)Chapter 4. Modeling of Milling Forces 79are equated to each other (Armarego and Brown, [6])cos c tan /‘tan(qi3 + i3) = . (4.32)tan— srno tan bFrom equations (4.27), (4.28) and (4.32), the following expression for the chip flow angle, is obtainedA sin — B cos— C sin cos ic + D cos2 1c = E (4.33)whereA = r cos a + cos tan /3B = tan/3sinosinbC = rsintan/3 (4.34)D = rtan,8tan’bB = sincosoThe chip flow equation (4.33) can be solved by numerical techniques for cutter geometry(i/’, a) and (r, /3) which are identified from orthogonal cutting tests. Detailed derivationsof the chip flow equation and a numerical solution procedure are given in Appendix A.The resulting variations of the chip flow angle, T1D, with the friction angle, cutting ratio,angle of obliquity and normal rake angle are computed and shown in Figures 4.8, 4.10,and 4.9.Figure 4.10 shows the effects of cutting ratio r and angle of obliquity b on.It canbe seen that Stabler’s rule may be a good approximation for only limited ranges of r and‘. Therefore, Stabler’s rule is a crude approximation and may result in errors when usedin milling force analysis. As experimentally observed by Shaw et al. [138], the solutionindicates that the chip flow angle reduces as the friction increases (see Figure 4.8). Inorder to obtain the real variations of chip flow angle, however, one should use cuttingdata since friction and shear angle are, in general, functions of rake angle. Hence, oneChapter 4. Modeling of Milling Forces 80rFigure 4.9: Variation of chip flow angle with cutting ratio for different values of frictionangle.70600,.50)400LI0.20C)10015 25 35 45Friction Angie (deg)Figure 4.8: Predicted variations of chip flow angle with rake and friction angles.80560< 4000C-)00.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Chapter 4. Modeling of Milling Forces80a)9- 60a)0)C40U0.0 20081Figure 4.10: The effects of inclination angle and cutting ratio on chip flow angle. Thisfigure indicates that Stabler’s rule is an acceptable approximation only in limited rangesof inclination angle and cutting ratio.cannot follow the constant a, r and 3 lines in Figures 4.8, 4.10 and 4.9 for a particularcutting operation which can be seen from the experimental results given by Shaw et. al[138], Brown et. al [29], Pal et. al [143] and Lin et. al [144]. That means orthogonalcutting data is necessary for the chip flow angle prediction as well.4.3.2 Procedure for Milling Force CalculationWith the chip flow angle prediction, the procedure is completed for milling forcecalculations. The method is summarized in Figure 4.11 where it is referred to as generalized milling force calculation. This is a general method as the same procedure canbe followed to calculate the cutting forces on any milling cutter geometry by using thesame orthogonal data. As an example, consider the ball end mill geometry shown inFigure 4.12. The cutting velocity and helix angle vary from finite values at the shankintersection to 00 at the tip of the ball. The three differential force components, dF, dFr5 15 25 35 45Inclination Angle, w(deg)Chapter 4. Modeling of Milling Forces 82Generalized Milling Force AnalysisFigure 4.11: Generalized milling force prediction algorithm. The procedure is repeatedat a number of points along the cutter flutes if the cutter has variable geometry thateffects the local cutting force coefficients.Chapter 4. Modeling of Milling Forces 83and dFa are oriented due to the ball which should be considered when calculating thecutting forces in x, y and z directions. The local values of the cutting velocity and helixangle are to be used in the cutting force coefficient calculations. Usually, the normal rakeangle is kept fixed on the flutes, however if the velocity rake is kept fixed, then the localvalues of normal rake angle should be considered. Finally, the differential cutting forcesin x, y and z directions can be integrated to obtain the total cutting forces. The ball endmilling forces have been modeled in [145, 146, 147, 148, 149]. Yang et. al [145, 146, 148]used the orthogonal data to predict the differential resultant force (not the tangential,axial and radial force components) in ball end milling. They did not consider the helix angle and oblique cutting model, which resulted in inaccuracies in their predictions.Lim et. al [149] used a calibration procedure to identify the cutting force coefficients inwhich the cutter was calibrated for different axial depths as the average chip thicknessis a function of the axial immersion in ball end milling. Yucesan and Altintas [147], onthe other hand, used an advanced numerical identification technique for ball end millingwhich gives accurate results with increased computation time. Therefore, modeling ofball end milling forces requires more calibration tests than the end milling. Preliminaryapplications of the generalized method to ball end milling gave quite promising results,however, this is kept outside the scope of this work.4.4 Simulation and Experimental ResultsSeveral orthogonal cutting and milling tests have been performed on a titaniumalloy (Ti6A14V) for the verification and comparison of the developed method with themechanistic model results. (Ti6A14V) is mainly used in aerospace applications, such as jetengine impellers, due to its high strength to weight ratio even at elevated temperatures.It accounts for 45 % of the total titanium production. Despite the extensive researchChapter 4. Modeling of Milling Forces 84dFadFrFigure 4.12: The orientation of differential milling force components on a ball end millflute. Variable milling coefficients may have to be used as the cutting velocity and helixangle vary along the cutter flutes. All three forces have components in x and y directions.Chapter 4. Modeling of Milling Forces 85efforts, titanium is still one of the most difficult materials to machine [150, 151]. It hasa very low thermal conductivity (one-sixth that of steel) which causes high temperaturegradients at the chip tool interface and results in low allowable cutting speeds. Also,the high strength is maintained at elevated temperatures and this opposes the plasticdeformation needed to form a chip. As it will be seen in the orthogonal cutting data,titanium’s chip is very thin which results in an unusually small contact area with thetool causing high contact pressures and temperatures.4.4.1 Cutting Conditions in Milling and Orthogonal TestsFull immersion milling tests were conducted with a carbide end mills with singleflute, 30° helix, 19.05 mm diameter. The axial depth of cut and the cutting speed were5.08 mm aild 30 m/min, respectively. The average chip thickness range of 0.008-0.1mm was considered. Helical end mills with different normal rake angles were used todetermine the robustness of the method for different geometries. The normal rake anglewas considered to be the angle corresponding to the rake angle in orthogonal cutting [29].Orthogonal turning tests were performed on titanium tubes to achieve ideal orthogonalcutting conditions . The outer diameter and the wall thickness of the tubes used were100 mm and 3.8 mm, respectively. The cutting speed range of 3-47 rn/mm and carbidetools with different rake angles corresponding to the normal rake angle on the end millswere used. The feed rate range of 0.005-0.1 mm was covered with 5 steps (0.005, 0.01,0.03, 0.07, 0.1 mm). Very low cutting speeds were considered in order to see the effect ofcutting speed and also to be able to use the same data base for different milling cuttergeometries such as ball end mills where the cutting speeds are close to zero at the ballend. For each case, tangential, F, and feed, F, cutting forces were measured by aKistler table dynamorneter mounted on the tool holder. In Figures 4.13.a and 4.13.b, the3The contact between the tooth nose and the finished surface results in a parasitic force in bar turning.Chapter 4. Modeling of Milling Forces 86variations of the measured forces with the uncut chip thickness h are shown for cuttingspeeds of V 28m/min and V = 47 rn/mm. As it can be seen from these figures, theforces do not vary with the cutting velocity significantly. This can be attributed to thecounterbalancing of the temperature and strain hardening effects on the shear stress.4.4.2 Analysis of Orthogonal Data: Identification of Cutting ParametersThe edge forces are shown in Figure 4.14 which are identified by extrapolating thecutting forces to zero chip thickness. The edge forces are close to each other for different cutting velocities and rake angles. Therefore, average values of the edge forces areused in the rest of the analysis. The mean and the standard deviation of the edge forceswere calculated as Kte=24 N/mm with a(Kje)=6.3 and Kre=43 N/mm with (Kre)7.3.The agreement between the edge force coefficients identified from milling tests and theorthogonal cutting suggests that the edge forces do not vary with the angle of obliquity.Also, the edge component of the force coefficients in the axial direction is very small andtherefore it can be neglected in the transformation of the orthogonal cutting data to theoblique model. These results have also been reported in [135, 136, 27, 130].The thickness of many chips were measured with a micrometer for each test and theaverage value of the measurements is used in calculating the chip ratio r. Figure 4.15shows the variation of r with the uncut chip thickness and rake angle. No significantvariation was observed with the cutting velocity. The relatively high cutting ratio (thinchip thickness), and thus higher shear angle, is a characteristic of all titanium alloys andresults in small contact areas between chip and tool. This increases the contact pressuresand temperatures contributing to the wear and chipping of the tool. As it can be seenfrom the figure, r exponentially varies with the chip thickness and the following equationChapter 4. Modeling of Milling Forces 87a (deg)800700.5600—. 500•10400 015- 3002001000• 0.12a (deg)800700.5600AB500•10a 4002°15LL 30020010000.12Figure 4.13: Measured cutting and feed forces in orthogonal cutting tests with differentrake angles (Material: Ti6A14V). a) V=28 rn/mm b) V=47 rn/mm0 0.02 0.04 0.06 0.08 0.1h (mm)V=47 rn/mmFcI..AA0Ff0 0.02 0.04 0.06 0.08 0.1t(mm)Chapter 4. Modeling of Milling ForcesEEzRake Angle (deg)Ez4,62.8(rn/mm)2.8V (rn/mm)884.615815Rake Angle (deg)10Figure 4.14: Edge forces as identified from the orthogonal cutting tests.Chapter 4. Modeling of Milling Forces()15Rake Angle(deg)89Figure 4.15: Variation of the measured cutting ratio r(= h/he) with chip thickness andrake angle.was identified by curve-fitting asr = r0har, = 1.755 — 0.028ca = 0.331 — 0.0082a(4.35)Whitfield [130] observed the opposite variations for steel (S1214 and CS1O4O), i.e.varies exponentially with the velocity and stays constant with the chip thickness. Thedrop in cutting ratio at small chip thicknesses has been noted in some previous works,but it has been neglected in the widely used shear angle relationships, e.g. Merchant[152], Lee and Shaffer [131] and Shaw et. al [133]. The reasons for this drop can beattributed to the reduced effective rake angle at small chip thicknesses due to the noseradius, rubbing and size effect phenomenon of metals. The cutting ratio obtained withhigh rake angles was compared with this equation and it was found that this equation0.005 0.010.03 0.07h (mm) 0.1Chapter 4. Modeling of Milling Forces 90is valid for the tools with rake angles 0 — 35°. In fact, equation (4.35) implies that thecutting ratio r strongly varies with the uncut chip thickness, whereas the variation withthe rake angle is quite small.Shear stress (r) and friction angle (3) are calculated from equations (4.24) and (4.23),and shown in Figures 4.16 and 4.17. As it can be seen from Figure 4.16, the shear stressdoes not vary significantly with the velocity and the rake angle. This is due to theopposite effects of the generated heat and the strain rate at the shear zone which areproportional to the cutting velocity. Therefore, an average value can be used for the shearstress. The average value and standard deviation of the shear stress were calculated asr=613 MPa with o(r) = 73. The friction angle, on the other hand, slightly varies withthe rake angle and the cutting velocity. The friction angle identified from the orthogonalcutting tests is the average value of the friction in the sticking and the sliding regionsbetween the chip and the rake face of the tool. The friction coefficient is higher in thesliding region which becomes longer as the rake angle is increased due to the reducedpressure on the rake face. Hence, the average value of the friction on the rake faceincreases with the rake angle. This is recognized as one of the main dilemmas in metalcutting for the cutting forces reduce whereas the friction increases with the rake angle [9].The friction angle slightly varies with the cutting velocity as well, but this variation ismainly in the low cutting velocity range (V < 10 m/min) and it is almost constant in thepractical cutting velocity range. Therefore, the variation of the friction coefficient withthe cutting velocity is more important in the analysis of ball-end milling as the cuttingvelocity approaches zero at the center of the ball. Hence, the variation of the frictionangle with the cutting velocity is neglected and the following equation for the frictionChapter 4. Modeling of Milling Forces 910C,)Cl)Cl)ci).U)Figure 4.16: The identified values of the shear stress at the shear plane from the orthogonal cutting tests.angle 3 is obtained by linear regression analysis:/3 = 19.1 + 0.29a (4.36)where the rake angle o and the friction angle /3 are in degrees. The average value of thefriction angle is /3 = 21.3° with o() = 3.1.The mean values of the percentage error between the identified and the calculatedvalues from the curve- fit equations are determined as(-0.07 %), /3: (1.6 %), (: -2.3 %), r: (3.8 %), Kie: (-10.9 %), Kre :( 5.1 %)The chip flow angle (i) for 30° helix angle is computed from the numerical solution ofequation (4.33) by using the orthogonal data, i.e. r and /3 for different rake angles. Figure4.18 shows that increases with the rake angle and decreases with the chip thickness, i.e.cutting ratio r, as illustrated in Figures 4.8 and 4.10. The cutting parameters identified8028 10 Rake Angle10 15 (deg)V (rn/mm) 3c)a)a)CC04-C)1.LIFigure 4.17: The friction angle calculated from the orthogonal cutting forces.Table 4.2: Cutting parameters identified for Ti6A14V from orthogonal cutting tests.from the orthogonal tests are summarized in Table 4.2.The orthogonal data has also been analyzed by using the exponential force model andused in [31]. The identified values of shear stress and the friction angle are scattered,and much higher friction angles are obtained, especially at low chip thickness, due to theunseparated edge force effects. However, the milling force coefficients calculated by theexponential force model have acceptable accuracy [31].Chapter 4. Modeling of Milling Forces 922Rake Angle(deg)285V (rn/mm)r = 613MPa= 19.1 + O.29ar = r0W’= 1.755— O.028aa = 0.331 — 0.0082aKte = 24 N/mmKre = 43 N/mmChapter 4. Modeling of Milling Forces 93F-’0)ci)-Dci)0)C0U—Rake Angle(deg)Average ChipThickness (mm)Figure 4.18: Predicted values of the chip flow angle for 300 inclination (helix) angle.4.4.3 Prediction of Milling Force CoefficientsThe data obtained from the orthogonal cutting tests and the calculated values ofchip flow angle were used in the transformation equation (4.29) to predict the millingforce coefficients, Krc and Ka. The same edge force coefficients, Kte and Kre, thatwere identified from the orthogonal data were used for milling. In a separate analysis, theedge and the cutting force coefficients were identified from the slot milling tests by usingsingle flute carbide cutters and a feedrate range of 0.0127-0.1 mm/tooth, and are givenin Table 4.3. Unlike the experimentally identified milling force coefficients, the predictedmilling force coefficients, K and Kac, vary with the feed per tooth as the cuttingratio r and, thus the chip flow angle , are functions of the uncut chip thickness. Thevariations of the predicted coefficients for a = 0 are shown in Figure 4.19.Chapter 4. Modeling of Milling Forces 9435003000Ktc + Krc A Kac2500 .cx=Qo 2000oç 1500°10004 4 450000 0.02 0.04 0.06 0.08 0.1 0.12Average Chip Thickness (mm)• Figure 4.19: Variations of the predicted milling force coefficients with the chip thickness.The predicted coefficients vary with the chip thickness due to the strong dependence ofthe cutting ratio on the chip thickness. (Material: Ti6A14V, V=30 rn/mm, a = 00)As it can be seen from the figure, very low r values result in high shear angles andforce coefficients at low chip thickness. However, for a first order approximation thisvariation may be neglected and the average values of the force coefficients can be used asthe edge forces are much higher than the cutting forces in low chip thickness zone. Thecritical chip thickness, defined by Yellowley [28], which is the chip thickness at whichthe edge force is equal to the cutting force, is useful in determining these zones. Fromthe orthogonal cutting data, the critical chip thickness can approximately be calculatedas 0.012 mm for the tangential direction. For the radial, or feeding forces, the criticalchip thickness is much higher ( 0.07 mm) due to high edge force and low cutting forcecomponents in this direction. The identified and the predicted milling force coefficientsare shown in Table 4.3. The average values of the predicted milling force coefficients (inthe average chip thickness range of 0.01-0.1 mm) are given. The agreement between thepredicted and the experimentally identified milling force coefficients in tangential andChapter 4. Modeling of Milling Forces 95the axial directions are satisfactory. The average values of edge forces in milling tests areKte = 25.2 and Kre = 44, which are very close to the values obtained from the orthogonalcutting data. The critical chip thickness in the axial direction is approximately 0.005 mm,and thus neglecting Kae in the predictions does not introduce too much error unless thechip thickness is very small. The error in the prediction of and K is approximatelywithin (+ 8 %) with mean values of (0.02 %) and (-2.2 %), respectively. If the Stablerrule is used in the calculations, i.e. i = 300, the mean values of the errors in and Kincrease to (-2.5 % ) and (-13 %), respectively. The increase in the error values associatedwith the Stabler rule would be higher for larger helix angles as predicted in Figure 4.10.The accuracy of prediction is low for high rake angles. As it can be seen from thetable, the predicted values of the radial force coefficient decrease from 646 MPa to 65MPa as the rake angle increases, from 00 to 20°. The radial or feeding force is equal tothe difference between the components of the rake face friction and the normal forces,in this direction. This can be expressed by the following equation for the orthogonalcutting:Ff = Fcosa — NsinaorFf = Ncosa(tan/3 — tan a) (4.37)where Ff is the feeding force, F and N are the friction and the normal forces on the rakeface, and /3 and a are the friction and the rake angles. As it can be seen from equation(4.37), the decrease in feed force with the rake angle depends on the variation of thefriction angle with the rake angle which is identified and given in equation (4.36). Thepoor agreement between the identified and the predicted values of Krc for the high rakeangles (> 15°) is due to the inaccuracy of the friction prediction which suggests that moreorthogonal data is required for high rake angles. Fortunately, in the radial direction, the• Chapter 4. Modeling of Milling Forces 96Table 4.3: Cutting force coefficients for different rake angles as identified and transformedfrom orthogonal data by using linear edge-force model. Material:Ti6A14V. Note that theaverage values of the predicted force coefficient values are given for the feedrate rangeof s = 0.01 — 0.1 mm/tooth as the transformed force coefficients vary with the chipthickness due to the exponential variation of the cutting ratio r with the chip thickness.The edge-force coefficients are in (N/mm) and cutting force coefficients are in (MPa).c Test Predicted(deg) Kte Ktc Kre Krc Kae Kac Kc Krc Kac0 29.7 1825 55.7 770 1.8 735 1963 646 7785 24.7 1698 42.9 438 5.5 591 1805 461 69912 22.7 1731 44.5 317 2.4 623 1619 253 60415 22.3 1630 37.9 340 2.1 608 1544 177 56220 26.8 1439 38.8 376 2.6 604 1434 65 500critical chip thickness is much larger than it is in the tangential direction, so that theinaccuracies in Kr have smaller impact on the overall force prediction accuracy. On theother hand, very high friction values are obtained at large rake angles and it makes theuse of high rake angles impractical in titanium machining.The milling force coefficients were also identified by the exponential force model (fromequation 4.13) and are given in Table 4.4. For comparison purposes, the predicted valuesof exponential force coefficients can be determined from the ones calculated by the linearedge-force model as follows:K = Kre/ha + (x = t, r, a) (4.38)After the cutting force coefficients are obtained for different average chip thickness ha,then KT, KR, KA,P, q and .s can be obtained by exponential curve-fit.Chapter 4. Modeling of Milling Forces 97Table 4.4: Cutting force coefficients for different rake angles as identified from millingtests by using exponential force model. Material: Ti6A14V. (The unit of the coefficientsis MPa.)a Milling Test(deg) 7? p KR q KA0 822 0.354 0.268 0.333 0.698 0.2505 799 0.326 0.165 0.402 0.457 0.09612 999 0.268 0.123 0.471 0.482 0.14615 540 0.404 0.134 0.432 0.509 0.15020 631 0.366 0.156 0.400 0.587 0.177IbChapter 4. Modeling of Milling Forces 984.4.4 Accuracy of Milling Force Calculation by Predicted CoefficientsThe accuracy of the milling force predictions by calculated cutting force coefficients(linear edge force model) has been tested for over 20 milling experiments. The experiments were performed using cutters with different rake angles (0, 5, 12, 15, 20) andnumber of teeth (1 and 4), axial depth of cut (5 and 7.5 mm), feed per tooth (0.0127,0.025, 0.05, 0.1, 0.2 mm/tooth) and radial depth of cut (slotting, up and down milling-half immersion). The percentage deviations of the average and the maximum cuttingforce predictions from the measurements are shown in Figure 4.20. About 80 % of theforce predictions in x and y directions have less than +10 % deviation and the maximumdeviation in all cases is less than 25 %. The predictions for the z direction, however, haveless accuracy as it can be seen from the figure. This can be attributed to the fact thatno edge force has been used for the z direction. Although, the edge force in z directionis very small compared to x and y directions as shown in Table 4.3, at a small chipthickness its contribution may become significant. Also, the noise in force measurementsmay affect the results as the cutting forces in z direction are relatively small (< 100 Nfor most of the cases considered in the statistical analysis).A sample of half immersion up milling and half immersion down milling tests areshown to illustrate the accuracy of the instantaneous force predictions. For up milling,the entry angle is 0° and exit angle is 90°, whereas in the down milling the entry angleis 90° and the exit angle is 180°. Figure 4.23 shows the measured and the linear edgemodel predicted milling forces for a half immersion-up milling cutting test using a 19.05mm diameter, four flute end mill. The feed per tooth is s = 0.05 mm/tooth and thenormal rake angle is 12°. The axial depth of cut and the cutting speed are 5.08mm and30m/min, respectively. The force predictions based on the coefficients identified fromChapter 4. Modeling of Milling ForcesFxmaxMean-3.930c 250-4-0G) 15U)-Q00259915C0-4-0ci)Cd).0010Io ) 0 u) () 0 U) 0(N - - I - ‘- (NI I% Deviation50o U) U) 0 U)I- •-% Deviationo U)ccJC0-4-0ci)U).0050403020100FymaxMean -0.1402 30-4-255 201550U) 0 it) it)-- I% Deviation025o U) 0 it) it) 0 I1)(N - - I% DeviationC0-I0ci)Cl)-o0151050FzmaxMean -14.7IIuit) 0 it) 0 it) it) 0 it) 0 it) 00(N (N - - - (N (N C’)I I% Deviation20C2154-0cj 10U).0050FzaMean -19 — —--U)Oit)it)QLOOU)OQ‘- ‘- I -— (N (N C’) ‘% DeviationFigure 4.20: The statistical error analysis of the milling force predictions. The percentageerror in the milling force predictions compared to the measured values were determinedfor over 20 different milling tests.Chapter 4. Modeling of Milling Forces 100the slot millings tests and those transferred from the orthogonal data by calculatingshow good agreement with the measured values. Figure 4.24 shows the predicted andmeasured forces for a half immersion-down milling test with a four flute end mill where= 0.0127 mm/tooth and the rake angle is 0°; the other conditions were the same asin up milling. The accuracy of the force predictions by using transformed data is almostthe same as the accuracy obtained by the milling test calibrated coefficients.Figures 4.23 and 4.24 show the cutting forces predicted by the exponential forcemodel. Although the accuracy of the milling force predictions by the exponential forcemodel is acceptable, it is low compared to the linear edge model predictions. The exponential force coefficient model, in general, gives better results in milling force predictionsby calibration than the ones shown in Figures 4.23 and 4.24. A relatively small axialdepth of cut results in a very high variation of the chip load on the flutes in every fluteperiod. This results in some inaccuracies as the average cutting force coefficient (whichcorresponds to the average chip thickness) is used in the exponential cutting force coefficient models. However, statistical analysis and the analysis of the instantaneous forcesshow that the accuracy of the cutting force predictions is quite satisfactory