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Modeling of mechanics and dynamics of boring Atabey, Fuat 2001

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MODELING OF MECHANICS AND DYNAMICS OF BORING By Fuat Atabey B.A.Sc. (Mechanical Engineering) Istanbul Technical University A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February 2001 © Fuat Atabey, 2001 In p resen t i ng this thesis in partial fu l f i lment of the requ i r emen ts for an a d v a n c e d d e g r e e at the Univers i ty of Brit ish C o l u m b i a , I agree that t he Library shal l m a k e it f reely avai lable fo r re fe rence and s tudy. I fur ther agree that p e r m i s s i o n fo r ex tens ive c o p y i n g of this thesis fo r scho lar ly p u r p o s e s may b e g ran ted by the h e a d o f m y depa r tmen t o r by his o r her representa t ives . It is u n d e r s t o o d that c o p y i n g o r pub l i ca t i on of this thesis for f inancia l ga in shal l no t b e a l l o w e d w i t h o u t m y wr i t t en p e r m i s s i o n . D e p a r t m e n t of T h e Un ivers i ty of Bri t ish C o l u m b i a V a n c o u v e r , C a n a d a D E - 6 (2/88) Abstract This thesis investigates the mechanics and dynamics of boring operations. The mechanics of boring operations deal with the prediction of cutting forces as a function of tool geometry, work material properties, and cutting conditions such as feed rate, radial depth of cut, and cutting speed. The dynamics of the process involve the modeling of interactions between the structural dynam-ics of a long, slender boring bar, with boring process mechanics. Evaluation of forces allows the prediction of static deflection errors, torque and the power required from the machine tool. Evalu-ation of the dynamic stability of the process leads to the prediction of the chatter vibration free feed rate, spindle speed, radial depth of cut, and tool geometry. The thesis shows that boring forces are strongly dependent on the tool nose geometry, side cutting edge angle, radial depth of cut, feed rate and cutting speed. The chip thickness distribution along the curved edge of the tool is rather complex. The chip close to the nose is thin, and becomes thicker along the curved edge as the radial depth of cut increases. The chip thickness dis-tribution is also affected by the feedrate. It is proposed that cutting forces are modeled as a function of total chip area and cutting coef-ficients. The chip area is divided into several distinct geometric regions, and the center of each area is identified. Friction and tangential cutting forces are formed at each region. Cutting forces are modeled at each region, and summed up to find the resultant friction and tangential cutting forces. Using an equivalent friction or lead angle, the friction force is projected in the radial and feed directions. This model allows the prediction of cutting forces in all three Cartesian direc-tions. The influence of tool setting errors for boring heads having multiple inserts are also consid-ered in the general model. Several experimental results are compared with the predictions based ii on the proposed mathematical model. The predictions are shown to have errors varying between 2% and 15%. The proposed model contributes to the improved prediction of boring mechanics. The fundamental mechanism behind chatter vibrations in boring process is also investigated. It is shown that the cutting coefficients, i.e. process gain, and directional factors, are dependent on the feed rate, radial depth of cut, tool geometry, and cutting speed. While the tool geometry and speed may be kept constant, vibrations modulate radial depth of cut, and leads it to be a time-varying process input parameter. This is the fundamental non-linearity in the process, which dif-fers from milling operations. The dynamic process is modeled in both frequency and time domains. However, the process non-linearity varies significantly during the process, preventing the application of classical linear chatter stability laws to the boring process. It is shown that the time domain modeling also suffers, mainly due to the digital integration of a significant number of tool deflection waves left on the boring surface. iii Table of Contents Abstract ii Table of Contents iV List of Tables vii List of Figures viij Acknowledgment xiii Nomenclature xiv 1. Introduction 1 2. Literature Review 5 2.1.Overview 5 2.2. Boring Force Models 5 2.3. Chatter Stability Analysis in Boring 9 2.4.Summary 19 3>. Force Prediction in Boring 3.1. Introduction 20 3.2. Mechanics of Boring 22 3.2.1.Boring Bar with one Insert 22 3.3. Force Prediction in Boring 26 3.3.1.Mechanistic Model 26 iv 3.3.1.1. Uncut Chip Area and Cutting Edge Contact Length Calculation 26 3.3.1.2. Experimental setup 31 3.3.1.3. Cutting Coefficient Identification 37 3.3.1.4. Experimental Verification of the Mechanistic Model 55 3.3.2. Orthogonal to Oblique Transformation Method 63 3.3.2.1. Orthogonal Cutting Test and Identification of Oblique Cutting Parameters 63 3.3.2.2. Prediction of the Oblique Cutting Forces 65 3.3.2.3. Experimental Verification of the Method 66 3.3.2.4. Experimental Verification of Orthogonal to Oblique Transformation Method 75 3.4.Summary 76 4. Process Faults in Boring 4.1 .Introduction 77 4.2.Mechanics of Multiple Inserted Boring Bar 79 4.3.Insert Runout in Radial and Feed (Axial) Directions 81 4.4. Deviation of the Boring Head from the Hole Center 86 4.5. Experimental Setup 89 4.6. Mechanistic Model Verification 91 4.6.Experimental Verification of the Mechanistic Model for Process Faults 92 4.8.Summary 99 5". Dynamic Modeling of Boring and Chatter Stability 5.1.Introduction 100 5.2.Dynamic Characteristics of the Boring Process 105 5.2.1. Regenerative Effect in Boring 105 5.2.2. Dynamic Cutting Force Prediction 114 v 5.2.3. Wave Generation on the Surface 117 5.3.Chatter Stability 126 5.3.1. Analytical Approach for Stability Solution 126 5.3.2. Chatter Stability Prediction in Time Domain 140 5.3.2.1. Tool Dynamics Model 145 5.3.3.Simulation and Experimental Results 147 5.3.3.1 .Experimental Results: 152 5.3.3.2. Time Domain Simulation Results: 160 5.4.Summary 162 6. Conclusions 163 Appendix-A 166 Appendix-B 177 Appendix-C 178 Bibliography 181 vi List of Tables 3.1 :Edge cutting force coefficients for the Valenite CCGT432-FH insert 41 4.1 :Experiments with runout in feed and radial directions; and are the radial depth of cuts of insert 1 and insert 2. 93 5.1 :Simulation parameters selected from the conducted chatter tests 105 5.2 :Prediction of the radial force considering the specified vibration history 134 - Modal Parameters of the Boring Bar Structure for the First Set of Experiments 5.3 :Modal parameters in radial direction 152 5.4 :Modal parameters in tangential direction 152 5.5 :Modal parameters in feed direction 152 5.6 :Experimental results of the first set 153 - Modal Parameters of the Boring Bar Structure for the Second Set of Experiments 5.7 :Modal parameters in radial direction 155 5.8 :Modal parameters in tangential direction 155 5.9 :Modal parameters in feed direction 155 5.10 :The results of the Second set of experiments 156 vii List of Figures 1.1: Schematic illustration of boring process 1 2.1: Geometries of orthogonal and oblique cutting 8 2.2 :Chatter stability lobes 13 2.3 Schematic illustration of the boring process 14 2.4 :Boring bar with flat surface 15 2.5 Regeneration of waves with different phase angles 16 2.6 Relationship between the process damping and relief angle 17 2.7 :Model of dynamic boring process presented in Zhang's thesis 18 3.1 :Single point cutting tool with corner radius and chip breaking groove 21 3.2 Schematic illustration of force directions in boring process 23 3.3 definition of the forces, cutting and geometrical parameters in boring process; friction force distribution along the cutting edge contact length 24 3.4 :Spiral path of boring tool, 25 3.5 :Four different uncut chip area configurations defined with depth of cut a, feed rate, c and corner radius of the tool R 27 3.6 :Uncut chip area calculation for the 1st configuration and the definition of the regions 29 3.7 :Other uncut chip area configurations considered in the area calculation model 30 3.8 :Workpiece-Al 6061-T6 used in the experiments 32 3.9 Schematic illustration of the experimental setup for force calibration 33 3.10 determination of the average tangential force value based on the collected data, V=75[m/ min], c=0.155[mm/rev], a=0.25[mm] 34 3.11 :Kennametal CPMT-32.52 K720 coated insert and A12-SCFPR3 steel shank boring bar 35 viii 3.12 :ValeniteCCGT432-FH Carbide PVD coated diamond insert with A-SCLPR/L boring bar 36 3.13 :Chip thickness variation along the corner radius of the tool 38 3.14 :Friction force distribution along the cutting edge 39 3.15 :Tangential, radial and feed force vs chip contact length 40 3.16 investigation of the dependency of the edge cutting forces on the cutting speed ; Material: Aliminum 6061-T6, Tool: Kennametal CPMT-32.52 K720 coated insert 42 3.17 determination of the centroid of region 1 46 3.18 :The Effective lead angle prediction 47 3.19 :Investigation of the variation of the effective lead angle modification factor 48 3.20 investigation of the Effective lead angle variation 49 3.21 :Graphical representation of the predicted-modified effective lead angle 50 3.22 -.Deviation of the effective lead angle along the cutting edge contact length 52 3.23 :Variation of the modification factor with the cutting edge contact length for a < R and a>R 53 3.24 :Friction force verification for a < R 56 3.25 :Friction force verification for a > R 57 3.26 :Tangential force verification for a<R and a > R 58 3.27 :Effective lead angle verification for a<R and a > R 59 3.28 :Radial force verification for a<R and a > R 60 3.29 :Feed force verification for a<R 61 3.30 :Feed force verification for a > R 62 3.31 Evaluation of the oblique cutting parameters for three regions of the uncut chip area 70 3.32 :Geometry of boring tool 71 3.33 :Valenite CTPGPL-16-3C tool holder and TPC-322J-VC2 insert 72 3.34 Experimental setup for the verification of the Orthogonal to Oblique Transformation Method 73 3.35 :Oblique tangential F r radial Fr and feed force Fy directions in each region and, dyna-mometer axes directions 74 3.36 :Comparison between the measured and predicted tangential, radial and feed forces using the orthogonal to oblique transformation method 75 4.1 :Radial and axial (feed) runouts on a two-insert Valenite boring head 78 4.2 :Force diagram of a boring bar with four inserts 80 4.3 :a-) Configuration 1: The amount of material removed from the workpiece when the radial runout of insert 1 is greater than 0 (er > 0) and feed runout of insert 2 is greater than feed rate (e^  > c), b-) Uniform uncut chip area for the condition without any insert runouts. 83 4.4 :Configuration 2: The amount of material removed from the workpiece when the radial runoutofinsertlisgreaterthanO(er > 0 )andfeedrunoutofinsert2islessthanfeedrate( 8 y < c ) 84 4.5 :Valenite boring head with twin cutter; Runout in radial and feed (axial) directions; The amount of material removed by each insert is shown by the shaded area in the bottom right part of the figures 85 4.6 :Schematic illustration of the deviations , of the boring head from the hole center 87 4.7 :The uncut chip area variation caused by deviations Ax, Ay 88 4.8 :Fadal VMC-2216 Machining Center 90 4.9 Resulting force in X direction and Feed force prediction for the condition of ax = 1.485[mm], a2 = 1.285mm] , er = 0.20[mm], Ey = 0.09[mm] , c = 0.06[mm], V = 150[m/mm] 94 4.10 Resulting force in X direction and Feed force prediction for the condition of <2j = 1.830[mm], a2 = 1.730mm] , er = 0.10[mm], = 0.12[mm] , c = 0.07[mm], V = I00[m/min] 95 4.11 Resulting force in X direction and Feed force prediction for the condition of x ax = 1.1 [mm] , a2 = 0.92[mm] , Er = 0.18[mm], £y = 0.14[mm] , c = 0.055[mm], V = 175[m/mm] 96 4.12 :Resulting force in X direction and Feed force prediction for the condition of al = 0.870[mm], a2 = 1.12mm] , Er = 0.25[mm], ef = 0.055[mm], c = 0.09[mm], V = 225 [m/min] 97 4.13 :The variation of the total force in X direction when considered the process has insert runouts in radial and axial directions, and deviation in both X and Y directions. 98 5.1 :Orthogonal plunge turning with regenerative chatter vibrations 102 5.2 :J31ock diagram of the regenerative chatter vibrations in the orthogonal cutting 104 5.3 : Algorithm for the determination of the phase angle for each revolution 107 5.4 :Transfer function of the boring bar in tangential, radial and feed direction 109 5.5 : Boring bar structure with two spring-mass and damping models of a single degree of free-dom system 110 5.6 :Regeneration of the waviness in boring process 111 5.7 :Surface rougness measurement of the workpiece,, a =0.75 [mm], c =0.12[mm/rev], y=184[m/min], n=1650[rpm], u)c = 849.7 [Hz], e = 324 [Deg 112 5.8 :The effect of the tangential vibrations in the regeneration of the waviness 113 5.9 :The uncut chip area variation for unstable cutting condition; s and d imply the static and dynamic cutting condition. 116 5.10 :Definition of the depth positions with 119 5.11 :Wave generation, phase angle e = 275°, chatter frequency u)c = 770 [Hz], black and white dots show the depths the tool make in the first 6 revolutions 120 5.12 identification of the surface finish geometrical parameters. Black dots show the depths cre-ated) yJitfoolritsinusoidaVibrationjtt = 1650[rpm] P = 31.5[mm]£ = 275° (0 C = 770[#z] 121 5.13 :Depth position determination algorithm in each wave period 122 5.14 :Flow chart to group the deeps for the identification of the inclination angles ax and oc2. 123 xi 5.15 :Simulated wave generation on the surface finish under the contidion of phase angle e = 170°, chatter frequency coc = 770 [Hz]; Black dots show the depths. 124 5.16 Simulated wave generation on the surface finish under the contidion of phase angle e = 90° , chatter frequency coc = 770 [Hz] 125 5.17 illustration of the uncut chip areas in Eq. (5.18) 128 5.18 :Block diagram representation of the boring process 129 5.19 :Transfer function measurement using the impact hammer test 131 5.20 :Block diagram representation of Equation (4.23) 132 5.21 :Dynamic radial force simulation, a=0.7 [mm], amplitude of the vibration=0.07[mm] 135 5.22 :The uncut chip area variations at the positions 1, 2, 3 and 4 136 5.23 :The uncut chip area variations at the positions 5, 6, 7 and 8 137 5.24 :The Variation of the radial cutting force coefficient and uncut chip area for the specified vibration history (Table 5.2) 138 5.25 :The variation of the radial cutting and edge cutting coefficients for the given vibration his-tory in Table (5.2) 139 5.26 :The Variation of the total radial force for the vibration history in (Table 5.2) 140 5.27 :Block diagram of time domain simulation model 142 5.28 :Algorithm of the time domain solution model 143 5.29 illustration of the tool position for an instant of time in time domain simulation 144 5.30 :The relation between the length of generated waves and process damping 150 5.31 Experimental setup-Chatter tests 151 5.32 :The results of the first set of experiments 154 5.33 Results of the second set of experiments 159 5.34 :Time domain simulation result, Set-1, Test#17, a=0.6[mm], V=250[m/min], c=0.1[mm/ rev], n=2476[rpm] 160 5.35 :Time domain simulation result, Set-1, Test#l, a=0.75[mm], V=75[m/min], c=0.1 [mm/rev], n=743 [rpm] 161 xii Acknowledgment I would like to express my sincere appreciation to my research supervisor Dr. Yusuf Altintas for his guidance, support, and encouragement throughout my research at the University of British Columbia. I am also indebted to Dr. Ismail Lazoglu, who supervised me while Dr. Altintas was away. My thanks are also due to my friends and colleagues in the Manufacturing Automation Laboratory at UBC, for their assistance, patience and friendship. I am deeply grateful to my parents for their constant support, patience, and encouragement during my graduate study. I dedicate this work to them. xiii N o m e n c l a t u r e depth of cut dynamic depth of cut total uncut chip area uncut chip area of Region 1 uncut chip area of Region 2 uncut chip area of Region 3 width of cut feed rate diameter of the hole vectorial friction force component for Region 1 vectorial friction force component for Region 2 predicted friction force for stable cutting conditions predicted friction force when the system has chatter vibrations regional force in x direction regional force in y direction regional force in z direction total force in x direction total force in Y direction total force in Z direction the chip thickness intended chip thickness the cut chip thickness oblique angle tangential cutting force coefficient xiv Kjrc friction cutting force coefficient KjrCi friction cutting force coefficient for Region 1 Kfrc^ friction cutting force coefficient for Region 2 Km effective lead angle modification factor Km modification factor for a < R Km modification factor for a>R Krc radial cutting force coefficient Kjc feed cutting force coefficient Kte tangential edge cutting force coefficient Kjre friction edge cutting force coefficient Kre radial edge cutting force coefficient Kje feed edge cutting force coefficient Lc total cutting edge contact length L C [ cutting edge contact length of Region 1 L C 2 cutting edge contact length of Region 2 Lp Distance between two depths in rotational direction when the system has chatter LT length of a wave in a period N the number of waves counted on the surface finish n spindle speed rc chip ratio r, r* residues in transfer function of the structure R corner radius of the tool T period of the spindle speed or period of one chatter wave V cutting speed u)c chatter frequency x:v y radial vibration a{, a2 angles used to describe the route of the tool during the chatter vibrations an normal rake angle cc0 orhogonal angle ay side rake angle ap back rake angle ar rake angle P a friction angle P n normal friction angle 8 phase shift between two successive revolutions <|)M normal shear angle (j)c shear angle <\>LBs predicted effective lead angle considering the system is stable for given cut conditions D d predicted effective lead angle when the system has chatter vibrations § L effective lead angle § L * predicted effective lead angle without modification \|/ approach angle of the tool Y|/ r side relief angle yL side cutting edge angle yc end cutting edge angle r\ chip flow angle Ty shear stress 0 • angular increment in the uncut chip area calculation 0 G angle of gravity center with respect to the center of the corner radius Aat the portion of the tangential displacement in the depth of cut variation xvi displacement of the tool in tangential direction displacement of the tool in radial direction xvii Chapter 1 Introduction Boring is a machining operation used to enlarge internal bore diameters of holes. Typical examples can be listed as engine cylinders, bearing mounting locations, inner surfaces of bearing rings, and gears. The holes are first opened either by drilling, or during the fabrication of blanks, using forging or casting technology. Depending on the size of the workpiece and hole diameter, either turning machines or large boring centers are used to carry out boring operations. Small parts, such as bearing rings and gears, can be mounted on the spindles of regular CNC lathes. The boring operation can be carried out with a single point tool mounted on a slender boring bar. The boring bar is attached to the tool carriage or turret, and linearly fed towards the hole of the rotating part mounted on the spindle chuck (Figure 1.1). Large workpieces, such as engine blocks, are mounted on a table. The boring bar is attached to the non-rotating spindle, and the circular motion is either provided by the contouring actions of the spindle carriage or table drives. Some machines, such as vertical lathes, have rotating tables. Figure 1.1: Schematic illustration of boring process 1 Chapter 1. Introduction 2 The boring bars are usually very flexible, due to large overhang length (L) to diameter ratio (D). The boring bar can be considered as a large cantilevered beam with cutting forces applied at the free end. The cutting force magnitude depends on the work material's hardness and the area of the metal chip cut instantaneously. The direction of the cutting force depends on the tool geome-try, and feed rate and radial depth of cut in boring operations. The slender boring bar elastically deflects under the excitation of the cutting force, which leads to changes in the chip area; hence the magnitude and direction of the cutting forces. In summary, the process has a closed loop dynamic system which may be stable or unstable depending on the process parameters, such as depth of cut and feed, and structural dynamics of the boring bar at its free end. When the process is stable, the system does not experience any vibrations, and this remains a desired operation. However, the cutting forces cause static deflection of the boring bar which may be larger than the tolerance of the hole surface. If the process is modeled mathematically, it may be possible to select a suitable tool geometry, boring bar cross section, and radial depth of cut and feed-rates, which do not violate the tolerance of the part due to static deflections. When the process is unstable, the structural modes of the boring bar are excited leading to self-excited chatter vibrations. The magnitude of the vibrations grows exponentially until the tool jumps out of the cut or breaks. Boring operations fail when chatter occurs, since this leads to poor surface finish and damage to the cutting tool. Chatter stability depends on the structural dynamics of the boring bar, the direction of dynamic cutting forces which are in turn dependent on the tool geometry, the work material hardness, radial depth of cut, feed rate and surface speed of the work-piece. If the process is modeled mathematically, it may be possible to avoid those cutting condi-tions that lead to unacceptable chatter vibrations before the part is machined. Although a significant amount of research has been conducted on general cutting mechanics and dynamics, boring has been studied less than conventional operations such as orthogonal cut-ting and milling. Unlike the dynamics in orthogonal cutting and milling operations, the dynamics Chapter 1. Introduction 3 of the boring process are quite non-linear. The process gain and directions of excitations depend on the process input parameters, such as feed when the system vibrates. This thesis presents time domain mathematical modeling of the boring process, with or with-out the presence of chatter vibrations. The thesis is organized as follows: Previous research is reviewed in Chapter 2. The modeling of cutting forces with tools having a nose radius and inclination angle is surveyed. The time and frequency domain modeling of gen-eral cutting operations, as well as boring operations that were limited, is surveyed. The funda-mental difficulties in modeling the boring operations are highlighted in Chapter 2. The chip geometry, the identification of force magnitude and directions, and process mechan-ics are modeled in Chapter 3. The chip area is evaluated by dividing the chip into several geomet-ric regions. It is assumed that the force is acting at the centroid of the chip area. The friction force is identified as a function of tool geometry and chip area, and it is resolved in the feed and radial directions. The tangential force is modeled as a function of chip area. Methods for both mechanistic and oblique cutting mechanics are presented in modeling the process mechanics. When the part is large, holes may be opened directly with a rotating boring head plunging into the hole. The boring heads usually have an even number of multiple inserts which are distrib-uted symmetrically. The symmetrical distribution cancels the radial forces, which minimize the radial deflections. However, it is not possible to place inserts accurately on the boring head. Inserts may have radial and axial deviations, which lead to uneven chip loads for each insert. As a result, the cutting forces are not uniform and the radial forces are not canceled completely. These are called process faults, and they are modeled in Chapter 4. The dynamics of the boring process are presented in Chapter 5. The closed loop dynamics of the boring process and the source of its fundamental non-linearity are discussed. The difficulties of solving boring chatter stability are highlighted. A time domain solution for boring chatter is presented briefly, and the difficulties involved in modeling the process physics are discussed. 