with the edgeforce models.The approach allows designing a common orthogonal cutting data base, which can beused to predict cutting forces in a variety of oblique machining operations. Integrationof such a database to NC tool path generation algorithms in CAD/CAM systems allowsprocess planners to generate optimal, chatter and tool breakage free tool paths. The database is also useful in analyzing the performance of different cutter design geometries priorto cutting tests.Chapter 4. Modeling of Milling Forces 1014.5 SummaryMechanistic milling force models have been reviewed. An improved mechanics ofmilling approach is presented. In this model, the chip flow angle is numerically calculatedwhich improves the accuracy of predictions. The model predictions are verified by numberof experiments, and compared with the mechanistic model predictions.Chapter 4. Modeling of Milling Forces 102Figure 4.21: Measured and simulated milling forces (linear-edge force model). Forceswere calculated by using the transformed milling force coefficients from the orthogonaldata (predicted) and the milling calibration tests (calibrated). (Material: Ti6A14V,half immersion-up milling, t=°O5 mm/tooth, a=5.08 mm, V=30 rn/mm; tool: 4 flutecarbide end mill, a = 12°, d =19.05 mm.)8000 45 90 135 180 225Rotation Angle (deg)Figure 4.22: Measured and simulated milling forces (linear-edge force model). (Material:Ti6A14V, half immersion-down milling, s=0.0127 mm/tooth, a=5.08 mm, V=30 rn/mm;tool: 4 flute carbide end mill, a = 0°, d =19.05 mm)800600400200o 0Ii--200-400-6000 45 90 135 180 225Rotation Angle (deg)270 315 360zci)0LI6004002000-200-400270 315 360Chapter 4. Modeling of Milling Forces 103zC)0LIFigure 4.24: Measured and simulated milling forces (exponential force model). (Material:Ti6A14V, half immersion-down milling, t=00127 mm/tooth, a=5.08 mm, V=30 rn/mm;tool: 4 flute carbide end mill, a = 0, d =19.05 mm)0 45 90 135 180 225 270 315 360800600400200ci)C.)0 0LI-200-400-600Rotation Angle (deg)Figure 4.23: Measured and simulated milling forces (exponential force model). Forceswere calculated by using the transformed milling force coefficients from the orthogonaldata (predicted) and the milling calibration tests (calibrated). (Material: Ti6A14V,half immersion-up milling, St=O.O5 mm/tooth, a=5.08 mm, V=30 m/min; tool: 4 flutecarbide end mill, a = 12°, d =19.05 mm.)8006004002000-200-4000 45 90 135 180 225Rotation Angle (deg)270 315 360Chapter 5Effects of Milling Conditions on Cutting Forces and Accuracy5.1 IntroductionTolerance requirements on machined parts limit material removal rates in finishend milling operations as end mill and workpiece deflect under milling forces causingdimensional errors. Surveys have indicated that typical metal removal rates in finishmilling operations are only a fraction of those which should be used for either minimumcost or maximum production rates as very conservative feedrates are used to ensure thatthe part will pass inspection [153, 154]. Therefore, analysis of dimensional accuracy inmilling is necessary to improve the productivity without violating the tolerances. In thischapter, a surface generation model with flexible end mills is introduced. The effectsof radial depth of cut and the feedrate on the cutting forces and the surface error areanalyzed. A method of constructing optimal cutting conditions which provide minimumdimensional error is demonstrated. A similar analysis is performed for end milling ofvery flexible plates in Chapter 7.5.2 Surface Generation by Statically Flexible End MillThe flexible cutter deflects under the periodically varying milling forces which aremodeled in the previous sections. As given in Chapter 3, the end mill is modeled as acantilever beam attached to the collet with linear springs in both x and y directions asshown in Figure 5.1.104Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 105zFigure 5.1: Statically flexible end mill model. The tool is modeled as a cantilever beamwith springs attached to its fixed end to account for the clamping stiffness. Due to thehelical flutes on the tool, an effective diameter is used in the deflection analysis.The tool is divided into a number of elements with equal length. The cutting forceproduced by an element of flute j is found from equation (4.8)= { - cos2(z) + Kr(2j(z) - sin2j(z))]k(5.1)= tSt [2(z) — sin 2(z) + Krcos2j(z)]kwhere zk represents the z axis boundary of the cutter at node k (see Figure 5.1). Theaxial boundaries are modified if they do not match with the nodes on the tool. Theelemental cutting forces are equally split by the nodes k and k— 1 bounding the toolelement i. The deflection in y direction at node k caused by the force applied at node m2k-n+1-2—1YChapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 106is given by the cantilever beam formulation as [113]____L\Fm6EJ(3h1m_)+ k ,O<Vk<Vm5(k,m) = (5.2)LFmV2___6EJ(3V1m)+ k ,l’m<l’kwhere E is the Young Modulus, I is the area moment of inertia of the tool, k is theexperimentally measured tool clamping stiffness in the collet and vk = 1—z, 1 beingthe gauge length of the cutter. The area moment of the tool is calculated by usingan equivalent tool radius of Re = .sR, where s ( 0.8) is the scale factor due to flutes[114]. In this deflection model, the contact stiffness between the workpiece and the tool isneglected. Deflections in the x direction can be found similarly. The total static deflectionat nodal station k is calculated by the superposition of the deffections produced by all(n + 1) nodal forces on the tooln+1 n+1= 5(k,m) ; 5,(k) = (k,m) (5.3)m=1 m=1The finished workpiece surface is generated by points on the helical flutes as theyintersect it. The surface generation occurs as the points on the helical flutes satisfy thefollowing immersion conditions,1 0 up — milling(5.4)( ir down — mzllzngThe axial coordinate (z) of the flute - surface contact point in axial direction z canbe determined as a function of the tool rotation angle, q, as,z() = up — milling_ _______(5.5)z() = +JpK down—millingNote that depending on the cutting geometry, there may be several flutes and contactpoints generating the surface and they can be determined from equation (5.5). In theChapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 107simulations, the cutter is rotated at 0 angular increments such that the contact point fora flute jumps from one node of the cutter to the next one, i.e. 0 = and Liz isthe axial length of a tool element. At node k, the deflection of the cutter in y direction,6(k), is imprinted on the surface as dimensional error e(k).e(k) = 6(k) (5.6)The dimensional surface error profile along the axial depth of cut is simulated by determining the error created at each node as the cutter is rotated at 0 angular intervals.5.3 Identification of Cutting Conditions for Minimum Dimensional SurfaceErrorDimensional surface error magnitudes are the primary concerns in finishing operations. It is common practice to use very low feedrates and radial width of cuts to obtainthe required accuracy. However, this approach presumes that the dimensional surfaceerror is proportional to the material removal rate (MRR). Researchers such as Wang[33] attempted to estimate the cutter deflection and surface error in end milling by usingMRR. The agreement with experimental values is poor as the real force components andtheir distribution on the cutter flutes are not considered. Therefore, a detailed analysisof the cutting condition- dimensional surface error relationship is necessary to improvethe process planning in end milling operations. This is partly accomplished by the forceand surface generation models presented in the previous sections and will be extended inthe following.The material removal rate in an end milling operation depends on the spindle speed. Itis better to use material removed per revolution MFR which is independent of the speedChapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 108as a measure of material removal. For end milling operations, MFR can be expressed asMPR— abstN (5.7)where a is the axial depth of cut (mm), b is the radial depth of cut (mm),.s is the feed pertooth (mm), N is the number of teeth, and MFR is the material removed per revolution(mm3/rev). Equation (5.7) indicates that the amount of material removed is linearlyproportional to a, b and s. The cutting forces and surface error, however, do not varylinearly with the feed per tooth, .s, and radial width of cut, b, due to two main reasons.First of all, the cutting coefficients given by equation (4.13) are nonlinear functions ofthe average chip thickness which depends on st and b. In addition, the cutting forcesare not proportional to the radial width of cut, b, as it can be seen from equation (4.8).Therefore, it may be possible to optimize b and .s such that surface errors are less thanthe tolerable values without reducing, even increasing, MPR. A suitable index for thispurpose is the ratio of maximum dimensional error, emax, to the MFR which will betermed as specific maximum surface error (SMSE).SMSE1b — emax(b,st),8t)— MPR(b,s)In other words, SMSE(b,.St) indicates the maximum error generated on the workpiecesurface in order to remove 1 mm3 material for a specific combination of b and s. Therefore, the optimization procedure reduces to determination of b and s values for which theSMSE is minimum. The analytical surface generation model described in the previoussection allows the identification of suitable cutting conditions (b,.St).The optimization procedure applies to both up and down milling operations. However,in up milling the effect of radial width, b, on the SMSE is enhanced due to a mechanism explained as follows. In up milling the components of tangential, F, and radial,Fr, forces in y axis are in opposite directions as shown in Figure 4.1. Some portions ofChapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 109the tangential and radial forces cancel each other, resulting in lower normal forces (F).The same cancellation mechanism results in lower feed forces, F, in down milling, whichdoes not contribute to surface generation. Up milling is usually not preferred in finishingoperations as it usually produces poor surface finish. In up milling, the chip thicknessis zero when a tooth starts cutting. In the transient part of the process, the tooth rubsover the surface and then ploughes the metal until the piled up work material is pushedup along the rake face to form a chip [155]. The surface finish is poor as the surfaceis generated during the transient part of the cutting process. However, if the surface isground after the finish milling operation, then the dimensional accuracy of the surfaceis much more important than the surface quality. The radial width of cut (b) has thehighest effect on the force cancellation. In peripheral milling, the exit angle, can beused to represent the immersion of cut as,= cos’(l—(5.9)For small radial widths (çe,v < 15°), the normal force is mainly composed of radial forcevectors, therefore it is negative in up-milling. For large radial engagements (qeT 90°),however, tangential forces contribute to F most and thus it becomes positive. Therefore,the sign of the average normal force,,changes from negative to positive as cex variesbetween (0°— 90°). The average normal force becomes zero for a particular exit anglewhich will be designated by q. At this point the peak normal forces are also minimum.However, the same comments cannot be made for the dimensional errors on the surfacesince they do not only depend on the magnitude of the force, but its distribution alongthe helical flutes as well. On the other hand, for relatively small axial depth of cuts itcan be assumed that the maximum surface error generated on the surface is proportionalto the maximum normal cutting force, Fymax An approximate analytical procedure maybe followed to demonstrate the optimal exit angle identification procedure. The averageChapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 110normal cutting force in peripheral up milling is given by,=—sin2q — Kr Sfl2 qex) (5.10)Since the magnitude of the mean force with respect to the exit angle is desired to beminimized, the square value should be differentiated with respect to exit angle Afterderivations, the following equation for optimal exit angle, , is obtained.— (0.5 + Kr°x) sin2c5, + 0.25 sin4q — Kr sin2g5+0.5Kr Sj2 2ç + K sin 2ç& sin2 — cos (5.11)17 2,Ao ()o_T1ki. sin Yex cos —Equation (5.11) is approximate as the variations of the cutting coefficients, K and Kr, areneglected in the differentiations. Equation (5.11) can be solved by the Newton-Raphsoniterative technique for different values of Kr as shown in Figure 5.2. As Kr increases,the optimal exit angle, q, increases. The relation can be expressed by an approximateequation= 60.8K,. — 15.5K (5.12)An average value for Kr should be used in equation (5.12). The analytical formulationpresented above is approximate but it is quite practical for process planning of peripheralmilling operations. If the average value of K,. is known for a material- helical end millpair, an approximate optimal exit angle can be directly determined from equation (5.12).The exact values of an optimal radial depth and feedrate can be determined by usingthe cutting force and surface generation models presented in the previous sections. Thevariation of maximum dimensional surface error is obtained by milling simulations for arange of radial depth of cuts and feed-rates. The maximum surface errors, peak normalChapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 1116050401000 0.2 1.4 1.6Figure 5.2: Variation of the optimum exit angle (for up-milling) with Kr.cutting forces and SMSE can be plotted as a function of cutting conditions. The feasiblecutting conditions can be determined from the maximum surface error vs. radial width-feed graphs according to the tolerance requirements. The procedure is illustrated byexperimental and simulation results in the next section.5.4 Simulation and Experimental ResultsSimulation and experimental results are given in two groups. The first group is toverify the force and deflection models presented . The second group demonstrates theidentification of optimal cutting conditions for minimum dimensional surface error.5.4.1 Cutting Force and Surface FinishSeveral half immersion up and down milling simulations and experiments have beencarried out to verify the surface finish model presented. The cutting forces were measured by a Kistler table dynamometer, and a dial gauge was used for the surface profile0.4 0.6 0.8 1 1.2Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 112measurements. The work material, 7075 Aluminum alloy, was milled by a four fluted highspeed steel (HSS) end mill with 30 degree helix angle. In both cases, 4 different feedratevalues were used at a spindle speed of 478 rev/mm or a surface speed of 28 m/min. Thecutting constants were identified for up milling and down milling separately. Commoncutting conditions for the tests and simulations presented here are summarized in Table5.1, under Case # 1 and Case 2.Sample simulated and measured cutting forces for a feedrate of st = 0.14 mm/toothare shown in Figures 5.3 and 5.4 for half immersion up and down milling tests, respectively. The agreement between the simulation and experimental force results are quitesatisfactory. It has to be noted that due to a reversed tangential cutting force vector,the magnitude of the normal forces are about 500 N higher in down milling than in thecase of up milling.The simulated and measured surface profiles for 4 different feedrates are shown inFigures 5.5 and 5.6. Here, the surface dimension errors are measured from the intendedreference finish surface, which was marked by using precision machining before the experiments. In the up milling operations, because the cutting forces push the cutter towardsthe finish workpiece surface, extra material is removed from the desired finish surfacecausing overcut conditions. As expected, the maximum surface error occurs at the mostflexible part of the end mill which is its tip. The dimensional errors reduce as the axialdepth location is closer to the cantilevered side of the helical end mill. The dimensionalerrors in down milling operations are shown in Figure 5.6. Because the normal cuttingforces deflects the cutter away from the workpiece, extra material is left on the finishsurface causing undercut conditions. The magnitudes of the dimensional errors in downmilling is larger than those in up milling for two reasons. First, the magnitudes of theChapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 113normal F cutting forces are larger in down milling. Second, in up milling the tool ispushed into the material which resists against the deflection. There is no resisting contact stiffness in down milling, because the tool deflects away from the workpiece towardsfree air. In half immersion down milling, the dimensional errors are almost shifted inthe y axis, whereas in up milling they reduce approximately in a linear fashion. This isexplained as follows: In down milling, cutting force intensities are large at higher axialdepth locations where the rigidity of the cutter is higher as well. This counterbalancingmechanism results in, somewhat close to, the constant deflection along the axial depthof cut. The opposite phenomenon occurs in up milling operations, and the large deflections are observed at the tip of the flexible end mill. The maximum difference betweenthe predicted and measured surface errors is about 15 % in both cases. As it can beseen from the figures, the surface error predictions are in satisfactory agreement with themeasurements.5.4.2 Selection of Optimal Cutting ConditionsUp milling experiments were performed on free machining steel (AISI 1040) by using a carbide end mill with a 300 helix to demonstrate the optimal selection of cuttingconditions. The spindle speed was 478 rpm. The cutting constants are given in Table5.1, under Case #3. The absolute values of maximum surface error, emax, generatedand )SMSEI are determined by the simulation for a range of radial width of cuts andfeedrates as shown in Figures 5.7 and 5.8.Figure 5.7 shows that the simulated maximum surface error remains almost constantuntil the radial width is about 3.3 mm. After 3.3 mm the surface error increases considerably. Therefore, the feasible region of a radial depth of cut and feed per tooth canbe determined from Figure 5.7 according to the tolerance requirements. The next stepChapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 114Table 5.1: Cutting conditions for up and down milling tests conducted for surface errorverification and identification of optimal milling conditions. Case # 1 and # 2: MaterialAl7075 with normal chuck. Case # 3: AISI 1040 with stiffer power chuck.CUTTING PARAMETERS Case #1 Case #2 Case #3Mode of Milling Up Down UpTool diameter - d0 (mm) 19.05 19.05 19.05Tool gauge length - 1 (mm) 54.5 54.5 50.0Clamping stiffness- k (N/mm) 10200 10200 25000Tool material HSS HSS CarbideChuck regular regular powerTangential force coeff.- KT (MPa) 546 437 1140p 0.246 0.343 0.28Radial force coeff.- KR ( - ) 0.270 0.180 0.470q 0.271 0.249 0.079Radial width of cut - b (mm) 9.525 9.525 1, 3.35, 5.5Axial depth of cut - a (mm) 19.05 21.59 19.05Immersion angle of cut-q (deg) 90 90 26.5, 49.5, 65Flute lag angle -ka (deg.) 66.2 75 66.2Number of nodes (n + 1) 50 50 50is the selection of optimal conditions from the SMSE graph shown iii Figure 5.8. Theoptimal conditions correspond to the minimum value of SMSE as explained before.Figure 5.8 shows that the optimal conditions describe a curve on the radial width-feedper tooth plane. The graph indicates that SMSE values are very large for small width ofcut and feed per tooth values. This can be attributed to the high percentage of parasiticedge components in the total cutting forces at a low chip thickness. To verify the resultspresented in Figures 5.7 and 5.8, a set of experiments were performed at three differentradial widths (1, 3.35, 5.5 mm) and feedrates at (0.01, 0.06, 0.1 mm/rev-tooth). Theabsolute values of maximum normal force, Fymax, and maximum surface error,obtained from experiments and simulations are shown in Figures 5.9 and 5.10. Thereis almost a linear proportionality between Fymax and emax because of a relatively smallChapter 5. Effects of Milling Conditions on Cutting FOrces and Accuracy 115axial depth of cut. The agreement between the experimental and simulation results isquite satisfactory. As it was observed from the previous graphs, Figure 5.10 shows thatat about b = 3.35 mm the surface errors are minimal. Compared to b = 1 mm, themaximum surface error is almost the same even though the MPR is more than tripled.Another important point is that at b = 3.35 mm, high feed per tooth values can beused without increasing the surface error significantly. First of all, use of high feed-ratesfurther increase the productivity or MFR. Furthermore, the surface finish quality canbe increased by using high feed per tooth values as very rough surfaces are obtained bysmall feedrates in up milling due to rubbing. A further increment on the radial widthto 5.5 mm results in approximately tripled dimensional error magnitudes. SMSE for thesame feeds and the radial widths is shown in Figure 5.11. Analysis of the Figures 5.10and 5.11 confirms that the optimal radial width is 3.35 mm for a feed-rate larger than0.06 mm.Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 116150010005000-500-1000-1500-2000-2500 -0 45 90 135 180 225Rotation Angle (deg)Figure 5.3: Measured and simulated milling forces for half immersion-up milling. (Material: 7075 aluminum alloy, St = 0.14 mm/tooth, a = 19 mm, V 28 m/min. Tool: 4flute HSS end miii, 300 helix, diameter d = 19.05 mm)250020001500100050000 45 90 135 180 225Rotation Angle (deg)270 315 360Figure 5.4: Measured and simulated milling forces for half immersion-down milling. (Material: 7075 aluminum alloy, s = 0.14 mm/tooth, a = 21.6 mm, V = 28 rn/mm. Tool: 4flute HSS end miii, 30° helix, diameter d = 19.05 mm)za)20LIsimulation experiment270 315 360Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 117Figure 5.5: Simulated and measured surface profiles for half immersion-up milling. (Material: 7075 aluminum alloy, a = 19 mm, V = 28 m/min. Tool: 4 flute HSS end mill, 300helix, diameter d = 19.05 mm.) (z=0 at the free end of the end mill.)Surface Error (microns)Figure 5.6: Simulated and measured surface profiles for half immersion-down milling.(Material: 7075 aluminum alloy, a = 21.6 mm, V = 28 rn/mm. Tool: 4 flute HSS endmill, 30° helix, diameter d = 19.05 mm)1815- 12E-c 900as300 50 100 150 200 250Surface Error (microns)2824E20161240-300 -250 -200 -150 -100 -50 0Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 118E ci:,. Th. ‘ ‘——-2-0Figure 5.7: Variation of the predicted maximum dimensional surface error due to tooldeflection with the radial depth of cut and the feed per tooth. For certain values of radialdepth of cut, the error is minimum although the material removal rate is not. This graphis useful in selecting the cutting conditions for improved surface accuracy. (Material:Free machining steel, a = 19 mm, V = 28 rn/mm. Tool: 4 flute carbide end mill, 300helix, diameter=19.05 mm)C..••0: cc0 ciSITSE(rntc-revmn,J)0246870/20.CD—‘o•—.CDCD—.CD-qa-,-Cl)C,)II•—-cc—..,c—<C.n,CDCD<c-t—C0CD(CqCD0-CDCDCl)CD ôCDcXcD’< —.CD CDQ----,CD O(I)• ——.$:2•CDeCDCDCDCD0—.cCD .•CD—.Ci)CDCD-...-.-.0<IO(-I ItC (t(.Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 12020001600. 1200xE>LA4000Radial Width of Cut (mm)Figure 5.9: Variation of measured and simulated maximum normal cutting force Fymavwith radial depth of cut for different values of feed per tooth. The tool deflections due tothe normal force are imprinted as dimensional errors on the finished surface. (Material:Free machining steel, a = 19 mm, V = 28 rn/mm. Tool: 4 flute carbide end miii, 300helix, diameter= 19.05 mm)0 1 2 3 4 5 6 7 8 9Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy250//150 /EELUC,)C’)121Simulationst0.1Experimentst=0.06200E10050st=0.01 ////st=0.01 st=0.06 — — — — st=0.100 1 2 3 4 5 6 7 8 9Radial Depth of Cut (mm)Figure 5.10: Variation of measured and simulated maximum dimensional surface erroremax with radial depth of cut for different values of feed per tooth. The predicted optimumradial depth of cut is verified experimentally. (Material: Free machining steel, up milling,a = 19 mm, V = 28 m/min. Tool: 4 flute carbide end mill, 30° helix, diameter=19.05mm)1210____________86420• I I I I I I0 1 2 3 4 5 6 7 8 9Radial Depth of Cut (mm)Figure 5.11: Variation of measured and simulated specific maximum surface error SMSEwith radial depth of cut for different values of feed per tooth. (Material: Free machiningsteel, up milling, a = 19 mm, V = 28 m/min. Tool: 4 flute carbide end mill, 30° helix,diameter=19.05 mm)Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 1225.5 SummarySurface generation model with a flexible end mill is presented. The effects of thefeedrate and the radial depth of cut on the surface accuracy and the milling forces are analyzed. An optimal milling condition identification method is presented. It is shown thatthe surface accuracy can be significantly improved without decreasing, even increasing,the material removal rate. The simulation results are verified experimentally.Chapter 6Static Structure-Milling Process Interaction6.1 IntroductionIn the previous chapter, milling forces and tool deflections are modeled independently. However, the cutting process and the structural deformations of the tool andworkpiece interact during milling. The dynamic interactions cause chatter and forcedvibrations which are analyzed in Chapter 8. The static interactions result in variationsin the cutting forces, and thus have to be considered for accurate force and deflectionpredictions. Sutherland et al. [24] and Armarego et al. [36] analyzed the staticallyflexible milling process by numerical models. They determined the chip thickness in aflexible milling system by considering deflections of the tool in successive tooth periods.The effects of deflections on the chip thickness and the cutter-workpiece engagementboundaries are determined together due to the numerical procedure followed. In thischapter, the effects of static deflections are considered in two groups: local and globaleffects. The local effect is the change in the chip thickness due to deflections which ismodeled by the statically regenerative milling force model, whereas the variation in thecutter-workpiece engagement boundaries is considered as a global effect of the deflectionsand it is analyzed in the flexible milling force model. An analytical formulation is givenwhich shows that the effect of deflections on the chip thickness diminishes very fast. Theregenerative milling mechanism is important for the adaptive control of the milling forcesby feedrate as analyzed in [22, 43]. However, for steady-state milling force and surface123Chapter 6. Static Structure-Milling Process Interaction 124error predictions, only the variation in the engagement boundaries or radial depth ofcut should be considered. Determination of the effective radial depth of cut under thedeflections is explained in this section, however, the application of the method is givenin Chapter 7 for plate milling analysis. Therefore, the analytical formulations for thestructure- milling process interactions given in the following sections eliminate the timeconsuming numerical solutions and give more insight to the real physics of the problem.6.2 Statically Regenerative Milling Force Model (Variation of Chip Thickness Due to Deflections)Figure 6.1 shows the generation of the tooth path by flute j at axial depth of z andtooth period number i. The tooth period count i is started from the instant when thereference flute’s tip is at zero immersion (j(O) = 0) in the beginning of the cut. i isincreased by one when the next flute reaches the same location, i.e. one pitch angle (q)later. As a result of the deflections of the tool in feed (x) and normal (y) directions, theflutes may lose contact with the workpiece at some points. Therefore, unlike the rigidmilling, the surface being cut by tooth j may not have been created by the previoustooth only. Here, notation will be used to denote the number of previous toothperiods which may have an influence on the presently cut surface at tooth period i byflute j. The value of chailges as the cutter rotates. It is assumed that the valueof remains the same for flute j along the axis z. From hereon, the notationwill be used in place of mj,j(q) for simplicity in the following formulation. In Figure6.1, O(z), O_,(z), O(z) and O_(z) are the undeflected and deflected tool centers ataxial location z, respectively. Broken and solid lines indicate the tooth paths of therigid and flexible cutters. Only tool deflections are considered in this section, however,Chapter 6. Static Structure-Milling Process Interaction 125YFigure 6.1: Statically regenerative chip thickness geometry in milling. The material lefton the cut surface due to the deflections is encountered by the following teeth. Thedeflected positions of the tool in the successive tooth periods have to be considered todetermine the chip thickness. Also, the contact between the cutter flutes and the workmay be lost at some points due to large deflections. This nonlinearity has to be consideredin the analysis as well.(i— M)tl flexibletooth path—(1 i’Y” rigidtooth pathi flexible toothpathB’Ri” rigid tooth pathxChapter 6. Static Structure-Milling Process Interaction 126the following formulation can easily be extended to include the workpiece deflections byadding them to the tool deflections. In previous sections, it was shown that the tooldeflections are dependent on the cutting force intensities, and for tooth period i they are6(,z) = ; (z) (6.1)where dF(q, z)/dz and dF(q, z)/dz are defined as cutting force intensities at the toothperiod i. Assuming that the axial and radial depths of cut and the cutting parameterKr remain the same under flexible and rigid cutting conditions, the unit cutting forceintensity functions are also identical in both regenerative and rigid cutting force models. Considering the changes in the chip load due to the regeneration mechanism, theregenerative cutting force intensities can be approximated as the product of the totalregenerative cutting force and the unit rigid cutting force intensitydF(qf,z) dF,(,z) (6.2)where and F1 are the regenerative cutting forces at the tooth period i, and df(g5, z)/dzand df(q5, z)/dz are the unit rigid force intensities given by equation (4.10). In the formulation, superscript R is used to indicate rigid cutting forces. In this analysis, F is notconsidered as it does not take part in the regeneration mechanism. However, F is affected by the deflections in x and y directions. Since the cutter deflection at a particularlocation is linearly dependent on the force magnitude, equation (6.1) can be scaled withtotal force magnitudes,= F (q) ; 6(gS,z) = (q5)6 (6.3)where the force magnitude independent unit deflections are defined asdfR(,z) . dfR(c,z)X — X’’PZ d ‘ “ — dChapter 6. Static Structure-Milling Process Interaction 127Due to cutter deflections, the chip thickness may change at each flute immersiollq, and previous deflections or regeneration marks symbolized by counter p. Due toregeneration, the tooth j experiences an actual chip thickness of A’B’ at tooth periodi, where A’ is on the arc cut at period i—t, and B’ is on the presently cut arc, seeFigure 6.1. Between two arcs, the cutter center is moved at amount of 1Ust distance infeed direction. From the exaggerated view of Figure 6.1 it can be argued that when thepoint A’ was being cut tooth period before, the immersion angle of the tool, wasnot exactly equal to the present immersion çj. However, a careful look at the geometrywould reveal that ç&j as R >> [tSt. Therefore, for the tooth period i the chipthickness at any z position on the tooth j can be expressed as followsz) = A’B’ = AB + BB’ — AA’orz) = Itst sin b(z) + (g5) coscos (z) + z) (6.5)sin qj(z)—sin qj(z)The unit deflections defined by equation (6.4) are the same for every tooth period as theydo not consider the regeneration in chip thickness. In equation (6.5), however, possibletool-workpiece contact losses are considered for tooth periods which are indicated byunit deflections with subscripts . This is called the basic nonlinearity in the regenerationmechanism, and it occurs when deflection magnitude BB’ is equal to or greater thanthe chip thickness h1,(q, z). Considering the direction sign from the geometry, the toolmaterial contact loss is modeled iii the formulation as.sin qj(z) + cosçj(z) = —h,(q5,z) (6.6)Chapter 6. Static Structure-Milling Process Interaction 128where h,(q) is given by equation 6.5. The above nonlinearity condition is reorganizedas— —h,(,z)—F) sin (z) + F) () cos(6.7)— —h,(g,z)yi — —F() () sin(z) + F)cos(z)The chip thickness which includes the effects of tool deflections and the regeneration,equation (6.5), is substituted in equations (4.1) and (4.6) to obtain the flexible cuttingforce intensities asdF,(,z)= K [cos(z) + Kr sinth(z)] {st sin(z)+- cos— sin(6.8)z)= K [sin(z) — Kr cos (z)] {‘St sin (z)+- cos(z)+[F(1 sinqj(z)}Equation (6.8) can be integrated along the in cut portion of the tooth to obtain the totalChapter 6. Static Structure-Milling Process Interaction 129cutting forces realized by the tooth() = —F (Xj,j () + F(_) ()cX() @)—(qy () + FX() @)7x(_),, (q5) +(6.9)(q) F ()cx,(g5— (q)+— +where the so-called flexibility terms are given by the following expressionsZ32(cb) 1= Ktf() (cos2 (z) + Kr sin 2(z))fZj,2(,b) /1= KtJ çsin2qj(z) + Kr Sifl2 cbj(b)) 6dzZj,i()(6.10)= Kj3’2 ( sin2(z) — Kr cos2 (z))= KtJ22 (sin2 (z) — Kr sin 2(z))z,j (ç)The flexibility terms of the (i — )th step are similar except and are to be replacedby and The above integrals are to be computed numerically by using the tooldeflection values at the nodal points. At this point an approximation will be introducedfor the sake of simplicity by assuming that at the tooth period i and immersion angle q,is the same for all of the teeth which are in contact with the workpiece. This meansthat, at a specific i and çS , if the tooth 0 is cutting the surface which was cut by 2 toothperiods before, then all of the teeth which are in contact with the workpiece are cuttingthe surfaces which were cut 2 tooth periods before. From equation (6.8), the flexibleq)Chapter 6. Static Structure-Milling Process Interaction 130cutting forces applied on each tooth can be summed up to obtain the following equation[A()] {F()} - [A()] {F()} + {F()} = {} (6.11)where the so-called flexibility matrix is given by[A()j = 1+ () a) (6.12)7(g) 1—amwhereN-i=j=oand the other flexibility terms are similar. The flexible and rigid force terms in equation(6.11) are defined respectively as,F) FR(){F(ci)} = ; {FR(q)} = (6.13)F(q) J F(q) JFor the surface generated by a flute which is free of cutting marks or regeneration,equation (6.11) reduces to the following form[A()]1 {F()}1 = {FR()} (6.14)from which the regenerative cutting forces for the very first tooth period can be obtainedas{F()}1 = [A()j1{F)} (6.15)Cutting forces calculated by equation (6.15) include only the current deflections of thetool. However, the surface left by the first tooth will be regenerated by the successiveteeth of the cutter. If the chip thickness for the first tooth reduces due to the deflections,the next tooth will have to remove the extra material. After computing the forces for theChapter 6. Static Structure-Milling Process Interaction 131first non-regenerative tooth period, (6.11) gives the flexible regenerative cutting forcesfor the successive tooth periods= [A()]’ {{FR()}- {F()} + [A()]1{F()}] (6.16)t can only be determined in an iterative manner due to two reasons. First, t cannotbe determined independently as the total flexible cutting forces are required in equation(6.5). The second reason is the nonlinearity defined on the deflections by equations (6.6)and (6.7). At each iteration step, ji is determined such that it gives the minimum chipload. The flexible cutting forces calculated in the previous tooth period can be used inequation (6.5) in the first iteration step. Modifications of deflections are followed by theredetermination of t in an iterative ioop until the convergence in and the flexible forcesare obtained. Then, the cutter is rotated by a certain incremental amount and the sameiterative loop is repeated.For an approximate analysis, the nonlinearity expressed by equation (6.7) can beneglected, i.e. it is assumed that the contact between the workpiece and tool is not lostduring the regeneration process and = 1. Then, the unit deflections are equal forsuccessive tooth periodsxi =L_1 6Yi—1 (6.17)It can be seen from equations (6.10) and (6.12), the flexibility matrix remains constant[A(g5)j = [A(qf)1 = [A(çi)]_1 (6.18)Substituting into equation (6.11) the following is obtained:{F()} = [A()]’ {F)} - [[A()]1- [‘1] {F()} (6.19)where [I] is the (2x2) identity matrix. The cutting forces for the first pass are= {A()]’ {F’)} (6.20)Chapter 6. Static Structure-Milling Process Interaction 132This can be substituted in equation (6.19) to obtain the forces in the second tooth period{F()}2 = [A()]’ [2 [I]- [A()]-’] {F)} (6.21)Similarly, forces in the 3rd and 4th tooth periods are obtained as{F()}3 = [A()]1 [3 [I]-3 [A()]’ + ([A)r1)2]{FR()}2 (6.22)[A()]’ [4 [I]-6 [A()]’ +4 ([(A()r’)- ([A()r1)3]{FR()}The following general form for the regenerative cutting forces is obtained by intuition:{F()}= [[I]- [[I] - [A()]1]t] {FR()} (6.23)Therefore, the regenerative cutting forces can be related to the rigid cutting forces analytically. The basic form of the regenerative cutting forces was proposed by Tlusty [156].Tomizuka et al. [157] derived the discrete form and Spence and Altintas [43] used it inthe adaptive control of milling forces. Equation (6.23) is more accurate than the previous models as the regeneration of the chip thickness along the helical flutes of the cutteris modeled, whereas in the previous models, a point contact between the tool and theworkpiece was considered.A zero helix cutter case will be used to demonstrate the behavior of milling forcespredicted by equation (6.23). For the zero helix case, the coefficients given by equation(6.10) can easily be determined by multiplying the integrand by the axial depth of cuta. Let us consider a particular rotation angle of = ir/2. Then,cr, = 0; = KtKra; a = 0; = KtarChapter 6. Static Structure-Milling Process Interaction 133For relatively short axial depth of cuts, the unit deflections can be approximated asconstant in the cut portion of the flute. Then, unit deflections become equivalent to thecompliance of the cutter= ; = (6.24)where k and lc are the stiffness of the cutter in x and y directions, at the tip of thecutter (or at the midpoint of the axial immersion for a better approximation). Also,assume that only one flute is in cut when qf = ir/2, e.g. 3 flute cutter in full immersion.Then,KtKra Kta7x=; = (6.25)andl+Ya 0[A(ir/2)] = (6.26)1Considering practical values of K, Kr, a and k it can be seen that Yx, Yy < 0.01. Thissuggests that , [A(ir/2)] [I] and {F(7r/2)} {FR(7r/2)}, and thus the regenerativecutting forces converge to the rigid cutting forces very fast. The result obtained fromthis particular case is generalized by the discussion given below.For a vibration free cutting process, the cutting forces must stabilize after a transientperiod. Here, the stability means that the forces converge to the force values obtainedin the previous tooth period. Therefore, after the steady-state is reached in the flexiblecutting forces, i is always 1 as each tooth has to cut the surface left by the previoustooth. Mathematically, this can be expressed as= {F(q5)}_1and= [A(q)1_1Chapter 6. Static Structure-Milling Process Interaction 134If these conditions are imposed on equation (6.11) the following equality is obtained= {FR()}Therefore, in a vibration free cutting process the flexible or regenerative cutting forcesconverge to the non-regenerative rigid cutting forces after some transients. This is anexpected result as the chip thickness removed with a flexible cutter and a rigid cutterare the same after some transients. However, as we analyzed in the surface generationmechanism, some material is left on the finished surface as surface error. This does notaffect the chip thickness as it is not on the machining surface (the surface between stand qex), but it changes the radial depth of cut as it will be analyzed in the next section.The formulation given above is programmed into the computer. The nonlinearitygiven by equation (6.7) is considered in the numerical solution. As an example, the simulated regenerative and rigid cutting forces in x direction are shown in Figure 6.2. Theaxial depth of cut and the immersion angle are 19 mm and 37.8°, respectively. A 4 flute,300 helix, 19.05 mm diameter end mill with a gauge length of 55 mm was used in thesimulations. The feed per tooth is 0.14 mm/tooth. The workpiece material is aluminumalloy for which K = 99OMFa and Kr = 0.52 were used. The cutter is divided into40 axial elements in the simulations. During the first tooth period, the cutter deflectsaway from the workpiece as the resultant cutting forces pull the cutter away from theworkpiece. Hence, the tool cuts a very small amount of the total chip load in the firsttooth period resulting in very small cutting forces. In the second tooth period, the cutterhas the additional load due to the material left behind during the first tooth period. Thecutter deflects again, but it removes much more material than the first period, as theaccumulated chip thickness is larger than the tool deflections in this period. As a result,the regenerative cutting forces approach to the rigid cutting forces and by the fourthChapter 6. Static Structure-Milling Process Interaction 1350-100-200z-300xU--400-500-6000 90 180 270 360Rotation Angle (deg)Figure 6.2: Simulated rigid and statically regenerative milling force in x direction. Thestatic regeneration mechanism diminishes in a few tooth periods which confirms the analytically obtained results. (K = 990 MPa, Kr = 0.52, a=19 mm, s = 0.14 mm/tooth,up-milling, exit angle: 37.8°. Tool: 4 flute, 300 helix, diameter=19.05 mm)period they become exactly equal to the rigid forces.The regenerative forces formulated can be used in modeling the transfer function ofthe statically compliant milling process for adaptive control strategies [43, 158]. Also, inmilling force predictions if the feed-rate and workpiece geometry (i.e. axial and radialdepths of cut) change frequently within 3 to four tooth periods, the regenerative forcemodel has to be used.6.3 Flexible Milling Force Model (Variation of Radial Depth of Cut Due toDeflections)Although the effect of deflections on the chip thickness diminishes very quickly, theradial depth of cut and thus the immersion boundaries vary due the deflections. ThisChapter 6. Static Structure-Milling Process Interaction 136variation is significant when the deflections are considerable compared to the radial depthof cut, so both tool and workpiece deflections are considered in this section. The variationin radial depth of cut and chip thickness due to tool deflections were considered togetherin the numerical models in [24, 36], which did not allow to separate the effects of eachmechanism. In this section, the steady-state effect of the deflections is formulated byconsidering the variation in the radial depth of cut.The exit angle, qex, in up milling and the start angle, , in down milling depend onthe radial width of cut, b. In Figure 6.3 the up and down milling geometry under theeffect of deflections is shown. In flexible milling, the radial depth of cut varies along theaxial direction, z, and feed direction x as shown in Figure 6.3 and is given bybf(z, ) = b + [6(z, ) — w(x, z, q)j (6.27)where b and bf are the nominal and effective radial depth of cut, respectively. 5,, and ware the tool and plate deflections in the normal direction (y). The tool deflection in the(+y) direction (or workpiece deflection in (—y) direction) results in an increased radialdepth of cut in up milling , and a reduced radial depth in down milling. This is thereason of for having + sign in equation (6.27), which is (+) for up milling and (-) fordown milling. The variable width of cut, bf(z, q), results in different exit or start anglesfor each flute which is in cut. In the following, the formulations for the exit and startangles in up and down milling operations will be given.Consider the up milling geometry shown in Figure 6.3 where the engagement of tooland workpiece for deflected and undeflected positions is shown. q denotes the exit anglefor the rigid milling. The exit angle under the deflections can be written as= cos1(1— ) (6.28)Chapter 6. Static Structure-Milling Process Interaction 137Down MillingFigure 6.3: Variation of radial depth of cut and immersion angles due to deflections inup and down milling.wflexible MiffingUp MillingyxRigid Mffiing¶ oyChapter 6. Static Structure-Milling Process Interaction 138Note that the start angle, ç5, does not change due to deflections. The cutting forces inflexible end milling can be calculated from equation (4.8) provided that the lower andupper limits of the incut portion of a flute, zj,i and zj,2, are determined under the effectof deflections. The relation between the upper and lower limits and immersion anglescan easily be obtained from equation (4.3), which shows the variation of the immersionangle, qj(z), in the z axis due to the helix. Then, the upper and lower limits are givenbyZj,2 =(6.29)zj = [ + (j - 1)p - e(Zj,i, )]As it can be seen from equation 6.29 the upper limit does not vary with the deflectionssince the start angle remains constant, q 0, as far as the contact between the work-piece and the tooth is not lost at z,2, i.e. bf(z,2) > 0. If the contact is lost, however, theupper limit becomes independent of the start angle and it can be calculated from equation (6.27). This is, however, unlikely to occur in up milling operations as the cuttingforces try to deflect the workpiece and tool towards each other if the radial depth of cutis not too small. In the case where the radial depth of cut is too small the radial forcesare dominant in the normal force which may try to separate the workpiece and the tool.Therefore, the existence of the contact between the work and the flute should be checkedand if necessary, the new value of z,2 should be determined before calculating the lowerlimit. If zj,2 is found to be less than zj,1 there is no need to update the lower limit asthe total contact between flute j and the workpiece is lost. The effect of deflections and,thus, the variation in the exit angle can be seen in the lower limit. The following implicitequation is obtained if ex from equation (6.28) is substituted into z,1 in equation (6.29)F(z,1)= z,1 — + (j — 1) — cos1(1 — bf(zj, ))] (6.30)The above equation can only be solved by iterative techniques as the radial depth bf isChapter 6. Static Structure-Milling Process Interaction 139also a function of the lower limit. In the Newton-Raphson method the solution in themtII iteration is updated asz1 = z1— (6.31)where— dF — dbf/dz,lz,1tan —(1— i)2wheredbf d5 dw=——(z,i) — (6.33)The values and the derivatives of the deflections at zj,1 can be interpolated from thedeflections of nodes k and k + 1 between which z,1 lies:dS(z,i) —__________________dz —(6.34)(z,1) = 6 ((k — 1)z) + d(z,1)(z,j — (k — 1)z)where Lz is the length of an axial element, /z = a/(n — 1). The workpiece deflectionand its derivative are approximated in a similar fashion. The iteration in equation (6.31)is started by zj,1 of the rigid milling. If z,1 becomes larger than zj,2 the contact betweenflute j and the workpiece is lost. This is imposed on the solution by letting z,1 = z,2when z,i > z,2.The variation of the start angle for down milling is shown in Figure 6.3. q is thestart angle of the rigid case. The exit angle which is equal to r does not vary under thedeflections if the contact is not lost, as shown in the figure. The comments made forthe upper limit in the up milling case apply to the lower limit in down milling, i.e., theexistence of contact between tool and workpiece should be checked first. Similar to theChapter 6. Static Structure-Milling Process Interaction 140exit angle in up milling, Cst in down milling can be expressed as followscst = — cos’(l — bf) (6.35)According to equation (6.29), z,1 remains fixed as the exit angle is always equal to r indown milling. If ckSt is substituted in equation (6.29) the following implicit equation isobtained for zj,2F(z,2)= z,2 — + (j — 1) — + cos’(l— )] = 0 (6.36)The solution for z,2 can be obtained iteratively as explained in up milling. There are,in fact, two iterative loops in the formulation presented above. In the inner ioop thelimits of integration are determined iteratively. The convergence is very fast in this loop.In the outer loop the cutting forces and defiections of the workpiece and the tool arecalculated by using the limits updated in the inner ioop. The convergence of this loopdepends on the magnitude of the deflections. The iteration process starts by using thevalues obtained in the rigid force model. The deflections are used to determine z,1 forup milling and z,2 for down milling iteratively, as explained above. The new values ofintegration limits are used in the cutting force equations, equations (4.8), to update theforces and deflections to be used in the inner loop again. This procedure is used forthe accurate prediction of dimensional form errors in milling very flexible parts which ispresented in the following chapter.6.4 SummaryThe variations in chip thickness and radial depth of cut due to cutter and workpiecedeflections are analyzed. It is analytically and numerically shown that the chip thicknessapproaches the intended value very fast for a stable milling process. A flexible millingChapter 6. Static Structure-Milling Process Interaction 141force model which determines the effective radial depth of cut under the deflections isdeveloped for static milling operations.Chapter 7Peripheral Milling of Plates7.1 IntroductionThe peripheral milling of flexible workpieces is complicated, where periodically varying milling forces excite the flexible cutter and workpiece structures both statically anddynamically. Static deflections produce dimensional form errors, and dynamic displacements produce a poor surface finish in milling. The dynamic cutting and stability analyses are given in Chapter 8. In this chapter, the static deflections of the plate and endmill under the milling forces and the resulting dimensional surface errors are considered. The structural models of the plate and the tool described in Chapter 3 and themilling force models given in Chapters 4 and 6 are used to develop a process simulationmodel which helps to produce acceptable tolerances in machining very flexible structures.A noted previous study on the peripheral milling of flexible structures was carriedout by Kline et al. [35]. Kline considered the milling of a clamped-clamped-clamped-free(CCCF) plate with a flexible end mill. He used the FE method to model the plate, andthe beam theory for the end mill. However, Kline’s CCCF plate was comparatively rigid,because it was clamped from the three edges leaving only one edge free for displacement. Also, he neglected the effect of the deflected plate and the tool on the immersionboundaries, which is significant in milling really flexible plates as illustrated in Chapter6 and in this chapter. Sagherian et al. [99] used a similar model to Kline’s in studying142Chapter 7. Peripheral Milling of Plates 143the