4 The thesis is concluded with a brief summary of contributions, difficulties in modeling the boring process, and recommended future research. Chapter 2 Literature Review 2.1. Overview In this chapter, a literature review of the boring process is presented. Existing force prediction models are briefly discussed, followed by the recent developments of the chatter stability theory in metal cutting. 2.2. Boring Force Models Generally, the mechanics of cutting processes are geometrically evaluated with two basic cut-ting process models, namely, orthogonal and oblique cutting (Figure 2.1). The difference between the two processes can be described with the orientation of the tool cutting edge with respect to the velocity vector of the process. In orthogonal cutting, the velocity vector is perpendicular to the cutting edge of the tool. This makes the cutting geometry simple due to its two-dimensional geo-metrical structure. Merchant [3] presented the basics of general 2-D orthogonal cutting mechan-ics. On the other hand, in oblique cutting, the cutting edge of the tool has an inclination angle i with the velocity vector (Figure 2.1). Oblique cutting has a three dimensional nature, thus, the relations between the cutting geometry and forces are more complicated. The most common geo-metrically complex cutting operations are usually defined with the aid of oblique cutting geome-try. In the past, extensive research has been devoted to the prediction of the cutting forces in machining, showing that the cutting forces could be defined as proportional to the uncut chip area A and width of cut b. The conventional formulation can be expressed as, 5 Chapter 2. Literature Review 6 F, = K + F,e = KtA + Kteb Fr = Frc + Fre = KrcA + Kreb (2-D Ff=Ffc + Ffe = KfcA + Kfeb where Ftc, Frc and F^c are cutting force components associated with shearing during the machining process. On the other hand, F(e, Fre and Fye are the edge cutting force components caused by the rubbing on the cutting edge and do not have any contribution to the shear deforma-tion in the cutting process. The orthogonal to oblique transformation method proposed by Armarego [22-23] is one of the methods used to predict the cutting forces in the boring process. In this method, the oblique cutting forces are predicted based on an existing orthogonal cutting database. If the data base has been previously developed, there is no need to perform any further calibration test. In this method, first the orthogonal cutting tests are performed in order to determine the orthogonal cutting parameters for a specific tool and workpiece pair. These parameters are then transformed to the oblique cutting geometry under specific rules [2]. The details of this method are explained in Chapter 3. This method is practical, and saves time by not requiring the performance of tests if the database has already been developed for the workpiece-tool pair intended to be used in the boring operation. However, it requires that the cutting mechanics of the tool be exactly defined with the oblique cutting geometry along the cutting edge. In other words, the tool should have a sharp cut-ting edge and a flat rake face. At present, most boring tools are manufactured with a nose radius and special chip breaking grooves along the cutting edge. Sometimes, the cutting geometry of these tools cannot be mod-eled with the existing oblique cutting models. For the force prediction of these tools, the mecha-nistic modeling approach is employed. Cutting force coefficients are empirically estimated for a Chapter 2. Literature Review 1 specific cutter geometry and workpiece material, relating force to the cutting parameters (i.e., depth of cut, feed rate and cutting speed) and other geometrical properties of the tool. One of the first cutting mechanics models is the one proposed by Kronenberg [5]. In this model, the tangential force Ft is proportional to the uncut chip area A. The radial and feed forces (Fr, Ff) are proportional to the tangential force. where Kx, K2 and K3 are the cutting coefficients that are functions of the tool geometry, tool and workpiece material, and cutting parameters. In this model, the corner radius of the tool is not considered. Hallam and Allsopp [6] proposed a method in which the uncut chip area is calculated by an integration method, which considers the depth of cut, feed rate, and corner radius of the tool as inputs. In this model, the cutting coefficients are assumed to be constant. The dependency of the cutting forces on the depth of cut showed good agreement with the experimental data, however , the model did not accurately predict the cutting forces as functions of feed rate. Sabberwal [9] later proposed that tangential and friction forces are proportional to the uncut chip area and the cutting coefficients are not constants but a function of the chip thickness. where b0, bx, dQ and d{ are empirical contants. Based on this method, a force prediction model for boring process has been developed by Subramani et. al.[16]. Sutherland et al. [17] mod-ified the model presented by Subramani, including the cutting speed and the effect of tool geome-try in the prediction of the cutting coefficients (Eq. 2.4). K.A (2.2) Chapter 2. Literature Review 8 Kfr = d0ady2hd3 (2.4) where, an and V are the normal rake angle of the tool and cutting speed, respectively. Orthogonal cutting geometry Chip Workp Oblique cutting geometry Chip-flow angle Workpiece Tool Flank face ing edge ination angle Figure 2.1 : Geometries of orthogonal and oblique cutting Chapter 2. Literature Review 9 2.3. Chatter Stability Analysis in Boring This section examines the background related to chatter stability of the boring process. In boring operations, the length to diameter ratio of the boring (L/D) bar (Figure 2.3) is usu-ally large and any dynamic force variation can easily excite the structure due to its low dynamic stiffness. If the cutting force is in resonance with one of the natural frequencies of the boring structure, vibrations become significant in determining surface finish quality. Any change in length to diameter ratio (L/D) has a substantial effect on the dynamic stiffness and the system stability. A steel shank plain boring bar can usually be used up to the value of 4.5-5 length to diameter ratio (L/D) for chatter free machining. However, for the large (L/D), cutting conditions are limited for a stable cutting process. Significant research efforts have been made in finding a method to increase the stability of the boring process. One of the approaches which researchers have been interested in is to increase the dynamic stability of the structure using passive vibration control techniques [36, 37]. In this technique, the vibration of the boring bar structure is absorbed using springs and dashpots. The forces generated by the passive vibration components tend to decrease the magnitude of vibra-tion. By means of this method it is possible to increase the stable operation range up to length to diameter ratios (L/D) of 5.5-6.0. The method of using passive vibration absorbers is commonly used by boring bar manufacturers. Passive vibration, on the other hand, has limitations in terms of system stability in large length to diameter ratios (L/D). Using an active dynamic observer makes it possible to perform a stable boring operation up to the length to diameter ratios (L/D) of 9 by supressing bar vibrations [24, 25, 26]. In these applications, a piezoelectric actuator is used as an active dynamic absorber and optimal control is applied to the system for the control of boring bar motion. These techniques help to increase the stability of the structure for a specific application, how-ever, due to some configuration problems and their high cost it cannot be applied to all boring operations. Chapter 2. Literature Review 10 This study focuses on the investigation of chatter stability due to self-excited vibrations. The main cause of self-excited chatter is the regenerative effect introduced by Tobias [29], [30], Tlusty [4] and Men-it [7]. If there is a relative vibration between the cutting tool and the workpiece, the tool leaves a wavy surface behind. In the next revolution, the tool encounters this wavy surface and removes material with a time-varying uncut chip area [29]. Periodic variation of the uncut chip area causes a variation in cutting forces. Thus, the structure is excited and chatter vibrations take place. Regenerative effect is caused by the phase shift between the waves generated on the cut sur-face (Figure 2.5). This phase shift can be defined as a function of spindle period T, and chatter frequency coc. e = mc-2nk (2.5) where k is the integer number of the waves on the cut surface in one full revolution. Figure 2.5 shows the variation of the chip thickness depending on the phase angle 8 between the succes-sive undulations. Zero phase angle produces constant chip thickness, hence, there is no regenera-tive effect, even though the system still has vibrations. When the phase angle e becomes n [rad], an extreme case of wave regeneration occurs. The oscillation of the chip thickness causes the forces to vary with its period, leading to unstable cutting conditions. The nonlinearity of the regeneration can be recognized when the tool jumps out of the workpiece [31]. There are some factors that increase the system stability. Among these factors, process damp-ing which is caused by the time-varying relief angle due to vibrations is important in low cutting speeds [33, 34, 35]. Figure 2.6 shows a tool moving to the right while it is oscillating. It should be noted that for low cutting speeds the lengths of the generated waves become short, causing the flank face of the tool to touch the cut surface. This creates a positive damping effect in the process due to the rubbing occuring on the flank face of the tool. In contrast, when the length of the wave is longer (as occurs at high cutting speeds) the relief angle of the tool becomes larger. In such a Chapter 2. Literature Review 11 case, the flank face does not come into contact with the surface and, hence, does not contribute to the damping of the system. As the modeling of process damping is rather difficult, its effect is not included in the analytical and time domain stability solution in this study. System stability is investigated and the force variation of a machining system, torque, bend-ing moment and surface finish can be obtained in time domain solution whose pioneering work was introduced by Tlusty [11, 32]. The other advantage of the time domain simulation of the pro-cess is that it is possible to investigate the nonlinearities of the process, such as the jumping of the tool from the workpiece and process faults. Chatter stability has been commonly expressed with stability lobe diagrams (Figure 2.2), which show the boundary between the stable and unstable cutting conditions in the form of axial depth of cut limit versus spindle speed (Figure 2.2). Tobias [30], Tlusty and Polacek [4] predicted this borderline considering the regenerative effect in a multi-degree of freedom system. Research dedicated to the stability analysis of the boring process is rarely found in the litera-ture. This may be because the dynamics of the boring process are rather complex compared to other machining processes. One of the first attempts to solve the stability problem in boring is the one by Zhang [18, 19]. His Ph.D. thesis analyzed the stability for two conditions; 1- The cutting condition with no over-lapping, 2- The cutting condition with overlapping. In both cases, the critical stiffness of the bor-ing bar is investigated under spiral cutting conditions, which are not representative of the boring process. In his model, for the purpose of facilitating the solution of stability analysis, equivalent width of cut and chip thickness are considered as system parameters instead of the direct use of depth of cut and feed rate (Figure 2.7). Dynamic chip load is calculated as, A = b(h0-h(t) + nh(t-T)) (2.6) where b, h0, h(t), h(t-T) and |J, are the width of cut, intended chip thickness, current chip thickness, the chip thickness at the previous tool position and overlapping factor, respectively. Chapter 2. Literature Review 12 Kuster et. al. [12] proposed a time domain solution model for the boring process. In the model they predicted the forces in three directions (i.e. tangential, radial and feed directions). They did not take into account the nonlinearity of the interactions between the current and previ-ous tool positions and all possible uncut chip area configurations depending on the previous tool positions. They investigated the stability limit by considering the vibrations in three directions of the boring bar; However, the presented experimental results to support the prediction of the stabil-ity limit were poor. S. Jayaram et al. suggested an analytical solution to the boring process [27]. In their model, the nose radius of the tool is not considered, and vibrations in the radial direction are neglected, even though they have the most significant effect in regenerative chatter in the boring process. In this model the boring bar is assumed to vibrate in the feed direction. However, it has been per-ceived from the transfer function measurements that the boring bar is relatively stiffer in the feed direction. In the model the boring process is analyzed in the same way as in the turning process. Good accuracy in the prediction of stability is presented; however, this validation is unrealistic since the dynamics of turning and boring processes are different. E.W. Parker investigated boring stability with a boring bar having rectangular cross-section [28]. The tool is attached to the tip of the boring bar with a ring and arranged at different angular position on the bar (Figure 2.4). Stability of the structure for different angular tool positions was investigated considering the model as a two-degree of freedom mass, spring damper system. The stability of the boring bar system shows dependence on the angular location of the tool on the ring of the bar and the optimum angular position of the tool was found. Chapter 2. Literature Review 2000 4000 6000 8000 Spindle speed [rpm] Figure 2.2 : Chatter stability lobes er 2. Literature Review Chapter 2. Literature Review Chapter 2. Literature Review Figure 2.5 : Regeneration of waves with different phase angles Chapter 2. Literature Review 17 Figure 2.6 : Relationship between the process damping and relief angle Chapter 2. Literature Review 18 Figure 2.7 : Model of dynamic boring process presented in Zhang's thesis [18] Chapter 2. Literature Review 19 2.4. Summary There has been very little research presented in the literature, which provided successful pre-diction of chatter stability in boring. Neither time domain nor frequency domain models were suc-cessful in even modest prediction of chatter stability in boring operations. Chapter 3 Force Prediction in Boring 3.1. Introduction Most boring tools used in industry are specially designed and manufactured with a nose radius, chip breaking grooves, a side cutting edge angle, side and back rake angles (Figure 3.1). The aims of these are to increase dimensional accuracy, to improve surface finish quality, and to extend the life of the tool in order to prevent failure. Prediction of the cutting forces and chatter vibrations enables the engineer to set the design parameters of the machine tool, cutting parame-ters, and fixture in an optimal fashion so that productivity can be increased by minimizing the machining cost per piece. Selecting inappropriate cutting parameters may cause damages on machine tool components, early tool wear and tool breakage, chipping, and poor surface finish quality, all of which are undesirable in manufacturing. One such method is to predict the cutting forces from an orthogonal cutting database using Orthogonal to Oblique Transformation Method [1]. Although this method is practical, it is not applicable for tools that have a chip breaking groove on the cutting edge. This is because the transformation can only be implemented for tools with a sharp cutting edge and flat rake face. Once an orthogonal cutting data base is developed for a tool-workpiece pair, orthogonal cutting parameters can be transformed to other complex tools (whose cutting geometry can be character-ized as oblique cutting). Thus, the need for calibration of each tool geometry is eliminated, and cutting forces can be predicted without performing any new experiments. This transformation is performed in a special manner that requires certain assumptions, which will be explained in detail in the following sections. 20 Chapter 3. Force Prediction in Boring 21 The second method for force prediction is the Mechanistic Identification Method, which has a simple formulation and results in accurate prediction for tools having complex cutting edge geometry. One of the drawbacks of the mechanistic approach is that more experimental data is needed to take the effect of all cutting and geometrical parameters on the cutting forces into account. Correlations between the parameters and cutting force coefficients are identified for each tool geometry. In the first part of this chapter, a mechanistic modeling approach for the prediction of the cut-ting forces in the boring process is described. The majority of this study lies in the prediction of the cutting force coefficients and effective lead angle for both stable and unstable cutting condi-tions. The methodology behind this approach is discussed. In the model, the expressions of the cutting force coefficients are estimated based on experimental data. Then, the model is experi-mentally validated. Chip br Figure 3.1 : Single point cutting tool with corner radius and chip breaking groove The second part of this chapter deals with the force prediction by utilizing the Orthogonal to Oblique Transformation Method. The identification of the cutting force coefficients based on the Chapter 3. Force Prediction in Boring 22 orthogonal cutting database and its application to tools with a nose radius are described in detail. Accuracy in the prediction of the cutting forces is experimentally investigated. 3.2. Mechanics of Boring The boring operation is performed in two different ways, depending on the machine type: 1-the boring bar rotates and is linearly fed into the workpiece with feed rate c, while the workpiece remains stationary; 2- The workpiece rotates as the boring bar makes a linear move into the work-piece with the feed rate c. The versatility of the operation can be increased with different types of boring bar and machines. In this chapter, a boring bar with a single insert under the second cutting condition, (described above), is considered. 3.2.1. Boring Bar with one Insert Figure 3.2 illustrates a schematic of the boring operation with a single insert. In this process, forces can be resolved into two components (Figure 3.3), namely the tangential force Ft, which acts perpendicular to the uncut chip area, and the friction force Fj-r, which is the sum of the forces acting perpendicular to the cutting edge. The direction of the friction force for each angular differ-ential element varies along the cutting edge contact length due to the nose radius, and is defined with effective lead angle tyL, which is the angle between the directions of the friction and the feed forces. Radial and feed forces (F r and Fy) are obtained by projecting the total friction force Fjr into the radial and feed directions. The X, Y and Z directions are referred to as the tangential, radial and feed directions, respectively (Figure 3.2). Fx = Ft Fy = Fr = FfrS™<bL FZ = F f = F f r C 0 S $ L (3.1) Chapter 3. Force Prediction in Boring 23 In the process, the combined linear motion of the boring bar with the feed rate per revolution and rotation of the workpiece causes the insert to travel along a spiral path. It is reasonable to assume that the pitch of this spiral is equal to the feed rate c. If the workpiece hole is rolled out onto a plane, the path followed by the insert can be shown in Figure 3.4. At any point on the path traveled, the distance between the current and preceding positions of the insert is equal to the feed rate c. L is equal to the circumference of the workpiece hole. n *- Tangent ia l force fr »~ Rad ia l force R »- F e e d force Figure 3.2 : Schematic illustration of force directions in boring process Chapter 3. Force Prediction in Boring Figure 3.3 : Definition of the forces, cutting and geometrical parameters in boring process; friction force distribution along the cutting edge contact length Lc Chapter 3. Force Prediction in Boring 25 3 Figure 3.4 : Spiral path of boring tool, L = nD Chapter 3. Force Prediction in Boring 26 3.3. Force Prediction in Boring 3.3.1. Mechanistic Model The mechanistic model approach relates the cutting and geometrical parameters (i.e. depth of cut a, feed rate c, cutting speed V, corner radius of the tool R, side cutting edge angle yL, end cut-ting edge angle y c) to the cutting forces. The basic force equation can be expressed as, F = KA (3.2) 2 where K and A are the cutting coefficient ([N/mm ]), which is also known as the propor-tionality constant in the literature, and the uncut chip area ([mm ]), respectively. In this expres-sion, K is not considered as a constant, but rather, as a function of cutting and geometrical parameters [9, 13, 14, 15, 16 and 17]. For the prediction of the cutting forces, the uncut chip area and the cutting coefficient need to be determined. In this study, two different inserts, with speci-fied geometry, are employed, and cutting force coefficients have been investigated within the desired range of cutting parameters. The effect of the tool geometry is considered in the calcula-tion of the uncut chip area. It should be noted that in the mechanistic modeling approach the cut-ting coefficients are predicted separately for each different workpiece-tool pair. 3.3.1.1. Uncut Chip Area and Cutting Edge Contact Length Calculation In the boring process, the uncut chip area varies as a function of the depth of cut a, feed rate c, side cutting edge angle yL and end cutting edge angle y c • For the uncut chip area calculation, four main different configurations (Figure 3.5) and nine various tool-workpiece interferences can be defined with respect to the cutting parameters (i.e. a, c and R)[23]. However, the most com-mon case encountered in boring applications is when the feed rate is less than the nose radius of the tool, due to the feed rate limitations for the given workpiece and cutting tool. Therefore, this study considers only five of nine interferences mentioned. Chapter 3. Force Prediction in Boring 27 Uncut material a < R a n d c < R a > R a n d c < R Figure 3.5 : Four different uncut chip area configurations defined with depth of cut a, feed rate, c and corner radius of the tool R Chapter 3. Force Prediction in Boring 28 Figure 3.5 also illustrates the relative positions of the insert at successive revolutions of the workpiece in four different configurations. Notice that the material left behind (uncut material) depends on the feed rate c and corner radius R and is expected to be large when the feed rate c is much greater. The uncut material also determines the surface finish quality. While large corner radii R and small feed rates c create good surface, small corner radii and large feed rates c cause the cut surface to be rough. The uncut chip area model contains five inputs, a [mm], c [mm/rev], R [mm], yL [Deg] and yc [Deg]. From these inputs, only the side cutting edge angle yL may have a negative value that defines the straight side edge of the tool to have an earlier contact with the workpiece rather than the coiner of the tool at the begining of cutting. The uncut chip area is calculated by discretizing the uncut chip area into small differential elements (Figure 3.6). The calculation is executed sepa-rately for regions defined in the uncut chip area. In the following, the calculation of the uncut chip area A and cutting edge contact length Lc are explained for only the first configuration. For Region 1, the uncut chip area of each differential element is approximated by subtracting the area of the triangle A0BB,, from the area of the circular ring sector A0DD, AOBBM = O.5|0fl|I.