milling of a CFFF type plate. Even though the axial depth of cut was considerablysmall, Sagherian et al. [99] reported significant defiections of the workpiece and resultingdimensional errors left on the finish surface. Altintas et al. [98] analyzed cutting forcesand deformations, both dynamically and statically, in the peripheral milling of such aflexible plate at a particular location by neglecting the time varying structural propertiesand the changes in the immersion boundaries. The true kinematics of dynamic milling[37] were employed in the model in order to track chatter vibration waves left on thesurface. Their study [98] showed that dimensional form errors produced by quasi-staticcomponents of the cutting forces were quite significant. Furthermore, although very lowcutting loads were used in milling the plates, the static deflection of the long slender endmill was found to be considerable. The previous studies did not investigate any methodwhich reduces the excessive form errors in milling very flexible structures.In this chapter, the simulation system developed for the milling of very flexible platesis explained. The plate structure is modeled by a developed FE code and the cutter isrepresented by an elastic beam. The variations in the plate structure and the partialdisengagement of plate from the cutter due to excessive deflections are considered whenpredicting the cutting forces and the deformed finish surface dimensions. The FiniteElement code and the milling force calculation routines have to be integrated due to theinteraction between the milling geometry and forces and the plate and tool deflections.A method of milling very flexible plates within the specified tolerance is developed byvarying the feed along the tool path. In the following sections, modeling of the plate andtool structures, the cutting force distribution and identification of varying immersionboundaries, the simulation of plate surface generation and constraint of dimensionalsurface errors by feed scheduling are presented. The surface generation and dimensionalaccuracy control methods are experimentally proven in machining very flexible plates.Chapter 7. Peripheral Milling of Plates 1447.2 Static Modeling of Plate MillingThe workpiece considered here is a clamped-free-free-free (CFFF) plate as shown inFigure 7.1. The plate is down milled by removing a small width of cut with a long slenderhelical end mill. As the material is removed and the tool changes its contact position,the stiffness of the plate changes both in the feed (x) and axial (z) directions. The toolstructure is modeled as a cantilevered beam and it remains fixed during the machiningprocess. The details of the structural models of the plate and tool are given in Chapter3. Here, the interaction of tool-plate structures and the machining process is presented.7.2.1 Structural Model of the PlateThe discontinuity in the plate thickness due to machining requires that the structureshould be considered as a three dimensional object. A finite element (FE) model ofthe plate is constructed using 8 node isoparametric elements with thickness control aspresented in Chapter 3. Approximate dimensions of the sample plates used in simulationsand experiments are 63.5mm x35mm (with 2.45mm and 6.3mm thickness). Plates aredivided into an equal number of (n) elements both in feed (x) and tool axis (z) directions.The number of elements is equal to the plate length divided by the cutter contact length= R sin q in the immersion zone, which simplifies the force and surface generationsimulation as explained later. Each node is constrained to have three translational degreesof freedom. The cutting forces acting on the nodes are all zero except at the toolworkpiece contact zone, which is represented by the elements in the immersion zone.The contact zone elements are bounded by nodal axes boundaries (A — A, B — B) whichare facing the end mill (Fig. 7.1.b). The cutter enters the workpiece in the down millingmode from the axial nodal line B — B where the plate has an uncut thickness (ta), andChapter 7. Peripheral Milling of Plates 145Figure 7.1: (a) Peripheral down milling of flexible plates, (b) Finite element model of theplate, (c) Corresponding nodal stations on the tool.z(a)—I b-’tucutter entryr%> tocutter exitA(b) (c)Chapter 7. Peripheral Milling of Plates 146exits from the plate at nodal axis A — A where the plate has an after-cut thickness (ta).The thickness changes linearly within the isoparametric elements as shown in Figure 7.1.Since the plate is most flexible in the normal direction, the normal F cutting forces areapplied to the plate at tool-plate contact nodes (A — A, B — B). The cutting forces arecalculated analytically within each element boundary and distributed to four face nodesequally along the tool’s z axis as explained below. The force distribution is carried outfor all elements representing the tool-plate contact zone in the z direction. The staticdeflections of the plate nodes are calculated by solving the following matrix equation,[K]{w} = {Q} (7.1)where [Kr] is the square stiffness matrix, and {w} is the displacement vector, and {Q}is the force vector whose elements are zero for all the nodes except the ones at the toolworkpiece contact zone. The elements of {Q} in the contact zone are equated to thenodal cutting forces, The simulation is carried out at elemental increments alongthe feed axis x, and the thickness of the elements are reduced along the tool-plate exitaxis A — A. The material removal is considered by updating the stiffness matrix at eachfeed location. The details of the Finite Element model of the plate are given in Chapter 3.7.2.2 Structural Model of the ToolAs described in Chapter 3, a slender helical end mill with a gauge length of 1 mm fromthe clamped chuck end is modeled as a cantilevered beam with an equivalent diameterof de = . d, where d is the diameter of the cutter and s is the scale factor due to helicalflutes. Experiments showed that s = 0.75 for d = 19.05mm diameter cutter, which isidentified as suggested by Kops et al. [114]. The stiffness of the tool clamping in thecollet is considered by assuming a linear spring between the rigid spindle body and the• Chapter 7. Peripheral Milling of Plates 147clamped end of the cutter in the collet. The cutter is divided into equally spaced axialnodes which correspond to the plate nodes in the axial direction (z). The same nodalforces {LF} are applied to the tool, but in the opposite direction to the plate nodes.The cantilever beam formulation is used to determine the tool deflections at the nodalstations. The contact stiffness between the workpiece and the tool is neglected. Thedeflection at node k caused by the force applied at node m is given by________/Fy,m6E1 (3VmZ’k)+ ,O<11k<Vm6Yk,m = (7.2)ZFm6E1 — , l’ <‘kwhere Vk = 1 — zk, E is the Young Modulus, I is the area moment of inertia of the tool,and k is the tool clamping stiffness in the collet. The total static deflection at nodalstation k is calculated by the superposition of the defiections produced by all (n + 1)nodal forces,n+15(k) = 6Yk,m (7.3)m=17.2.3 Cutting Force Distribution-Rigid and Flexible Force ModelsMilling forces are calculated as described in Chapter 4. Immersion angles are measured clockwise from the normal (y) axis to a reference flute j = 0, which has immersionq at its tip z = 0. On flute j, a differential chip element at axial location z has immersionangle 4(z) = q+jq5—k&z, where = (tan ‘çb)/R and ‘k is the helix angle. The tangential and radial forces acting on the flute element are resolved in x and y directions, andintegrated analytically along the in-cut axial element /c of the flute j which correspondsto the finite element k on the plate. The axial boundaries of the element k are nodalChapter 7. Peripheral Milling of Plates 148stations k — 1 and k,k) = —sK jk [sinj(z) cos (z) + Kr sin2 dzZk_ 1Zk (7.4)k) = stKtJ [sin2 qj(z)— Kr sinj(z)cosj(z)] dzZk_1where zk represents the z axis boundary of the cutter at node k. If a flute element is not inthe contact or in cutting zone, it contributes a zero cutting force. The axial boundariesare modified if they do not match the nodal stations on the tool. The cutting forcescontributed by all flutes are calculated and summed to obtain the total instantaneousforces acting on element k. For an end mill with N number of flutes,N-i N-izF(q, k) = k), k) = k) (7.5)j=O j=OThe cutting forces k) are split by the nodal stations k — 1 and k boundingthe tool element k. The same forces are equally split in the opposite direction by thefour nodes of the corresponding plate element k which are facing the tool. The cuttingforces are distributed in a similar fashion to the remaining plate nodes in the contactzone and at the tool’s nodal stations. The force computation and distribution model ismore accurate than the digitally integrated forces concentrated at the force center of thestructure [35].For experimental verification, the total cutting forces applied to the whole tool orplate are calculated by summing the elemental forces.= k), F()=k) (7.6)Chapter 7. Peripheral Milling of Plates 149The effect of tool and workpiece deflections on the chip load and cutting force calculations have been neglected so far. The method in this form will be termed as therigid model. The analysis is further improved to include the effect of defiections on theimmersion and the cutting force distribution in the flexible model which is formulatedin Chapter 6. Sutherland et al. [24] developed a numerical model which determines theactual in cut portions of the flutes and the chip thickness under the tool deflections byemploying a regenerative chip thickness algorithm. In the analytical formulation givenin Chapter 6, it is shown that for a static peripheral milling process, which is free ofchatter vibrations, the chip thickness predicted by the regenerative model converges tothe chip thickness in the rigid model after several revolutions of the cutter. The flexibleforce model needs to consider only the variations in the immersion boundaries, i.e. start,cst, and exit, q, angles of the cut along the tool - plate contact zone. çb depends onthe radial width of cut in down milling:= — cos’(l— -) (7.7)where b is the actual radial width of cut due to deflections in down milling operations,• as shown in Figure 6.3. It varies along the z—axis and as the cutter rotates,b(z, ) = b — 6(z, g) + w(x, z, (7.8)where b is the desired radial width of cut, 6 and w are the normal cutter and platedeflections, respectively. Note that due to the different geometrical orientation of downand up milling operations, the signs in front of the deflection terms in equation (7.8) areopposite for up milling, as explained in Chapter 6. (7.8) can be used to determineHowever, rather than the start angle, the z—axis boundaries are required to calculatethe cutting forces in x and y directions. The varying immersion dependent upper, z,2,Chapter 7. Peripheral Milling of Plates 150and lower, z,1 axial limits of the immersion for flute j are given byZj,2 =(7.9)Zj,i= + jçp — ex)zj,1 does not change for down milling as qex remains constant. The equation given forzj,2 is solved by iterative techniques as q is a function of z. If çb.9 given by equation(7.7) is substituted in equation (7.9) the following is obtained1—1 t1fZJ,2= 7 c-I-Jcp—7r+cos (1— ) (7.10)Iiwhere z,2 is solved with the Newton-Raphson iterative algorithm. The deflection values inbetween the nodes can be determined by interpolating the nodal deflections. The iterationis started with the axial limit z,2 of rigid tool and workpiece. After the convergence inzj,2 is obtained, its compatibility with the forces and the deflections which were used inthe iteration is checked. If they are not compatible, zj,2 is updated by using the recentvalues of deflections. The same procedure is repeated at every angular step.7.3 Simulation of Peripheral Plate MillingIn milling, the cutting forces are periodic at the flute passing frequency. The distancebetween the two nodal stations on the cutter is equal to the plate element height, whichis constant /z = a/n where a is the axial depth of cut. For a helix angle of b, the lagangle between the two axial nodal stations is 0 = In the rigid model, the cuttingforce pulsation for a flute pitch interval (q,) is calculated at helix lag angle incrementsof 0 by rotating the tool, and stored in memory for application to the plate-tool contactnodes later. For this, first the engagement limits are determined as explained by Table4.1 and Figure 4.2. Then, the elemental and total cutting forces in x and y directionsare calculated from equations (7.4-7.6). The static deflections at the axial nodal stationsChapter 7. Peripheral Milling of Plates 151of the tool are also calculated once, and stored in memory for the surface generationsimulation as the tool dynamics do not change during milling. In the flexible model,however, both the force and deflections are modified continuously as the tool and platedisengage due to deflections. The flexible model therefore significantly differs from theprevious rigid model [35] and the improved rigid model presented here. The cutter isfed along the x axis, and its centerline is positioned at the axial nodal line A— A. Theposition of the cutter is frozen, and the tool is rotated with 0 angular increments. Thecutting forces are distributed to the plate nodes on the nodal lines A— A, B — B at eachangular increment 0. The upper engagement limits for every tooth in cut ,zj,2, is updatedfrom equation (7.10) by using the calculated plate and tool deflections. This requires aniterative solution, as equation (7.10) is implicit in z,2. However, the convergence isquite fast unless the deflections are extremely large. Then, the new values of the upperengagement limits are used in the cutting force calculation in an iterative manner untilconvergence in the milling forces is obtained. The convergence depends on the deflectionsand the tolerance value used in the iterations. In order to speed up the convergence byreducing the stiffness of the iterations, a weighted milling force value obtained in theprevious two iterations are used, i.e. F = (3F_1 + F_2)/4, where i is the iterationnumber.7.3.1 Plate Surface GenerationThe finished plate surface is generated by points on helical flutes as they intersectthe contact nodes at the exit nodal axis A — A where the instantaneous immersion is r indown milling, i.e. q(j, k) = q + jq — kz = ir. Starting with the flute having immersion= ir, it generates the surface at the bottom contact node (z = 0). Simultaneously,the following flute touches a nodal point whose height is z = When the cutteris rotated a b angular increment in the force simulation, the flute jumps to the secondChapter 7. Peripheral Milling of Plates 152node, and the following flute climbs to the node above its previous position on A — A.Depending on the width (b) and axial depth of cut (a), there may be more than oneflute generating the surface simultaneously. Using the cutting force distribution at eachcutter rotation increment 0, the deflections of the nodal points touched by the flutesat the exit axis A— A are calculated from the developed Finite Element routine, andrepresented as w(x, k), where x is the cutter center coordinate in feed direction x. In theflexible force model, tool and workpiece defiections are updated until the convergenceis obtained. Since the forces on the cutter and plate are applied in opposite directions,they either deflect away or towards each other leaving an overcut or undercut surface.Therefore, in down milling the final error on the surface node k at cutter feed location xise(x, k) = 6(k) — w(x, k) (7.11)By repeating the simulation in 0 angular intervals over one flute passing period (q5,), allthe nodal points at the finished plate surface are traversed and the surface errors arerecorded. The cutter is shifted to the next axial nodal line along the feed axis x. Thestiffness matrix of the plate is updated by reducing the thickness at the exit nodal lineA— A, and the simulation is repeated. The solution is continued until the tool leaves theplate. Peripheral milling of other flexible components with different boundary conditionscan be simulated by the developed algorithm.7.3.2 Control of AccuracyIt is possible to constrain the magnitude of form errors within the specified tolerancesby predicting the feed along the tool path. The maximum error left on the workpiecesurface is different at each location along the feed direction as a result of the movingposition of the force and the removed material from the plate. For a dimensional toleranceChapter 7. Peripheral Milling of Plates 153value (t) on the workpiece surface. it is possible to schedule the feedrate along the feedaxis in order to meet the required accuracy. At each feed location x, the maximumsurface error is determined by using the feedrate obtained in the previous step. The newvalue of the feedrate can be approximated ast.st(x,m) = st(x,m— 1)max [e(x)Jwhere m indicates the iteration step and max[e(x)J is the maximum dimensional errorat the feed location x. At the first feed location, x = 0, the iteration is started by aguessed feedrate value. The iterations are continued until a convergence is obtained inthe feedrate. The cutting coefficients are updated at each step according to the new valueof the average chip thickness. At the following feed locations, the feedrate determined inthe previous feed location is used to start the iteration. After the feedrate is scheduled,the machining time for the operation can be calculated. The machining time is veryimportant as flexible plates require very small feedrates resulting in high machiningcosts. By using the simulation program, the effects of the tolerance and other cuttingconditions can be analyzed to find feasible conditions as demonstrated in the followingexamples.7.4 Simulation and Experimental ResultsA number of simulations and experiments have been carried out to demonstrate thecapabilities of the models. Two of the peripheral down milling test results together withthe applied feedrate scheduling are presented below. The plate dimensions, cutting testsand simulation conditions are summarized in Table 7.1.The second plate, which is more flexible in the y direction, is similar to the compressorblades in a jet engine. A spindle speed of 478 rpm was used in both cases. TitaniumChapter 7. Peripheral Milling of Plates 154Table 7.1: Cutting conditions for experiments 1 and 2. (Material: Titanium AlloyTi6A14V)CUTTING PARAMETERS Case #1 Case #2KT (MPa) 275 207p 0.6 0.67KR 0.525 1.39q 0.18 0.043Uncut Plate thickness-t,, (mm) 6.3 2.45Radial width of cut - b (mm) 1.3 0.65Axial depth of cut- a (mm) 35 34Immersion angle of cut-c.s (deg) 30.3 21.3Max. uncut chip (tm) 25.2 2.9Flute lag angle-kua (deg.) 121.6 118.1Simulation angle step (deg) 8.7 6.6Plate mesh size (n x n) 14 x 14 18 x 18(Ti6A14V) alloyed plates were down milled by a single fluted, 19.05 mm diameter carbideend mill with a helix angle of b = 300 in dry cutting conditions. The tool gauge lengthis 55.6 mm and k0 was measured to be 19800 N/mm. Single flute milling tests eliminatethe effects of runout. In practice, the carbide end mills are ground with the tool holderin order to eliminate the run-out. Otherwise, tools with the run-out are inefficient inmachining plates at low chip thickness. Flute sections with a shorter radius may not cutthe plate at all, whereas the following flutes experience a larger chip thickness, leadingto larger force [159, 24, 45] and hence unacceptable deformations on the very flexibleplate. The Young’s Modulus of the tool and workpiece materials are 620 GPa and 110GPa, respectively. Titanium alloys are usually used in aerospace applications because oftheir high strength to weight ratio, e.g. (825 MPa/4.4 g/cm3) yield strength to densityratio for Ti6A14V compared to (530 MPa/7.84 g/cm3) for AISI-1045 cold drawn steel.However, the modulus of elasticity of Ti6A14V is about the half of that of steel, whichChapter 7. Peripheral Milling of Plates 155results in low stiffness in addition to the highly flexible plate geometry. During the experiments, the plate was rigidly clamped to a Kistler table dynamometer to measure feedand normal cutting forces. The experiments were carried out on a vertical CNC millingmachine. A spindle mounted dial gauge was used for surface measurements after eachexperiment. Chatter vibrations were present during the experiments, but their influenceon the measurements were filtered by passing all the measurements through a 100Hz lowpass filter. Note that even though the chip thickness in plate machining is very small thestatic surface form errors are very large due to the high flexibility of the plates.Case 1:In this test, it is shown that the improved rigid force model can predict the cuttingforces and the surface errors sufficiently when the plate is relatively rigid. The platethickness is to be reduced from 6.3mm to 5mm by using a feedrate of s = 0.05mm feedper flute which is constant along the feed direction. A sample window of the simulatedinstantaneous forces for one cutter revolution shows the agreement between the predictions and the measured values (see Figure 7.2). Note that the square wave-like cuttingforces are somewhat distorted due to the filtering.In Figure 7.3, the simulated and measured dimensional surface errors are shown. Dueto the relatively high rigidity of the plate, both rigid and flexible models give almostthe same force and surface error predictions. The form errors are the smallest at thecantilevered bottom of the plate, but they are not zero because of tool deflections. Tooldeflections at the plate bottom remain almost the same along the feed axis x, and thesimulation and experimental measurements give approximately 50pm form error here.The form error magnitudes increase towards the free end of the plate (z— axis), whereChapter 7. Peripheral Milling of Plates 156600500400z300LL 2001000Rotation Angle (deg)Figure 7.2: Experiment #1 -ting conditions: dry cuttingmm/tooth, cutting speed=30mm.A sample window of simulated and measured forces. Cutin down-milling mode, a =35 mm, b=1.3 mm, st =0.05rn/mm. Tool: carbide end miii, 1 flute, 300 helix, d=19.050 45 90 135 180 225 270 315 360Chapter 7. Peripheral Milling of Plates 157the tool stiffness and plate flexibility increase. At the upper portions of the plate, bothsimulation and measurements indicate an increasing trend in the surface form error amplitudes in the feed direction. This is due to the decreasing stiffness of the plate as aresult of material removal.The simulation and experimental dimensional errors are in satisfactory agreementwith each other. This shows that if the deflections are relatively small then the rigidmodel can predict the surface finish with satisfactory accuracy. When the plate is veryflexible, the influence of deflections on the immersion of the cut along the cutter axismust be considered as demonstrated in the following test.Case 2:Plate thickness is to be reduced from 2.45mm to 1.8mm by using feedrate of s = 8zmfeed per tooth. Experimentally measured and simulated dimensional surface errors areshown in Figure 7.4. The surface form error at the cantilevered edge of the plate is dueto tool deflection, and it is approximately 3Otm on both the simulated and measuredsurfaces. The maximum surface errors predicted by the rigid model are approximately15Ozm more than the experimental values, which can be clearly observed from the detailed views of the deformations shown in the beginning, middle and at the end sectionsof the plate (Figure 7.5). The error in the rigid model predictions is due to the immersionthus force distribution changes as illustrated by the results in Figures 7.6 and 7.7. Thesurface error predictions obtained using the flexible model are in very good agreementwith the experimental values as shown in Figure 7.4 and 7.5.The average cutting forces exponentially increase and decrease in the cutter entry158Figure 7.3: Experiment #1- (a) Simulated, (b) Measured surface finish dimensions. Platedimensions: 63.5x35 mm, uncut plate thickness: 6.3 mm, cut plate thickness: 5 mm.Chapter 7. Peripheral Milling of Plates(:1EL0C,(30C(t(a)C),CC)C,(b)--I:-—‘, ,c:,__.— .-159Figure 7.4: Experiment #2- (a) Rigid model, (b) Flexible model, (c) Measured surfacefinish dimensions. Plate dimensions: 63.5x34 mm, uncut plate thickness: 2.45 mm, cutplate thickness: 1.8 mm.(b)Chapter 7. Peripheral Milling of PlatesL0c(a) ,Db -4-(C)c0cC)C.-Chapter 7. Peripheral Milling of Plates 160353025N6003530E 25NE30E 25N20a)c1o0800Surface Error (mic)Figure 7.5: Experiment #2 Predicted and measured surface profiles near the beginning,middle and exit feed stations.0 100 200 300 400 500100 200 300 400 5000 200 400 600Chapter 7. Peripheral Milling of Plates 161100. 80zC,C)UciCd1>200zC)0U-Rotation Angie (deg) (b)Figure 7.6: Experiment #2- (a) Measured and simulated average forces, (b) A samplewindow of measured and simulated cutting forces. Cutting conditions: dry cutting indown-milling mode, a =34 mm, b=0.65 mm, s =0.008 mm/tooth, cutting speed=30rn/mm. Tool: carbide end miii, 1 flute, 300 helix, d=19.05 mm.Feed Direction -x (mm) (a)45 90 135 180 225 270 315 360Chapter 7. Peripheral Milling of Plates 162172170- 168166164C,)162< 160158160 180 200 220 240 260 280 300Rotation Angle (deg)Figure 7.7: Experiment #2 - Flexible force model predicted variation of ct in the beginning, middle and close to exit feed stations. For the rigid case =l58.7° which variesdue to the deflections of the plate and tool.and exit transients. The flexible model predicts the average force satisfactorily, whereasthe rigid model results are about 25 % higher than the measured values. After the cutterreaches the steady state immersion angle ç6. = cos1(1 — b/R) = 21.3°, the average forceshave a slow decreasing trend toward the exit. This is attributed to the changes in theimmersion of cut q due to deflections. Figure 7.7 shows the variation of the start angleof cut, 4st, predicted by the algorithm described in the flexible force model. In the rigidcase = 180—= 158.7°. Increase in q5 means a reduction in effective immersionq which varies as the tool rotates and moves along the feed direction. As the tool rotates, the immersed portion of the flute moves up where the plate deflections are larger.This results in higher qst values. As the cutter approaches the end of the cut, the platethickness is reduced, and so is the stiffness. The plate deflects further away from the toolin the y direction causing a reduction in the effective immersion of the cut. Decreasingimmersion results in reduced cutting forces. q3 remains constant when the intersectionpoint of the flute with axial line B — B reaches the tip of the plate. The variation of g8Chapter 7. Peripheral Milling of Plates 163is shown at the beginning, i.e. when x = 0, middle and close to the end of the cut justbefore the exit transients. Note that the shape of the variation at each location is verysimilar to the surface profile at that point. A sample window of measured and simulatedinstantaneous cutting forces (Fig. 7.6.b) shows that the flexible model force predictionsare quite satisfactory compared to the experimental values. Especially for the rotationangles where the immersed portion of the flute is close to the tip of the plate, the rigidmodel-normal force predictions are more than twice the measurements and the flexiblemodel predictions. It is evident that the previous [35] and the improved rigid modelpresented here can not predict the form errors and the cutting forces when the plate isvery flexible.Control of Accuracy:The same plates have been machined with a varying feed predicted by the model fora specified form error limit. The scheduled feed per tooth, s, for Case 1 and Case 2 areshown in Figure 7.8. The maximum surface errors allowed are 80 im and 250 um forCase 1 and 2, respectively. The very flexible plate in Case 2 does not allow to obtaina higher accuracy as it is analyzed later in detail. Due to the relatively high rigidity ofthe plate in Case 1, both the flexible and the rigid models give the same results. In Case2, however, the rigid model estimates an almost three times smaller feed than the flexible model does. Also, as a result of interaction between the deflections and the cuttingforces, the variation of the feedrate along the feed direction is not smooth. The simulatedand the measured surface errors obtained by using the scheduled feedrates are shown inFigures 7.9 and 7.10 for Case 1 and 2, respectively. Flexible model predictions wereused in Case 2 as they are expected to be more accurate. As the feedrates are determined iteratively in the simulations, the resulting surface error may be slightly differentChapter 7. Peripheral Milling of Plates 1640.06U • U0.050.042 0.03E .E.—.00.01.0 I I I0 10 20 30 40 50 60Feed Direction -x (mm)0.0030.0025 U U U • • U U0.002 •• Flexible Model0.0015 .• Rigid Modelg0•01 ..•..•..0.0005 • • • U. U0 I I I I I0 10 20 30 40 50 60Feed Direction -x (mm)Figure 7.8: Scheduled feedrates for Experiment 1 and Experiment 2 for surface errortolerances of 80 um and 250 m, respectively.Chapter 7. Peripheral Milling of Plates()Ec(3(I)165Figure 7.9: Experiment 1 with scheduled feedrate for 80 pm tolerance- a) Simulatedsurface errors b) Measured surface errors(a)-‘C)cc—--(b)‘0Chapter 7. Peripheral Milling of Plates 166- ctFigure 7.10: Experiment # 2 with scheduled feedrate for 250 um tolerance- a) Simulated(flexible model) surface errors b) Measured surface errors(3EC—0C.(313ri—.C(a)—(b)(3EC-0C-.(54—C-Chapter 7. Peripheral Milling of Plates 167100902 8070a)E 60—• 5020100Figure 7.11: The variation of the machining time with tolerance value in Case # 1 and# 2. The cutting conditions are the same as described before except for the feedratewhich is scheduled to achieve a required tolerance on the surface.than the tolerance value depending on the error value accepted in the iterations. Thatis why the maximum predicted surface error fluctuates around 250 1um in Figure 7.10.The measured surface errors indicate that the required accuracy is achieved by usingthe feedrate scheduling strategy, which would not be possible when the rigid model isused in milling very flexible plates. Depending on the desired accuracy sometimes theresulting scheduled feedrates may be very small. This was the situation in Case 2. Oneshould realize that it may not be possible to achieve the desired accuracy always. If theworkpiece is very flexible, the edge forces may be high enough to create large deflectionseven if the feedrate is very small, even zero. The small feedrates result in long machining times increasing the cost. Before deciding on the cutting conditions and tolerancevalue one should have an idea about the effect of those parameters on the machining time.In Figure 7.11 the variation of the machining time obtained from the simulations vs.the specified dimensional tolerance is shown for Case 1 and 2. They have exponential0 100 200 300 400 500Tolerance (mic)Chapter 7. Peripheral Milling of Plates 16820 2-- b=0.65 mm (varying R)16 • 1.6- -- R=9.5 mm (varying b)- -E- E12.- 1.2E------DU8 •. -- 0.8A- -AI I I 0.40 200 400 600 800 1000Machining Time (mm)Figure 7.12: The simulated variation of the machining time with tool radius and radialwidth of cut in Experiment 2 for tolerance of 250 ,um on the surface.shapes similar to general cost vs. accuracy relations. It can be seen from the figure thatfor Case 1 the tolerance can be decreased from 300 ,um to 80 tm without increasingthe machining time significantly. Reducing the tolerance constraint from 80 um to 50tim, however, increases the machining time more than 6 times. Therefore, 80 m wasa very good tolerance selection for Case 1. In Case 2, the accuracy can be improvedby 150 m by reducing the tolerance from 500 1um to 350 m without any significantincrease in the machining time. Increasing the accuracy 100 itm more results in a tripledmachining time. It should be noted that machining times were calculated for a one flutedcutter and 478 rpm spindle speed. They can be significantly reduced by increasing thenumber of flutes and the spindle speed. The effects of tool diameter and radial widthof cut for this case are shown in Figure 7.12, where the radial width of cut is 0.65 mmand the tool radius is varied. In the cases where the radial width of cut is varied, thetool diameter is 19.05 mm. In all cases the desired cut thickness of the plate is 1.8mm and the tolerance is 250 tim. The tool diameter and the radial width of cut definethe immersion angle and average chip thickness. In down milling, peak normal force FChapter 7. Peripheral Milling of Plates 169increases as the immersion angle increases if the cutting coefficients are assumed to beconstant. However, the variation of the cutting coefficients is too sharp, especially at alow average chip thickness. Also, the radial width of cut affects the flexibility of the plateas the final thickness is fixed. These different factors compete against each other andthe optimal values can be obtained from the analysis of Figure 7.12. As it can be seenfor 250 um accuracy and 0.65 mm radial width of cut, the minimum machining time isobtained by 15.88 mm (5/8”) diameter tool. The charts which are prepared by the platemilling simulation system can be used in process planning.7.5 SummaryA simulation system for the peripheral milling of very flexible, plate type structuresis developed. The plate is cantilevered from the base and is free at the other three edges.The tool is modeled as a cantilevered beam, and stiffness loss in the collet is considered.A finite element model with a varying stiffness due to metal removal is used for the platestructure. The changes in the immersion, both in the feed and cutter axis directions, areconsidered in the model. The developed model allows satisfactory prediction of cuttingforces and dimensional surface errors due to deflections of the tool and plate structuresduring milling. It is shown that unless the changes in the plate-tool contact boundariesare considered, the form errors and the cutting forces can not be predicted satisfactorily inthe peripheral milling of very flexible plates. The developed model allows the predictionand scheduling of feeds along the tool path, and keeps the form errors within the specifiedlimit.Chapter 8Analysis of Dynamic Cutting and Chatter Stability in Milling8.1 IntroductionIn the milling process, cutter, workpiece and machine tool structures are subjectto periodic and transient vibrations due to the intermittent engagement of cutter teethand periodically varying milling forces. These effects, however, can be minimized by theproper selection of cutter geometry and spindle speeds to avoid resonances and largeimpact loadings. A more important vibration type in machining is self-excited chattervibrations which cause instabilities resulting in poor surface finish and dimensional accuracy, chipping of the cutter teeth, and may damage the workpiece and machine tool.The dynamic milling process-structure interaction is particularly important in millingthin-walled workpieces due to a highly flexible workpiece and slender end mill. In thischapter, the chatter stability of milling is analyzed. A general formulation is developedfor the analytical prediction of milling stability and it is applied to several common caseslike the milling of flexible workpieces.The fundamental chatter theory has been developed by Tobias [3] and Tiusty [5].Tobias considers the variations in the cutting forces due to the dynamic variations ofchip thickness and cutting velocity. Chip thickness can vary due to either a modulatedsurface left from the previous pass (outer modulation or wave removing) or vibrationsof the tool towards the cut surface (inner modulation or wave cutting); whereas cutting170Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 171velocity can vary due to the vibrations of the tool in the cutting velocity direction. Then,he considers the total amount of damping in the cutting system, which is a summationof the structural damping and the damping generated due to variations in the cuttingvelocity and the chip thickness, to assess the stability. On the other hand, Tiusty showsthat the cutting system has higher stability against the mode coupling chatter mechanism(excitation energy generated due to a variation in the cutting velocity) and considers theregeneration mechanism as the dominant mode of chatter, but includes the process damping generated due to the flank contact. The phase between inner and outer modulationsis the most important factor of the regeneration mechanism. It determines the amount ofperiodic variation in the chip thickness and depends on cutting conditions and dynamiccharacteristics of the structure. Both approaches by Tobias [3] and Tlusty [5] can beused to obtain the stability limit (maximum allowable width of cut without chatter) as afunction of cutting velocity which is referred to as the stability diagrams or stability lobes.A stability analysis of the milling process is much more complicated than the orthogonal cutting case. This is mainly due to the rotating milling cutter, multiple cutting teethand the dynamically coupled multi degree-of-freedom cutting system. The directionalcoefficients (direction cosines to determine the component of the cutting force and theoriented transfer function in the chip thickness direction) vary as the cutter rotates. Inthe early milling stability analysis, Tiusty [5, 77] applied his orthogonal cutting-stabilityformula by considering an average direction and number of flutes in cut. Therefore, heused constant directional coefficients which are calculated in the average direction. Thereis no theoretical basis for this approximation and the accuracy is not predictable. Dueto these reasons, later Tlusty et al. [75, 69, 88, 160] and others [37, 98, 161, 162] haveused time domain simulations extensively in milling stability prediction. However, timedomain simulations in chatter are computationally very expensive. In order to obtainChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 172a milling stability diagram, hundreds of simulations have to be carried out at differentspindle speeds by increasing the axial depth of cut and observing the convergence ofthe displacements for many revolutions of the cutter. The computational time is evenmore critical for the case of milling flexible workpieces where the dynamic characteristicsof the workpiece change due to machining which requires that the chatter simulationsbe repeated at a number of stations along the feed direction. The basic nonlinearity inchatter [75] (loss of contact between the cutting tooth and the workpiece due to largevibration amplitudes) can be best modeled by time domain simulations, however this is• not important in predicting the onset of chatter [82]. Opitz et al. [163, 94] replaced theperiodic coefficients with their average values over the time interval during which thetooth of the cutter is in contact with the workpiece. Then, the oriented transfer functionis calculated by using the average directional coefficients which reduces the coupled dynamics to a single degree-of-freedom case. However, no theoretical justification is givenfor the method.The first comprehensive theoretical analysis of milling stability has been performedby Sridhar et al. [78, 79, 80]. They formulated the dynamic milling forces for a straighttooth cutter. They used a numerical stability algorithm which is based on the numerical evaluation of the system’s state transition matrix. Minis et al. [82, 81] applied thetheory of periodic differential equations on the milling dynamics equations. They usedthe Nyquist stability criterion to determine the stability limits. The algorithm dependson the numerical evaluation of the eigenvalues as the axial depth of cut is increased untilthe stability limit is reached. Lee et al. [164, 165] also used the Nyquist criterion todetermine the stability limits numerically.In this chapter, a comprehensive formulation of dynamic milling forces is given byChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 173neglecting process damping, i.e. the relationship between the milling forces and the chipthickness is represented by a simple proportionality constant. This is a valid assumption ifthe cutting velocity is relatively high (> 100 m/min). The milling cutter and workpieceare modeled as multi degree-of-freedom structures and the dynamic interaction alongthe axial depth of cut is considered in the formulation, unlike the point contact modelsused in all of the other chatter formulations. The effects of the variations in cutter andworkpiece dynamics in the axial direction are considered in the model. This is necessaryfor the accurate modeling of dynamic milling forces in milling flexible workpieces andhas been neglected in the previous models. The stability analysis is performed by usingtwo different approaches both of which give the same result. First, the classical periodicsystem theory is applied to dynamic milling. Second, a stability analysis which is based onthe physics of the dynamic milling is used. It is realized that the second method is simplerand gives physical insight to the milling dynamics. An analytical method is developed topredict the stability limit by deriving a relationship between the chatter frequency andthe spindle speed, for the first time in milling. The application of the general formulationto some special cases and the accuracy of the predictions are illustrated through examples.8.2 Formulation of Dynamic Milling ForcesFigure 8.1 shows a crossection of an end mill tooth (j) vibrating and removing awavy surface cut by the previous tooth (j — 1). u3 and vj are the rotating tangential andnormal directions at the tip of tooth j, and they can be expressed in terms of the fixedcoordinate system x and y as follows:u3 = —xcos(z)+ysin4(z) (81)vj = —xsingj(z)—ycosq!j(z)Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 174Workpiecevibration marksleft by tooth )//xvibration marksleft by tooth (i-i)Figure 8.1: Dynamic milling process.Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 175or= —cos(z) sin(z) x(8.2)( v J —singj(z) —cosq5j(z) L ‘ Jwhere4(z) =k — tan?1b- R2ir—N is the number of teeth on the cutter, R is the cutter radius and g = Qt is the rotationangle of the end mill (measured with respect to tooth number j = 0), Q being therotational speed of the cutter in (rad/sec).• 8.2.1 Dynamic-Regenerative Chip ThicknessThe chip thickness can be written as a summation of the static and the regenerativechip thickness as follows:z)= (v — v) — (v — v) + s sin q(z) (8.3)where v, v, and v, v are the dynamic displacements of the cutter and workpiecein the v direction for current and previous tooth passes (for the rotation angle of thecutter q, at the axial depth z), respectively. According to the reference system shownin Figure 8.1, the current cutter displacements in the positive v direction decrease thechip thickness, whereas the positive cutter displacements in the previous pass increasethe chip thickness. It should be noted that, at this stage the cutter and workpiecedeflections are considered to be in +vj direction, although, in general, they are in theopposite directions. However, this will be imposed when dynamic displacement-millingforce relations are included to the formulation. Substituting for the rotating coordinatesChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 176from equation (8.1), the following is obtained:z) [(xe — x)— (x — 4)] sin qj(z) (8 4)+ [(Ycy) — (ywy)]COSj(z)+8tSiflj(Z)where x, y and x, Yw are the cutter and workpiece displacements in the current pass inthe x and y directions (for the rotation angle of the cutter ç and at the axial depth z),respectively. Similarly, x, y° and 4, y are the cutter and the workpiece displacementsat the same point in the previous tooth period.8.2.2 Differential Dynamic Milling ForcesDynamic differential milling forces can be obtained by using the dynamic chip thickness given in equation (8.4). The formulation of these forces is similar to the static millingforce model given in Chapter 6, except the chip thickness dynamically varies in this case.The effect of vibratory cutting on the milling force coefficients will be neglected. Themilling forces in the x and y directions will be considered as the spindle and the cutterare quite rigid in the z direction compared to x and y. From Figure 8.1 the tangentialand the radial forces can be resolved in x and y directions as:dF3(q,z) = [—dFt(,z)cos(z) — dFrj(q,z)sinqj(z)]g(qj(z))(85)dF(,z) = [dFt(,z)sin(z) — dFrj(,z)cosj(z)]g(j(z))wheredFt3(ç,z) = Kh(q,z)dzdFrj (q, z) = KrdF3and g(gj(z)) determines whether the tooth is in cut. Mathematically, it can be expressedas follows:g(j(z)) = 1 st <(z) <e 1 (86)g(j(z))=O or j(Z)>ex JChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 177Then,dF,(c,z) = —Kjh(,z)[cos(z) + Kr sin q(z)]g((z))dz(87)dF(,z) =— Kr cos(z)]g((z))dzSubstituting equation (8.4) into (8.7), the following explicit form is obtained for thedifferential milling forces:dF(,z) = —Kt(cosq(z) + Kr sin qS(z)) {[(x — x)— (x — x)]sin3(z) + [(yc - y) - (y - y)]CO5j(Z) + sjsin(z)}g((z))dz(8.8)dF2 (ç, z) = Kt(sin çj(z)— Kr cos (z)) {[(x — x)— (x — x)]sin çf(z) + [(yc— °) — (Yw— Y,)] cosqj(z) + stsinq!(z)}g(q(z))dz8.2.3 Total Dynamic Milling ForcesThe widely used milling chatter models developed by Tlusty et al. [5, 75, 77] andTobias [3], and a relatively recent method by Minis et al. [82, 81], consider a point contactbetween the milling cutter and the workpiece. The cutter is modeled as a 2 degree offreedom structure which is lumped at the tip of the tool. The variation of the cutterdynamics in the axial direction and the helix angle are neglected in these models whichmay cause inaccuracies in the stability limit predictions, especially in end milling wherethe axial depth of cut is usually high and the variation of the tool dynamics within theincut portion of the end mill (or mode shape in a cutter vibration mode) is significant. Inorder to consider the real dynamic interaction between the milling cutter and workpiece,the cutter and workpiece are divided into a number of elements in the axial direction, asin the case of the static tool and workpiece deflection analysis given in Chapters 5 and7. This is because the cutter and workpiece dynamics are usually identified by modaltests at a number of points (or elements). Also, the resulting integrals for the dynamicChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 178milling forces cannot be taken analytically even if the dynamics of the structure canbe determined by analytical means. Then, the differential dynamic milling forces areintegrated within an element i to determine the elemental or nodal milling forces for theflute j as:zi+1 z1+1F2(i)= j dF(b,z)dz ; F(i) = j dF(q,z)dz (8.9)where z and z1 are the lower and upper z coordinates of the axial element. After theintegration, equation (8.9) can be arranged as follows:F3(i) = F.(i)— J(t[Ca,() + KrCbj() + Cc,(i) + KrCdj()J(8.10)F3(i) = F(i) + Kt[Cb,(i) — 1(rCcz,() + Cd,(i) — IrCcj()]whereZjf 1Caj()= jCZi+1Cb3(i)= j (8.11)C,(i) jZi+ iCd(i) jwhere/x = (x, — x) (x — x)(8.12)=F and F, in equation (8.10) are the milling forces, due to mean chip thickness (St sin qj(z)).The integrals in equation (8.11), which are to be integrated within each axial element,contain the dynamic displacements of the cutter and workpiece. These displacements canbe assumed to be constant within each element as the dynamic characteristics and theresponse of tool and workpiece structures are determined at discrete nodal points bothby modal tests and the Finite Element solutions. Also, the unit step function g(qj(z))Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 179• will be evaluated at the lower boundary of each axial element, z, and will be assumed tobe invariant within the element when integrating equations (8.11). This does not degradethe accuracy as it will be shown later, Ca, C6,C, Gd are further integrated for one fullrotation of the cutter in the stability analysis. Then, the integrals can be determinedanalytically as follows:Caj() = + 1) — cos2q(i)]C6 (i) x(i)g(i)[ZZ + —(sin 2q(i + 1) — sin 2q(i))]C,(i) zy(i)gj(i)[Zz — —(sin 2(i + 1) — sin 2q(i))}Cd(i) + 1) — cos2(i)]whereLSx(i) = [x(i) — x(i)]— [x(i) — x(i)]qj(i) = j(q,zj)gj(i) = g(b(z))=m is the number of axial elements, x(i),x(i),.. etc. are the nodal displacements ofthe tool and the workpiece for the considered rotational angle of the cutter. The aboveequations can be simplified by expanding qj(i), qj(i + 1),(cos 2q(i + 1) — cos 2(i)) and(sin 2(i + 1) — sin 2q3(i)) as follows:==(814)== gf(i)—Consider Taylor series expansion of cos 2(a + 3) and sin 2(a + 3) around 6:cos 2(o + 3) = cos 2a — 2/3 sin 2a — 2/32 cos 2a +8 15sin2(cr+/3) = sin2a+2/3cos2c—2/3cos2 +...Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 180Then, the following is obtained by using the first order Taylor expansions:cos2q3(i+ 1)— cos2g3(i) = cos2(q(i) — k/z) — cos2c23(i)= cos2q(i) + 2kz sin 2q(i) — cos2ç(i)= 2kzsin2(i)(8.16)sin2(i + 1) — sin2(i) = sin2(q(i) — k,/z) — sin2(i)= sin 2(i) — 2kz cos 2q(i) — sin 2gS(i)= —2kzcos2ç5(i)The above simplification is necessary in the stability analysis in order to obtain anexplicit expression for the axial depth of cut, or in this case element thickness. If thisis not done at this stage, an implicit stability equation in the axial depth of cut willbe obtained at the end of this analysis. The accuracy of the first order Taylor seriesapproximation to the above trigonometric expressions depends on the magnitude of theelemental lag angle (k,Lz). The lag angle becomes smaller as the number of elementsis increased, thus for a sufficiently high number of elements, the accuracy of the firstorder approximation is high. Another way of increasing the accuracy is to employ ahigh order Taylor series expansion which results in higher powers of Liz. (Note that,in equation 8.16, second order expansions contain (z2).) Equation (8.13) takes thefollowing simplified form when (8.16) is substituted:Caj() = z/x(i)g(i)sin2q(i)Cb(i) = zLx(i)g(i)(1 — cos2(i))C2(i) = zzy(i)g(i)(1 + cos2(i))Cd(i) = zy(i)g(i)sin2q(i)After the comparison of equations (8.11) and (8.17), it is realized that the first orderTaylor series approximation gives the same result with neglecting the helix angle withinChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 181the axial element. This can be obtained by multiplying the integrands in equation (8.11)by the axial element thickness Liz. It should be noted that the helix effect is consideredbetween the elements or nodes in equation (8.17). This approach is similar to the one usedin the static milling force model by Kline et al. [35]. However, as discussed above, a highnumber of elements can be used to increase the accuracy. The dynamics of the cutterand workpiece are usually available at a few discrete points along the axial direction.One may think that dummy elements can be used between the main nodes at whichthe dynamic response is known, for the sake of increasing the accuracy in helix anglemodeling. However, in the stability analysis of the dynamic milling forces, it will beshown that the average values of the directional factors in equation (8.17) are used, andthus the helix angle disappears. Therefore, increasing the number of elements to improvehelix angle modeling-more than the variation of the cutter and workpiece dynamics inthe axial direction requires-does not increase the accuracy of the overall formulation.Finally, substituting equation (8.17) into (8.10) the following is obtained for the nodaldynamic milling forces:F(i) = F(i)+a,(i) a(i) Lx(i) (8.18)( F,,(i) J ( F;.