|Ofl'|I.sin8l. (3.3) AODD,i = °-59<*2 <3-4> A\,i-AODD',i~AOBB',i (3-5) The total area of region 1 is calculated as a subtotal of the area of each differential element as, n ^ i = £ A U (3-6) i = l Region 2 is assumed to be a rectangle, although one side of it (i.e. side KE) has a slight curva-ture caused by the corner radius of the previous tool position. Its area can be approximated as, A2 = \MG\\KM\ (3.7) Chapter 3. Force Prediction in Boring 29 Region 3 is a simple triangle and its area is calculated as, A 3 = 0.5|tfM||LM|sinY£ (3.8) Finally, total uncut chip area is found by adding together these areas for each region. A = Ax +A2 + A3 (3.9) The cutting edge contact length is calculated by considering only Region 1 and 2. Region 3 does not have any contribution to the total contact length. The total cutting edge contact length is, Lc = Lcl+Lc2 (3.10) n where L C [ is the contact length of Region 1 and equal to L c l = ^ | D D ' | ( (n is the number of element in Region 1). Similarly, Lc^ is the contact length of Region 2 and equal to the length of MG. The uncut chip areas A and cutting edge contact lengths Lc are calculated with the same manner for other configurations shown in Figure 3.7. Figure 3.6 : Uncut chip area calculation for Configuration 1 and the definitions of the regions Chapter 3. Force Prediction in Boring y < 0, a > R Sin(yL) ^ ^ ^ ^ M | n i II 111 (, 1 1 Figure 3.7 : Other uncut chip area configurations considered in the area calculation model Chapter 3. Force Prediction in Boring 31 3.3.1.2. Experimental setup Calibration tests have been conducted with Aluminum 6061-T6 disks (Figure 3.8) on a Hard-inge Superslant turning center that has a high precision positioning accuracy of 0.0005 [mm]. A Kistler 9257A three axis dynamometer, which has a maximum 5000 [N] measurement capacity in each direction, has been used for the force calibration. For the use of the boring bar on the machine a special tool holder had to be designed (Figure 3.9). The force signals were sampled at 1000 [Hz] and amplified by Kistler 5010B1 Dual Mode Charge Amplifiers prior to being digitized by data acquisition software CutPro-MALDaq. One thousand data points were collected in one second and the average value of this data is used in the development of the mechanistic model (Figure 3.10). Two different inserts have been used in the experiments: - Kennametal CPMT-32.52 K720 coated insert with A12-SCFPR3 steel shank boring bar o with 0 side cutting edge angle yL (Figure 3.11) o - Valenite CCGT432-FH 80 Carbide PVD coated diamond insert with A-SCLPR/L boring bar with -5° side cutting edge angle yL. In the experiments, in order to avoid chatter vibrations, the boring bar was clamped onto the tool holder with a short length to diameter ratio (L/D=2.5). Specifications of the inserts and bor-ing bars are shown in Figures 3.11 and 3.12. Two sets of experiments for each insert were conducted with different combinations of the cutting parameters within the ranges of 0.05-0.19 [mm/rev] feed rate c, 75-275 [m/min] cutting speed V, 0.25-3.25 [mm] depth of cut a. The first set of experiments was used to determine the empirical constants in the equations for the estimation of the cutting force coefficients. The sec-ond series of experiments were carried out to further examine the validity of the mechanistic model. Chapter 3. Force Prediction in Boring 32 25 [mm] Figure 3.8 : Workpiece-Al 6061-T6 used in the experiments Chapter 3. Force Prediction in Boring Figure 3.9 : Schematic illustration of the experimental setup for force calibration Chapter 3. Force Prediction in Boring 34 V=75 [m/min] c=0.155 [mm/rev] a=0.25 [mm] 801 1 1 1 1 1 1— i r Measured tangential force, F Measured-average tangential force, R 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time [sec] Figure 3.10 : Determination of the average tangential force value based on the collected data, V=75[m/min], c=0.155[mm/rev], a=0.25[mm] Chapter 3. Force Prediction in Boring 35 Chapter 3. Force Prediction in Boring 36 Figure 3.12 : Valenite CCGT432-FH 80° Carbide PVD coated diamond insert with A-SCLPR/L boring bar Chapter 3. Force Prediction in Boring 37 3.3.1.3. Cutting Coefficient Identification In this section, the development of the mechanistic force model is described in detail, i-) Identification of the Edge Cutting Force Coefficients: Tangential and friction cutting forces Ftc, F^rc are assumed to be proportional to the uncut chip area A [9], This expression leads to the definition of the cutting force coefficients. Previous researchers reported that these coefficients vary with chip thickness h, cutting speed V, and geometrical properties of the tool, such as the side rake angle, the back rake angle and the relief angle [9, 16 and 17]. In this study, the previously proposed models have been modified such that cutting parameters, a, c, and V and the associated cutting edge contact length Lc, the uncut chip area A, and centroid of the uncut chip area geometry Q are used in the prediction of the cutting force coefficients and forces for the selected inserts (i.e. Kennametal CPMT-32.52 K720 and Valenite CCGT432-FH). In addition, the effective lead angle § L is predicted in order to find the direction of the friction force F^r. Then, the radial and feed forces (Fr and Fj) are calculated as compo-nents of the predicted friction force. Instead of chip thickness h, the depth of cut a and feed rate c are used in the model. This is because the cutting edge is an arc, due to the corner radius of the insert, rather than a straight line as in milling and orthogonal plunge turning. Hence, the chip thickness is not a constant but varies along the cutting edge contact length (Figure 3.13). (3.11) F, (3.12) Chapter 3. Force Prediction in Boring 38 In the mechanistic modeling approach, the tangential cutting force coefficient Ktc is assumed to be a function of A [mm ] and V [m/sec], whereas the friction cutting force coefficient Kjrc is implemented as a function of the cutting edge contact length Lc and cutting speed V. It is pro-posed that the friction force is strongly dependent on the cutting edge contact length Lc, as the friction cutting force is generated by the friction on the cutting edge. Cutting forces on the tool during machining consist of two components, which are the actual and the edge cutting forces [1] (Eq. 3.13). The actual cutting forces (Ftc and Ffrc) are induced by shearing on the shear zone. The edge cutting forces (Fte and Ffre) are caused by rubbing and ploughing on the cutting edge and do not contribute to the cutting process. The edge cutting force components are functions of the cutting edge contact length Lc and the edge cutting force coeffi-cients (Kte and Kfre) that represent the force for unit cutting edge contact length (i.e. l[mm]). Ft = Ftc + F(e = KlcA + KteLc (3.13) Ffr = V i + Ffrc2 + V = V A + V A + KfreLc (3-14) Chapter 3. Force Prediction in Boring 39 where Ax and A 2 are the uncut chip areas of Region 1 and Region 2 (Figure 3.14). Ftc, Fy r C i and Ffrc are assumed to be proportional to the uncut chip area A. Unlike Ft , Fj-r is con-sidered as having two components acting in two separate regions of the uncut chip area, Region 1 and 2 (Figure 3.14) The reason for this separation is that for a small depth of cut the corner radius has a significant effect on the direction and magnitude of the friction force distribution. On the other hand, for large depths of cut, the straight edge of the insert, on which the magnitude and direction of the friction force distribution are constant, is predominant in the cutting operation. For comparatively large depths of cut, the direction of the total friction force <j)L tends to approach the side cutting edge angle Y/ of the insert. In such a case, the magnitude ratio of F r /Fy decreases. For example, if the side cutting edge angle yt is zero, friction force Fyr does not con-tribute to Fr from Region 2, when a is selected larger than the nose radius R. Distribution of the Friction Force Figure 3.14 : Friction force distribution along the cutting edge In order to determine the edge cutting force coefficients Kte, Kre and Kfe, 24 experiments have been conducted at constant 1.5[mm] depth of cut and 150[m/min] cutting speed but varying Chapter 3. Force Prediction in Boring 40 feed rate c from 0.025 to 0.19 [mm/rev]. Thereafter, linear regression was performed between the calibrated forces and the cutting edge contact length Lc. The following describes the method for predicting the edge cutting forces and their corresponding coefficients. 300 r ^250 200 T3 CD £ 150 TD C CO CO TD CO rr C CD cn c .CO 100 50 O Measured tangential force <> Measured feed force V Measured radial force 0 s*^ t Q ^ ) o , o o \ *<* ) A <>jiJ>-F : = 9 3 3 r .47 L c -1 7 7 9 . 9 o; • ^Tv i 1.96 1.97 1.98 1.99 2 2.01 2.02 2.03 2.04 2.05 2.06 Chip contact length, LJmm] 1.9556 Figure 3.15 : Tangential, radial and feed force vs chip contact length Performing linear regression leads to the following tangential force equation (Eq. 3.15). The first term on the right hand side corresponds to the actual tangential force component Ftc, and the second term is the tangential edge cutting force component Fte. F, = 2438.5LC-4721.4 [N] Lc> 1.9556[mm] (3.15) where, Lc is the cutting edge contact length. This equation is valid only when Lc is equal to and greater than 1.9556 [mm], which corresponds to zero feed rate for a=1.5 [mm] depth of cut. Chapter 3. Force Prediction in Boring 41 c = 0 [mm/rev] refers to the rubbing process on the cutting edge. Substitution of Lc = 1.9556 [mm] into the Eq. (3.15) gives F(e for the given a (1.5 [mm]) and zero feed rate. (3.16) F<e = V = 1.9556 = 4 7 - 4 1 ^ The tangential edge cutting coefficient Kte is determined dividing Fte by the total cutting edge contact length Lc F r N i KT=-£ = 24.24 — L [_mm_ (3.17) Kte represents the tangential rubbing force per unit cutting edge contact length Lc. Follow-ing the same procedure, the radial and feed edge cutting forces Fre and Ffe and corresponding edge cutting force coefficients Kre and Kfe are found as follows. Table 3.1: Edge cutting force coefficients for the Valenite CCGT432-FH insert Expression . ^ . = 2438.5Lt.-4721.4 Measured edge cutting forces Ffe' Fre> Ffe i [N] 47.41 Chip contact length, Lc [mm]' r9556 Edge cutting ]i force coefficients [N/mm] 24.24 Fr = 417.74LC-800.40 16.53 1.9556 8.45 Ff = 933.47LC-1779.9 45.58 1.9556 24.79 As the friction force F^r is the compound of Ff and Fr, its edge cutting force and coefficient is calculated as follows. fre = 1 2 2 Fre + Ffe (3.18) Kf — *JK fe + K re — v / r e _ ^ fe +  re = 26.1928[Mmm ] (3.19) As can be noted that Lc does not change significantly with feed rate c, thus the effect of the c on edge cutting forces is negligible. The dependence of the edge cutting forces on the cutting speed V has also been investigated with a series of experiments. In these experiments, depth of cut was taken constant 1.0[mm] but various feed rates, within the range of 0.015 to 0.14 [mm/ Chapter 3. Force Prediction in Boring 42 rev], at three different cutting speeds V (75, 150 and 250 [m/min]), were employed. This investi-gation has been carried out with a Kennametal tool, and therefore, the edge cutting force coeffi-cients cannot be compared with those presented above for the Valenite insert. The results are shown below in Figure 3.16. As noticed, the edge cutting forces at three different cutting speeds have almost the same magnitude and can be assumed to be independent of the cutting speed. V=150[m/min] a=1 [mm] 0.02 0.06 0.1 0.14 Feed Rate, c [mm/rev] 0 0.02 0.06 0.1 0.14 Feed Rate, c [mm/rev] 0 0.02 0.06 0.1 0.14 Feed Rate, c [mm/rev] V=75[m/min] a=1[mm] V=150[m/min] a=1[mm] 0.06 0.1 0.14 Feed Rate, c [mm/rev] 0 0.02 0.06 0.1 0.14 Feed Rate, c [mm/rev] 110 100 z 80 LL <D 60 O O LL 40 ~a T3 to 20 cr V=250[m/min] a=1[mm] 0 0.02 0.06 0.1 0.14 Feed Rate, c [mm/rev] V=75[m/min] a=1[mm] V=150[m/min] a=1[mm] 0.02 0.06 0.1 0.14 Feed Rate, c [mm/rev] 0.02 0.06 0.1 0.14 Feed Rate, c [mm/rev] 0 0.02 0.06 0.1 0.14 Feed Rate, c [mm/rev] Figure 3.16 : Investigation of the dependence of the edge cutting forces on the cutting speed V; Material: Aliminum 6061-T6, Tool: Kennametal CPMT-32.52 K720 coated insert ii-) Identification of the Cutting Force Coefficients for Valenite CCGT432-FH Insert Chapter 3. Force Prediction in Boring 43 Experimental calibration tests showed that Ktc and Kfrc change exponentially with uncut chip area A, cutting edge contact length Lc and cutting speed V. This variation can be repre-sented with the following equations. Ktc = / 0 A V 2 (3.20) Kfrc, = e LcV Kfrc2 = e Lc2V (3-21) where, b 0 , b v b2, niQ, mx, m2, n0, nx and n2 are empricial constants, and estimated using the least squares method based on the experimental data (See Appendix C). For least squares esti-mates, cutting coefficients calculated from the experimental data were used. After processing the data, the following equations were obtained. Ktc = ^.9477A-0.0853v-0.2750 ( 3 2 2 ) K f r C i = /1965 L c -0.6737 y _o.4210 ( 3 . 2 3 ) „ 9.6152. -0-0241 0.7597 - . . Kfrc2 = e Lc2 v (3-24) Once the friction cutting force coefficients KfrCi and Kjrc^, the uncut chip area A and cutting edge contact length Lc are determined, the radial Fr and feed forces Ff are predicted as compo-nents of the friction force Ffr. However, this requires that the friction force direction, which is defined with the effective lead angle § L , to be known. The prediction of the effective lead angle consists of two steps. As mentioned above, the uncut chip area A is divided into two regions and the friction cutting force p has been expressed separately (FfrCi and FfrCi), for each region for an accurate prediction. In the above equations, the cutting force coefficients change inversely as functions of the uncut chip area A, cutting edge contact length Lc, and cutting speed V. The variations of the cut-ting force coefficients with uncut chip area and cutting edge contact length represent the effects of the tool geometry on the cutting forces. The relationship between the cutting force coefficients Chapter 3. Force Prediction in Boring 44 and, the uncut chip area A and cutting edge contact length Lc is nonlinear, due to the corner radius of the tool and the chip breaking groove along the cutting edge. On the other hand, the decrease of the cutting force coefficients with cutting speed is induced by the reduction in the maximum shear strength of the material at high speeds. The material tends to soften and shear strength diminishes at high speeds, causing the cutting forces to lessen. As noted, the effect of each parameter in the equations of each cutting force coefficient is different. For the depths of cut which are larger than the corner radius R, the effect of the cutting edge contact length on the fric-tion cutting force coefficient is not as significant as compared to the case in which the depth of cut is less than R. This may be because the straight side of the tool dominates the cutting pro-cess for the larger depths of cut. Thus, the friction cutting force coefficient Ay r C j does not change much with the depth of cut, due to the homogeneity of the friction force distribution on the straight side of the tool. Since the friction force acts perpendicular to the cutting edge and is also proportional to the uncut chip area for each differential element along the cutting edge, it can be predicted by assum-ing that each component of the friction force passes through the gravity center of each related region (Figure 3.18). The friction force component of each region is added up vectorially, and the total friction force Fjr is obtained. It should be noted that these vectorial friction force compo-nents are presented with the notation F*y r i, F*yrj (Figure 3.18) and are not identical to the ones in the Eq. (3.14). Fyr) and Fj in Eq. (3.14) are the components contributing from each region to the total friction force, and do not have vectorial meanings. The determination of the centroid of the uncut chip area is shown in the following equation, as an example. The calculation is executed with respect to the origin of the corner radius C 2 for a given tool position (Figure 3.17). Chapter 3. Force Prediction in Boring 45 n i = 1 (3.25) where A-, 0,-, and AT are the area of a differential element, angular position of the differen-tial element and the total area of the region, respectively. Based on the definition of the friction force direction, the regional lead angle <t>L in Region 2 can be assumed to be equal to the side cutting edge angle yt of the tool along the straight line of the cutting edge. The total effective lead angle is determined from the sum of two friction force vectors (Figure 3.18). After processing the data, analysis has shown that there were certain discrepancies between measured and predicted effective lead angles based on the above approach (Figure 3.14). This may be because the friction force F^r is not really acting perpendicular to the cutting edge. This could be caused by the chip breakage groove along the cutting edge, or the assumption of the per-pendicularity of the friction force to the cutting edge is not accurate. The difference between the measured and predicted effective lead angle (j)L has been investigated with two sets of 5 experi-ments, varying the cutting edge contact length Lc and cutting speed V (Figure 3.20). This inves-tigation has revealed that the effective lead angle shows linear variation with V and Lc. Hence, it can be tuned in the calculation with a modification factor Km that is also a linear function of Lc and V. where § L * is the predicted effective lead angle based on the regular procedure described above and § L is the final modified-predicted effective lead angle. For the same five experimental conditions, the variation of the modification factor Km depending on the cutting speed V and the (3.26) Chapter 3. Force Prediction in Boring 46 cutting edge contact length Lc is depicted in Figure 3.19. Graphical representation of the modi-fied-predicted effective lead angle 0L is shown in (Figure 3.21). Figure 3.17 : Determination of the centroid of region 1 Chapter 3. Force Prediction in Boring 4 7 Chapter 3. Force Prediction in Boring 48 Figure 3.19 : Investigation of the variation of the effective lead angle modification factor Km; Km v.s V and Lc Chapter 3. Force Prediction in Boring 49 Figure 3.20 : Investigation of the Effective lead angle variation; § L v.s Lc and V Chapter 3. Force Prediction in Boring 50 Chapter 3. Force Prediction in Boring 51 With the information obtained from the above investigation, the modification factor can be represented with the following expression. Km = q0 + q1Lc + q2V (3.27) where q0, qx and q2 are the empirical constants and determined for the conditions a < R and a > R separately from the experimental data by performing the least squares method. This results in the following linear equations. Kmi = 1.0743 - 0.3567(10)_3LC + 0.9763(10)~V , for a<R (3.28) Kmi = - 0.0163 + 0.6299LC + 0.0013 V , for a > R (3.29) The above identified modification factors show that, for a<R, the effective lead angle <\>L has an almost constant deviation along the cutting edge contact length (i.e. the effects of Lc and V are negligible) (Eq. 3.28) . However, for a >R the deviation has a strong dependence on Lc and V. This may arise from the nature of the chip flow. For a<R, the chip attempts to flow towards the center of the corner radius with an almost constant deviation. However, along the straight edge, the chip is forced to flow away, towards the outside of the contact in the radial direction and with a continuously diverging friction force, causing the chip to curl. The diver-gence of the chip exhibits a continuous increase along the cutting edge and reaches a maximum at the end of the contact length. A schematic representation of the directional variation of the friction force is shown in Figure 3.22. It should be noted that the cutting edge contact length in Eq. (3.29) is the total contact length including Region 1. The variations of the modification factors Km^ and K are presented in Figure 3.23 for 0.25...0.8 [mm] and 0.8..3.25 [mm] range of the depth of cuts a, 0.1 [mm/rev] of constant feed rate c and 150[m/min] of constant cutting speed V. In these figures, the variation of the modifi-cation factors is examined changing cutting edge contact length Lc only. In the figures, the modi-fication factor K for the condition a < R decreases insignificantly with the cutting edge contact Chapter 3. Force Prediction in Boring 52 length Lc. The effect of Lc on Km^ is less than 1%, and is therefore negligible for the condition a<R. On the other hand, the modification factor Km^ for the condition a > R exhibits strong dependence on the cutting edge contact length Lc varying from 1.0 to 2.57 for 0.8... 3.25[mm] depth of cut range. The reason behind this has been explained in the previous paragraph. Region 1 * L 6 > ( l ) L 5 > ( t ) L 4 > ( t ) L 3 = ( l ) L 2 = ( t ) L 1 Figure 3.22 : Deviation of the effective lead angle <\>L along the cutting edge contact length er 3. Force Prediction in Boring 1.0888 1.0888 1.0887L 1.0886 J 1.0886 a=0.25.. 0.8 [mm], c=0.1 [mm/rev], V=150[m/min ] 1.088. i.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Contact length L c [mm] a=0.8.. 3.25 [mm], c=0.1 [mm/rev], V=150 [m/min] 1.4 1.5 2 2.5 3 Contact length L c [mm] 3.5 Figure 3.23 : Variation of the modification factor Km with the cutting edge contact length for a < R and a > R Chapter 3. Force Prediction in Boring 54 It would also be possible to predict the effective lead angle through the direct use of cutting parameters, a, c and V. The goal of developing this approach is to find a generalized cutting force prediction model, so that the cutting forces can be predicted for both stable and unstable cut-ting conditions . Detail of this will be presented in chapter 5. Once the above procedure has been completed, the radial and feed force predictions are made based on the predicted friction force F^r and effective lead angle <])L as follows. Fr = Ffrsm$L Ff=FfrcosQ>L (3.30) Radial and feed actual cutting force components become, Prc = Fr-Pre (3-3D Ffc = Ff-Ffe (3.32) Corresponding cutting force coefficients are obtained as, Krc = *f (3.33) Kfc = F-f (3.34) iii-) Cutting Force Coefficients of Kennametal CPMT-32.52 K720 Using the procedure presented above, the cutting force coefficients and effective lead angle modification factor for Kennametal CPMT-32.52 K720 insert have been obtained as follows. „ 8.0428 .-0.1696t/-0.2512 O C N Ktc = e A V (3.35) v 1.1522 T -0-6093 0.2189 , Kfrc{ = e Lcx v (3-36) „ 9.3082, 0.0541 0.5470 Kfrc2 = e Lc2 v (3-37) The effective lead angle modification factors, Chapter 3. Force Prediction in Boring 55 Km = 1.2963 + 0.0604L -0.0006V, a<R (3.38) Km = -0.4138 + 0.7021LC + 0.0025V, a >R (3.39) Edge cutting force coefficients were determined by performing the linear regression method shown in Section 3.3.1.3. Kte = 13.777 [N/mm], Kre = 13.036 [N/mm], Kfe = 19.572 [N/mm], Kfre = 23.516 [N/ mm] It can be noted that the trend of the variation of the cutting force coefficients and the effective lead angle modification factor depending on the parameters (i.e. cutting edge contact length Lc, cutting speed V, uncut chip area A) is similar to the ones for the Valenite insert, but only the empirical constants of these parameters are different. This is caused by the difference between the geometry of the two tools. The differences are; - The Valenite tool has a 5° side cutting edge angle yl while the Kennametal insert's is 0 ° . - The Valenite tool has a 7° relief angle, while the Kennametal tool's is 11°. - The forms of the grooves on the two inserts are different. All differences were considered in the determination of the empirical constants by perform-ing the least-squares method. 3.3.1.4. Experimental Verification of the Mechanistic Model In order to validate the mechanistic model, a different set of experiments was conducted under different cutting conditions from those used for the implementation of the mechanistic model expressions. The presented model in the previous sections (3.3.1.1, .... 3.3.1.3) results in good force pre-diction with under 10% absolute average error, for both Valenite and Kennametal inserts. The tan-gential force Ft and friction forces for a < R (Fy r l) and a > R (Ffr2) are predicted with 99.5% , 93.5% and 98.4% correlations, respectively. The results of cutting force prediction for the Valen-Chapter 3. Force Prediction in Boring 56 ite insert are presented in the following figures. Cutting conditions and prediction results are also shown in the tables in Appendix A. Figure 3.24 : Friction force verification for a < R Chapter 3. Force Prediction in Boring 57 Border lines of -10[%] and 10[%] error Reference line of 0 [%] error 145 135 125 I 115 £ l 0 5 o £ 95 I 85 T3 O T 3 CD 75 65 55 45 35 25 I I ! I I I i ' T V ™T~ —y 1 - 7 /• , - * v jr : v / • . jr : * y ,'••-<> •JS • / * Xls 1* : X / %<' X 00 / : / X - 'X ' jT X%o'l/ : L - >V • • . . . . . 0 / , ' / / * X . . . . . . • S •* . . . . . . / S o / ; ; ; ; ; ; / o' s$ * y . . . . . . . . * X\ Jr * S /• * • * s 1 1 i t i i i i i i i 25 35 45 55 65 75 85 95 105 115 125 135 145 Measured Friction force F f r [N] 350 c o T 3 CD o T 3 CD 1_ CL 3301-310 290 270 250 230 210 190 170 150 • 1 1 • • • y r • / • / • / .: A / : • / • • / V / >• j r • > r : • • / 0 * * • • - s * * * / w * • • !> 150 170 190 210 230 250 270 290 310 330 350 Measured Friction force F f r [N] Absolute average error [%]=6.41 Figure 3.25 : Friction force verification for a > R Chapter 3. Force Prediction in Boring 58 Border lines of -10[%] and 10[%] error Reference line of 0 [%] error • v • • * • 0 • • • 0 or * . ' 0 So > * • / A C < • s * '/* _l I 1_ _l I I l_ 20 40 60 80 100 120 140 160 Measured tangential force F [N] 1 I , ' ' o • • • > • • • • S oS (S :f. 0. • y • • • • * * 0,' 0 / S 0 0,' yT • • • y V, s > * 175 200 225 250 275 300 325 350 375 400 425 450 475 500 Measured tangential force F [N] Abso lu te average error [%]=6.11 Figure 3.26 : Tangential force verification for a<R and a > Chapter 3. Force Prediction in Boring 59 Border lines of -10[%] and 10[%] error Reference line of 0 [%] error <D 55 O ) c CO "O CO ® 50 CD > T5 CD ! £ CD 4 5 y 4 0 • • • • <>y' * 4* / 0 * > 0 o ,'' * 0 / / * • * • y ' * * • y 0 / o / Abso lu te average error [%]=3.96 | 35 40 45 50 55 60 65 Measured effective lead angle (j). [Deg] Measured effective lead angle 4 [Deg] Figure 3.27 : Effective lead angle verification for a<R and a > R Chapter 3. Force Prediction in Boring 60 55 50 45 Border lines of -10[%] and 10[%] error Reference line of 0 [%] error CD « 40 "D CO TD CD I 35 CD 30 25 H mm • • • • • ^ ^ ^ ^ ^ J Absolute average error [%]=7.02 25 60 30 35 40 45 Measured radial force F [N] 50 55 55 50 45 .« 40 TJ CO i— T3 a 3 5 o TD CD 30 25 20 4*  • 4* 4* 4* 4* 4* 4* 0 y • 4* • 4* 4* 0 :ym/ y • y • / : 4* y * y 4* / • 4* M y (, / ,''o / o.' / y A /i '' 4* ' 0 y : /so,*' 0 $y<$> y • 4* > ^ 0 ^ ^ ^ ^ Absolute average error [%]=7.76 20 25 30 35 40 45 50 Measured radial force F [N] 55 60 Figure 3.28 : Radial force verification for a < R and a > R Chapter 3. Force Prediction in Boring 61 Border lines of -10[%] and 10[%] error Reference line of 0 [%] error Measured feed force Ff [N] Figure 3.29 : Feed force verification for a < R Chapter 3. Force Prediction in Boring 62 Border lines of -10[%] and 10[%] error Reference line of 0 [%] error TJ CO 0) TJ £ o TJ CO 145| 135 125 115 105 95 85 75 65 55 45 35 25. 1 1 V 1 >> x x' X X'O y ' ' : yX 0 oX. X s •. X • X 0 . / . . A • • • ' o X •* • • > X* X v r s s s > V > r . j / s ' s . * X X > X ' > / • / X * X s ' ^ x / s / ' s / " / i 1 1 1 1 1 25 35 45 55 65 75 85 95 105 115 125 135 145 Measured feed force Ff [N] TJ CO £• TJ £ o TJ CO 350 330 310 290 270 250 230 210 190 170 150 l l v > • • • / • X • > / > r • ' * >r • / • / - X ; • • • • • • • • • • • • • • • • • • • • s * * t tL 150 170 190 210 230 250 270 290 310 330 350 Measured feed force Ff [N] Absolute average error [%]=5.02 Figure 3.30 : Feed force verification for a > /? Chapter 3. Force Prediction in Boring 63 3.3.2. Orthogonal to Oblique Transformation Method As mentioned in the earlier section, another method for predicting cutting force coefficients and forces in boring operations is to apply the Orthogonal to Oblique Transformation Method, as suggested in [1], [22] and [23]. The important parameters of the tool, which influence the force prediction and chip flow in the boring process, are the corner radius R, side cutting edge angle Y;, end cutting edge angle yc, back rake angle ap and side rake angle (Figure 3.32). Among these parameters, rake angles have an important effect on controlling the direction of the chip flow and strength of the tool tip. Positive values of the rake angles reduce the cutting forces and tempera-ture created on the tool. The corner radius of the tool causes oblique cutting parameters to change around the cutting edge. The effects of these parameters on the cutting forces can be examined with the Orthogonal to Oblique Transformation Method. The corner radius R also contributes to this process by making the tool tip stronger; It also has an effect on the surface finish quality. A larger corner radius results in a better surface finish, with less unwanted material left on the cut surface. 3.3.2.1. Orthogonal Cutting Test and Identification of Oblique Cutting Parameters In the prediction of the cutting forces of a tool which has an oblique cutting geometry, the orthogonal cutting force coefficients Ktc, Kte, K^c, K^e and cutting parameters, such as chip ratio rc, friction angle Pfl, shear angle <|)c, shear stress xs, are identified for the specified workpiece-tool pair by performing orthogonal cutting tests. These parameters are then transfered to the oblique cutting geometry and cutting forces are predicted based on the method proposed in [1],[2], [22] and [23]. This is the fundamental concept of the Orthogonal to Oblique Transforma-tion Method. Orthogonal cutting tests are carried out with a tool which has specific rake and relief angles. In the tests, the workpiece rotates while the tool is fed linearly into the workpiece with the speed of the feed rate c [mm/rev]. The experimental setup is illustrated as in Figure 3.34. In orthogonal cutting tests, a tube material is utilized, instead of the shaft workpiece. It should also be noted that Chapter 3. Force Prediction in Boring 64 the approach angle of the tool is supposed to be zero, in order for the process to be defined as orthogonal cutting. Tests are performed with varying feed rates c, at constant cutting speed V and width of cut b (corresponding to tube thickness). After the completion of the tests, linear regression is performed on the measured tangential and feed forces for the identification of the cutting and edge cutting force coefficients Ktc, Kte, K^c, K^e. This regression yields, Ft = Ktcbh + Kteb (3.40) Ff = Kfcbh + Kfeb (3.41) where b and h are the width of cut and chip thickness, which is equal to the feed rate c. Later, the value of other parameters rc, (3a, §c and xs for each orthogonal cutting test are deter-mined with the following equations [2]. rc - A (3.42) Pfl = ar+atan^ (3.43) ( r.cos(a„) \ 0C = atanN-S r-^ — (3.44) U - r c s i n ( a r ) J T = [F/ccos(<|)c)-F/csin((|)c)]sin(<|)c) s bh where hc and ar are the cut chip thickness and the rake angle, respectively. Once the above parameters are obtained for each experimental condition, their calculated average values are used in the transformation method. These average values of the orthogonal cutting parameters devel-oped at different cutting speeds V, rake and relief angles \\ir of the tool constitute the orthog-onal cutting data base. For any tool whose cutting geometry can be defined as oblique cutting, this data base can be used for the force prediction without conducting any further calibration test. Chapter 3. Force Prediction in Boring 65 Orthogonal cutting parameters, shear stress xs, friction angle (3a, and chip ratio rc have already been identified by Ren [21] for carbide tools and P20 steel material as functions of feed rate c and cutting speed V. Correlations are given as follows. TS = 507.0 + 1398.76c + 0.327 V (3.46) Pa = 33.69- 12.16c-0.022 V (3.47) rc = 0.227 + 2.71c + 0.00045 V (3.48) where feed rate c , cutting speed V , shear stress TS and friction angle Pa are in the units of N [mm/rev], [m/min], and [Deg], respectively. Edge cutting coefficients Kte, Kre and Kfe mm have also been identified by Ren as presented in the following. Kte = 0.1199(10fV-0.1487V+76.85 (3.49) Kfe = 0.1366(10)~V-0.2007 V+97.98 (3.50) Kre = KteSin(i) (3.51) where V and / are the cutting speed and oblique angle, respectively. 3.3.2.2. Prediction of the Oblique Cutting Forces Once orthogonal cutting parameters are obtained, they are transferred to the oblique cutting geometry with the following assumptions [2]. 1- The orthogonal shear angle is equal to the normal shear angle in oblique cutting.tyc = 2- The normal rake angle in oblique cutting is equal to the rake angle in orthogonal cutting 3- The chip flow angle is equal to the oblique angle n_ = i 4- The friction coefficient (3a and shear stress xs are the same in both orthogonal and oblique cutting for a given cutting condition. Oblique tangential, radial, and feed cutting forces acting on the tool and cutting force coeffi-cients are expressed in terms of cutting and geometrical parameters (i.e. width of cut b, chip Chapter 3. Force Prediction in Boring 66 thickness h, oblique angle i, oblique shear angle § n , shear stress xs, friction angle (3„ and nor-mal rake angle an). Cutting forces are expressed in the general form of, Ft = Ktcbh + Kteb (3.52) Fr = Krcbh + Kreb (3.53) Ff = Kfcbh + Kfeb (3.54) where oblique cutting force coefficients are defined as, X . cos(B -cc„) + tam'tanr|sinB„ Ktc = • \ • (3.55) x sin(6 - a ) KfC = ^ i r — • < = ( 3 - 5 6 ) J C sin (p COS l I 2 2 2 A / C O S ((|)„ + P„-a„) + tan r)sin p„ x. cos(P„-a„)tan/-tanrisinP„ Krc = • 1 • (3.57) Y" /^cos (())„ +P„ - a„) + tan Tisin Pn In general, Kte and are determined in the evaluation of the orthogonal cutting test results. Because there is no radial force component measured in orthogonal cutting tests, Kre is not known. However, experimental investigations have shown that the radial cutting edge force F(e is very small and therefore negligible in the transfomation method. 3.3.2.3. Experimental Verification of the Method Selecting an insert (Valenite CTPGPL-16-3C) with a nose radius, a sharp cutting edge, a flat rake face (Figure 3.33), the Orthogonal to Oblique Transformation Method can be described for the boring process as follows. In this study the same uncut chip area configurations as in the mechanistic model has been considered, and a necessary force prediction program has been devel-oped. Only the first configuration is presented in this section in order to demonstrate the proce-dure. The uncut chip area is divided into three regions (Region 1, 2 and 3, Figure 3.31). Region 1 Chapter 3. Force Prediction in Boring 67 then is discretized into equal angular segments 6; in order for the transformation method to be applied. The reason for the discretization of the uncut chip area is that the oblique cutting geome-try parameters vary in Region 1 along the cutting edge, due to the corner radius of the tool. How-ever, in Region 2, the uncut chip area is uniform, and the oblique cutting parameters do not change with location. Thus, Region 2 is not discretized, but considered as one element in the eval-uation. Similarly, Region 3 is considered as a whole element. For the selected Valenite tool, the back rake angle ap and side cutting edge angle yt are zero, hence, the cutting in Region 2 and 3 cannot be characterized as an oblique cutting because there is no inclination angle between the cutting velocity vector and cutting edge of the tool. In such a case, the cutting process in Regions 2 and 3 is described as an orthogonal cutting , although the cutting in Region 1 is still oblique. The uncut chip area and the oblique cutting geometry parameters for each discretized element are calculated with the following equations derived from the geometrical relations [2]. - REGION 1 - Uncut chip area, At Eq. (3.5) - Approach angle \yr = jQ, where j is the counter of differential elements and 9 is the angu-lar increment of each differential element. - Orthogonal angle cc0 = atan(tanayCos\|/A.+ tanapsin\|/r), where a^ -, ap and \(/r are the side rake angle, back rake angle and side relief angle respectively. - Oblique angle i = atan (tan a p cos \j/ r + tanOySin\|/r), - Normal rake angle an = atan (tan oc0 cos z'), cc0 is orthogonal rake angle. - Chip ratio Eq. (3.48) r cos (x - Normal shear angle (j)„ = atan— — , a„ is the normal rake angle. l - r c s i n a M - Friction angle Eq.(3.47) - Normal friction angle Pn = atan (tan Pa cos/) - Shear stress xs Eq. (3.46) Chapter 3. Force Prediction in Boring 68 Substituting the necessary parameters into Equations (3.55), (3.56) and (3.57) tangential, radial, and feed cutting force coefficients, Ktc, Krc and Fyc for each differential element are determined. As can be noticed, the oblique cutting parameters change around Region 1 due to the varia-tion of the approach angle \\ir. Oblique radial and feed forces in Region 1 do not match the direc-tions of the dynamometer (Y and Z) and need to be projected into dynamometer directions in order to obtain the total global forces in the three directions (Figure 3.35). For each differential element, oblique tangential, radial, and feed forces are expressed as, Then, the total forces in the dynamometer directions are calculated by adding together the forces acting on each differential element. N (3.58) F. xl (3.59) N F, 2 [F A .sinO,.)-F r l > , .008(6,.)] (3.60) i= 1 N F. ^[F / 1,.cos(0,.) + F r l .s in( e . ) ] (3.61) i = 1 -REGION 2 For Region 2 the same equations are used in the force prediction except the following. - Uncut chip area Eq.(3.7) - Approach angle \[/r = -yL, yL is the side cutting edge angle of the tool. Chapter 3. Force Prediction in Boring Cutting forces in the dynamometer directions are found similarly. 69 Fx2 = Ft2 Fy2 ~ Ff2sin(-yL)-Fr2COS(-yL) Fz2 = Ff2COS(-yD + Fr2sin(-Y0 (3.62) (3.63) (3.64) -REGION 3 - Uncut chip area Eq. (3.8) Y/ - Approach angle \\fr = -— For this region the edge cutting force components are assumed to be zero due to the zero con-tact length. Cutting force components are obtained by using the same equations presented above. (3.65) Ft3 = KteA3 Fr3 = Kre*3 = KfeA3 These oblique cutting forces contributed by region 3 are obtained as, Fx3 = Ft3 Fy3 = F / 3sin(-Y L/2)-F r 3cos((-Y L)/2) Fz3 = F / 3cos((-YL)/2) + F r 3sin(-Y L/2) The total forces in dynamometer directions, X, Y and Z are found as, Fx = Fx\ + Fx2 + Fx3 FY = Fy\ + Fy2 + Fy3 FZ= ^ 1 + ^ 2 + ^ 3 (3.66) (3.67) (3.68) (3.69) (3.70) (3.71) Chapter 3. Force Prediction in Boring Figure 3.31 : Evaluation of the oblique cutting parameters for three regions of the uncut chip area Chapter 3. Force Prediction in Boring S i d e rake ang le Figure 3.32 : Geometry of boring tool Chapter 3. Force Prediction in Boring 72 Tool holder Clamp screw Figure 3.33 : Valenite CTPGPL-16-3C tool holder and TPC-322J-VC2 insert Chapter 3. Force Prediction in Boring Figure 3.34 : Experimental setup for the verification of the Orthogonal to Oblique Transformation Method Chapter 3. Force Prediction in Boring 74 Figure 3.35 : Oblique tangential Ft, radial Fr and feed Fj force directions in each region and, dynamometer axes directions Chapter 3. Force Prediction in Boring 75 3.3.2.4. Experimental Verification of Orthogonal to Oblique Transformation Method For the verification of the method, 7 experiments were conducted with the material P20 mold steel shaft by using a Valenite TPC 322J Uncoated VC2 grade CTPGL-16-3 C Left-hand tool holder at different cutting speeds and depth of cuts, but with constant 0.05[mm/rev] feed rates. The specifications of the tool and tool holder are given in Figure 3.33. For the calibration, the experimental setup was the same as the one used in the development of the mechanistic model, except for the tool holder and workpiece. The experimental results are shown below in Figure 3.36 (The cutting conditions, the measured and the predicted forces are also presented in a table in Appendix B). Tangential force is predicted with under 10% average error; however, prediction error in radial and feed forces rises to 25% in some cases. L L - 4 0 0 CD H 3 0 0 o I 2 0 0 S 100 § 0.8 I-z 4 0 0 v M e a s u r e d F o r c e o P red i c ted F o r c e . . ,V7 u 8 1.2 1.4 1.6 1.8 2 2.2 Cut t ing e d g e contact length, l_ c [mm] 2.4 2.6 0 3 0 0 o 2 0 0 T J CD CD 0 100 1 1 1 i V V o 55?. O i ^ (V) 1 0.8 2 ,400,— 1.2 1.4 1.6 1.8 2 2.2 Cut t ing e d g e contact length, L c [mm] 2.4 2.6 CD 3 0 0 o L? 2 0 0 I 100[ co 0 1 0.8 o 8 5 1.2 1.4 1.6 1.8 2 2 .2 Cut t ing e d g e contact length, L c [mm] 2.4 2.6 Figure 3.36 : Comparison between the measured and predicted tangential, radial and feed forces using the orthogonal to oblique transformation method Chapter 3. Force Prediction in Boring 76 3.4. S u m m a r y In this chapter, a mechanistic force prediction model has been developed for two inserts with a corner radius and grooves for chip breakage. This development allows for the force prediction of stable and unstable cutting conditions. The mechanistic model requires a vast amount of cutting data for each new cutter geometry, and, more importantly, the data cannot be generalized for application to other tools, as there is no explicit relationship between the tool geometry, cutting conditions, and the cutting force coefficients. The mechanistic model approach does not provide any physical insight into the process, such as shearing stress, or friction in the process. In the sec-ond part of the chapter, an orthogonal to oblique transformation method is utilized for the force prediction in the boring process. For the verification of this method, a previously developed orthogonal cutting database has been used. Once the orthogonal cutting data base is developed, this method does not require any further experiments for the force prediction. Both methods are experimentally verified with good accuracy. Chapter 4 Process Faults in Boring 4.1. Introduction Process faults are the most common problem associated with the use of multiple-inserted tools, causing the loss of accuracy in machining operations such as milling and boring. The main advantage of using a multiple-inserted boring bar is that the workpiece can be machined with a high feed rate, which is the number of insert times the desirable feed rate for one insert (Nxc ). High feed rates result in increased productivity in manufacturing. Because the inserts are arranged with symmetrical angular positions on the boring head, the forces on the inserts in the X and Y directions cancel each other, and the total force in these directions becomes zero. Hence, it is pos-sible to obtain better tolerances with multiple-inserted bars with a large operational length-to-diameter ratio (L/D). Process faults in the multiple-inserted boring process are defined as any deviation of the bor-ing head from the hole center, as well as insert runout. As mentioned in the previous chapter, the boring process is performed on an existing hole produced by preceding processes such as drilling and punching. In other words, boring is usually at least the second process applied to a workpiece. If the boring operation is not performed on the same machine as the one used for the previous operation, the center of the boring head and the hole should be aligned for an accurate process. In any case, if the boring head has a deviation, the depth of cut varies continuously around the rota-tional axis. Presumably, the cutting forces will also follow this depth of cut variation. Although tool manufacturers produce very precise multiple inserted tool holders, the inserts may contain offsets in the radial and feed directions when they are secured through tightening of the insert screw. This is called insert runout. In this case, the insert having a radial offset rotates 77 Chapter 4. Process Faults in Boring 78 with a larger radius with respect to the axis of the boring head, hence removing more material than the other inserts. Similarly, the insert having an offset in the feed direction moves into the workpiece ahead of the other ones and removes more material (Figure 4.1). CD o o" 3 Figure 4.1 : Radial and axial (feed) runouts on a two-insert Valenite boring head In this chapter, process faults in the boring process are investigated and cutting forces are pre-dicted by employing the mechanistic model developed in Chapter 3. A Valenite boring head (Fig-ure 4.1) with two inserts (the same insert as the one used in the implementation of the mechanistic model in Chapter 3) has been utilized in the experiments for the verification of the force predic-tion. Chapter 4. Process Faults in Boring 79 4.2. Mechanics of Multiple Inserted Boring Bar In the multiple-inserted boring process, each insert has three cutting force components, tan-gential, radial, and feed forces (Ft, Fr and Fy). In the evaluation of the mechanics of multiple-inserted boring bars, the tangential and radial forces are combined to give a resulting force FR • (/ is the insert number). The direction of FR for each insert changes with the rotation of the spindle and produces the total force in the X or Y directions. The total feed force acting on the bar is the summation of the feed forces on each insert. Here, the equations of the total force in the X and Y directions are derived by considering a boring head with four inserts. They can later be generalized for any boring head that has a differ-ent number of inserts. Referring to Figure 4.2, 0 is an angle to define the direction of FR •, 0 = 90-P (4.1) where P is the rotation angle of the boring head in the clockwise direction. The direction of the resultant force FR • for each insert is determined with the following equations (4.2) and (4.3). a, 3 = 0 + atan (Fr \ F (4.2) tt2,4 = a t a n V '2,4 J -0 (4.3) where the indices 1, 2, 3 and 4 imply the insert numbers. The total forces in the X and Y direc-tions are calculated as, Fx = F^cosa, + F^cosaj - F^3cosa3 - F^cosa 4 (4.4) Fy = FR sin aj - FR sin a 2 - FR sin a 3 + FR sin a 4 (4.5) Ideally, Fx and Fy are expected to be zero for cutting conditions without any process faults, because the resulting forces FR • on each insert should be equal and cancel each other for any Chapter 4. Process Faults in Boring given cutting parameters. Process faults produce force differences on the inserts causing a peri-odic variation in the total force in the X and Y directions (Fx, F ). The period of Fx and Fy is equal to the period of the spindle (T = 60/n), where n is the spindle speed in [rpm]. Cutting torque and power are obtained based on the tangential force Ft on each insert as, N (4.6) 2 Ld' t i= 1 1000 (4.7) i= l where D is the diameter of the hole in [mm], N is the number of inserts on the boring bar, n is the spindle speed in [rpm] and Ft • is the tangential force on i th insert. / Ro ta t i on a n g l e Figure 4.2 : Force diagram of a boring bar with four inserts Chapter 4. Process Faults in Boring 81 4.3. Insert Runout in the Radial and Feed (Axial) Directions When the inserts on the boring head have runout in the radial or feed directions (or both), the amount of material being removed by each insert becomes different. This causes an unbalance in the total force in the X or Y direction. If there were no insert runout, the total force measured in the X or Y direction would be expected to be zero, as the tangential and radial cutting forces (Ft, Fr) acting on each insert are equal and in opposite directions, thus canceling each other (Figure 4.5). For the Valenite boring head intended to be used in this study, there are more than 12 possible uncut chip area configurations to be considered, depending on which insert has the offset in the radial or feed directions. For example, one of the offsets may be on the first insert as the second one is on the second insert, or one insert may have both radial and feed offsets at the same time. In this study, only two configurations, which are defined in detail below, are considered. The condi-tions of these configurations are as follows. Configuration 1: Runout in the feed direction is on the first insert and greater than the desired feed rate for one insert e^ > c, where c is in the unit of [(mm/rev)/insert]), andrunout in the radial direction is greater than zero, er > 0, and is on the second insert (Figure 4.3). Configuration 2: Runout in the feed direction of the first insert is less than the desired feed rate for one insert, £y< c, where c is in the unit of [(mm/rev)/insert]), and runout in the radial direction of the second insert is greater