(i) J a(i) a(i) ( y(i) Jwherec = (8.19)In equation (8.18), the matrix elements .., a which relate the dynamic displacements to the dynamic milling forces will be called directional dynamic milling coefficientsChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 182and they are given as:a(i) = —g[sin2g5(i)+ Kr(l — cos2(i))]a,(i) = —g3[(1 + cos2(i)) + Krsin2qj(i)] (820)a(i) = g[(1 — cos 2(i))— Kr sin 2(i)]a,(i) = g3[sin2q5( )— Kr(1 + cos2q3(i))jThe total milling forces in each element can be obtained by summing up the cuttingforces on each flute:N—i N—iF(i) = F,(i) ; F,(i) = F,,(i) (8.21)j=O j=OThen, equation (8.18) takes the following form for the total forcesF(i) = F(i) + a(i) a(i) Lx(i) (8.22)( F,(i) J ( F(i) J a(i) a(i) y(j) JwhereN-ia(i) = (8.23)j=oand are similar. Equation (8.18) can be written for every axial element, whenthese equations are combined the following matrix equation is obtained:{F} {F}+[a] [a] (8.24)( {F} J ( {F:} J [a] [a] ( {zy} Jor in more compact form{F} = {F8} + c[A(t)]{z} (8.25)whereI {x} 1{zSj = (8.26)Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 183The force and the displacement vectors contain the elemental force and displacements,e.g. {F}T = {F(1), F(2), ..., F(rn)}. [a], [a,], [a] and [a] are diagonal matriceswith elemental values of a(i), a,(i) etc. in the diagonal, e.g.a(i) =(8.27)j#i J[A(t)] will be referred to as the directional dynamic milling coefficient matrix which isperiodic at the tooth passing frequency w. Unlike the previous models [3, 5, 77, 81],equation (8.24) gives the dynamic milling forces by including the effects of the helixangle and the variation in the tool and the workpiece dynamics along the axial direction.Before the stability analysis of the milling process can be performed, the structuraldisplacement- milling force relations should be substituted in equation (8.24). The millingforces, {F}, {F}, will be dropped from equation (8.24) from this point on as they arenot to be considered in the stability analysis of milling since the dynamic milling forcesare generated due to dynamic displacements of the tool and workpiece. Also, accordingto linear system theory, external forces are neglected in the stability analysis of linearsystems [166]. However, the static parts of the milling forces should be considered if anonlinear analysis is to be done. The basic nonlinearity in machining chatter is the loss ofcontact between the cutter and the workpiece due to high amplitude chatter vibrations[75]. However, this does not affect the stability limit prediction for which the onsetof chatter vibrations is considered. Other nonlinearity which is especially important inflexible workpiece milling is the variation of cutter-workpiece immersion boundaries dueto static deflections. This is analyzed in Chapter 6 where it is shown that the radialdepth of cut varies along the tool axis. In these cases, the use of average radial depth ofcut in the stability analysis may be an acceptable first order approximation. For exactanalysis of this type of nonlinearity, time domain solutions are required. The next stepChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 184in the formulation is to express the dynamic displacements due to the dynamic millingforces.8.2.4 Dynamic Displacements of Cutter and WorkpieceThe dynamics of the cutter are described by the following linear differential equation:{r} = [G(D)J{F} (8.28)where the vectors {r} and {F} contain the cutter displacements and the milling forcesin x and y directions, respectively:I{x1 I{F}{r} = ; {F} = (8.29)I {yc} J I {F} J[G(D)] is the dynamic flexibility matrix of the cutter and has the following components:[G (D)] {G (D)][G(D)] = (8.30){G(D)] [G(D)]where {G (D)j, (D)] represent the direct-dynamic flexibility matrices of the cutterin x and y directions, [G(D)], [G(D)] are the cross-dynamic flexibility matrices andD is the differential operator d/dt. In general, the flexibility matrices have the followingform:[Gj = [[M]D2 + [BCX1D + (8.31)where [B] and [K] are the mass, damping and stiffness matrices of the cutterin the x direction, and the other flexibility matrices are similar. It should be noted thatthe flexibility matrices are not transfer functions but linear operators. The size of theflexibility matrices are mxrn, m being the number of nodes on the tool and the workpiecealong the axial direction. The workpiece displacements, {r}, can be written similarlyas:{r} = —[G(D)]{F} (8.32)Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 185The (-) sign in equation (8.29) is due to the fact that the milling forces applied on the• cutter and the workpiece are in the opposite directions. The displacement vector and theflexibility matrix have the same form with those of the cutter:{r} ;= [G(D)] [G(D)] (8.33)({y} J [G(D)] [G(D)]The dynamic displacements of the cutter and the workpiece in the preceding tooth period,{ r}, {r}, can be expressed as= {rw(t-T)}=e{r} (8.34)where T = 2ir/N is the tooth period and e_TD represents the time delay operator.Then, {x} and {Ly} can be obtained as follows:{Lx} = ({x} — {x})— ({x} — {%}) = (1 — e_TD)({xc} — {x}) (8 35){y} = ({yc} - {y}) - ({Yw}- {y}) = (1-TD)({} - {Yw})The following is obtained if the displacements of the cutter and workpiece given byequations (8.28) and (8.33) are substituted in (8.35):I {x}{z.S} = = (1 —e_TD)([Gc(D)] + {G(D)]){F} (8.36)I {Ly} JBy substituting equation (8.36) into (8.25), the following eigenvalue equation is obtained(the static part of the milling forces is ignored as discussed before):{F} = c(1 —e_TD)[A(t)][G(D)]{F} (8.37)where [G(D)] = [G(D)] + [G(D)], [A(t)] is defined in equations (8.24) and (8.25), andc is defined in (8.19). The above expression was more or less derived by Sridhar et al.[78], but for a structure which had two orthogonal modes. However, the derivation givenChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 186node (m+1)node 1Figure 8.2: Node numbering on cutter and workpiece.here is for multi degree-of-freedom workpiece and tool structures. It should be notedthat [G(D)] and [G(D)} contain the dynamic flexibility of the cutter and workpieceonly in the immersed portions of the structures. As shown in Figure 8.2 , the numberingof the nodes (degree of freedom) start from the free end of the cutter, i.e. node 1for the cutter is at the bottom of the tool. Workpiece nodes have the same numberscorresponding to the ones on the cutter. The stability of milling is governed by equation(8.37) which models multi-degree-of- freedom cutter and workpiece dynamics and helicalcutting flutes. The periodic terms in [A(t)] are due to the time varying directional milling• coefficients between the local chip thickness directions and the milling forces, and theyhave been the main difficulty in milling stability analysis as the standard methods ofstability cannot be used for the time varying systems. For that, Tlusty et al. [5, 167, 77]approximated the directional coefficients and the stability limit in the direction of theChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 187average resultant force. Similarly, Sankin [168] evaluated the varying coefficients for themost critical directional orientation. Opitz et al. [163, 94] used the time average valuesof the directional coefficients. There is no theoretical justification for these approachesand the accuracy is not predictable. This was later appreciated by Tlusty et al. [75, 76]and Smith [169, 70] who used time domain simulations for the milling stability analysis.A comprehensive model of milling dynamics was developed by Sridhar et al. [78, 79, 80]for a special and general case of milling, however they used numerical procedures for thestability analysis. Following the standard procedure for the stability of periodic systems,Minis et al. [82, 81] used the Fourier analysis and the concept of parametric transferfunctions for the analysis of a two degree-of-freedom milling system8.3 Stability AnalysisThe chatter stability of milling which is governed by equation (8.37) will be allalyzedby two methods. First, a mathematical stability analysis will be presented by using thetheory of periodic systems. In the following section a brief history and theory of periodicsystem stability are given. The second method of the milling stability analysis is based onan interpretation of the physics of dynamic milling. Both methods yield the same result,however the second approach is much easier, shorter and gives more physical insight intothe problem.8.3.1 Stability Theory of Periodic SystemsEquation (8.37) is identified as a linear periodic-differential difference equation.There exist several methods to study the stability of periodic or delayed (difference)equations. However, the existence of both periodic and delay terms in equation (8.37)increases the complexity of the problem. Periodic differential equations arise in manyChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 188fields of physics and engineering, including problems in mechanics, astronomy, wave propagation, electric circuits, quantum theory of metals and the stability theory of certainnonlinear differential equations [170, 171]. Homogeneous, linear, second order differentialequations with periodic coefficients are called Hill’s equations, named after Hill due tohis important and lasting contributions to their theory. Hill derived and analyzed thefollowing general form of Hill’s equation in his study on the motion of the lunar perigee[172] which was completed in 1877, but published in 1886y” + [6 — 2Eq(t)]y = 0 (8.38)where q(t) is a real periodic function. If q(t) = cos 2t the above equation is known asMathieu’s differential equation, introduced by Mathieu in 1868, when he determined thevibration modes of a stretched membrane having an elliptical boundary [173]. The classical stability analysis of Hill’s equation is the procedure of the infinite determinant whichutilizes the Floquet theorem and the Fourier series expansion of the periodic functionq(t) [170, 174, 173, 175]. A similar procedure will be followed for the stability analysisof the milling, i.e. equation (8.37). There exist other effective methods for the stabilityanalysis of differential time varying systems including other series type solutions (Besseland McLaurin), perturbation and Liapunov methods [170, 173, 176, 177, 178, 179, 174,171, 180]. The stability of these systems depends on the relative magnitudes of the staticrestoring coefficient (designated by 6) and the time varying one, (c), and it can be determined from the stability intervals in , 6 plane (usually referred to as Strutt diagram)[174, 173, 170, 181]. The addition of the periodic restoring force may cause instabilityor make the otherwise unstable system stable, depending on its frequecy and amplitude.This is illustrated by Cooley et al. [175] on some examples of second (stable to unstable)and third order (unstable to stable) systems. For a physical example, consider oscillations of a pendulum. An inverted pendulum which is unstable at its vertical equilibriumChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 189position can be stabilized, whereas a regular pendulum which is stable around its vertical equilibrium position may become unstable by moving the pivot point harmonicallyin the horizontal or vertical direction. Almost all the work done in this area deals withthe second order systems due to their importance in dynamic systems analysis. Theliterature mentioned here considers mostly single degree of freedom systems, except [179]which presents a wide spectrum of methods for the analysis of second order-multi degreeof freedom systems.8.3.2 Stability Analysis of Milling Using Periodic System TheoryMilling stability equation, (8.37), has a delay term, eT), due to the regeneration mechanism. Delay or differential difference equations arise in many problems ofphysics, engineering, economics and biology [182, 183]. In an early work, Minorsky[184] analyzed the effect of delay on the self-excited oscillations and stability. Stability of delayed systems has been studied in many works, some of them are cited here[185, 186, 183, 187, 188, 189]. Comprehensive analysis of differential difference equationscan be found in [182].The stability method used in this section is based on the Fourier analysis and theconcept of parametric transfer functions introduced by Zadeh [190] and utilized by Rozen[191]. The Fourier analysis has traditionally been used in the analysis of periodicsystems [170, 173, 179, 175] and recently by Hall [180]. Cooley et al. [175] used theFourier analysis to examine the stability of a class of systems with a sinusoidally varyingparameter.Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 190Equation (8.37) is re-written{F} = c(i —e_TD)[A(t)][G(D)}{F} (8.39)where [A(t)] is periodic in time. The contribution of Minis and Yanushevsky [81] is toapply the stability theory of periodic systems, i.e. the Fourier analysis and parametrictransfer function concept, to the above dynamic milling force expression-for a two degreeof freedom model-presented by Sridhar et al. [80].Floquet’s theorem (G. Floquet, 1883) [176, 179, 81] states that for a second orderdifferential equation with periodic and piecewise continuous coefficients like the millingdynamics equation (8.39), the solutions have the following form:{F(t)} = eAt{P(t)} (8.40)where the function {P(t)} is periodic with period T (tooth period). Therefore, the millingsystem is stable if the real part of all exponent A’s are negative. In order to obtain explicitstability conditions, the characteristic equation of the system is derived using the Fourieranalysis. The periodic function {P(t)} is expanded into the following Fourier series:{P(t)} = {Pk}e (8.41)k=-oowhere the tooth frequency w = 2ir/T = NfZ is the fundamental frequency of [A(t)] andi = In order to obtain the kth Fourier coefficient {Pk}, equation (8.41) is multipliedby e_t and integrated. Then, the following is obtained for the Fourier coefficients byusing the properties of orthogonal functions{Pk} = JTkt (8.42)Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 191Equations (8.40) and (8.41) are substituted into equation (8.39) to obtain:{F} = c(1 - eTD) [A(t)] [G(D)] (ext {Pk}e1t)k=-oo(8.43)= c(1 — e_TD) [A(t)] [G(D)] {Pk}e(itk=-ooThe result of [G(D)] operating on the exponential function is determined by theshifting theorem of linear differential operators [192]:G(D)eat = eatG( ) (8.44)Then, by using the shifting theorem in equation (8.43) and considering that {Pk} isa constant vector, the following equation is obtained:{F} C e(kw)t(l — e_T)[A(t)][G( + ikw)]{Pk} (8.45)k=—oowhere eT(ik) e7’ is substituted as Tw = 2ir. From equations (8.40) and (8.45), itis concluded that{P(t)} = c(1 — e_T) eikwt[A(t)][G(A + ikw)]{Pk} (8.46)k=-co[A(t)] can be expanded into a Fourier series as it is also periodic at the tooth frequency w.When substituted, equation (8.46) will contain a double Fourier series. This is performedby multiplying both sides of equation (8.46) by 1/Te_t and integrating from 0 to T:{Pr} = c(1 — e_T) [Wr_k(A + ikw)]{Pk} (r, k 0, +1, +2,...) (8.47)k=-oowhere[Wr_k(A + ikw)] = [Ar_k][G(\ + (8.48)[Ar_k] is the (r — k)th Fourier coefficient of [A(t)]:1 T[Ar_k] = j [A(t)]e_(r_)dt (8.49)Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 192The linear algebraic system given by (8.47) can be written in the following matrix form:{P0} [W0(\)] [W_1(.\ + iw)] [W1.— iw)] . . {P0}{P1} [W1)] [W0+ iw)] [W2X— iw)] . {P1}{P1} = c(1 - eT) [W1()] [W2(+ iw)] [W0(- iw)] {P1}(8.50)Equation (8.50) has nontrivial solutions if the determinant is zero:det[6Tk[I] — c(1 — e_T)[Wr_k(\ + ikw)]] = 0 (8.51)where 6rk is the Kronecker delta (i.e. 5rk = 1 if r = k, 6rk = 0 if r k), and [I] is the(2mx2m) identity matrix.Equation (8.51) is the characteristic equation of the closed loop milling system.This equation is an infinite determinant which is a characteristic of periodic systems[172, 170, 175, 174]. For the system to be stable, all the roots (eigenvalues) of the characteristic equation must have negative real parts. Approximate roots of this infinite ordercharacteristic equation can be obtained by solving its truncated versions. Hill [172], forexample, considered only a 3x3 determinant (first order approximation) in his analysis.Minis and Yanushevsky [81] derived an expression similar to (8.51) for the two degreeof-freedom milling system model they used. They numerically solved the truncated eigenvalue equation by increasing axial depth of cut and determining the sign of the real partof ). Although this is faster than time domain simulations of milling chatter, still manyiterations have to be done before the stability diagrams can be constructed. Also, thepresented application of the periodic system theory on the milling stability does notChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 193provide physical insight to the dynamic milling process, e.g. relationships between chatter frequency, spindle speed and transfer functions. The author performed the stabilityanalysis by considering the physics of dynamic milling which is presented next.8.3.3 Milling Stability Analysis Based on the Interpretation of Physics ofMilling DynamicsThe stability analysis can be performed in a much simpler way by considering thedynamic displacements and milling forces at the stability limit. Start with the dynamicmilling equation without the mean forces {F8}:{F}= c[A(t)1{} (8.52)where{} = ({r} - {r}) - ({r} - {r}) (8.53){r} and {r} are the cutter and workpiece displacements. At the chatter stabilitylimit, cutter and workpiece will vibrate with frequency Of course, this is true forthe zero order approximation, as the solution must include the response to the integermultiples of w. This is due to the fact that the dynamic milling forces are produced bythe structural vibrations, and vice versa. Then, these forces must also respond to theperiodic variations of the directional milling coefficients matrix [A(t)] which relate thedynamic displacements to the dynamic milling forces. However, w, stays constant duringa particular milling operation. This is explained by Tlusty [77]:“The chatter frequency cannot change instantaneously as the vibrations decayor increase slowly, time is needed for transition.”If there are close vibration modes, the chatter frequency may shift to another vibrationmode if the process is disturbed, e.g. by changing the spindle speed. Then, for constantChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 194cutting conditions, chatter frequency can be taken as constant.Zero Order Approximation:For the zero order approximation, i.e. when the vibrations at the chatter frequencyw are considered oniy, the cutter and workpiece displacements can be determined from{r} [G(ic)]{F}et , {r} = —[Gw(iwc]{F}e1”t (8.54)where the harmonics of the tooth passing frequency (ku) have been neglected. Thephase difference between the tooth vibrations in two successive periods is w’T. Then,substituting equations (8.53) and (8.54) with {r°} = e_T{r} into equation (8.52) thefollowing is obtained:{F}e = c(1 — e_T)[Ao][G(iwc)]{F}eit (8.55)where [G] = [Ge] + [Gm]. [A0] is the averaged value of [A(t)j in a tooth period, i.e. theconstant term in the Fourier series expansion of [A(t)]:[A0] =jT[A(t)]dt (8.56)The higher Fourier coefficients of [A(t)] are neglected as the response of the millingforces, cutter and workpiece to the periodic variations of the directional milling coefficients are not considered. Then, equation (8.55) becomes{F} = c(1 —e_T)[Ao][G(iw)]{F} (8.57)The characteristic equation is obtained as follows:det[[I]— c(1 — e_T)[Ao][G(iwc)]] = 0 (8.58)Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 195This is basically the truncated version of the characteristic equation (8.51) for (r, k =0). However, it should be noted that equation (8.51) contains the eigenvalue A whereasit has been replaced by (iwo) in equation (8.58). Although the difference seems to bethe simple substitution of A = ±iw0 for the limit of stability (marginal stability), as itwill be shown later, this substitution leads to the analytical determination of the chatterstability limit in milling. Also, it is realized that, compared to the previous method,the mathematical analysis presented in this section is shorter and simpler. Sridhar etal. [78, 79, 80j and Minis et al. [82, 81] numerically determined the chatter limit bycontinuously increasing the axial depth of cut in the simulation and determining thecorresponding eigenvalue A for 2 degree-of-freedom milling dynamics models.Higher Order Approximations:If the periodic components of [A(t)] are considered in the solution, then the responseof the dynamic forces to these should be included:iT[A(t)] = [Ar]e [Ar] j [A(t)]e_irwtdt(8.59)00 00{ F} = {F}e’ = {Fk}eitk—oo k—00Hence, the chatter frequency will be superimposed on the integer multiples of thetooth frequency. It should be noted that the oscillations of the forces with the multiplesof tooth frequency is not because of the sinusoidal variation of the mean chip thickness orintermittent engagement of the cutter teeth as the mean cutting forces are not consideredhere. It is because of the periodically varying directional coefficients contained in [A(t)].These periodic variations can be regarded as disturbances on the development of chatter.Whether these periodic fluctuations in the system characteristics can help to increaseChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 196the chatter limit compared to a constant dynamics chatter process (like in turning)by disturbing the phase between the inner and outer modulation (like the methods ofspindle speed variation [85, 86, 87] and irregular tooth pitch [83, 84]) will not be analyzedhere, however, will be discussed in some examples. By using superposition principle thedynamic displacements can be written as{r}= k=-00+ ikw)]{Fk}eit(8.60){r} = — [G,(iw +Substituting into (8.52) the following is obtained:00 00= c(1 — eiT) [A(t)] [G(iw + ikw)] {Fk}e’k—oo k=—c,o(8.61)= c(1 — e_iT)k00[A(t)] [G(i + ik)] {Fk}etIn the above equation, the Fourier coefficients of [A(t)] can be kept under the samesummation with the others by multiplying both sides of the equation by 1/Te_iTwt andintegrating from 0 to T, the following is obtained:{Fr} = c(1 — e_T) [Wr_k(c + ikw)]{Fk} (r, k = 0, +1, ±2,...) (8.62)where[Wr_k + ikw)] = [Ar_k][G\ + ikw)] (8.63)[Ar_k] is the (r — k)th Fourier coefficient of [A(t)]:[Ar_k] = (8.64)Equation (8.62) has nontrivial solutions if the determinant is zero:det[Srk[I] — c(1 — e_T)[Wr_k(A + ikw)]] = 0 (8.65)Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 197where 6rk is the Kronecker delta and [I] is the (2mx2m) identity matrix. Equation (8.65)is the same as (8.51) and, therefore, the simplified analysis of the milling stability withoutusing the parametric functions gives the same result obtained by classical procedure.8.3.4 Truncation of the Characteristic Equation of Dynamic MillingFrom the simplified stability analysis, or by substituting ) = iw, in equation (8.65),the characteristic equation at the stability limit takes the following form:det[6rk[I]— S[Ar_k] [G(iw + ikw)]] = 0 (8.66)where s = c(1 — eiT), [I] is the identity matrix with size of 2mx2rn. Equation (8.66)defines an infinite determinant and must be truncated to determine the stability limit.For example, the zero order approximation is obtained when r = k = 0 in equation(8.66):det[[I]— s[Ao][G(iw)]] = 0 (8.67)where [A0] contains the mean values of the corresponding periodic directional millingcoefficients:1 T T [a] [a][A0] = j [A(t)]dt = j dt (8.68)[a] [a]where [a], are diagonal matrices with the elemental directional coefficients in themain diagonal:[apr]kl = apr(k) 1 = k 1 (p,r=x,y)[apr]kl=0 lk JandN-iapr(k) = a(k) (p,r=x,y)j=0Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 198and apr (k) are given in equation (8.20). Before evaluating the integral defined in equation(8.68), consider the following change of variables:4(k) g + jq. — kpzk = (t + jT) — kzk =— kzk (8.69)where r = t + jT, Nw is the angular speed of the spindle in (rad/sec) and w is thetooth passing frequency. Then, (8.68) becomes[A0] = [a (t + jT)] [ax (t + jT)] dtT o o (t + jT)] [ay, (t + jT)]1 N—i ç(j+i)T (8.70)= j [AQr)]drj01 NT= j [A(r)]drTurning back to angular domain by substituting °k = — kzk the following is obtained:1 NTI—k,/,zk[A0]= f [A(Ok)]dOk (8.71)T—kzkUsing T = 2ir/N, equation (8.71) becomes[A0] = Nf21r-kzk = N f2-kzk [a(Ok)] [a(Ok)]dO (8.72)2ir—k,zk 2ir—kzk [a(Ok)] [a(Ok)]The elements of the diagonal matrices [a] etc. are written from equation (8.20) asa(k) = —g(Ok)[sin2Ok + Kr(1 — cos2Ok)]a(k) = —g(Ok)[(l + cos2Ok) + Kr sin 20k]873a(k) = g(Ok)[(1— cos2Ok) — Kr sin 2Ok]a(k) = g(Ok)[sin20k — Kr(1 + cos2Ok)]Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 199whereg(O) = 1 <Ok < (8.74)g(Ok) = 0 Ok < st or °k > cex JThus, all the elements of the matrices [a] etc. are non zero oniy in the interval (4st—qex).Then, the limits of integration become q and q!ex, independent of the element axialposition zk. According to this result, there is no effect of the helix angle on the chatterstability as predicted by this formulation. Therefore, elements in diagonal of [a] becomeequal to each other. Then, equation (8.72) takes the following form:[A0] = (8.75)2ir c4°)(O)[I] c4?(0)[i]where [I] is the mxm identity matrix, andfcrr=j apr(0)dO (p,r=x,y) (8.76)kstFinally, the following is obtained by integrating equation (8.73):if ]ex= L°2° 2KrO + Kr sin 20]1 r 1ex= {—sin2O—28-f-Krcos2Oj(8.77)1 ex= [_sin2O+20+Krcos20]1 er= {_cos2O_2KrO_Krsin2O]Then, the zero order approximation to the characteristic determinant, given by equationChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 200(8.67) takes the following form:N 2 [‘Im [‘]m [G(i)1 [G(iw)]det [112m — = 0 (8.78)27r c4 [dim 0) [dim [G(iw)] [G(i)]where [G] = [G] + [G3,]. Equation (8.78) can be further simplified to the following:det[[I]2m — = 0 (8.79)where s = c(1 — e_T), [W0] can be regarded as the oriented transfer function in millingand is given as follows:a} [G(i.)