than zero, zr > 0 (Figure 4.4). The corresponding uncut chip areas are illustrated in Figures 4.3 and 4.4. These uncut chip areas also represent the amount of material removed by each insert. As can be seen from these fig-ures, the uncut chip area has rather an irregular shape due to the runouts (er and Ey). The uncut chip area A and cutting edge contact length Lc for each insert are calculated in a similar fashion as performed in Chapter 3, Section 3.3.1.1 in order to predict the forces. In the process faults model, the depth of cut a is defined with the depth of cut of the insert that has the radial runout er. The phase difference between the inserts is T/N in time and 2n/N Chapter 4. Process Faults in Boring 82 in angular position, where N is the number of inserts on the boring head and Tis the period of one revolution. If a Valenite boring head with two inserts is considered, the phase becomes T/2 and 7t in time and angular position, respectively. When there are more than 2 inserts on the boring head, the number and complexity of the uncut chip area configurations increase depending on the distribution of the runout among the inserts. In such a case, for an accurate force prediction model to be achieved, each single uncut chip area configuration needs to be defined based on the inserts' runouts and their corresponding directions. Chapter 4. Process Faults in Boring 83 Figure 4.3 : a-) Configuration 1: The amount of material removed from the workpiece when the radial runout of insert 1 is greater than 0 (er > 0) and feed runout of insert 2 is greater than feed rate (8y > c), b-) Uniform uncut chip area for the condition without any insert runouts. Chapter 4. Process Faults in Boring 8 4 Figure 4.4 : Configuration 2: The amount of material removed from the workpiece when the radial runout of insert 1 is greater than 0 (£,. > 0) and feed runout of insert 2 is less than feed rate (ef < c) Chapter 4. Process Faults in Boring 85 Figure 4.5 : Valenite boring head with twin cutter; Runout in radial and feed (axial) directions; The amount of material removed by each insert is shown by the shaded area in the bottom right part of the figures Chapter 4. Process Faults in Boring 86 4.4. Deviation of the Boring Head from the Hole Center As explained in the introduction, the boring operation is performed in conjuction with other processes. Consequently, the centers of the boring head and the hole, which is produced in the preceding operation, need to be aligned for an accurate process. When the center of the boring head has a deviation with respect to the hole center, the depth of cut varies around the rotational axis of the boring head. The uncut chip area A and the cutting forces (Ft, Fr and Fj-) follow the same variation in the relationships formulated in Chapter 3 (Eq. (3.13), (3.14), ( 3.22), (3.23) and (3.24)). The depth of cut reaches a maximum when the tool reaches the clockwise rotational posi-tion of, where Ax and Ay are the deviations in X and Y directions. The depth of cut variation caused by these deviations occurs only in the first pass of the boring process, after which the centers of the boring head and hole align. The depth of cut for each insert is different at each angular loca-tion around the spindle axis because of misalignment and insert runouts (Figure 4.6). Depending on the magnitude of the misalignment, the configuration of the uncut chip areas of each insert may not be constant, and may jump to another configuration during the revolution. In other words, the configurations of the uncut chip areas could be different at rotational increments in the simulation. Once this happens, the simulation process must be able to select the corresponding configuration so that the uncut chip area A, effective lead angle § L , cutting force coefficients (Ktc, Kfrc, Krc and Kfc) and forces (F(, Fr and F A can be calculated accurately. P = P.-7C (4.8) where (3; is the angle where the depth of cut a is the minimum and calculated as, (4.9) Chapter 4. Process Faults in Boring 87 As previously mentioned, the deviation of the boring head affects the force variation only during the first pass. If the deviation is large compared to the intended depth of cut a, the force variation caused by the runouts and the deviation may cause the system to forced vibrations. Figure 4.6 : Schematic illustration of the deviations Ax, Ay of the boring head from the hole center Chapter 4. Process Faults in Boring 8 8 0 go 180 270 0-360 ; i ; a ! a 3 a0: Rotational Axis Process with deviation 'Insert Figure 4.7 : The depth of cut variation caused by deviations Ax, Ay (a is the intended depth of cut) Chapter 4. Process Faults in Boring 89 4.5. Experimental Setup The experimental setup used for the force calibration consists of components similar to those used for data collection in the mechanistic model development in Chapter 3. The same insert o (i.e.Valenite CCGT432-FH 80 Carbide PVD coated diamond insert) for which the mechanistic model was developed has been used in the experiments for the verification of the force prediction in the presence of runouts in the radial and feed directions. Two Valenite CCGT432-FH inserts were attached to a Valenite boring head (Figure 4.1). The experiments were conducted on a FADAL VMC-2216 machining center (Figure 4.8). A Kistler 9255B-605027 3 axis dynamometer, Kistler charge amplifiers, and CutPro-MalDAQ data acquisition software, were used to generate, amplify, and digitize the force signals. The total resulting and feed forces were measured in the X, Y and Z directions, respectively. In the preparation of the experiments, the dynamometer and vise were first mounted on the table of the FADAL machining center. Then the hollow cylindrical workpiece, which is the same as the one used in the development of the mechanistic model in Chapter 3 (See Figure 3.7), was held on the vise. The runouts of the boring head were measured using the dial gauge with 0.01 [mm] precision. During the experiments, the workpiece is stationary but the boring head rotates and moves into the workpiece at the speed of feed rate, c [(mm/rev)/insert]. The signals produced by the dynamometer were amplified by charge amplifiers and sent to the data acquisi-tion software . Chapter 4. Process Faults in Boring Figure 4.8 : Fadal VMC-2216 Machining Center Chapter 4. Process Faults in Boring 91 4.6. Mechanistic Model Verification Force variation has been simulated around the rotational axis and experimentally verified for the case in which the boring head has only runouts in the radial and feed directions (er and 8y). However, the case in which runouts and deviation are considered together was only simulated, not validated. The cutting forces were predicted as a function of the uncut chip area A, cutting edge contact length Lc, and cutting speed V, in the mechanistic model developed in Chapter 3. As stated ear-lier, the mechanistic model also has the ability to predict the cutting forces for the case in which the uncut chip area does not have a uniform but rather an irregular and complex shape. The shapes of the uncut chip area become irregular with the incorporation of insert runouts (i.e. er and Ef) and deviation of the boring head from the hole center (Ax and Ay). The uncut chip area A and cutting edge contact length Lc need to be calculated for the force prediction. Force prediction considering the runouts of the inserts is executed in the following order. - Angular increments for one full revolution are set for the force simulation. - The uncut chip areas (Ax and A 2 ) are calculated separately for each insert (insert 1 and insert 2). This calculation is performed in the same way as in the development of the mechanistic model. - Centers of gravity of the uncut chip areas corresponding to each insert, Gx and G 2 , are determined in the way presented in Chapter 3, Eq.(3.25). - The modification factor Km is predicted using Eq. (3.28) and (3.29). - Effective lead angles (J)L for each insert are predicted based on the calculated Gx , G2 and Km Eq. (3.26). - Cutting edge contact lengths of each insert (LC [ and L C J are calculated using the method in Chapter 3, Section 3.3.1.1. - Tangential cutting force coefficient Ktc and tangential force Ft are predicted (Eq.(3.13) and (3.22)). Chapter 4. Process Faults in Boring 92 - Friction cutting force coefficients KfrCi, Kjrc^ and friction force Ffr are predicted (Eq. (3.14), (3.23), (3.24)). - Radial and feed forces,Fr and Fy, are calculated with Eq. (3.26) and (3.30). - Total resulting cutting forces in the X and Y directions, Fx and Fy, at each angular incre-ments are predicted with Eq. (4.4) and (4.5). Due to the rotation of the spindle, the amplitude of the resulting forces varies with the same period of the spindle, T. 4.7. Experimental Verification of the Mechanistic Model for Process Faults For the verification of the mechanistic model, experiments were conducted with different insert runouts in both radial and feed directions. The cutting conditions were selected so that the uncut chip area of the inserts stays in the two configurations mentioned in section 4.3. In these selections, the radial and feed runouts were distributed separately (i.e. radial runout was on the first insert while the feed runout was on the second insert or vice versa). Four experiments were conducted under the conditions specified in section 4.3. Figure 4.13 shows the simulated total force Fx in the X direction when the boring head has both runouts in the radial and feed directions er and 8y, and deviations defined by Ax and Ay. Note that when the deviation increases the peaks of Fx shift to the right and the left. This can be explained as follows: the uncut chip area for each insert has an irregular shape, due to the radial and feed runouts. When the deviation of the boring head is also included, the variation of the depth of cut of each insert is affected, and the engagement of the inserts with the workpiece vary in the depth of cut direction as shown in Figure 4.7. The uncut chip areas of each insert have sim-ilar irregular shapes. Since there is no linear relationship between the cutting force and depth of cut variation, the effect of the deviation may reflect the force variation as positive and negative shifts as shown in Figure 4.13 (i.e.the maximum peak of the force diagram shifts to right-hand side while the minimum peak moves to the left, with an increase of the boring bar deviation). When deviations in both directions are zero (Ax = 0 and Ay = 0), the total force in the X direc-Chapter 4. Process Faults in Boring 93 tion Fx makes its minimum at the rotational position of 64° while it has its maximum value at 244° angular position. Note that there is a 180° phase shift between the two peaks for this condi-tion. When both Ax and Ay are set to 0.05[mm], Fx has its maximum and minimum at the posi-tions of 60° , and 250°. The angular phase shift is 190° in this case. Similarly, for the deviations of Ax = 0.10 [mm] and Ay = 0.10 [mm], the maximum and minimum forces are on angular positions of 57° and 256°, increasing the phase shift to 199°. This investigation states that, the deviation of the boring head changes the amplitude of the total force in the X direction (Fx) as the phase shift between the maximum and minimum forces increases. The conditions in which the experiments were performed for the verification of the force pre-diction are shown in the following Table 4.1. The results of this investigation are presented in Fig-ures 4.9, 4.10, 4.11 and 4.12. As seen, for the first three tests, the forces were predicted with a good accuracy of under 10% error. However, in the last test the error is over 10%. This may be due to a mistake made at the stage of the measurement of runouts. As a result of this experimental verification, it can be stated that the mechanistic model can be used for the force prediction of a cutting condition that generates an irregular and complicated uncut chip area shape. Table 4.1: Experiments with runout in both feed and radial directions; ax and a2 are the radial depth of cuts of insert 1 and insert 2. Exp. # V [m/min] c [mm/rev-insert] 8 f [mm] 8 r [mm] a-|[mm] &2 [mm] 1 150 0.0600 0.09(insert 2) 0.20 1.485 1.285 2 100 0.0700 0.12(insert 2) 0.10 1.830 1.730 3 175 0.0550 0.14(insert 2) 0.18 1.100 0.920 4 225 0.0900 0.055(insert 1) 0.25 0.870 1.120 where a{ and a2 are the depth of cuts of the insert 1 and 2, respectively. As stated earlier, in the model, the intended depth of cut and radial runout are considered with respect to the insert that has the largest depth of cut. In other words, if the depth of cut of the insert 2 is greater than the Chapter 4. Process Faults in Boring 94 depth of cut of the insert 1, i.e. a2 > ax, the intended depth of cut is assumed to be a2 and radial runout er, which is calculated as er = a2 - ax, is on the insert 2, or vice versa. It should be also noted that the radial and feed (axial) runouts are on different inserts in the above experimental conditions. — Predicted Force — Measured Force 0.2 0.4 0.6 Time [sec] 0.8 — Predicted Force — Measured Force • • \ \ Average rrieasi jred Ff = 112.39 [N] Ff =113.67 [N] 0.2 0.4 0.6 0.8 Time [Sec] The amount of material removed by each insert o i i a x 5 : ai : TJ CtJ oc Insert 2 Feed direction Figure 4.9 : Resulting force in X direction Fx and Feed force Fj prediction for the condition of ax = 1.485[mm], a2 = 1.285mm] , er = 0.20[mm], £ f = 0.09[mm] , c = 0.06[mm], V = 150[m/min] Chapter 4. Process Faults in Boring 95 The amount of material removed by each insert Insert 1 Figure 4.10 : Resulting force in X direction Fx and Feed force Fy prediction for the condition of «j = 1.830[mm], a2 = 1.730mm] , er = 0.10[mm], Ef = 0.12[mm] , c = 0.07[mm], V = 100[m/mm] Chapter 4. Process Faults in Boring 96 — Predicted Force — Measured Force Average measured F{=7 8.34 [N] Predicted Ff III I . =67.39 [N] 0.2 0.4 0.6 Time [sec] 0.2 0.4 0.6 0.8 Time [sec] Figure 4.11: Resulting force in X direction Fx and Feed force Fj prediction for the condition of ax = 1.1 [mm] , a2 = 0.92[mm] , Er = 0.18[mm], 8y = 0.14[mm] , c = 0.055[mm], V = 175[m/min] Chapter 4. Process Faults in Boring 97 1 5 0 — Predicted Force — Measured Force Average measured F x = + 72.92 [N] ft 0.4 0 .6 T i m e [sec] LL CD O CD CD 1 9 0 1 7 0 1 5 0 1 3 0 1 1 0 9 0 7 0 5 0 0 — Predicted Force — Measured Force Average measured Ff = 125.92 [N] Predicted Ff=91 .28 [N] 0.2 0 .4 0 .6 0 .8 Time [sec] The amount of material removed by each insert Figure 4.12 : Resulting force in X direction Fx and Feed force Fy prediction for the condition of ax = 0.870[mra], a2 = 1.12mm] , 8 r = 0.25[mm], 8y = 0.055[mm], c = 0.09[mm], V = 225 [m/min] Chapter 4. Process Faults in Boring Predicted Fx (Runouts:8r , 8 f ) Li.': Predicted Fx (Runouts:8r,Sf + Deviations: Ax ,Ay) Rotational Angle [Deg] Figure 4.13 : The variation of the total force in X direction when considered the process has insert runouts in radial and axial directions, and deviation in both X and Y directions.er = 0.1 [mm], 6y = 0.12 [mm], V = 100 [m/min], a = 1.83 [mm],c = 0.07 [(mm/rev)/Insert], Axj = 0[mm], AJC 2 = 0.05 [mm],Ajt3 = 0.10 [mm], Ay{ = 0 [mm], L\y2 = 0.05 [mm] and Ay3 = 0.10 [mm] Chapter 4. Process Faults in Boring 99 4.8. Summary In this chapter, the effects of process faults on cutting forces have been investigated. Using the mechanistic model developed in Chapter 3, the cutting forces have been predicted for the ver-ification of only two uncut chip area configurations. This validation also shows that, in the mech-anistic model, relating the cutting forces not only to cutting parameters, but also to the geometry of the uncut chip area allows the prediction of cutting forces for a cutting condition that causes the material to be removed with an irregular uncut chip area shape. Chapter 5 Dynamic Modeling of Boring and Chatter Stability 5.1. Introduction Chatter vibrations are an undesirable phenomenon in machining operations. If uncontrolled, they result in poor surface finish and dimensional accuracy, may damage machine tool compo-nents, cause early tool wear, chipping and failure of the cutting tool and generate undesirable noise. Various factors cause vibrations in machining. Vibrations are categorized as forced or self-excited. Forced vibrations can be generated by unbalanced rotating components of the machine tool, and backlash in the transmission gear. Chatter vibrations are self-excited vibrations induced by the regenerative effect, which can be explained with a simple orthogonal cutting process such as plunge turning (Figure 5.1). In this process, the tool moves into the rotating workpiece mounted between the chuck and the tail stock, with the feed rate of c which is also equal to the intended chip thickness hQ. The workpiece tends to oscillate under the feed force Fy, due to its flexibility in the feed direction. The relative vibrations between the tool and the workpiece generate waves on the cut surface. In the following pass, the tool encounters a wavy surface and removes a chip with time-varying thickness, h . Since the feed force is proportional to the uncut chip area, it (the feed force) follows the same periodic variation and excites the structure, causing chatter vibra-tions. The regenerative effect is caused by the phase difference £ between two successive revolu-tions. If the phase difference is zero, as encountered when the frequency of the spindle speed n [rev/sec] and the chatter frequency have an integer ratio, self-excited force and chatter vibra-100 Chapter 5. Dynamic Modeling of Boring and Chatter Stability 101 tions may not be generated. The relationship between the spindle speed n and chatter frequency coc is expressed with the following equation. 60co. c C = N+±- (5.1) n 2n where the chatter frequency coc is in [Hz] and the spindle speed n is in [rev/min]. N is the integer number of waves and E/2n is the fraction of a wave. When the phase angle e is zero, the chip thickness does not vary even though there are still vibrations taking place. The most drastic variation in chip thickness occurs when the phase angle becomes 180° (Figure 5.1). In the model, the feed force Fy is expressed as a linear function of the width of cut b, chip thickness h, and the feed force cutting coefficient Kp assuming that the cutting coefficient does not vary significantly for the selected range of the cutting parameters and is constant. In reality, the cutting force coefficient is not constant, but a function of the instantaneous chip thickness and cutting speed [9]. This approach has been modified in Chapter 3, and cutting coefficients have been established as exponential functions of the uncut chip area A, cutting speed V, cutting edge contact length Lc and centroid of the uncut chip area. At high speeds, the material being cut tends to soften due to excessive heat, and the shear stress decreases causing the forces to lessen. How-ever, for the low speed range, the variation of the cutting force with the cutting speed and the chip thickness may be negligible, as considered in Equation (5.2). This approach simplifies force pre-diction modeling and system stability analysis. The dynamic variable of the orthogonal cutting system is the chip thickness h(t) that incor-porates the vibrations of the tool at current and previous revolutions, y(t) and y(t - T) Eq(5.4). F/r) = KjA(t) (5.2) A(t) = ah{t) (5.3) h(t) = h0-y(t)+y(t-T) (5.4) Chapter 5. Dynamic Modeling of Boring and Chatter Stability 102 where a, hQ and A are the depth of cut, the intended chip thickness and the uncut chip area, respectively. ho " V Dynamic-uncut chip area Static-uncut chip area - i h h Tool Tool n e=o Chapter 5. Dynamic Modeling of Boring and Chatter Stability 103 Figure 5.2 shows a block diagram representation of the chatter vibrations, in which all the dynamic parameters of the system are presented in the Laplace domain. In the model, the input of the system is the intended chip thickness hQ and the output is the current vibration y(s). The vibration of the tool during the previous revolution is represented with a delay term, —s T e y(s) = Ly(t-T), where T is the period for one full revolution. The stability is analyzed based on the frequency response method. The mathematical derivations yield the following equa-tion, (5.5) [4]. a u m = 2tf/?(a>c) ( 5 - 5 ) where aUm is the maximum allowable critical depth of cut for the stable cutting condition, and G(coc) is the real part of the transfer function of the structure 0(s). As it is difficult to model the nonlinearities of the system, such as process damping, the tool jumping out from the work-piece, dependency of the cutting force coefficients on cutting parameters, and multiple regenera-tion, the chatter stability is examined based on the linear stability theory introduced by Tobias[30], Tlusty[4] and Merrit[7]. Nonlinearities of the process can only be taken into account when a time domain simulation method is used. One of the disadvantages of the time domain solution method is that the process is considered with small time increments, and therefore requires a long time to compute a result. On the other hand, analytical modeling is considerably faster and is preferred in practice. In the past decades, extensive research has been devoted to the chatter problem in machining, but detailed work on chatter stability in boring has been rare. This may be because the tool used in the boring process is a single point cutting tool with a corner radius, such that the cutting geome-try in the boring operation is more complicated compared to other machining operations. In this chapter, the dynamic characteristics and the regenerative effect in the boring process are presented in detail. The stability solution of the boring process in both frequency and time Chapter 5. Dynamic Modeling of Boring and Chatter Stability 104 domains is discussed from all perspectives, based on the information introduced in the following sections. y0(s) K f a Ff(s) Ms) y(s) y(s) Inner Modulation Outer Modulation Figure 5.2 : Block diagram of the regenerative chatter vibrations in orthogonal cutting Chapter 5. Dynamic Modeling of Boring and Chatter Stability 105 5.2. Dynamic Characteristics of the Boring Process 5.2.1. Regenerative Effect in Boring The regenerative effect in the boring process is different from the one in milling and turning processes. The boring bar structure is significantly stiffer in the feed direction, which aligns with the boring bar axis. Chatter vibrations are therefore caused mainly by flexibility in the radial and tangential directions (Figure 5.4). For the purpose of conceptual evaluation, the boring bar struc-ture can be modeled by two orthogonal single degree of freedom systems in the radial and tangen-tial directions. A schematic illustration of the boring bar structure is shown in Figure 5.5. The position of the tool in the boring operation is defined with only the current instantaneous vibration y(s) in the radial direction. a(s) = a0 + y(s) (5.6) where a(s) and a0 are the instantaneous dynamic and intended radial depths of cut. The next position of the tool in the feed direction will be ahead of the previous position by the amount of the feed rate c. Due to the phase angle, only the periodic uncut chip area interactions, which depend on the tool positions in the successive revolutions, produce the regenerative effect. The regenerative effect in boring can be explained with Figure 5.6. The figure depicts the simulated tool positions by assuming sinusoidal chatter vibrations with 0.1 [mm] amplitude. In the simula-tion, only one period of sinusoidal path, which also implies the generation of one wave, is consid-ered. The parameters for the simulation have been selected from experiments conducted under the conditions defined in Table 5.1. Table 5.1: Simulation parameters selected from the conducted chatter tests Depth of Cut, a [mm] Feed Rate, c[mm /rev] Cutting Speed, V[m/min] 0.8[mm ] 0.10 225 Spindle Speed, n [rpm] Measured Chatter Freq. [Hz] Measured Phase Angle, 8 [Deg] 2228 771 274.7 Chapter 5. Dynamic Modeling of Boring and Chatter Stability 106 Figure 5.6 shows 6 full and a fraction of the 7th revolutions. As it can be noticed that the depth of cut a and the uncut chip area A have different but periodic variations at each revolution, due to the phase angle 8 between two successive revolutions. The difference in depth of cut vari-ation at each revolution continues until the number of revolutions times phase angle adds up to 360°; when the subtotal of the fractions of the waves e/27t becomes one full wave (one period, 2TT ). The depth of cut variation for each revolution is then repeated in the same fashion. This sec-ond periodic repetition is illusrated in Figure 5.7, with the surface roughness measurement of a workpiece on which the chatter test was performed with another specified cutting condition (depth of cut a=0.75[mm], feed rate c=0.12[mm/rev], cutting speed V=184[m/min], spindle speed n=1650[rpm]). In this test, the chatter frequency is measured as coc= 849.7 [Hz] and the phase angle is obtained as E S 324° using Equation (5.1). The surface roughness measurement is per-formed with a surface roughness analyzer. It should be noted that the measurement has been car-ried out on a linear reference path in the feed direction (or on the same angular position of the hole circumference). In other words, the tool positions shown in the figure are taken on this line. In this test, the process is expected to have a different depth of cut variation in each period of the first 10 revolutions, due to a 324° phase angle. After 10 revolutions, the phase angle completes itself to 360° (or 0°) , and the same depth of cut variations seen in the first 10 revolutions will be repeated. The determination of the number of the revolution N (corresponds to 10 in the above example) when the phase angle e becomes zero, and the corresponding phase angles at each rev-olution, are shown in the following algorithm (Figure 5.3). This algorithm determines wave gen-eration characteristics based on spindle speed n [rpm], and measured chatter frequency coc. It is therefore assumed that the test has been conducted under specific cutting conditions, and the chat-ter frequency coc has been determined from FFT of the measured force (or acceleration) data. Using Eq.(5.1), the phase angle e, which is assumed to be the initial phase angle at the 1st revolu-Chapter 5. Dynamic Modeling of Boring and Chatter Stability 107 tion of the process with chatter vibrations, is first calculated. Then, the phase angle e at each rev-olution is found. This regeneration process keeps repeating itself until the end of cutting. Input: Phase angle, 8 Calculate: the phase angle in each revolution. E , = j.e Yes 8 = 8 : i = floor(8j /360) 8 = 8 n - i.360° Find: The remainder (CC) after the division of 8 into 360° j=j+1 This algorithm finds the total number of revolution N for the completion of the phase angle to 360 [Deg] (i.e. after N revolutions the tool will start the (N+1)th revolution in the phase angle of 1st revolution. After N+1 Jth revolution, the same depth of cut (or uncut chip area) variation perceived in the first N revolutions will be repeated upto 2N th revolution. Figure 5.3 : Algorithm for the determination of the phase angle for each revolution At each instantaneous position of the tool, the uncut chip area may have an irregular and complex form, due to the corner radius of the tool. For example, if the tool is frozen at each revo-lution when it reaches a reference line, the corresponding uncut chip area takes the forms shown in Figure 5.6. The amount of material removed by the tool at each revolution differs, and may have a complicated form in some cases. Chapter 5. Dynamic Modeling of Boring and Chatter Stability 108 So far, the significant differences between regeneration of the waviness in boring and, turning and milling have been highlighted. The following question ensues: Do the vibrations in the tan-gential direction influence the regeneration of the waviness in addition to those in the radial direc-tion? As it can be recognized from the transfer function measurement (Figure 5.4), the boring bar in the tangential direction is as flexible as in the radial direction. However, its contribution to the depth of cut variation is comparatively smaller than that in the radial direction. Consequently, the influence of vibrations in the tangential direction can be assumed to be negligible in the evalua-tion of the regenerative effect in boring. The influence of the tangential displacements can be examined with Figure 5.8. Considering the boring bar with the tangential and radial vibrations (Ax and Ay), the total depth of cut varia-tion is found to be, where a, ad, Ay, Ax, R and Aat are the intended depth of cut [mm], dynamic depth of cut [mm], the displacement of the tool in radial direction [mm], the displacement of the tool in tan-gential direction [mm], the radius of the hole [mm] and the variation of the depth of cut caused by the displacement of the tool in tangential direction, respectively. The influence of the displacements in the tangential direction on the depth of cut variation can be examined with an example. Ax = 0.1 [mm] displacement in the tangential direction, when the radius of the hole is 40 [mm], leads to 1.25 x 10 [mm] depth of cut variation, Aat. How-ever, the same displacement in the radial direction has a direct effect on the depth of cut variation with the same magnitude. Therefore, it is reasonable to neglect Aat in order to simplify the prob-lem. ad = a + Ay + Aat (5.7) The portion of the tangential displacement in depth of cut variation is, (5.8) Chapter 5. Dynamic Modeling of Boring and Chatter Stability 109 x 10" Transfer function in radial direction 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency [Hz] x 10"6 Transfer function in tangential direction x 10 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency [Hz] 7 Transfer function in feed direction 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency [Hz] Figure 5.4 : Transfer function of the boring bar in tangential, radial and feed direction Figure 5.5 : Boring bar structure with two spring-mass and damping models of a single degree of freedom system Chapter 5. Dynamic Modeling of Boring and Chatter Stability 111 | 0 .95 > 0.9H I 0 .8 m t 0 .7 •< >~ - >> . Lv?VV 'fe s • . . . \ 0 2 3 4 Length [mm] Feed direction Figure 5.6 : Regeneration of the waviness in boring process Chapter 5. Dynamic Modeling of Boring and Chatter Stability 112 Surface Roughness Measurement Surface finish with chatter marks Surface roughness analyzer cfhn Tool positions \ . \ / / g Tool position at the current revolution Feed direction c=0.12 [mm] 1.20 [mm] ( 1.20 [mm] Figure 5.7 : Surface rougness measurement of the workpiece, a =0.75 [mm], c=0.12[mm/rev], V=184[m/min], n=1650[rpm], coc s 849.7 [Hz], e = 324 [Deg Chapter 5. Dynamic Modeling of Boring and Chatter Stability Figure 5.8 : The effect of the tangential vibrations in the regeneration of the waviness Chapter 5. Dynamic Modeling of Boring and Chatter Stability 114 5.2.2. Dynamic Cutting Force Prediction The uncut chip area has a uniform shape during rigid cutting conditions. However, it varies drastically and may have an irregular and complicated form when the tool begins to vibrate (Fig-ure 5.9). In some cases, more than three revolutions before the current one may need to be taken into consideration for the uncut chip area calculation and radial force prediction. Chapter 3 described the calculation of the uncut chip area, considering its exact geometry for cutting condi-tions with free chatter vibrations. The radial force was obtained as an exponential function of the cutting parameters Eq. (5.9). The prediction of the radial force requires the uncut chip area to be separated into two regions for more accurate results (Figure 3.15). Regions 1 and 2 are defined on the corner radius and the straight side of the tool, respectively. Based on the equations derived in Chapter 3, the radial force is determined as a component of the friction force. Fr = FfrSin(Q>L) Fr = [K^Al+K^A2 + K^(LCi + LC2)]Sin^L) „ r, 8.1965 T -0.6737 T/-0.4210. . . 9.6152 T -0.0241 T/-0.7597. . „ , r r . . Fr = [(« Lc, v LC2 V )A2 + Kfre(LCi + LC2)]sm(<\>L) (5.9) The radial cutting force is found as, Frc = Fr-KreLc (5.10) Then, the radial cutting force coefficient is obtained by dividing Frc into A. (5.11) Variations in the depth of cut for stable and unstable cutting conditions are not equivalent in terms of the uncut chip area and force variation, due to the nonlinearities of the process caused by the corner radius of the insert R. As explained in the previous section, cutting force variations are completely dependent on the instant interaction between the current and previous tool positions at the same angular position of the hole circumference. Using the mechanistic model presented in Chapter 5. Dynamic Modeling of Boring and Chatter Stability 115 Chapter 3, the friction force is predicted based on the uncut chip area geometry, rather than by using the cutting parameters directly. This consideration was achieved by taking the contact length Lc of the tool with the workpiece into the derivation, and assuming that the total friction force passes through the centroid of each related region (Region 1 and 2). This approach ade-quately covers the complication of the uncut chip area geometry encountered in unstable cutting conditions, and enables the prediction of the cutting forces with good accuracy. In Figure 5.9, the graphical representation of the dynamic friction force distribution along the cutting edge contact length Lc and uncut chip area variation are shown in comparison to those in the regular static case. The friction force is distributed proportionally to the uncut chip area of each differential element along the cutting edge contact length. The figures at the bottom (5.9-bl and b2) show how much error could occur between the direct use of the cutting parameters (static base) and use of the uncut chip area geometry (dynamic base) for the same depth of cut variation in the prediction of the effective lead angle (j)L. In the figure, indices s and d represent static and dynamic instances. Figure 5.9a shows a reg-ular static uncut chip area form, in which the current and previous tool positions are the same. In this case, the chip thickness is reduced towards the tip of the tool, while its elemental effective lead angle increases with respect to the center of the corner radius. The friction force distribution conforms to the chip thickness reduction, which varies proportionally to the uncut chip area of each differential element along the cutting edge contact length. The friction force has the same magnitude at each differential element, and the elemental effective lead angle is the same on the straight side of the uncut chip area. Aditionally, the interaction of two positions that represent the dynamic variation of the depth of cut generates the uncut chip area 3 (Figure 5.9b). For this case, if the static model is considered, the effective lead angle and the total friction force, § L _ S , Fjr_s will be over-predicted (Figure 5.9-bl). Consequently, the radial and feed forces ( F r _ s and F j _ s ) will be over-predicted as well. Figure 5.9-b2 shows the actual friction force distribution and the total friction force Frr_d with corresponding effective lead angle § L _ d - Presumably, the error Chapter 5. Dynamic Modeling of Boring and Chatter Stability 116 varies depending on the current interaction of the tool positions. Another schematic comparison is illustrated in Figure 5.9-c, in which the current tool position interacts with the tool position of the second revolution before the current one. This interaction produces the uncut chip area 4. Similar to this last case, when the current depth of cut becomes greater than the previous one, the total friction force Fy r and effective lead angle (])L are under-predicted when the static-based model is considered. i i Stable cutting condition i • Unstable cutting condition U J Distribution of the dynamic-friction force along the cutting edge a-) b-) c-) Figure 5.9 : The uncut chip area variation for unstable cutting condition; s and d imply the static and dynamic cutting conditions. Chapter 5. Dynamic Modeling of Boring and Chatter Stability 117 5 . 2 . 3 . W a v e G e n e r a t i o n o n t h e S u r f a c e The number of waves left on the cut surface between subsequent revolutions is simply calcu-lated withEq. (5.1). 60 o v e N = c-- — n 2n In the boring process, the surface waviness shows different characteristics depending on the phase angle e and the chatter frequency coc. The number of waves left on the cut surface should be equal to N + — after a cutting operation. However, when counted, it was found to be different 2n than expected. This can be explained as follows. Depending on the interactions of the tool positions that are functions of the phase angle e, different undulated patterns are created on the cut surface. This is explained in Figure 5.11, which shows the simulated surface finish for a wave period of 12 successive revolutions with the follow-ing system parameters: rc=1650 [rpm], D=31.5 [mm] (Diameter of the hole), a = 0.1[mm], e = 275°, coc = 770 [Hz] The period and length of a wave are calculated as, T = -L LT= ^2%RT (5.12) u)„ 60 'c where R is the radius of the circular hole and n is the spindle speed in [rpm]. In the first revolution, the tool reaches maximum depth at point-1 on its sinuosidal route (Figure 5.11 and 5.12). Having completed one full revolution, the tool moves inside the work-piece with the feed rate and reaches the maximum depth again, which is Lp [mm] ahead of the previous depth, at point 2 in the same period. Lp is the distance between two successive depths (in rotational direction) and defined as, e<7t=>L = eRT p (5.13) e > TC => L p = (2n-e)RT Chapter 5. Dynamic Modeling of Boring and Chatter Stability 118 Other geometrical identification parameters may be ct, L-, aj and a 2 , which identify the exact positions of the depths, and the inclination of the wave pattern created on the surface finish (Figure 5.12). These identification parameters can be determined with the algorithm illustrated with the flow charts (Figure 5.13 and 5.14). In these algorithms, the exact phase angle for each revolution is defined with respect to the phase angle in the first revolution. The tool makes the maximum depth when the period angle reaches to — for each revolution. Based on this defini-tion, the angle 0, which indicates the distance of the tool depth Ln from the beginning of the period, is introduced, where n implies the number of revolutions. The positions of the tool depths in the feed direction are calculated with the following equation (5.14). where R is the radius of the hole. Once, Ln and cn are calculated, the depths are grouped and installed as a row of a matrix (as vectors) so that the inclination angles of the wave grooves can be obtained by using the specific elements of the vectors (Figure 5.12). Two inclination angles on the surface finish that describe the route of the tool with respect to the edge of the workpiece are defined as, cn = (n-l)c + (5.14) (5.15) (5.16) where i is the group number, q is the element number of each vector. Hence, L ' 9+1 ' U1 L„ and c are the second and first elements of any position vector defined by a group matrix. Chapter 5. Dynamic Modeling of Boring and Chatter Stability 119 i y (Radial direction) ( 8 t / / / / / / / 1 1 t % s~ \ s / \ 1 \ I \ I \ I \ \ \ .. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ / \ / \ / \ f / / X (Tangential-rotational direction) \ \ \ / -< 271 3 • Depths Figure 5.10 : Definition of the depth positions with 0 The process continues as formulated and produces the illustrated patterns for the given parameters. When counted, the number of waves is usually higher than expected because the tool depths are arranged in grooves, which are seen as waves. In addition to the first one, two more examples exposing different patterns can be depicted for different parameters (Figure 5.15 and 5.16). Chapter 5. Dynamic Modeling of Boring and Chatter Stability 120 Figure 5.11 : Wave generation, phase angle 8 = 275°, chatter frequency coc = 770 [Hz], black and white dots show the depths the tool make in the first 6 revolutions Chapter 5. Dynamic Modeling of Boring and Chatter Stability 121 group 1 Figure 5.12 : Identification of the surface finish geometrical parameters. Black dots show the depths created by the tool in its sinusoidal vibration; n = 1650[rpm] ,D = 31.5 [mm], e = 275°, coc = 770[Hz] er 5. Dynamic Modeling of Boring and Chatter Stability Input: Phase angle, s Feed rage, c Length of the period, L T Circumference of the hole, C Calculate: the phase angle in each revolution. 8 n = n.s n =n +1 Figure 5.13 : Depth position determination algorithm in each wave period Chapter 5. Dynamic Modeling of Boring and Chatter Stability Input: L n for each revolution Set: Flag = 0 i = 1 group Figure 5.14 : Flow chart to group the deeps for the identification of the inclination angles and ct2. Figure 5.15 : Simulated wave generation on the surface finish under the contidion of phase angle e = 170°, chatter frequency coc = 770 [Hz]; Black dots show the depths. Chapter 5. Dynamic Modeling of Boring and Chatter Stability 125 Figure 5.16 : Simulated wave generation on the surface finish under the contidion of phase angle 8 = 90° , chatter frequency coc = 770 [Hz] Chapter 5. Dynamic Modeling of Boring and Chatter Stability 126 5.3. Chatter Stability 5.3.1. Analytical Approach for Stability Solution This section presents an analytical approach for the dynamic boring process, in which the regenerative effect is modeled considering only the vibrations in the radial direction. As detailed in the preceding sections, the geometry and dynamics of the regenerative effect in boring are different. Nonlinear variation of the radial force with the instantaneous radial vibration, and the preceding vibration history of more than one revolution, makes the modeling of chatter in the boring process more difficult than other existing models, such as milling and turning. The main consideration in the analytical solution is how the nonlinear dynamic characteristic parame-ters, namely radial cutting force coefficient Krc, and uncut chip area A, are involved in the dynamic boring model. The first stage has been to make certain assumptions that will facilitate the analysis, and linearize the relation between the parameters Krc, A, and Lc, which are employed in the radial force prediction and the vibration of the boring bar y. The following assumptions are made to simplify the process. - The amplitude of the chatter vibrations is assumed to be within a range of %10 of the intended depth of cut a. - The radial cutting force coefficient Krc is determined with Equation (5.11) and is assumed not to change and remain constant for the given range of the vibration amplitude. Only the influ-ence of the uncut chip area variation is considered in the radial force variation. - The uncut chip area calculation is simplified with the following linear equation, [10] A = ca (5.17) where c and a are the feed rate and depth of cut. Based on the assumptions made above, the chatter vibrations in the boring process can be represented with the following block diagram, in which three gain factors, Kl, K2, and K3, are used to linearize the dynamic uncut chip area vari-ation (Figure 5.17). The input of the system is the intended radial depth of cut a0. yx, y2 and y 3 Chapter 5. Dynamic Modeling of Boring and Chatter Stability 127 are the vibrations measured on the same angular position of the hole circumference, in the current, previous, and the second revolution before the current one, respectively. The dynamic depth of cut ax is obtained by adding the current vibration yx to the intended depth of cut a 0 (Eq.(5.6)). In this case, the vibration direction into the workpiece is taken as positive. The effect of the previous vibrations (y2 and y3) on the uncut chip area variation are considered with the linear relation pre-sented in Equation (5.18). In the model, gain factors Kx, K2, and K3 are supposed to be tuned throughout the depth of cut range with small segments. For each segment, 20 different combina-tions of the tool vibrations within the vibration amplitude range (10% of the depth of cut) are con-sidered, and the corresponding uncut chip areas are calculated. Based on the calculated uncut chip areas and the expression (5.18), linear regression has been performed for the identification of the gain factors. A = c[K1(a0-y1) + K2(a0-y2) + K3(a0-y3)] (5.18) A = KlAl+K2A2 + K3A3 (5.19) In this evaluation, the dynamic uncut chip area A is determined based on the vibration his-tory of two preceding revolutions considered by the gain factors Kx, K2, K3, and the aspect of the static behaviour of the process, in which the uncut chip area is approximately calculated with Equation (5.17). The geometrical representation of the evaluation is indicated in Figure 5.17. It should be noted that Ax, A2, and A3 are the corresponding static uncut chip areas for the depth of cuts ax, a2, and a3. Chapter 5. Dynamic Modeling of Boring and Chatter Stability 128 ] Stable cutting condition i Unstable cutting condition Figure 5.17 : Illustration of the uncut chip areas in Eq. (5.18) Chapter 5. Dynamic Modeling of Boring and Chatter Stability Figure 5.18 : Block diagram representation of the boring process Chapter 5. Dynamic Modeling of Boring and Chatter Stability 130 The dynamic chip area can be written in Laplace domain as follows. A(s) = c i K ^ a ^ - y ^ - K ^ a ^ - y ^ - K ^ a ^ - y ^ s ) ] } (5.20) The dynamic uncut chip area produces the dynamic radial cutting force with the following expression. Frc(s) = KrcA(s) (5.21) The radial cutting force induces vibration on the boring bar, y(s) = Frc(s)^(s) (5.22) where d>(^ ) is the transfer function of the system, which is implemented based on the dynamic characteristics of the boring bar structure obtained by performing impact hammer tests (Figure 5.19). For the conceptual evaluation, the structure with a single mode can be considered with the following transfer function (Eq. (5.23)). The implementation of the transfer function is explained in detail in the next section. 1 / m Frc(s) s 2 + 2C,(nns + co^  Substitution of A(s) and Frc(s) into Equations (5.21) and (5.22) yields, (5.23) y(s) = -<sT -2iT Kx-K2e -K3e cKr®(s) (5.24) The previous vibrations, y2 and y3, are defined with delay terms in the model. The resulting transfer function between the vibration of the structure y(s) and intended depth of cut a0 can be obtained as follows. y(s) = a0(s) Kx-K2 - £ 3 ] cKrc<S>{s) 1 + -sT -2sT Kx-K2e -K3e (5.25) cKWs) The characteristic equation of the above transfer function is, Chapter 5. Dynamic Modeling of Boring and Chatter Stability l + \K1-K2e-sT-K3e~2sT cKr*(s) = 0 131 (5.26) Output (Acceleration) Hammer Input (Force) Amplifier "^Accelerometer Machine Structure -1 Borinq bar U Amplifier CutPro-Modal Analysis CO, t R M J +\+ ModeS 1 2 Fieguency 1.8112E*03 1.3?15E*03 Dampr.g 2.0909E-02 1.5924E-02 Residue (Re) | B.9127E-0B 3.3*29E-0e| leadueflml 4.4770C 07 B 3821E-08 Displacement [m/N] Measurement #1 Displacement [m/N] 1500 2000 2500 3000 3500 Frequency[Hz] Figure 5.19 : Transfer function measurement using the impact hammer test Chapter 5. Dynamic Modeling of Boring and Chatter Stability 132 After the linearization of the system, the stability of the process can be examined with the fre-quency response method, which is performed by replacing s with ;'to in the transfer function of the system. The magnitude and phase angle of the transfer function (())(/to)) can be illustrated by graphical plots that provide significant insight. The block diagram representation of Eq. (5.22) is shown in Figure 5.20. The input of the sys-tem is sinusoidal radial force Fr(s), and the resulting output is the radial displacement of the structure y(s), which differs from the input waveform only in amplitude and phase angle . Fr(s) Transfer function in radial direction Radial force $.(s) y(s) Radial vibration Figure 5.20 : Block diagram representation of Equation (4.23) The transfer function is described in the frequency domain as, <D(/co) = G+jH (5.27) where G is the real and jH the imaginary part of the transfer function. From the characteris-tic equation of the transfer function (Eq. 5.26) the critical borderline of feed rate value is obtained as, C l i m ~ Kr[G(K1 - K2cos(0)cT) - K3cos(2acT)) - H(K2sin(wcT) + K3sin(2(HcT))] ( 5 ' 2 8 ) clim is the maximum allowable critical feed rate for a cutting condition with free chatter vibration. Phase angle of the structure \|/ is, tf(cri) sm(arT)(K7 + 2^cos((D,7/)) tany = ——°- = v c / v 2 ^ v c " (5.29) G ( c o c ) [-K l+K2cos((£>cT) + K3cos(2d)cT)] Chapter 5. Dynamic Modeling of Boring and Chatter Stability 133 Equation (5.1) is expressed in the form of, tocT = 2nN+z (5.30) where Q)c and T are the chatter frequency in [rad/sec] and period of the one full revolution. Substituting Eq. (5.30) into Eq. (5.29) yields, sm(z)[K2 + 2A"3cos(e)] \|/ = atan (5.31) .