} + [G(iw)] oj [G(iw)] + cj,) [G(iw)}[W0(zw] = (8.80)a) [G(iw)] + c4,) [Gyr(ic)] 4} [G(iw)] +Before the analytical solution of the zero order approximation, higher order Fouriercoefficients of [A(t)] will be given. For the first order approximation r, k = 0, +1, thefollowing truncated determinant is obtained:[I]— .s[Ao] [G(iw)] —s[A_i] [G(iw + iL.)] —s[A1][G(iw — i)]det—s[A1][G(iw)] [I] — .s[Ao][G(iw + ü.)] —s[A2][G(iw — iw)]—s[A][G(iw)] —s[A_2][G(ic + iw)] [I]— s[Ao] [G(iw — iw)](8.81)where [Aq] represents the qtI Fourier series coefficient:1 rT[Aq] =— j [A(t)]e”tdt (8.82)ToThe procedure is very similar to the one followed for [A0]. Similar to the resultobtained in equation (8.72), equation (8.82) takes the following form after the substitutionof 0:[Aq] =- fex[A(0)]iN8dO(8.83)27r stChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 201where = Nw has been substituted. As in the case of zero order approximation, [Aq]becomes independent of the axial position (or element number):Jm zyN [a [Il [11m 1[Aq] =— I (8.84)2ir () [Il (q)im [Ij jThen, [Wr_kj takes the following form:[Wrkl = [Ar_k][G(Wc + ikw)j— F (r_k)[(j)] + a(k)[G(iwk)] (r_k)rG (iwk)j + (r_k)[(j)] 1xx—[a()) [G (iw)] + a()[G (iwk)] (r_k)rG (iwk)] + (r_k) [C (iwk)] jyx I xy(8.85)where wk = w + kw. Expanding e9 = cos qNO — i sin qNO and using trigonometricidentities, the following general form for the coefficients.., o are obtained:ir 1exP2° I(q) = — J< eiqNO +c1&’° — ce jxxir ]cex= { — CKrjC’° +c1e9+ c2exy(8.86)ex= [coKreN0 +c1e’0+c2e29]jf ex(q) Jç_iNO—c1e’0+ C2etP29lj‘1stChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 202wherep1=2+Nq , p2=2—Nq2 Krjc1= (8.87)ivq Pi— Kr + iC2—P2Also, are equal to the complex conjugates ofrespectively. Equations (8.84) and (8.86) can be used in truncations of the characteristicdeterminant (8.66) to find the chatter limit.The developed stability formulation predicts no effect of helix angle on the chatterlimit. Of course, this is true when the cutter teeth are equally spaced, the helix angleis constant along the flutes and the same on each tooth. If the helix angle were lotapproximated within axial elements (equations 8.16- 8.17), matrices [a], .., [afl,] wouldhave the elemental lag angle k,&/z in the arguments of the trigonometric terms given in(8.77). This would require an iterative solution for the chatter limit. On the other hand,theoretically, the element thickness can be taken as infinitesimal in which case the cutteredge in each axial element approaches to a straight line (zero helix). When the resultinginfinite sized matrices were integrated in equation (8.72), the dependence on the axialposition would be lost again leaving no effect of helix.8.3.5 Summary of the Calculation of Milling Stability for the General CaseThe stability of dynamic milling is predicted by solving the infinite determinantequation:d6t[6rk[I1 — SEAr_k] [G(zW + ikw)]] = 0whereChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 203örk: Kronecker delta[I]: 2mx2m identity matrixrn: number of axial elementsi=s = c(1 — eT)c = LzKLSz: axial element thicknessK: tangential milling force coefficientc: chatter frequencyw: tooth (passing) frequency[G] = [Ge] + [Gm][Ge], [Gm]: cutter and workpiece transfer functionsand [Ar_k] which is the (r — k)t Fourier coefficients of the periodically varying directional milling coefficients are given by[Ar_k] = (8.88)27r dr—k) [I]m dr—k) [I]mwhere a, .., a are given in equation (8.86). For the zero order approximation, thedeterminant takes the following form:det[[I]— s—[W0(iw)]] = 0where [W0] contains the oriented transfer functions and is defined in equation (8.80).8.3.6 Accuracy of the Chatter Limit Prediction by the Truncated Characteristic EquationIn the rest of the analysis, the zero order approximation of the characteristic equationwill be used to develop an analytical milling stability condition. In general, the accuracyChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 204876120153045607590Rotation Angle (deg)Figure 8.3: The effect of number of teeth on peak to peak (AC component) value ofdirectional coefficient a. (Half immersion-up milling with Kr = 0.5)of the truncated characteristic determinant depends on whether the Fourier coefficientsof [A(t)] with significant amplitudes are included in the truncated determinant [174].However, it should be noted that the chatter vibrations occur at a specific chatter frequency, and the effect of the directional coefficient oscillations in the harmonics of toothpassing frequency may not be significant for the stability limit. The Fourier coefficientsof [A(t)] are closely related to the number of teeth in cut. The higher the number ofteeth in cutting is, the smaller the overall variation of the directional coefficients contained in [A(t)] becomes. As an example, the variation of the directional coefficientgiven by equation (8.20) is shown in Figure 8.3 for 4,8,12 and 24 teeth cutters. A halfimmersion-up milling case is considered with Kr = 0.5. As it can be seen from the figure,the AC component reduces considerably compared to the average value (i.e. zero orderapproximation) as the number of teeth is increased. However, as it will be shown byexamples, the zero order approximation predicts the stability limit accurately, includingthe cases where AC components are high.Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 2058.3.7 Solution of the Characteristic Equation to Determine the Chatter Stability Limit in MillingIn this section oniy the zero order truncation of the characteristic equation willbe considered as it leads to an analytical method for milling stability. Therefore, thefollowing eigenvalue equation is to be considered:det[[I] + A[A0][G(iw]] = 0 (8.89)where the eigenvalue A is as followsA = = —Ktz(l — e_T) (8.90)2ir 47rIt should be noted that axial depth of cut is equal to a = mz, where ZXz is thethickness of axial elements. The unknowns in equations (8.89) and (8.90) are the chatterfrequency w and the axial depth of cut for the defined number of teeth N, radial depthof cut b, spindle speed (to determine the tooth period T) and milling force constants Kand Kr. Equation (8.90) can be solved for two unknowns as it is a complex equation.However, aiim cannot be directly obtained from equation (8.90) as it does not appearexplicitly, but /z does. The procedure is explained as follows. In the formulation of thedynamic milling forces, the variation of the end mill and the workpiece dynamics in theaxial direction were modeled by dividing the total axial depth to number of elements.The dynamics of the structures were assumed to be constant within each element. Theelement thickness /z is selected depending on how strong the variations of the dynamicsof structures in the axial direction are, and the available modal test data or finite elementgrid solutions. The difficulty with equation (8.89) is that before the critical axial depthof cut can be determined, the number of elements in the axial direction, m, must beknown to construct the matrix [W0] which contains the oriented transfer functions of thecutter and workpiece, respectively. However, nZz is equal to the axial depth of cut aiimChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 206which is being solved for. Therefore, the solution has to be started with an arbitrary mor a — rn/z. If w and A are known, then equation (8.90) gives the critical depth Zlimor aiim = m/ziim. [f aiim > a + /-z or aiim a z, the new value of m has to bedetermined asm’ = aijm//z = m/zijrn//z (8.91)Then, equation (8.89) should be solved again by using m’ to determine the new valueof the critical depth. The presented general milling stability formulation can be usedto analyze the stability of flexible workpiece milling as it considers the variation in theworkpiece dynamics in the axial direction which has been neglected in the other stabilitymodels. This procedure is illustrated on a plate milling example given in section 9.4.3.Also, the algorithm is given in Figure 8.13. It is realized that a critical radial depth ofcut bum for chatter stability can also be determined from the same equation if the axialdepth of cut is specified. The radial depth of cut is implicitly contained in the elementsof [A0] as it determines the start or exit angles (, q).Derivation of Chatter Frequency-Spindle Speed Relation• In order to be able to the calculate eigenvalue of equation (8.89) the chatter frequencyw must be known. The chatter frequency, in general, depends on the spindle speed andthe structural dynamics parameters. The number of waves between subsequent cuts (orthe number of vibration cycles within one tooth period) can be expressed ask + /2ir = wT (8.92)where k is the largest possible integer such that < 2ir. In other words, there are k fullvibration cycles and a fraction f/27r of a cycle between the subsequent passes at the samepoint. In orthogonal cutting-chatter stability theory, the phase difference at the limit ofChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 2071.61.51.41.11Figure 8.4: Variation of chatter frequency with spindle speed as predicted by the orthogonal cutting chatter theory.stability () is determined as [5]d2 —1= 2ir — 2tan’(2d (8.93)where d = and w and are the undamped natural frequency and damping ratioof the considered vibration mode. Equations (8.92) and (8.93) define a relationship between the chatter frequency w and the tooth period T. This is plotted in Figure 8.4 interms of the frequency ratios, w/w and w/w, for = 0.05. The corresponding spindlespeed n (rpm) can be determined as n = 60/(NT). It should be noted that it is easierto determine the tooth period T from equation (8.92) when the chatter frequency w isspecified whereas the reverse requires the solution of equation (8.93) which is implicitin w. A similar equation for milling, which is complex due to the periodically varyingdirectional coefficients, has not been derived before. In this thesis, chatter stability limitsfor milling are derived using the stability formulation presented in the previous sections.For that, the analysis of chatter frequency-spindle speed relation is necessary and derived0 0.2 0.4 0.6 0.8 1w/wnChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 208in the following.The critical axial depth of cut in terms of the element thickness can be obtained fromequation (8.90) asZ1jm =— rA (8.94)— e_cT)Substituting A = AR + iA1 and e_T = cos wT — i sin wT in equation (8.94)4ir AR+iAIZ1im =— NK [1 — (cos — i sin wT)] (8.95)from which the real and imaginary parts are separated as follows:47r AR(1— coswT) + AisinwT .A1(l— coswT)— ARSiIIWCTLZlim NK (1— coswT)2+ sin2 + (1— coswT)2+ sin2(8.96)/Zlim is a real number, then the imaginary part of equation (8.96) must vanish:— coswT) — ARsinwT = 0 (8.97)or,AR 1—cos (8.98)A1 +/1— cos2wTThe following quadratic equation is obtained after rearranging the above equation:(1 +2)coswT—2c swT+(1 — K2)— 0 (8.99)One of the solutions of equation (8.99) is the trivial solution of = 0 + 2k7r whichcorresponds to the case of no regeneration. The nontrivial solution is1 K2coswT= 2 (8.100)or,wT = cos’( ) + 2k (8.101)Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 209Equation (8.101) defines a relationship between the chatter and the tooth frequencies.This equation is the same as the general form of the regeneration expression given in(8.92), in this case1—= cos1(+(8.102)The following equation for /Zlim is obtained if coswT from equation (8.100) andsin = — cos2wT are substituted into the real part of equation (8.96) (imaginarypart vanishes):Z1jm = 2(1 + 1/a) (8.103)Therefore, given the chatter frequency, the chatter limit in terms of the element thicknesscan be determined from equation (8.103). As explained in the beginning of this section,the corresponding chatter limit can be determined as aiim = m/ziim. If rn’ determinedfrom equation (8.91) is different than m, then the eigenvalue equation (8.89) is solvedagain by using m’ number of axial elements. In general the value of the eigenvalue A canonly be determined by numerically solving the eigenvalue problem defined in equation(8.89). It should be noted that this is the only part in the developed stability analysiswhich requires a numerical solution. However, this is necessary only for the most generalcase of the milling stability. As it will be shown in the following sections, the numericaleigenvalue solution is not necessary for most of the practical cases. These are the caseswhere the variation of the cutter and workpiece dynamics in the axial direction can beneglected, i.e. m = 1. The corresponding tooth period T or spindle speed m to thecomputed chatter limit can be determined from equation (8.101). When computing thevalue of e, it should be noted that ‘ = 27r— c is also a solution to the (cos1) functiongiven in equation (8.102). In general, the actual value of the angle can be determinedfrom the signs of the x (real) and y (imaginary) components of the vector which defines it,Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 210i.e. in which quadrant the vector is. However, this is not possible for (cos) function. Oneway of solving this problem is to substitute both solutions into equation (8.96): the onewhich gives real /Ztim only, is the solution. Another solution is obtained by consideringthe angle defined by the eigenvalue A in the complex plane. The angle subtended by thisvector is= tan’ = tan’ (8.104)Then, by substituting = 1/tan into equation (8.100), the following is obtained:= —cos2 (8.105)The solutions of the above equation iswT = (+ir + 2) + 2k7r (8.106)Considering the solutions obtained from (8.101), the solution is found to be:= + 2k7r (8.107)where= — 2 (8.108)It should be noted that this solution is more convenient as the actual value of = tan1can easily be obtained from the sign of AR: ‘ = cp + r, if AR < 0. Therefore, by usingthe milling stability formulation developed in this thesis, the relationship between thechatter frequency and the spindle speed in milling is obtained for the first time. Thisalso makes the analytical prediction of milling chatter possible.8.4 Solutions of Milling Stability Equation for Special CasesThe developed general stability formulation considers the dynamic displacementsof the milling cutter and workpiece in x and y directions, and the variation of theirChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 211dynamic characteristics in the axial z direction. The variation of the dynamics in theaxial direction is important in milling very flexible workpieces and may affect the stabilitylimit. This has not been considered in the previous milling chatter models, and it ismodeled in the developed formulation in this thesis. The stability limit and phase (orspindle speed) formulae require the solution of the eigenvalue A for the stability equation[[I] + A[A0j[G(iwj]. For the general case, this has to be performed numerically which isthe only numerical part in the formulation. In this section, the general formulation willbe applied to special and practical cases of milling. For these cases, the solution can beobtained completely analytically. Although a numerical eigenvalue solution is very faston computers, the analytical solution provides direct relationships between the millingconditions, stability limit and chatter frequency.8.4.1 Milling of a Single Degree-of-Freedom WorkpieceConsider the single degree-of-freedom dynamic milling model shown in Figure 8.5.This model represents the milling of a flexible part with a relatively rigid cutter. Itshould be noted that, although the workpiece is modeled as a single degree-of-freedomstructure, it can have more than one vibration mode in the considered degree-of-freedomdirection. Then, milling of thin walled components such as impellers, thin webs etc. canalso be analyzed by this model if the considered axial depth of cuts are small so thatthe dynamics of the workpiece can be assumed constant within the cutting depth. As itwill be shown by some experimental and simulation results in the following sections, thechatter free axial depth of cuts for flexible workpieces are very small unless low cuttingspeeds are used. With moderate and high cutting speeds, the total depth is removedlayer by layer, using very small depths [95]. In practice, usually a constant axial depth ofcut is used throughout, however the dynamics of the workpiece are different for differentlayers. Therefore, by the accurate prediction of the stability lobes the number of passesChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 212can be reduced significantly.For the system shown in Figure 8.5 which has single axial element, i.e. m = 1,= aiim. Then, the eigenvalue equation (8.89) becomes:1— -Ktaijm(1 — e cT)a,yGy(iwc) = 0 (8.109)where c is the directional coefficient given by equation (8.77). The superscript “(0)”to denote the zero order approximation will be dropped from the directional millingcoefficient matrix [A0] from this point on for the sake of simplicity. G is the transferfunction of the workpiece in the y direction:2 /kG (i) ‘ “ (8.110)2—fl C ‘Y fly Cwhere and are the stiffness, damping ratio and undamped natural frequency ofFigure 8.5: Single degree-of-freedom milling system model.Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 213the workpiece in the y direction. If the workpiece has more than one vibration mode inthis direction, then the transfer function G can be determined by modal superpositionas:— r2 w2 ij1 Jfly C “Y3 flY”Cwhere the subscript j denotes the parameter of the th vibration mode. From equation(8.109), the chatter limit is obtained asaiim= N1 (8.111)— ecT)As aiim is a real number, the imaginary part of the complex term G(i)(1 — e_iT)must vanish. Hence, the imaginary parts of G(iw)e’Tand G(iw) should be equal toeach other, as shown in Figure 8.6. Then, the real parts will have opposite signs resultinginG(i)(1 — e_cT) = 2Re[G(iw)] (8.112)The real part of the transfer function G(iw) is as follows:1 1-d2= k,(1 — d)2 +4d (8.113)where d = Substituting (8.112) in equation (8.109)aiim T (8.114)From Figure 8.6, the phase is obtained as in the orthogonal cutting chatter stabilitytheory:d2= 27r — 2tan12cl (8.115)Then, the spindle speeds (n) corresponding to the considered chatter frequency in thedifferent lobes (k) can be found fromk + 6/27r = (8.116)Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 214Figure 8.6: The phase angle and transfer function at the chatter stability limit. Inmilling, chatter frequency may be higher or lower than the modal frequency dependingon the cutting conditions. In both cases G(1 —6T) = 2Re[G].ImN-ReG -iwTGeChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 215where n = 60/(NT). If the workpiece is flexible oniy in the x direction then the criticalaxial depth is determined similarly as:aiim= N1 (8.117)These equations derived for milling here are similar to the stability condition developed by Tiusty [5] for single tooth cutting with nonvarying directional coefficients, e.g.turning and boring. Equations (8.114) and (8.117) suggest that the chatter frequencycan be smaller or higher than the natural frequency depending on the signs of the directional milling coefficients a, and c. If a > 0, then Re[G(i)] must be positive,i.e. < w,, in order to have aiim > 0, and vice versa. In the classical orthogonalcutting stability theory of Tlusty, the chatter frequency is always larger than the modalfrequency. This is because of the presumed positive direction for the feed cutting forcewhich can only be negative for large and impractical rake angles (due to an increasedcomponent of the rake face normal force in the feed direction). However, in milling, thedirectional coefficients are not constant as proven here...,o represent averageddirectional milling coefficients, and depending on the radial immersion of the cutter andKr, their sign may change. Minis et al. [193] showed that even in turning, the chatterfrequency at the limit of stability can be smaller or larger than the modal frequencydepending on the orientation of the tool holder (right or left-handed), thus the transferfunction. The variations of a with the immersion angle for up and down milling anddifferent Kr values are shown in Figure (8.7). c is always negative for down milling,thus the chatter frequency is always higher than the natural frequency of the structurewhereas in up milling, it depends on the exit angle. Similar to the milling force in they direction, o, is smaller for up milling resulting in higher stability limits. As it canbe seen from these graphs, the variation of the directional coefficient n and thus thestability limit aiim with the radial immersion is not linear. These graphs can be used toChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 2161.000.500.00-0.50>‘>‘-.-1.50-2.00-2.50-3.00-3.501800.00-0.50-1.00-1.50-2.00-2.50-3.00-3.50180stFigure 8.7: Variation of the directional milling coefficient a with the immersion anglein up and down milling. The sign and magnitude of the directional coefficients determinethe chatter frequency and stability limit, respectively.0 30 60 90 120 150ex0 30 60 90 120 150Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 217determine the optimal radial depth of cuts for chatter free milling.Example: Milling of a SDOF StructureThe stability formula developed in this section is applied to a milling operation investigated by Opitz [163], Sridhar et al. [80] and Minis et al. [81]. The flexibility of theworkpiece in the x direction is represented by a single degree-of-freedom system with thefollowing parameters:= 7.152x106 N/rn (39700 ib/in), = 0.0417, = 355 rad/secA straight tooth cutter with N = 10 teeth is used and the start and exit anglesare 67° and 139°, respectively. Kr = 0.577 and a is calculated from (8.77) as (-1.6).Equation (8.117) was used to calculate the stability limit for different values of chatterfrequency w. Then, the corresponding spindle speeds are found from equation (8.92).Stability lobes are shown in Figure 8.8 in terms of the product of critical depth of cutaiim and tangential cutting force coefficient K. Also shown in Figure 8.8 is the datafrom [81] and [80]. Minis et al. [81] used zero order approximation, but determined thestability limit for a given spindle speed by increasing the axial depth of cut until themarginal stability was obtained. Sridhar et al. [80] used the analog simulation data from[163] to compare with their numerical stability solution results. All three solutions showexcellent agreement except around two peak stability limits. Around the peak stabilitypoints, the chatter frequencies are very close to the natural frequency of the structuresince Re[G(iw] —* 0 as —*.At these frequencies the variation of Re[G(i] with thefrequency or spindle speed is quite steep, and therefore a fine spindle speed or frequencyresolution is necessary for accurate results. It should be noted that the only analyticalChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 2186050000100 400Figure 8.8: Comparison of the predicted chatter limit with the published data for thesingle degree-of-freedom milling system example.solution is the original contribution proposed here. The resolution can be made as fineas required without increasing the computation time significantly. Hence, the numericalsolution procedures used in [81] and [163] may have resulted in the differences aroundthe peaks.8.4.2 Milling of a Flexible Structure with a Flexible End Mill-Single AxialElementConsider the milling system model shown in Figure 8.9. This is the most general caseof a milling system when only one axial element is considered. As in the previous case,the cutter and workpiece can have more than one vibration mode in the two consideredorthogonal directions, x and y. Then, the general stability equation takes the following150 200 250 300 350Spindle Speed (rpm)Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 219WorkpiecekcxFigure 8.9: Milling system model with two degree-of-freedom cutter and workpiece.Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 220form:c G(ic4) 0det I [I] + A = 0 (8.118)c c 0 G(iw) )where .., c are given in equation (8.77). G, = + , G and are thecutter and workpiece transfer functions which may include contributions of more thanone vibration mode. The cross-transfer functions and have been neglected forthe sake of algebraic simplicity. From equation (8.118) the following quadratic equationis obtained for A:a0A2 + a1A + 1 = 0 (8.119)wherea0 = — abZ) (8.120)a1 = c,G(iw) +Then, the eigenvalues A are obtained as:A = ——(a1+ — 4a0) (8.121)The critica’ axia’ depth of cut can be obtained by substituting A into equation (8.103):aiim = —(1 + 1/ic) (8.122)where ic = . Corresponding spindle speed (n = 60/NT) to a considered chatterfrequency can be found from equations (8.104), (8.107) and (8.108) as:= +2k7rwhere=— 2çoand=tan —=tan —ARChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 221Example: Milling of a Rigid Workpiece with 2 DOF End MillThe developed stability method is applied to an end milling operation investigated byWeck, Altintas and Beer [162] at the Machine Tool Laboratory of Technical Universityof Aachen. The workpiece material is aluminum alloy AlZnMgCu 1.5 which is machinedby using 30 mm diameter, three fluted helical end mill with 300 helix angle and 110 mmgauge length. The measured milling force coefficients are K 600 MPa and Kr = 0.07(too low radial force coefficient Kr implies that the rake angle on the flutes is very high).The dynamic parameters of the end mill were determined from modal tests as:k = 5590 N/mm = 0.039 w = 3788 rad/sec (603 Hz)k = 5715 N/mm = 0.035 w = 4161 rad/sec (666 Hz)Hence, the cutter has one dominant vibration mode in each direction. Chatter testswere conducted at different radial depth of cuts by using a feed per tooth of st = 0.07mm/tooth. Axial depth of cut and spindle speed increments were 0.5 mm and 200 rpm,respectively. Discrete time domain chatter simulations were also used to determine thestability limit. Both experimental and time domain simulation results for quarter immersion up milling and slotting tests are shown in Figure 8.10 which is taken from [162].In chatter simulations, the process was assumed to be unstable when the peak values ofthe vibration grow in thirty consecutive oscillation periods. For each spindle speed, thisprocedure has to be repeated by increasing the axial depth of cut until the stability limitat that speed is obtained. Small increments of spindle speed and axial depth should beused in the simulations to accurately obtain the stability limits, especially the pockets inthe lobes. Then, depending on the spindle speed range of interest and the desired accuracy, hundreds of simulations may be necessary to obtain a single stability lobe diagram.Also, the effect of cutting conditions on the stability limit can only be analyzed after theChapter 8. Analysis of Dynamic Cutting and Chatter Stability in MillingE222Figure 8.10: Experimental and time domain simulated stability limits for end millingtests. (data by Weck, Altintas and Beer, 1993)................—-——-•ts• pq*w•••i••••• 4.a4.ha.? ad0000 ooo0t0000I0p0000b=R12Legendo :Stablecut• :Lightchatr• :ChatterSimulated stability0 1000 2000 3000 4000 5000Sprndle speed n (rev/mm)0I I I100 200 300Cutting speed v (rn/mm)40010000a)935101000 2000 3000 4000 5000Spindle Speed (rpm)Figure 8.11: Analytically predicted stability lobes for the case for which the experimental• data and time domain simulations are shown in Figure 8.10.diagrams are obtained. The developed stability method was used to obtain the stabilitydiagrams shown in Figure 8.11.The stability limits were calculated at 100 frequencies (or spindle speeds) for eachlobe, thus a total of about 800 frequencies. The total computation time to obtain adiagram on a IBM 486-66 computer was about 10-15 seconds. As it can be seen fromthe figures, the analytical predictions and the time domain simulation results are in goodagreement although the cutter has only three flutes resulting in the directional coefficients with high AC/DC ratio.Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 223b=R12b=2RAnother example of a 2 DOF milling system is taken from Smith and Tlusty [70].Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 224This is an half immersion-up milling of an aluminum workpiece by a 4 in. diameter shell mill with 8 teeth. The modal properties of the cutter were determined as follows:Mode Frequency (Hz) Stiffness (N/m) Damping RatioX 1 260 2.26x108 0.122 389 5.54x107 0.04Y 1 150 2.13x108 0.12 348 2.14x107 0.1The results of the analytical method predictions and the time domain simulations performed by Smith and Tiusty [70] are shown in Figure 8.12. As it can also be seen fromthis figure, the zero order approximation is in excellent agreement with the time domainsimulations.8.4.3 Milling of a Flexible Structure with a Rigid End Mill-Varying Dynamics in Axial DirectionIn this section, the application of the general stability formulation to the milling ofplate-like workpieces is given by considering the variation of the workpiece dynamics inthe axial direction. For this case, the stability equation (8.89) becomes:det ([I] — Ktz(1 — e_iT)cryy[Gy(iwc)]) = 0 (8.123)The transfer function of the workpiece in the y direction can be obtained by modalsuperposition as:k[G(iw)] = [u]H (iwo) (8.124)j=1Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 2250.0120.01c 0.0080.0060.0040.00200.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80Spindle Speed /10000 (rpm)Figure 8.12: Analytical and time domain stability limit predictions for a case analyzedby Smith and Tiusty.whereH (i) = . (8.125)— w2 + 2zw,wand [u3] = {uy,}{uy}T is the j undamped modal matrix of the workpiece, normalizedfor unity modal mass . The stability diagrams for flexible workpiece milling can begenerated by using the general stability equation (8.103). However, the formulation isfurther simplified if the modes of the workpiece are well separated. In this case, thestability analysis can be performed around the most flexible mode of the structure byneglecting the contributions of the other modes. Then, for the rth mode, equation (8.123)becomes:det ([I]— A[ttyr]) = 0 (8.126)1The structure is assumed to be proportionally damped.k=1 k=0Analytical• Simulation (Smith & Tiusty)ChatterStableChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 226whereA = Kjz(1 — e_iT)cyyHyr(iw) (8.127)The imaginary parts in equation (8.127) must vanish to obtain real /z. The eigenvalueA is a real number as the [uyr] in equation (8.126) is a real matrix. Then, similar toequation (8.111), the following equation must hold at the limit of stability (see Figure8.6)(1 — e_T)Hy = 2Re[Hyr(ic)] (8.128)The corresponding phase angle and the spindle speeds can be determined same as the single degree-of- freedom case given by equations 8.115 and 8.116. Substituting in equation(8.127), the following is obtained for the critical depth of cut in terms of the elementalthickness /.Ziim:AZiim = (8.129)e[ ,r(zc]As explained before, the critical depth of cut is obtained as aiim = mzjjm, m is thenumber of axial elements. For an arbitrary m and axial depth of cut a = m/z, ifaiim a + /z or aiim a — z, the new value of m has to be determined asm’ = aijm//..z = m/zljm/LSz (8.130)The solution is started with m = 1. The procedure is outlined in Figure 8.13 and themethod is illustrated with a plate milling example in the following.Example: Peripheral Milling of a Plate-MDOF ModelStability limits in milling a cantilever plate (Ti4A16V) with dimensions (63.5x44x3.8mm) is simulated. The 19.05 mm diameter end mill, which is assumed to be rigid, has4 flutes, is used in the down milling mode to finish the plate surface. A very smallradial depth of cut with Qst = 175° (which allows relatively high critical depth of cuts soChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 227Consider M differentchatter frequencies aroundthe rth modal frequency.Start with single axialelementSolve the elgenvalue problemby using (m) nodes in the modalmatrix Iuyrl (mode shapesat the in-cut nodes only).Determine the elementalstability limit by using theknown values of N, Kandc which depends on radialdepth of cut and K rCalculate the limiting axial depthof cut.Check whether the calculatedlimiting axial depth of cut is insidethe part of the workpiece structureconsidered in the determinationof eigenvalue A.Determine the correspondingspindle speeds (n) in differentlobes (k).Figure 8.13: Stability limit calculation algorithm for flexible workpieces with varyingdynamics in the axial direction.n=60/(NT) (rpm)k+Re—arctanlmIHvr(wc)]Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 2280.0140M12o o 0080.0060.0040 0.002010000 15000 20000 25000 30000 35000 40000 45000 50000Spindle Speed (rpm)Figure 8.14: Predicted effect of vibration mode shape on the stability limit. The rigidityof the plate increases in the axial direction towards the fixed end allowing higher stableaxial depth of cuts than the ones predicted by neglecting this variation.that more than one axial element is in cut) used in the simulations to show the effectsof mode shapes on the stability limits. K 1500 MPa, Kr = 0.7 were used. Thedeveloped Finite Element code was used to determine the normalized modal vectors andfrequencies of the plate. Figure 8.14 shows the simulated stability lobes. Also shownis the predictions by using the single axial element at the tip of the plate. The rigidityof the plate increases along the axial direction towards the fixed end, thus as more axialelements are considered, the predicted stability limit increases. In the stability pocketshown in the figure, the MDOF model prediction is about 50 % more than the SDOFmodel prediction.Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 2298.5 Dynamic Peripheral Milling of PlatesCutting tests have been performed on cantilever plates in order to analyze the dynamic milling process and verify the developed milling stability model. Results of severaltests with a titanium (Ti6A14V) cantilever plate of dimensions (63.5x44x3.8 mm) arepresented here. The plate is down-milled by an 8 fluted, 300 helical carbide end mill with19.05 mm diameter. In all cases, feed per tooth of St = 0.008 mm and radial depth of cutof 0.325 mm, which corresponds to 15° immersion angle (or = 163°, q = 180°), wereused. The modal parameters of the plate were determined from the developed FilliteElement program. First three undamped modal frequencies for the unmachined plateare 1667, 3057 and 7074 Hz. The stability diagram around the first mode is shown inFigure 8.15. Milling force coefficients of K = 2000 1VIPa and Kr 0.72, and dampingratio of = 0.05 were used in the simulations. The dominant first mode of the plate isconsidered, and chatter stability diagram is computed using the single degree-of-freedomstability model presented in section 8.4.1. Due to high flexibility, the stable axial depthof cuts are very small in the peripheral milling of this plate. The highest stability limit isobtained approximately at 6500 rpm, in the pocket between the lobes k = 1 and k = 2.The largest pocket (between k = 0 and k = 1) could not be obtained as the maximumspindle speed of the milling machine was 10000 rpm. The tooth passing frequency with6500 rpm is 867 Hz which is very close to the half of the first undamped modal frequency,i.e. 1667/2=833 Hz. The peak stability limits correspond to the tooth passing frequencieswhich are integer divisions of the considered modal frequency of the structure. Therefore,at these spindle speeds while the stability against chatter is maximized, forced vibrationsmay become excessive.2These are average values for Ti6A14V as presented in Chapter 4. Note that edge force componentsshould not be included as they do not depend on the chip thickness, thus produce dc force componentswhich are not considered in chatter analysis.Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 2300.6E0.50.40.3aa)90.2C).1L-01002000 3000 4000 5000 6000 7000 8000 9000 10000Spindle Speed (rpm)Figure 8.15: Stability diagram for the cantilever titanium Ti6A14V plate down milled by8 flute carbide end mill with 19.05 mm diameter. Plate dimensions: 63.5x44x3.8 mm.Radial depth of cut b=zO.325 mm.In the first test 0.4 mm axial depth of cut was removed at 6500 rpm. As shown inFigure 8.15, this axial depth is slightly lower than the stability limit, thus a stable cut isexpected. The cutting forces in the x and y direction were measured by a Kistler table dynamometer whose original bandwidth was 4000 Hz. However, due to the flexibility of theclamping between the dynamometer and the milling machine table and the mass of theplate which is mounted on the dynamometer, the real bandwidth is about 1900 Hz in they direction and 1750 Hz in the x direction. Also, the sound generated during machiningwas recorded by an ordinary microphone which has a wider reliable bandwidth. Figure8.16 shows the cutting force spectrums in x and y directions. The fundamental frequencyof the cutting force, w = 867 Hz, can be clearly seen in the spectrums. The second peakat about 1600 Hz is close to the first modal frequency of the plate and second harmonicof the cutting forces. In order to clarify the source of the vibrations at this frequency,Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 231the following analyses are given. First, consider the sound signal and its spectrum shownin Figure 8.17. In general, chatter vibrations gradually grow and then diminish whenthe vibration amplitudes become so large that the contact between cutting tooth andworkpiece is lost. This pattern is not seen in the sound signal. The plate is milledat 5000 rpm with the same axial depth of a = 0.4 and radial depth of b = 0.325 mm.The force spectrums are shown in Figure 8.18. Although at 5000 rpm, a = 0.4 mm ishigher than the predicted stability limit shown in Figure 8.15, the maximum amplitudesof the force spectrums occur at the tooth passing frequency, = 667 Hz, and its secondharmonic. In Figure 8.19 the cutting forces in y direction are compared for n = 6500rpm and n = 5000 rpm. Although n = 5000 rpm is in the unstable zone, the peak to• peak force amplitudes are higher for n = 6500 rpm. This is due to the fact that n = 6500rpm spindle speed causes resonance in the plate. Therefore, it can be concluded that theforced vibrations in plate milling are as critical as the chatter vibrations.The cutting forces for n = 6500 rpm and a = 2 mm are shown in Figure 8.20. In thiscase the chatter behavior can be clearly seen as 2 mm and is well above the stability limit.The cutting force amplitudes grow to very large values and diminish in an exponential-like manner after the contact between the cutting teeth and the workpiece is lost. Thesame behavior can be seen in the sound signal shown in Figure 8.21. The force spectrums shown in Figure 8.22 indicate that the forced and chatter vibrations exist together.The chatter free axial depth of cuts are very small in plate milling. Therefore, eithermany cuts with small axial depths have to be used to machine the plate without chatter orvery slow cutting speeds should be used to utilize the high process damping generated atthose speeds. In order to show the effect of cutting speed and the process damping on thechatter behavior, 3 different spindle speeds, n = 6500, 400, 250 rpm, were used to removeChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 2325.) 4-o__________0 800 1600 2400 3200 400040302520151050 Li . I I I0 800 1600 2400 3200 4000Frequency (Hz)Figure 8.16: Cutting force spectrums in x and y directions for n = 6500 rpm and a = 0.4mm.Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 233—1-2-3-4432100II25 50 75 100 125Tooth Period1500.70.60.50.40.30.20.10.00 800 1600 2400 3200Frequency (Hz)4000Figure 8.17: Sound sigilal and its spectrum at n = 6500 rpm, a = 0.4 mm.Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 2345=0 800 1600 2400 3200 400010:_____ ________________________0 800 1600 2400 3200 4000Frequency (Hz)Figure 8.18: Cutting force spectrums in the x and y directions for n = 5000 rpm anda = 0.4 mm.Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 2356040200-20-40-60n=5000 rpmI L i iiiiiii I III________________________________________________________II80 120 160Tooth PeriodFigure 8.19: Cutting forces in the y direction for n = 5000 rpm anda = 0.4 mm.6040200-20-40-600 40 80 120 160nUUrpiTI j200‘II ‘I I’I I II ‘II liii0 40 200n = 6500 rpm forChapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling40022040-140-320-500500333167-0-167-333-5000 50 100 150 200 250 300 3500 50 100 150 200 250 300 350Tooth Period236Figure 8.20: Cutting forces in x and y directions for n = 6500 rpm and a = 2 mm.Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 237•1Frequency (Hz)20151050-5-10-15-203.02.52.01.51.00.50.00 25 50 75Tooth Period100 125 1500 800 1600 2400 3200 4000Figure 8.21: Sound signal and its spectrum for n = 6500 rpm and a = 2 mm.Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 238141210‘—, 4200 400060e) 50—E 0201000 4000Frequency (Hz)800 1600 2400 3200800 1600 2400 3200Figure 8.22: Cutting force spectrums for n = 6500 rpm and a = 2 mm.Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 239the whole depth of a = 44 mm on the plate with the same radial depth of cut b = 0.325mm and feedrate s = 0.07 mm. Figure 8.23 shows the sound spectrums for three spindlespeeds. It should be noted that although the predicted chatter limit is higher at n = 6500rpm compared to n = 250 rpm, no chatter is observed at n = 250 rpm due to processdamping. Figure 8.24 shows the cutting forces for n = 6500 rpm and n = 250 rpm. As itcan also be seen from this figure, much lower peak to peak force amplitudes are obtainedwhen the resonances are avoided, i.e. harmonics of the tooth frequencies do not matchwith the modal frequencies of the plate, and the high process damping zone is utilized.Then, the peripheral milling of plates at slow spindle speeds with the full axial depth ofcut is more efficient than removing material layer by layer using high spindle speeds withvery small axial depths. In addition to the higher machining times and machining marksleft on the surface in layer cutting, the forced vibrations are excessive in plate millingwhen cutting speeds close to high stability pockets are selected.8.6 SummaryA comprehensive dynamic milling formulation and an analytical chatter stabilitymodel are developed. The cutter and workpiece are modeled as multi degree-of-freedomstructures. A general formula which is able to predict chatter free milling conditions,i.e. axial and radial depths of cut, and spindle speed, is derived. The formulation isapplied to several special cases including the milling of flexible workpieces. The modelpredictions are verified by experimental data and time domain chatter simulations. Also,dynamic milling tests show that the developed model can be used to determine chatterfree axial depth of cuts in plate milling.Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 2402.52.01.51.00.50.00.200.15— n=6500 rpmFrequency (Hz)Figure 8.23: Sound spectrums for different spindle speeds showing the effect of processdamping on chatter stability, a 44 mm302520151050n=250 rpm0.100.050.000 2000 4000 6000 8000 10000Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 2416004002000-200-400-600-40-60-80-100-120-140-160-180Tooth PeriodFigure 8.24: Effect of spindle speed and the process damping on the cutting forces.0 2 4 6 8 10 120 2 4 6 8 10 12a = 44 mm.Chapter 9ConclusionsThe peripheral milling of very flexible titanium alloy plates under static and dynamiccutting conditions has been analyzed and modeled in this thesis. The plate is modeledusing 8 node isoparametric finite elements, and the end mill is represented by an elasticbeam. The structural models of the plate and tool are integrated with the models ofperipheral milling mechanics and dynamics developed in the thesis. The models includeaccurate and unified formulation of milling mechanics, accurate chip thickness and forcecalculations by considering the partial disengagement of flexible cutter and workpiecestructures. The system developed is able to predict the irregular distribution of millingforces and the dimensional form errors left on the finished surface due to static deformations. A form error constraint algorithm, that schedules the feed along the plate whosethickness continuously varies due to metal removal, is developed. Another algorithmhas been developed to identify optimal radial depths of cut which allow the metal removal rate to be maximized but constrain the normal forces which cause dimensionalform errors. The methods developed are based on the models of peripheral milling andsurface generation proposed in the thesis. The dynamic interaction between the flexibleworkpiece and the flexible cutter is formulated by using multi-degree of freedom models. A stability model of self excited chatter vibrations in milling has been formulated,and solved, using a novel analytical approach which eliminates the use of time domainnumerical simulations of chatter in milling. All models developed and proposed in thisthesis have been verified experimentally.242Chapter 9. Conclusions 243The contributions of this thesis can be summarized in the following main categorieswith the related results and conclusions:• The peripheral milling of very flexible titanium plates is studied in depth for thefirst time. There is very little previous work in this area, and in the previousinvestigations the plates were either more rigid or as rigid as the cutter itself. Thehigh flexibility of the plates causes their separation from the cutter, requiring verysmall chip loads and the development of different structural and cutting modelsthan those used for moderately rigid structures.— A complete simulation system has been developed to predict the milling forcedistribution, the static deformation of the very flexible plate and the cutter structures as well as their interaction with the milling process. Forces,surface form errors and milling chatter stability limits are predicted and experimentally proven for the first time for very flexible multi-degree of freedommilling systems. The physical models of milling mechanics, chip thickness regeneration, surface form error, process-structure interaction, constraint basedoptimization of cutting conditions, and stability for chatter vibrations are developed as parts of the plate milling simulation study.— The milling force coefficients, which relate the milling forces to the chip thickness, are expressed using existing mechanistic curve fitting and calibrationtechniques. Alternatively, the same coefficients are obtained using an improved oblique cutting model with a more accurate chip flow estimation. Ithas been demonstrated that the milling force coefficients can be evaluated withsatisfactory accuracy by applying oblique cutting transformation on a set ofChapter 9. Conclusions 244standard two dimensional orthogonal cutting test parameters. This methodeliminates the use of calibration tests for each milling cutter geometry.— It is proven both analytically and by simulation that the effect of static regeneration of chip thickness on cutting forces diminishes in a few tooth periods,and the regeneration changes only the dynamic milling forces as it is the primary cause of chatter vibrations. The changes in the static cutting forcesare not due to chip thickness regeneration but rather due to changes in theimmersion boundaries between the flexible tool and workpiece.— An optimal value of radial depth of cut which minimizes the normal millingforces and dimensional surface errors is formulated for up milling operations.It is shown that much higher feedrate values can be used at predicted radialdepths of cut without increasing the forces and deflections significantly, thusimproving the productivity.— The plate is modeled by 8 node isoparametric finite elements, the cutter isrepresented by a continuous elastic beam and the stiffness of tool clamping tothe collet is approximated by a linear spring. The thickness of the elementsis updated as the material is removed in the feed direction. The integratedmodeling of the three interacting structures during peripheral milling is themost comprehensive found in literature.— The cutting forces and form errors in peripheral milling of the most flexibleparts ever considered were predicted most accurately. It is shown that in platemilling, in order to predict the cutting forces and form errors accurately, thestructure-milling process interactions have to be modeled.— An algorithm has been developed to constrain the maximum form errorscaused by the static deflections of very flexible plates, flexible end mills andChapter 9. Conclusions 245collets. The algorithm is integrated with the plate milling simulation system,and it automatically identifies the feeds along the tool path.• A complete analytical stability formula is derived for the first time for multi degree-of-freedom helical milling operations. Traditionally, milling chatter analyses havebeen performed using time domain numerical simulations to allow the consideration of periodic excitation and variation of directional coefficients in milling. Thefollowing formulations are introduced to the milling dynamics literature:— Directional dynamic milling coefficients are introduced in the chatter formulation. The coefficients are expressed as functions of cutting conditions (axialand radial depths of cut, cutting speed), milling force coefficients, and cuttergeometry (helix angle, diameter, number of flutes).— An analytical relationship between milling conditions, chatter frequency andspindle speed is derived for the first time. This is obtained by two differentapproaches. First, a novel stability analysis of milling dynamics which isbased on the interpretation of the physics of the process is developed. Second,the stability theory of periodic systems is applied to milling dynamics. Bothapproaches yield the same results; however, the former is more practical andgives more physical insight to the problem.— For the first time, radial depth of cut explicitly appears in an accurate millingstability formulation. Hence, both chatter free radial or axial depths of cutcan be directly obtained. It is shown that an optimal radial depth of cut whichmaximizes the chatter free material removal rate can be identified using theintroduced directional dynamic milling coefficients.Chapter 9. Conclusions 246— The stability method is applied to the most general case of milling flexible workpieces by flexible end mills which have multi degree-of-freedom andchanging dynamics along the cutter axis. The effects of variation of workpiecedynamics on the stability limit are illustrated.— The stability formula is applied to various practical cases and verified experimentally. 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Then, substituting equation (A.4) into (A.1), and (A.3) into(A.2):262Appendix A. Chip Flow Angle Formulation 263— tanqsn+tan/3co 17c— 1—tan8tan/3cos(A.5)— tani/’cosa— tan—sinatanbT COS 71c cos antan q8n = (A.6)cosij’— rcosilcslnanSolving equations (A.5) and (A.6) together:r cos 7)c cos acos— r cos 71c sin a(A.7)— tancosa—tan/3cosTi(tanTi—sinatan’)tan 7c — sin a tan b + tan /3 tan L cos a COS 7cThe above equation can be put into the following form:AsinTi — Bcosq — CsinTicosTi +Dcos2’q= F (A.8)whereA = rcosa+cosL’tan/9B = tan/3sinansin/’C = rsinatanL3 (A.9)D = rtan/3tanF sinbcosaEquation (A.8) is numerically solved for Newton-Raphson algorithm is used for thesolution. The convergence is quite fast for reasonable values of friction angle and thecutting ratio. The solution does not converge for very high values of the friction angle(/3> 600) which are not possible physically.

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