[- Kx + K2cos(z) + K3cos(2e)] Equation (5.31) is a nonlinear and requires an iterative solution to find the phase angle e. The variation in the radial cutting force coefficient in the presence of chatter vibrations is investigated with the following parameters: depth of cut, a = 0.7[mm], feed rate, c = 0.1 [mm/rev] and cutting speed, V = I50[m/min]. The displacements of the tool for five successive revolutions were considered to be in the range of %10 of the depth of cut, a, (i.e. T-0.07 [mm]). av a2, a3, aA, and a5 are the instant depth of cuts of the current, previous, third, fourth, and fifth revolutions at the selected 8 positions of a wave period as shown in Figure 5.21. The corresponding vibrations are shown with the same indices with the notation of y. For the selected tool positions, calculated uncut chip areas A, cut-ting edge contact lengths Lc, predicted radial cutting forces Frc and its coefficients Krc, are shown in Table 5.2. Figures 5.22 and 5.23 depict the corresponding uncut chip area shapes, (i.e. the amount of material removed by the tool) at the selected 8 positions of the 5th revolution of the simulated process. As can be noticed, the uncut chip area A varies drastically at these particular positions of 5 sucessive revolutions during each revolution. Figures 5.24, 5.25 and 5.26 show the actual values of the parameters determined, based on the mechanistic model and their average values calculated by considering only the static nature of the process (second assumption). These average values are determined for the test conditions given above. As can be seen in the figures, the error between the predicted dynamic radial cutting force coefficients Krc and average values may go up to 20[%] in some cases. Chapter 5. Dynamic Modeling of Boring and Chatter Stability 134 Table 5.2 : Prediction of the radial force considering the specified vibration history POINT-1 POINT-2 POINT-3 POINT-4 POINT-5 POINT-6 POINT-7 POINT-8 a 1 [mm] 0.6345 0.6673 0.7264 0.7577 0.77 0.7322 0.6641 0.63 a 2 [mm] 0.7211 0.7601 0.7661 0.7427 0.7018 0.6396 0.6381 0.6953 a 3 [mm] 0.7678 0.739 0.6805 0.6468 0.6302 0.6615 0.7295 0.7695 a 4 [mm] 0.686 0.644 0.6318 0.6518 0.6909 0.7563 0.765 0.7119 a 5 [mm] 0.6308 0.6552 0.7123 0.7481 0.7688 0.7444 0.6773 0.6317 L c [mm] 0.5852 0.5932 0.9344 1.3840 1.4312 1.2760 0.9728 0.6823 A [mm 2 ] 0.0251 0.0266 0.0489 0.0878 0.1338 0.1317 0.0710 0.0303 Predicted K r c [N/mm 2] 394.31 343.27 331.95 381.49 391.63 385.43 410.59 413.55 Predicted F r c [N] 9.88 9.14 16.25 33.51 52.39 50.77 29.16 12.52 Predicted F r [N] 14.83 14.15 24.15 45.21 64.49 61.55 37.38 18.29 Predicted V [Deg] 29.27 26.82 31.88 43.76 52.10 49.60 41.91 32.37 Predicted K f r c [N/mm 2 ] 631.44 625.71 460.68 353.57 345.68 373.46 448.37 569.38 Predicted F f r c [N] 15.82 16.65 22.55 31.06 46.25 49.19 31.84 17.24 Predicted F f r [N] 30.33 31.36 45.72 65.37 81.73 80.82 55.96 34.15 Chapter 5. Dynamic Modeling of Boring and Chatter Stability 135 — 1st Revolution — . — . 2nd Revolution — — — — 3th Revolution — . 4th Revolution 5th Revolution .5 1 1.5 2 2.5 3 3.5 4 4.5 5 Length of a period [mm] Tangential (rotational) direction Figure 5.21 : Dynamic radial force simulation, a=0.7 [mm], amplitude of the vibration=0.07[mm] Chapter 5. Dynamic Modeling of Boring and Chatter Stability 136 Position-1 Position-2 a, 0.6345 [mm] a2 0.7211 [mm] a 3 0.7678 [mm] a4 0.6860 [mm] a5 0.6308 [mm] a1 0.6673 [mm] a 2 0.7601 [mm] a 3 0.7390 [mm] a4 0.6440 [mm] a5 0.6552 [mm] Position-3 Position-4 0.7264 [mm] a2 0.7661 [mm] a 3 0.6805 [mm] a4 0.6318 [mm] a5 0.7123 [mm] a1 0.7577 [mm] a2 0.7427 [mm] a3 0.6468 [mm] a4 0.6518 [mm] a5 0.7481 [mm] Figure 5.22 : The uncut chip area variations at the positions 1, 2, 3 and 4 Chapter 5. Dynamic Modeling of Boring and Chatter Stability 137 Pos i t ion-5 a , 0.7700 [mm] a 2 0.7018 [mm] a 3 0.6302 [mm] a 4 0.6909 [mm] a 5 0.7688 [mm] P o s i t i o n - 7 a . , 0.6641 [mm] a 2 0.6381 [mm] a 3 0.7295 [mm] a 4 0.7650 [mm] a 5 0.6773 [mm] Pos i t ion -6 a , 0.7322 [mm] a 2 0.6396 [mm] a 3 0.6615 [mm] a 4 0.7563 [mm] a 5 0.7444 [mm] Pos i t i on -8 a 1 0.6300 [mm] a 2 0.6953 [mm] a 3 0.7695 [mm] a 4 0.7119 [mm] a 5 0.6317 [mm] Figure 5.23 : The uncut chip area variations at the positions 5, 6, 7 and 8 Chapter 5. Dynamic Modeling of Boring and Chatter Stability 138 420 | 410 " 400 ~ 390 1 380 o o CD 370 o o) 360 c 3 CO CO rr 330. 0.14 0.12 No Chatter Chatter 1 1 1 1 1 1 ; 405.40[N] / / / / i : 0- 5 « / / / / i / • . . . » <? 6 / 0 • : ; » : ; \ j . : : ' : \. .' : i ' « ' i ' & : ! < • * v * ^ » 1 "03 , | 1 1 1 1 E E 0.1 < co Cl) Q. O = 0.06 c 0.04 0.02 0.5 1 1.5 2 2.5 3 3.5 Length in a period [mm] 0 0.5 1 1.5 2 2.5 3 3.5 4 Length in a period [mm] 4.5 5 1 I 1 ! o5--/ • i • i 1 6 '"""9 \ \ 1 . i r. i: / : / : \ \ \ \ \ / \ i i \ \ \ \ \ \ 0. 0652 [ m m 2 ] ; / V > \ \ / ; / / \ \ \ \ \ \ • • 2 / -o 1 I V \ b 8 4.5 5 Figure 5.24 : The Variation of the radial cutting force coefficient and uncut chip area for the specified vibration history (Table 5.2) Chapter 5. Dynamic Modeling of Boring and Chatter Stability 139 13, 13 £1 11 <D 10f o 1 o O) 9 c e 3 8 o cn "D 7 Q) 7 "1 03 6 DC No Chatter Chatter 4 o-10.23 [N] 2/ 5 b - - - o _1 I I l_ >8. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 L e n g t h in a pe r i od [mm] 55 50 45 h ==• 40 CD O O) c O 35 30 25 CO T J CO DC K 20 15 10b i 1 1 r o§-j.._. 28.51 [ N ] / 3,' 1 2 , ' —o N8 O J I I I I i_ 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 L e n g t h in a pe r i od [mm] Figure 5.25 : The variation of the radial cutting and edge cutting coefficients for the given vibration history in Table 5.2 Chapter 5. Dynamic Modeling of Boring and Chatter Stability 140 70 60 => 50 8 40 cc T3 co rr 30 20 10. No Chatter 2 , -o OS 4/ o 38.75 [N] 6 H r Chatter .1.8. P 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Leng th in a pe r iod [mm] Figure 5.26 : The Variation of the total radial force for the vibration history in (Table 5.2) 5.3.2. Chatter Stability Prediction in Time Domain This section presents the development of the digital simulation method for the boring process which is to predict the occurrence of chatter vibrations. The time domain model enables us to understand aspects of the regenerative effect and chatter, with a good insight into the dynamic behaviour of the boring bar structure. It is also possible to consider the nonlinearities of the pro-cess in chatter vibrations. One of the nonlinearities is the tool jumping out of the cut, due to vibra-tions with large amplitude. In this case the cutting forces become zero for a short period. The second nonlinearity is caused by the nature of the cutting geometry of the boring process. As explained in detail in the preceding sections, when the tool vibrates, the uncut chip area varies drastically due to the corner radius of the tool, and more than three revolutions may be involved in the evaluation of the parameters necessary for the force prediction. All possible nonlinear interac-tions of the tool positions are considered in the time domain solution. Chapter 5. Dynamic Modeling of Boring and Chatter Stability 141 The block diagram of the time domain solution model is shown in Figure 5.27. The model consists of 3 sub-models, namely a chip load geometry model, a tool dynamics model in radial direction, and a force model. The inputs of the model for simulation are: the radial depth of cut a, feed rate c, and cutting speed V. Based on these inputs, the uncut chip area A, cutting edge con-tact length Lc, and effective lead angle (j)c, which are the necessary parameters for the radial force prediction, are determined. From these parameters, the friction cutting force F^rc and edge cutting force components Fjre are first predicted. Radial force is then calculated from the total predicted friction force F^r, and the effective lead angle § L . Eventually, the radial force generates vibra-tions, changing the radial position of the tool (depth of cut). This process continues with the spec-ified order. The steps of the time domain solution can be summarized as follows. - The workpiece is considered to be rolled out, like a rectangular block, and discretized with small elements. In this discretization, the elements are selected small enough relative to the corner radius of the insert. - Based on the exact kinematics of the boring process, the position of the insert and corre-sponding coordinates defining the cutting edge are calculated and subtracted from the coordinates of the workpiece surface profile. - The simulation starts with the above evaluation for the inputs, which are, depth of cut, feed rate, and cutting speed. Depending on these inputs, the uncut chip area, cutting edge contact length, and effective lead angle, which are necessary for the prediction of the radial force, are determined by utilizing the mechanistic model presented in Chapter 3. - The predicted radial force excites the structure and generates vibrations, and these vibra-tions are calculated. - For each time-increment, the position of the tool (the current depth of cut), the uncut chip area A, the cutting edge contact length Lc, and the effective lead angle § L , are updated, based on the calculated vibration and current interaction between the insert and workpiece surface profile. Chapter 5. Dynamic Modeling of Boring and Chatter Stability 142 Figure 5.29 illustrates a scheme of the simulation, showing the amount of material being removed from the workpiece for an instant of time. As can be noticed in the figure, the fourth revolution before the current one is being taken into account for the determination of the simulation parame-ters and the radial force prediction. - The workpiece surface profile is updated after the evaluation of the tool displacement at each time-increment. - The process continues in the specified course for each increment of the workpiece rotation. The detailed algorithm steps of the time domain simulation is illustrated in Figure 5.28. FORCE MODEL Lc(t) Depth of cut Feed rate + Cutting s p e e d V TOOL DYNAMICS MODEL IN RADIAL DIRECTION , 8y(t) sin <hL(t) Figure 5.27 : Block diagram of time domain simulation model Chapter 5. Dynamic Modeling of Boring and Chatter Stability 143 E n t e r : Process parameters Depth of cut, a [mm] Feed rate, c [mm/rev] Cutting speed, V [m/min] 1 r R e a d : *.cmp data file containing the experimental modal analysis result. \ 1 II I d e n t i f y : Transfer function, <j>(s) 1 \ S e t : Simulation parameters: number of divisions (grids), time increment, number of desired revolution \ C r e a t e : workpiece.bin data file consisting grids on the workpiece surface profile \ D e f i n e : The tool geometry as grids J • S e t : The initial condition: time, cutting force, vibration • • | S t a r t : The simulation || Figure 5.28 : Algorithm of the C a l c u l a t e : The parameters: the uncut chip area (A), cutting edge contact length (L c), Effective lead angle ((|>L) C a l c u l a t e : The cutting forces and vibration for the initial condition (Runge Kutta) I U p d a t e a n d S a v e : The workpiece surface profile and tool position \ \ C h e c k : If the tool is still in contact with the workpiece (jumping out of the workpiece) \ U p d a t e : Parameters: A, L c and • • C a l c u l a t e : The cutting forces and vibration. time domain solution model Chapter 5. Dynamic Modeling of Boring and Chatter Stability Figure 5.29 : Illustration of the tool position for an instant of time in time domain simulation Chapter 5. Dynamic Modeling of Boring and Chatter Stability 145 5.3.2.1. Tool Dynamics Model Using the experimental modal analysis method [2], the transfer function of the structure is obtained in the following form. m = m= y ^Klhl vv > f(s) la 2 .„ (5.32) where n is the total number of modes in the system and k represents each of these modes, a and P are the parameters derived based on the following equation. A second order system is represented in Laplace domain as, = = \t™ F(s) 2 2 The partial fraction expansion of the transfer function can be written in the following order. Us) = — + = - a + P * z (5.33) s-sx s-s* / + 2C(0„5 + co„ where the residues are, r = a +jv r* = a-jv and 5j and s2 are the complex conjugate roots of the characteristic equation of the transfer function. sl = -&n+j(Qd s2 = - C c o „ _ / ( O d The parameters, a and p are expressed as, a = 2 ( C t o n a - w d v ) p = 2a (5.34) Having performed the impact hammer test, the measured data is processed in CutPro (modal analysis software) for the determination of the dynamic characteristics of the boring bar system structure. The outputs of this process are: the damping ratio C,, natural frequencies con, equivalent stiffness k and mass m of each mode of the structure, and the residue values (r and r*) shown Chapter 5. Dynamic Modeling of Boring and Chatter Stability 146 above. Using these parameters, the transfer function of the structure is built based on the Eq. (5.32). To obtain the time domain solution, the nested programing method is conducted and the transfer function of the structure in radial direction Eq. (5.32) is modified into the state space rep-resentation with the observable canonical form [8]. The displacement in the radial direction can then be expressed as, Y(s) = k$Fr(S)-2&J(s)] + \[aFr(s)-(i)2nY(S)] (5.35) s s State variables are defined as follows, X^s) = ±[aFr(s)-(02nY(s)] (5.36) X2(s) = i[BF r(j)-2Ca) By(j)+X 1(5)] (5.37) The Eq.(5.35) can be written as, Y(s) = X2(s) (5.38) Substituting Eq. (5.38) into Eq.(5.36) and (5.37) leads to the state and output equations in a vector-matrix form as follows. *1 0 2 + p X>2 I x2 a (5.39) Y = X2 (5.40) Fourth order Runge Kutta equation is used in the numerical integration of the process [20]. The general equation of Runge Kutta is presented as, Xk+ 1 = Xk + \(kl + 2k2 + 2k3 + h) (5-41) Chapter 5. Dynamic Modeling of Boring and Chatter Stability 147 The slopes of each state variable equation are defined as follows. k \ , 1 = W- « f e + aFr) k2,l = h{-^l{X2 + l k l , l } + a F r ) k3,l = h{-®l[X2 + i\k2,l)+aFr) *4>1 = h(-(a2n(X2 + k31) + aFr) where h is the time interval. The first value of the state variable is obtained as,. 1 = Xh + i + 2 h i + 2K i + h, i) (5-42) Similarly, the slopes for the second state variable X2, kh2 = /i(^i-2CconX2 + f3Fr) h 2 = + \h i) " 2<x(x2 + \klt 2) + PF r ) h, 2 = + \k2> + [-2^n[x2 + \k% 2)) + PF r ) k4>2 = h((Xl + k3tl)-2ti(on(X2 + k3f2) + ^Fr) Finally, the second state variable that is also equal to the radial vibration (Eq. 5.40)) is accomplished with the following equation. (5.43) 5 . 3 . 3 . Simulation and Experimental Results In order to investigate the stability's dependence on the cutting parameters, and verify the time domain solution model, two series of experiments were performed. For each test, the initial surface of the workpiece was cleaned and the diameter set to the desired specification. Chatter Chapter 5. Dynamic Modeling of Boring and Chatter Stability 148 vibrations during the test were detected by means of accelerometers (PCB 353B11 SN 65847, SN 68836 and Kistler SN C128797) and a microphone attached to the boring bar and to the turret of the lathe. Acceleration signals were amplified by charge amplifiers (Kistler-Type 5114), and later digitized by data acquisition software CutPro-MalDAQ. In these experiments, the boring bar was first mounted on the turret with a critical length, in order to catch a critical border between the stable and unstable cutting conditions within the spec-ified cutting parameter ranges. If the boring bar is held on the turret with a short length, the sys-tem may be stable for all cutting conditions; if it is mounted with a long length, only unstable cutting conditions would be observed. In the first set of experiments, workpieces with 32.14 [mm] hole diameter were employed, the depth of cut was selected less than corner radius of the tool (a < 0.8 [mm]), and the feed rate was kept constant for all conditions with the value of 0.1 [mm/rev]. At the beginning, a high cut-ting speed was chosen, and gradually reduced by varying the depth of cut until the critical border of cutting speed was observed. As seen in Table 5.6, there is a certain chatter vibration stability border between the cutting speeds 112.5 and 125 [m/min]. The effect of the depth of cut in this critical border seems to be insignificant. However, it has a notable effect on the system stability for cutting conditions with larger depth of cut ranges. In the second set of experiments, both the depth of cut and the feed rate were changed, and the system stability showed dependency not only on the depth of cut, but also on the feed rate (Table 5.10). Linearizing assumptions expect system stability to remain independent of the radial depth of cut, which is the input of the system. However, results of the experiments conducted so far have contradicted this expectation, showing dependence of the system stability on radial depth of cut. In the experiments, in order to have the identical analysis for the system stability, the hole diameter of the workpiece was kept the same. Different hole diameters may affect the stability as regards process damping, even though the cutting parameters (depth of cut, a, feed rate c and cutting speed V) are the same. The process tends to damp due to the friction associated with the Chapter 5. Dynamic Modeling of Boring and Chatter Stability 149 tool flank face and workpiece interference as a result of the short waves left on the cut surface (Figure 5.30). For a different hole diameter D, but the same cutting speed V, spindle speed n [rpm] will be different. At each revolution, the phase angle e (for the same chatter frequency coc and cutting speed V) varies. Hence, due to the different phase angle for the same cutting condition (i.e. the same depth of cut, a, feed rate c and cutting speed V), the depth of cut variation will not be the same. Consequently, the experiments performed with a different hole diameter may not be identical. The relation between the integer number of waves, spindle speed, chatter frequency, and phase angle, is expressed with the Eq. (5.1) Figures 5.34 and 5.35 show time domain simulation results for experiments 1 and 17 in the first series of experiments (Table 5.6). Both simulation results exhibit unstable cutting conditions, even though no chatter is expected for the condition of experiment 1. Note that the radial vibra-tions in experiment 1 are even more severe. This may be explained as follows. The radial cutting force changes inversely with the cutting speed V, based on the mechanistic model expression pre-sented in Chapter 3. Thus, the lesser cutting speed V produces more cutting force, which causes large displacement on the boring bar structure. On the other hand, friction on the rake face of the tool increases with low cutting speed. This additional friction causes extra damping in the process and prevents chatter vibrations. Furthermore, if the process tends to vibrate, the length of the wave becomes short at low speeds, and rubbing between the flank face of the tool and wavy sur-face also contributes to damping of system (Figure 5.30). The wave length becomes shorter at low cutting speeds. The flank face of the tool is then in continuous contact with the material surface, adding an extra positive damping (known as process damping) effect to the system. As can be seen in Figure 5.30, with short waves the relief angle of the tool may be zero, whereas with long waves it is greater than zero, and has no significant effect on the damping of the system. Modeling process damping is difficult, and is therefore not included in the time domain simulation model. Chapter 5. Dynamic Modeling of Boring and Chatter Stability 150 The results of the second set of experiments are presented in Table 5.10 and Figure 5.33. These experiments were conducted at various cutting speeds (from 75 to 175[m/min]), depths of cut (from 0.25 to 2.5[mm]), and two feed rates (0.075 and 0.125[mm/rev]). As can be noticed, there is a certain stability border of depth of cut at 100[m/min] of cutting speed. After reaching 100 [m/min] cutting speed, the system exhibits unstable cutting conditions regardless of the depth of cut and feed rate. The effect of the feed rate is also seen in the figures. For a feed rate value of 0.075 [mm/rev] feed rate the system has no chatter vibrations until the depth of cut reaches 1.25[mm]. However, the maximum allowable depth of cut is 0.65[mm] when the feed rate is set to 0.125[mm/rev]. These series of experiments show that the system stability of a boring process depends on both depth of cut and feed rate.. Figure 5.30 : The relation between the length of generated waves and process damping Chapter 5. Dynamic Modeling of Boring and Chatter Stability 151 Fy • Fr (Radia l force) Fz >- Ff (Feed force) Figure 5.31 : Experimental setup-Chatter tests Chapter 5. Dynamic Modeling of Boring and Chatter Stability 152 5.3.3.1. Experimental Results: - Modal Parameters of the Boring Bar Structure for the First Set of Experiments Table 5.3 : Modal parameters in radial direction Mode# Freq. [Hz] Damping Ratio [%] Residue (Real) Residue (Im) Stiffness [N/m] Mass [kg] 1 483.02 9.33 1.026623E-05 -2.784654E-04 5.473229E+06 0.5942 2 753.03 2.62 3.041659E-05 -4.995113E-04 4.737671 E+06 0.2116 Table 5.4 : Modal parameters in tangential direction Mode# Freq. [Hz] Damping Ratio [°/<j Residue (Real) Residue (Im) Stiffness [N/m] Mass [kg] 1 665.67 2.26 7.90253E-06 -5.778915E-04 3.62E+06 0.2069 Table 5.5 : Modal parameters in feed direction Mode# Freq. [hk] Damping Ratio[% ] Residue [Real] Residue [Im] Stiffness [N/m] Mass [kg] 1 432.82 10.59 1.082661E-05 -4.111103E-05 3.326233E+07 4.497 2 754.56 2.49 5.757860E-06 -1.490884E-05 1.590511E+08 7.076 er 5. Dynamic Modeling of Boring and Chatter Stability Table 5.6 : Experimental results of the first set Exp.# V [m/min] n [rpm] C [mm/rev] a [mm] D[mm] Result 8 [Deg] C0C [H ] 1 75 743 0.10 0.75 32.14 Nochatter 2 100 990 0.10 0.30 32.14 Nochatter 3 100 990 0.10 0.60 32.14 Nochatter 4 75 743 0.10 0.30 32.14 Nochatter 5 87.5 867 0.10 0.60 32.14 Nochatter 6 87.5 867 0.10 0.30 32.14 Nochatter 7 112.5 1114 0.10 0.30 32.14 Nochatter 8 112.5 1114 0.10 0.60 32.14 Nochatter 11 125 1238 0.10 0.30 32.14 Nochatter 9 225 2228 0.10 0.75 32.14 Chatter 273.5 771 10 125 1238 0.10 0.50 32.14 Chatter 243.3 798 12 175 1733 0.10 0.60 32.14 Chatter 261.2 772 13 175 1733 0.10 0.30 32.14 Chatter 223.8 769 14 150 1486 0.10 0.60 32.14 Chatter 239.2 784 15 150 1486 0.10 0.30 32.14 Chatter 239.2 784 16 225 2228 0.10 0.30 32.14 Chatter 225.0 766 17 250 2476 0.10 0.60 32.14 Chatter 263.5 773 18 275 2724 0.10 0.60 32.14 Chatter 282.3 762 Chapter 5. Dynamic Modeling of Boring and Chatter Stability Feed rate, c=0.1 [mm/rev] D=32.14 [mm] © • . J§ i A J f - %9 \ P w % P w V ©••0"C J • ©••© ( P • V • Chatter O No Chatter 0 50 100 150 200 Cutting speed, V [m/min] Figure 5.32 : The results of the first set of experiments 250 300 Chapter 5. Dynamic Modeling of Boring and Chatter Stability 155 - Modal Parameters of the Boring Bar Structure for the Second Set of Experi-ments Table 5.7 : Modal parameters in radial direction Mode# Freq. [Hz] Damping Ratio [%] Residue (Real) Residue (Im) Stiffness [N/m] Mass [kg] 1 764.53 2.40 -9.42390E-07 -1.264124E-04 1.900543E+07 0.8236 Table 5.8 : Modal parameters in tangential direction Mode# Freq. [Hz] Damping Ratb [%] Residue (Real) Residue (Im) Stiffness [N/m] Mass [kg] 1 764.51 3.43 -6.59550E-06 -1.574528E-04 1.526302E+07 0.6615 Table 5.9 : Modal parameters in feed direction Mode# Freq. [Hz] Damping Ratio[%] Residue (Real) Residue (Im) Stiffness [N/m] Mass [kg] 1 281.23 9.52 -1.16473E-07 -1.266268E-07 7.009194E+09 2244.8153 2 1369.57 7.84 3.06011E-07 -1.73772E-06 2483663312 33.5402908 Chapter 5. Dynamic Modeling of Boring and Chatter Stability 156 Table 5.10 : The results of the second set of experiments Exp.# V [m/min] n [rpm] C [mm/rev] 3 [mm] Result S [Deg] C0 C [Hz] 1 175 754 0.075 0.6 Chat ter 212.9 774 2 175 1869 0.125 0.75 Chat ter 247 .3 769 3 175 775 0.075 0.9 Chat ter 220.6 770 4 175 1763 0.125 1.5 Chat ter 196.4 780 5 150 700 0.075 0.5 Chat ter 185.1 776 6 150 687 0.075 1 Chat ter 215.4 774 7 150 665 0.075 1.5 Chat ter 203.0 771 8 150 491 0.125 0.25 Chat ter 209.7 774 9 150 497 0.125 0.5 Chat ter 115.2 773 10 150 487 0.125 1 Chat ter 134.5 766 11 125 596 0.075 0.5 Chat ter 294.8 773 12 125 585 0.075 0.75 Chat ter 175.4 775 13 125 571 0.075 1 Chat ter 290 .0 769 14 125 1160 0.075 1.5 Chat ter 260 .7 768 15 125 641 0.125 0.75 Chat ter 252.2 766 16 125 557 0.125 1 Chat ter 223.6 767 17 125 623 0.125 1.5 Chat ter 173.9 763 18 112.5 527 0.075 0.5 Chat ter 280.8 771 19 112.5 517 0.075 0.75 Chat ter 323.1 766 20 112.5 504 0.075 1 Chat ter 214 .3 761 21 112.5 489 0.075 1.25 Chat ter 267.2 764 22 112.5 471 0.075 1.5 Chat ter 293 .5 760 2 3 112.5 399 0.125 0.5 Chat ter 121.8 767 24 112.5 394 0.125 0.75 Chat ter 179.1 765 2 5 112.5 386 0.125 1 Chat te r 216.4 763 26 112.5 377 0.125 1.5 Chat ter 41.1 761 27 100 866 0.075 0.45 N o Chat ter 28 100 840 0.075 0.55 No Chat ter Chapter 5. Dynamic Modeling of Boring and Chatter Stability 157 Exp.# V [m/mln] n [rpm] C [mm/rev] 3 [mm] Result £ [Deg] C0 C [Hz] 29 100 813 0.075 0.65 No Chat ter 30 100 783 0.075 0.8 No Chat ter 31 100 748 0.075 1 No Chat ter 32 100 711 0.075 1.25 No Chat ter 33 100 669 0.075 1.65 Chatter 122.7 762 34 100 776 0.125 0.4 No Chat ter 35 100 802 0.125 0.55 No Chat ter 36 100 767 0.125 0.65 No Chat ter 37 100 740 0.125 1 Chat te r 311.4 763 38 100 703 0.125 1.25 Chat te r 311.4 760 39 100 662 0.125 1.65 Chat te r 154.4 755 40 100 569 0.125 2.5 Chat ter 201.2 745 41 87.5 464 0.075 0.5 No Chat ter 42 87.5 455 0.075 0.75 No Chat ter 4 3 87.5 443 0.075 1 No Chat ter 44 87.5 429 0.075 1.5 Chat te r 156.1 761 45 87.5 341 0.125 0.5 NoChat te r 46 87.5 336 0.125 0.75 N o Chat ter 47 87.5 329 0.125 1 No Chat ter 48 87.5 321 0.125 1.5 No Chat ter 49 87.5 295 0.125 1.75 Chat te r 48 .8 748 50 75 498 0.075 0.5 No Chat ter 51 75 488 0.075 0.75 No Chat ter 52 75 472 0.075 1 N o Chat ter 53 75 365 0.075 1.25 Chat te r 29.6 767 54 75 453 0.075 1.5 Chat te r 143.0 758 55 75 324 0.125 0.5 No Chat ter 56 75 320 0.125 0.75 No Chat ter Chapter 5. Dynamic Modeling of Boring and Chatter Stability 158 Exp.# V [m/m in] ll [rpm] C [mm/rev] 3 [mm] Result 8 [Deg] CO c [Hz] 57 7 5 312 0 .125 1 Cha t t e r 3 3 2 . 3 764 58 7 5 2 9 5 0 .125 1.5 Cha t t e r 2 6 8 . 5 751 59 7 5 256 0 .125 1.25 Cha t t e r 2 3 6 . 3 758 Chapter 5. Dynamic Modeling of Boring and Chatter Stability 159 2 1.8 1.61-I1-4 CO S " l 2 o -C Q. 1 CU D 0.8 0.6 0.4 Feed rate, c=0.075 [mm/rev] 60 2.5k E E S 1 o a. CD Q .5 0.5 • Chatter O No Chatter 4 » • • • • • • c ) • _ f - \ . . . r\ r *\ A ( "\ • o K o c • • A . _ o o o • o • • 80 100 120 140 Cutting speed, V [m/min] Feed rate, c=0.125 [mm/rev] 160 180 O O O O O O O O O • Chatter O No Chatter 60 80 100 120 140 160 Cutting speed, V [m/min] 180 200 Figure 5.33 : Results of the second set of experiments Chapter 5. Dynamic Modeling of Boring and Chatter Stability 5.3.3.2. Time Domain Simulation Results: 160 co C C o o CD CO 8 =5 < 5 1500 1000 500 0 -500| -1000 -15001 EXPERIMENTAL RESULT-CHATTER FFT of the acceleration 0.2 0.4 0.6 Time [sec] 0.8 1 4 0 ^ 120 — 100 c c c •£ 80 o o "•5 CD o r . CO .!= 60 •D 40 Q _ CD CO 8 -s < 2 20 0< ! ! 1 7 7 3 [Hz] 0 1000 3000 Frequency [Hz] 5000 SIMULATION RESULT-CHATTER . A FFT of the radial vibration 0.4 0.6 Time [sec] CO o II II CO o . g £ > -5 I • 1 I 7 5 1 . 6 1 | 1 1 Hzl ' i .... 000 3000 Frequency [Hz] 5000 Figure 5.34 : Time domain simulation result, Set-1, Test#17, a=0.6[mm], V=250[m/min], c=0.1 [mm/rev], n=2476[rpm] Chapter 5. Dynamic Modeling of Boring and Chatter Stability 161 1500 1000 500 0 -500 -1000 -1500 EXPERIMENTAL RESULT-NO CHATTER 140 FFT of the radial vibration 0 0.2 0.4 0.6 0.8 Time [sec] co E, c c o o _CD ^ CD CO 8 =5 << CO 120 100 80 60 40 20 0 0 1000 3000 Frequency [Hz] 5000 SIMULATION RESULT-CHATTER A n FFT of the acceleration "0 0.5 1 1.5 2 2.5 3 Time [sec] co o I! . i g CO o . g 2 > TJ 736.25 [Hi 2] I : 5000 Frequency [Hz] Figure 5.35 : Time domain simulation result, Set-1, Test#l, a=0.75[mm], V=75[m/min], c=0.1 [mm/rev], n=743[rpm] Chapter 5. Dynamic Modeling of Boring and Chatter Stability 162 5.4. Summary This chapter described the self-excited chatter vibrations originating from the regenerative effect and the dynamic characteristics of the boring process in detail. The mechanism of wave regeneration in the boring process differs from other models and has a nonlinear nature, due to the dynamic characteristics of the boring process. In this chapter, an approach for the analytical solu-tion of the chatter stability was presented, but not experimentally validated. A time domain solu-tion model was also implemented to predict cutting forces and chatter stability limits. Only radial vibrations were taken into account in the time domain solution, since vibrations in the tangential direction do not have a significant effect in wave regeneration. A time domain simulation model was experimentally examined, and some discrepancies were noticed between predicted and experimentally observed cutting conditions, which were most likely due to the effects of process damping. The time domain solution model is capable of considering the nonlinearity of the pro-cess, such as the loss of tool contact caused by chatter vibrations during machining. On the other hand, the time domain simulation model can be used for the static process force prediction, if the vibrations are neglected. Chapter 6 Conclusions This thesis investigates the dynamics of boring operations. The mechanics of boring opera-tion deal with the prediction of cutting forces as a function of tool geometry, work material prop-erties and cutting conditions such as feed rate, radial depth of cut, and cutting speed. The dynamics of the process involve the modeling of interactions between the structural dynamics of a long, slender boring bar with boring process mechanics. Evaluation of forces allows the predic-tion of static deflection errors, torque and the power required from the machine tool. Evaluation of dynamic stability of the process allows the prediction of chatter vibration free feed rate, spindle speed, radial depth of cut, and tool geometry. The thesis shows that boring forces strongly depend on tool nose geometry, side cutting edge angle, radial depth of cut, feed rate and cutting speed. The chip thickness distribution along the curved edge of the tool is rather complex. The chip is thin close to the nose, and becomes thicker along the curved edge as the radial depth of cut increases. The chip thickness distribution is also affected by the feed rate. The mechanics of boring are investigated. The shape of the chip is modeled as a function of tool nose radius, side cutting edge angle, radial depth of cut, and feed rate. The chip area is divided into regions. One of the regions is the nose radius area where the chip starts with zero thickness and increases as a function of nose radius and feed. Along the straight inclined edge, the chip thickness is constant and the chip area is defined by a rectangle. The area of each region and its center of gravity is evaluated. The cutting pressure along the edge of the tool is identified, using either orthogonal to oblique cutting transformation, or mechanistically calibrated force coefficients. Oblique transformation uses the shear stress, shear angle and friction coefficient of 163 Chapter 6. Conclusions 164 the material identified from orthogonal machining tests. They are mapped to oblique or curved cutting edge at digitized regions. The differential cutting pressures are summed up along the cut-ting edge, which leads to total cutting forces in each region. Alternatively, the cutting forces at each region are evaluated using mechanistically identified cutting coefficients which relate the chip area or contact length to the cutting force magnitudes. The cutting forces in all regions are summed up to find resultant friction and tangential cutting forces. Using an equivalent friction or lead angle, the friction force is projected in the radial and feed directions. The model allows the prediction of cutting forces in all three Cartesian directions. When the boring bar with multiple inserts is used, the chip load distribution to all inserts is modeled. The radial and axial setting errors of the inserts are integrated with the process mechan-ics model. A significant number of experiments have been conducted in the boring of Aluminum 6061 alloy. It is shown that the predicted and measured cutting forces are in good agreement with the predictions provided by the proposed mechanics models. The predictions are shown to have errors varying between 2% to 15%. The proposed model contributes to the improved prediction of bor-ing mechanics. The fundamental mechanism behind chatter vibrations in boring is also investigated. It is shown that the cutting coefficients, i.e. process gain, and directional factors, are dependent on the feed rate, radial depth of cut, tool geometry, and cutting speed. While the tool geometry and speed may be kept constant, vibrations modulate radial depth of cut, leading to time-varying pro-cess input parameter. The vibrations of the tool in the radial direction change the effective radial depth of cut. The tool travels over the previously machined surface marked by the vibrations gen-erated during the previous revolutions. The changes in the chip area become very abrupt and sig-nificant, difficult to track even in the time domain. The changes in the chip area affect cutting force coefficients (e.g. process gains) and directional factors (e.g. strength of modal parameters in each direction). This is the fundamental non-linearity in the process, which differs from milling Chapter 6. Conclusions 165 operations. The author presents the block diagram model of the process in the frequency domain, and illustrates the nonlinear influence of process changes. In addition, the process is simulated in the time domain. However, it is shown that even the time domain modeling suffers, mainly due to the digital integration of a significant number of tool deflection waves left on the boring surface. Future work should focus on the development of an improved time domain solution for chat-ter in the boring process. If a more simple tool geometry is used, it may be possible to minimize the drastic nonlinear changes in the cutting coefficients and directional factors, which may lead to the approximate solution of chatter stability in the frequency domain. Appendix A : Experimental Results of The Mechanistic Model Verification Friction force verification for a < R Exp# c [mm/rev] a [mm] V [m/min] Measure Fjr[N] Predicted Fjr [N] Error [%] 1 0.0600 0.7750 112.5 51.02 60.40 -15.52 2 0.1700 0.6500 155.0 72.06 72.92 -1.17 3 0.1800 0.4500 180.0 57.65 56.06 2.82 4 0.1000 0.5500 190.0 47.24 48.14 -1.86 5 0.1650 0.5500 220.0 58.99 56.34 4.70 6 0.0650 0.4750 230.0 35.37 30.59 15.61 7 0.1850 0.6750 250.0 68.92 64.85 6.28 8 0.0725 0.7750 270.0 47.88 43.94 8.97 9 0.1350 0.4250 235.0 44.99 38.49 16.89 10 0.0675 0.7250 252.5 45.29 41.37 9.47 11 0.1375 0.3250 97.5 47.19 54.88 -14.01 12 0.0580 0.5600 114.0 41.43 38.79 6.81 13 0.1110 0.6660 152.0 58.15 61.24 -5.05 14 0.1400 0.7770 123.0 75.79 66.71 13.62 15 0.1200 0.6000 156.0 56.57 51.51 9.81 The absolute average error [%]=8.83 166 167 Friction force verification for a > R Exp# c [mm/rev] a [mm] V [m/min] Measured f r^ [N] Predicted F r^ [N] Error [%] 1 0.0400 2.3750 85.0 103.60 118.94 14.80 2 0.1300 1.5650 100.0 137.20 144.48 5.30 3 0.1450 0.8650 120.0 80.30 87.28 8.70 4 0.0700 3.1000 130.0 170.67 168.15 -1.47 5 0.0750 1.8750 135.0 111.63 109.40 -2.00 6 0.1600 1.6500 145.0 156.34 144.73 -7.43 7 0.0550 0.9500 165.0 52.86 52.43 -0.82 8 0.0900 1.0000 170.0 65.20 65.49 0.44 9 0.1100 2.2500 200.0 139.74 134.17 -3.98 10 0.0375 3.1000 205.0 101.66 118.71 16.78 11 0.0550 2.1500 210.0 85.08 95.63 12.41 12 0.0775 1.2500 240.0 74.78 66.25 -11.42 13 0.0925 1.9500 137.5 118.18 124.79 5.60 14 0.0575 2.1500 142.5 111.97 107.88 -3.65 15 0.1275 2.7750 127.5 205.79 212.94 3.47 16 0.0465 1.3500 217.5 57.78 61.33 6.14 17 0.1150 1.4500 195.0 95.72 94.64 -1.14 18 0.1500 3.0000 87.5 304.90 309.10 1.37 19 0.0825 2.3000 187.5 118.98 122.25 2.75 20 0.0950 1.2750 262.5 75.72 70.91 -6.35 21 0.1085 3.1000 152.5 175.90 198.08 12.61 22 0.0666 1.7350 172.5 77.85 89.83 15.39 23 0.1475 2.3750 185.0 168.54 170.48 1.15 24 0.0825 1.4750 125.0 98.38 95.88 -2.54 25 0.1330 1.3330 188.0 101.58 96.46 -5.04 26 0.1440 1.6660 222.0 109.78 113.91 3.76 27 0.0444 1.8880 177.0 70.27 83.27 18.50 28 0.1666 2.0000 88.0 210.86 228.58 8.40 29 0.0888 1.8300 266.0 98.81 92.81 -6.08 30 0.0600 2.0000 181.0 96.03 98.87 2.96 The absolute average error [%]=6.41 168 Tangential force verification for a < R and a > R Exp# c [mm/rev] a [mm] V [m/min] Measured Fj [N] Predicted FJ [N] Error [%] 1 0.0400 2.3750 85 149.72 166.00 10.87 2 0.1300 1.5650 100 228.07 236.29 3.60 3 0.0600 0.7750 112.5 78.51 77.12 -1.78 4 0.1450 0.8650 120 132.24 147.12 11.25 5 0.0700 3.1000 130 263.07 270.56 2.85 6 0.0750 1.8750 135 177.37 179.47 1.18 7 0.1600 1.6500 145 282.52 265.75 -5.94 8 0.1700 0.6500 155 128.63 122.83 -4.51 9 0.0550 0.9500 165 77.56 81.38 4.91 10 0.0900 1.0000 170 111.99 112.46 ' 0.41 11 0.1800 0.4500 180 100.07 91.64 -8.43 12 0.1000 0.5500 190 70.97 72.46 2.11 13 0.1100 2.2500 200 251.90 250.66 -0.49 14 0.0550 2.1500 210 135.49 156.12 15.23 15 0.1650 0.5500 220 98.67 97.54 -1.15 16 0.0775 1.2500 240 111.71 116.34 4.14 17 0.1850 0.6750 250 119.20 121.77 2.16 18 0.0450 0.3500 260 33.74 33.11 -1.87 19 0.0725 0.7750 270 65.12 74.22 13.96 20 0.0925 1.9500 137.5 197.48 211.93 7.32 21 0.0575 2.1500 142.5 163.38 170.79 4.54 22 0.1275 2.7750 127.5 361.39 368.23 1.89 23 0.0465 1.3500 217.5 86.06 95.50 10.97 24 0.1150 1.4500 195 169.18 176.44 4.30 25 0.1500 3.0000 87.5 493.83 484.04 -1.98 26 0.0825 2.3000 187.5 199.28 214.51 7.64 27 0.1350 0.4250 235 59.25 68.62 15.82 28 0.0675 0.7250 252.5 67.30 68.48 1.76 29 0.0950 1.2750 262.5 136.10 131.73 -3.21 30 0.1085 3.1000 152.5 326.84 349.67 6.98 169 Exp# c [mm/rev] a [mm] V [m/min] Measured Fj [N] Predicted ^ [N] Error [%] 31 0.0666 1.7350 172.5 129.12 149.25 15.59 32 0.1475 2.3750 185 322.18 328.29 1.90 33 0.1375 0.3250 97.5 64.95 66.38 2.20 34 0.0825 1.4750 125 159.11 156.98 -1.34 35 0.0444 1.8350 102.5 119.03 136.00 14.26 36 0.0580 0.5600 114 50.33 58.63 16.48 37 0.1110 0.6660 152 84.80 94.00 10.85 38 0.1330 1.3330 188 176.11 182.59 3.68 39 0.1440 1.6660 222 221.96 226.59 2.09 40 0.0444 1.8880 177 113.50 127.96 12.73 41 0.1666 2.0000 88 354.18 363.62 2.67 42 0.0888 1.8300 266 165.30 172.09 4.11 43 0.1400 0.7770 123 115.55 130.40 12.85 44 0.1200 0.6000 156 82.54 90.44 9.57 45 0.0600 2.0000 181 160.09 157.68 -1.51 The absolute average error [%]=6.11 170 Effective lead angle prediction for a < R Exp# c[mm /rev] a[mm ] V[m/min ] Measured \_ [Deg] Modified-Predicted ^ •_ [Deg] Error[%] 1 0.0600 0.7750 112.5 37.47 37.78 0.81 2 0.1700 0.6500 155.0 46.70 46.98 0.58 3 0.1800 0.4500 180.0 58.30 58.23 -0.12 4 0.1000 0.5500 190.0 48.55 50.25 3.49 5 0.1650 0.5500 220.0 55.48 52.37 -5.61 6 0.0650 0.4750 230.0 49.40 53.40 8.09 7 0.1850 0.6750 250.0 46.13 46.52 0.85 8 0.0725 0.7750 270.0 41.42 38.62 -6.76 9 0.1350 0.4250 235.0 57.48 58.49 1.75 10 0.0675 0.7250 252.5 42.83 40.82 -4.71 11 0.1375 0.3250 97.5 62.87 63.92 1.67 12 0.0580 0.5600 114.0 52.07 48.16 -7.49 13 0.1110 0.6660 152.0 48.19 44.48 -7.70 14 0.1400 0.7770 123.0 41.89 39.71 -5.22 15 0.1200 0.6000 156.0 46.00 48.09 4.55 The absolute average error [%]=3.96 171 Verification of the effective lead angle for a > R Exp# c [mm/rev] a [mm] V [m/min] Measured L [Deg] Modified-Predicted T* L[Deg] Error[%] 1 0.1300 1.5650 100.0 21.25 17.34 -18.41 2 0.0700 3.1000 130.0 11.35 9.85 -13.15 3 0.1600 1.6500 145.0 20.18 16.88 -16.38 4 0.0900 1.0000 170.0 30.79 26.07 -15.33 5 0.1100 2.2500 200.0 13.05 14.11 8.11 6 0.0550 2.1500 210.0 14.38 15.81 9.91 7 0.0775 1.2500 240.0 23.26 24.21 4.08 8 0.0925 1.9500 137.5 15.42 15.66 1.57 9 0.0575 2.1500 142.5 13.99 15.04 7.52 10 0.1275 2.7750 127.5 11.36 10.44 -8.08 11 0.0465 1.3500 217.5 25.36 23.03 -9.18 12 0.1150 1.4500 195.0 22.27 20.23 -9.16 13 0.0825 2.3000 187.5 13.45 14.16 5.23 14 0.0950 1.2750 262.5 24.05 24.03 -0.11 15 0.0666 1.7350 172.5 16.10 18.12 12.53 16 0.1475 2.3750 185.0 12.60 12.63 0.23 17 0.0825 1.4750 125.0 17.47 19.28 10.34 18 0.1330 1.3330 188.0 21.01 21.06 0.27 19 0.1440 1.6660 222.0 18.54 18.09 -2.44 20 0.0444 1.8880 177.0 14.72 17.41 18.31 21 0.1666 2.0000 88.0 15.17 13.66 -9.96 22 0.0888 1.8300 266.0 18.85 18.24 -3.24 23 0.0600 2.0000 181.0 15.86 16.40 3.36 The absolute average error [%]=8.12 172 Radial force verfication for a < R Exp# c [mm/rev] a [mm] V [m/min] Measured Fr [N] Predicted Fr [N] Error [%] 1 0.0600 0.7750 112.5 36.74 30.29 -17.57 2 0.1700 0.6500 155.0 53.07 52.86 -0.39 3 0.1800 0.4500 180.0 47.70 51.55 8.07 4 0.1000 0.5500 190.0 36.09 36.09 0.01 5 0.1650 0.5500 220.0 46.42 46.94 1.13 6 0.1850 0.6750 250.0 46.75 47.85 2.35 7 0.0725 0.7750 270.0 29.07 27.29 -6.11 8 0.1350 0.4250 235.0 32.46 39.14 20.59 9 0.0675 0.7250 252.5 28.13 27.34 -2.79 10 0.1375 0.3250 97.5 48.84 46.49 -4.81 11 0.0580 0.5600 114.0 30.59 31.13 1.75 12 0.1110 0.6660 152.0 45.65 39.94 -12.52 13 0.1400 0.7770 123.0 44.54 47.47 6.56 14 0.1200 0.6000 156.0 37.05 42.14 13.71 The absolute average error [%]=7.02 173 Radial force verfication for a > R Exp# c[mm /rev] a[mm ] V[m/min ] Measured F r [N] Predicted F r [N] Error[% ] 1 0.1300 1.5650 100.0 49.73 43.17 -13.19 2 0.0700 3.1000 130.0 33.57 28.85 -14.06 3 0.0550 0.9500 165.0 29.79 24.24 -18.63 4 0.0900 1.0000 170.0 33.37 28.85 -13.56 5 0.1100 2.2500 200.0 31.56 32.80 3.93 6 0.0775 1.2500 240.0 29.54 27.24 -7.79 7 0.0925 1.9500 137.5 31.42 33.78 7.49 8 0.0575 2.1500 142.5 27.06 28.07 3.71 9 0.1275 2.7750 127.5 40.53 38.69 -4.54 10 0.0465 1.3500 217.5 24.75 24.06 -2.80 11 0.1500 3.0000 87.5 58.49 47.66 -18.52 12 0.0825 2.3000 187.5 27.68 29.98 8.30 13 0.0950 1.2750 262.5 30.86 28.95 -6.21 14 0.1475 2.3750 185.0 36.78 37.38 1.64 15 0.0825 1.4750 125.0 29.53 31.73 7.44 16 0.1330 1.3330 188.0 36.42 34.76 -4.56 17 0.1440 1.6660 222.0 34.91 35.46 1.58 18 0.1666 2.0000 88.0 55.19 54.13 -1.92 19 0.0888 1.8300 266.0 31.93 29.12 -8.79 20 0.0600 2.0000 181.0 26.25 27.98 6.60 The absolute average error [%]=7.76 174 Feed force verfication for a < R Exp# c [mm/rev] a [mm] V [m/min] Measured F f [N] Predicted F f [N] Error [%] 1 0.0600 0.7750 112.5 47.94 41.08 -14.31 2 0.1700 0.6500 155.0 50.01 46.81 -6.39 3 0.1800 0.4500 180.0 29.46 22.44 -23.81 4 0.1000 0.5500 190.0 31.86 28.76 -9.75 5 0.1650 0.5500 220.0 31.93 31.90 -0.08 6 0.0650 0.4750 230.0 19.91 20.13 1.11 7 0.1850 0.6750 250.0 44.94 44.72 -0.48 8 0.0725 0.7750 270.0 32.95 37.35 13.36 9 0.1350 0.4250 235.0 20.69 18.88 -8.76 10 0.0770 0.3000 242.5 12.27 9.96 -18.82 11 0.0675 0.7250 252.5 30.34 34.39 13.35 12 0.0580 0.5600 114.0 23.85 27.53 15.45 13 0.1110 0.6660 152.0 40.83 41.00 0.43 14 0.1400 0.7770 123.0 49.66 58.45 17.72 15 0.1200 0.6000 156.0 35.79 36.34 1.55 The absolute average error [%]=9.69 175 Feed force verification for a > R Exp# c [mm/rev] a [mm] V [m/min] Measured F f [N] Predicted F f [N] Error [%] 1 0.1300 1.5650 100.0 127.87 137.88 7.83 2 0.0700 3.1000 130.0 167.33 165.66 -1.00 3 0.0550 0.9500 165.0 43.66 46.49 6.46 4 0.0900 1.0000 170.0 56.01 58.79 4.96 5 0.1100 2.2500 200.0 136.13 130.10 -4.43 6 0.0775 1.2500 240.0 68.70 60.39 -12.11 7 0.0925 1.9500 137.5 113.92 120.14 5.46 8 0.0575 2.1500 142.5 108.65 104.17 -4.12 9 0.1275 2.7750 127.5 201.76 209.40 3.78 10 0.0465 1.3500 217.5 52.21 56.42 8.05 11 0.1500 3.0000 87.5 299.24 305.40 2.06 12 0.0825 2.3000 187.5 115.71 118.52 2.42 13 0.0950 1.2750 262.5 69.15 64.74 -6.38 14 0.1475 2.3750 185.0 164.48 166.33 1.12 15 0.0825 1.4750 125.0 93.84 90.47 -3.59 16 0.1330 1.3330 188.0 94.83 89.98 -5.12 17 0.1440 1.6660 222.0 104.08 108.25 4.00 18 0.1666 2.0000 88.0 203.51 222.08 9.12 19 0.0888 1.8300 266.0 93.51 88.12 -5.77 20 0.0600 2.0000 181.0 92.37 94.83 2.66 The absolute average error [%]=5.022 176 Appendix B: Experimental Results of The Orthogonal to Oblique Transformation Method Orthogonal to oblique transformation method tangential, radial and feed force verifications Tangential force verification Exp. # V [m/min] c [mm/rev] a [mm] Measured Ft [N] Predicted Ft [N] Error [%] 1 240 0.050 1.000 203.626 184.106 -9.586 2 150 0.050 0.500 106.207 112.656 6.072 3 150 0.050 0.750 152.219 154.959 1.800 4 150 0.050 1.250 224.270 238.594 6.387 5 100 0.050 1.500 301.647 291.984 -3.203 6 100 0.050 1.750 346.888 335.193 -3.372 7 115 0.050 0.625 126.027 138.553 9.940 Radial force verification Exp. # V [m/min] c [mm/rev] a[mm]: 1 Measured Fr [N] Predicted Fr [N] Error [%] 1 240 0.050 1.000 87.825 68.770 -21.696 2 150 0.050 0.500 75.737 73.340 -3.165 3 150 0.050 0.750 99.956 79.238 -20.727 4 150 0.050 1.250 82.262 79.230 -3.686 5 100 0.050 1.500 111.172 85.900 -22.732 6 100 0.050 1.750 115.137 85.900 -25.393 7 115 0.050 0.625 96.351 82.090 -14.801 Feed force verification Exp. # V [m/min] c [mm/rev] a [mm] Measured Ff [N] Predicted Ff [N] Error [%] 1 240 0.050 1.000 134.309 107.200 -20.184 2 150 0.050 0.500 60.718 62.944 3.666 3 150 0.050 0.750 109.580 95.680 -12.685 4 150 0.050 1.250 154.594 145.700 -5.753 5 100 0.050 1.500 238.613 183.000 -23.307 6 100 0.050 1.750 279.764 209.000 -25.294 7 115 0.050 0.625 88.636 83.380 -5.930 Appendix C -Determination of the Emprical Contants in Mechanistic Model Least squares method: Ktc = /0AV2 (6.1) \og(Ktc) = logO*0) + * i l o g ( A ) + b2log(V) (6.2) The terms are redefined as follows. K = log(Ktc) C = l o g ( A ) D = \og(V), (6.3) K = b0 + blC + b2D (6.4) b0, bx and b2 in Eq. (6.4) should be selected such that the square of distances between the predicted curve and measured data are minimized. The deviation between the points and curve is, dt = Kt - O 0 + blCi + b2Dt) l<i<n (6.5) n 2 Hence, ^ d i is minimized. i = 1 f(b0,bvb2) = d2 + d22 + d23 + +d2n = £ [Ki-(b0 + blCi + b2Di)]2 (6.6) i = 1 The numbers Kt, Ci and D{ are all constants. The partial derivatives of the / are, J £ = -2 2 [Ki-(bQ + blCi + b2Dt)] (6.7) 9*0 i = l J £ = - 2 £ C ^ . - O Q + ^ Q + ^ D . ) ] (6.8) i = l 177 178 (6.9) »= 1 When both partial derivatives are set to zero, the following linear equations are obtained. n n E** = 2 ( * 0 + * i C , + v > * ) i = 1 j = 1 (6.10) i = i i = i (6.11) i = 1 / = 1 (6.12) We can solve these three equations for b0, bx and b2. The resulting curve Ktc = eb°AblVb2 always minimizes the sum of the squares of the deviations. The equations can be expressed in matrix form. n n 1 C, Dt~ h X Kfi • I ^ c) CtDt i= 1 i= 1 Dt Cpt D] b2 n Kt n "l C,. Dt * 0 T= X Kfi • z = s ^ c] cpi , w = h i = l Kpi i — 1 Dt Cpi D] b2 (6.13) (6.14) W = ZlT (6.15) 179 VQ U\ The resulting curve Ktc = e A V always minimizes the sum of the squares of the devia-tions. Bibliography [I] E. Budak, Y. Altintas, E. J. A Armarego, 1996, Prediction of Milling Force Coefficients From Orthogonal Cutting Data, ASME Vol. 118, pp. 216-224 [2] Y. Altintas, 2000, Course Book, Manufacturing Automation, Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design, Cambridge, ISBN 0521659736 [3] Merchant, M.E., Basics Mechanics of the Metal Cutting Process, Journal of Applied Mechanics ASME, pp A-168 to A-175, 1944 [4] J. Tlusty and M. Polacek, The Stability of Machine Tools Against Self Excited Vibrations in Machining, International Research in Production Engineering, ASME, pages 465-474, 1963 [5] Kronenberg M. Machining Science and Application, Pergamon Press Inc., 1966 pp 209-225 [6] R. A. Hallam and R. S. Allsopp, The Design, Development and Testing of a Prototype Bor-ing Dynamometer, Int. J. Mach. Tool Des. Res. Vol. 24, No. 1, 1962, pp. 241-266 [7] H.E. Merrit. Theory of Self-Excited Machine Tool Chatter. Trans. ASME Journal of Engi-neering for Industry, 87:447-454, 1965 [8] K. Ogata, Modern Control Engineering, Prentice Hall, 1990 [9] Sabberwal, A.J.P, Chip Section and Cutting Force During the Milling Operation, Annals of the CIRP, 1960, pp, 197-203 [10] O.Burak Ozdogonlar W.J. Endres An Analytical, Non-Dimensional Representation of Chip Area for Corner-Radiused Tools Under Depth of Cut and Feed Variations, ASME 1998 Manufacturing Science and Engineering Division, Vol. 8. pp251-258 [II] J. Tlusty Machine Dynamics, McMaster University, [12] F. Kuster; IWF, ETH Zurich, Cutting Dynamics and Stability of Boring Bars, Annals of the CIRP Vol. 39 pp361-366, January 15,1990 [13] Kline, W. A., R.E.Devor, and J.R. Linberg, The Prediction of Cutting Force in End Milling with Application to Cornering Cuts, Int. Jour. Mach. Tool Des. and Res. Vol. 22 No.l 1982, pp 7-22 180 liography 181 Sutherland, J.W, and R.E.Devor, and S.GKapoor, A Mechanistic Model for the Prediction of the Force and Surface Error Prediction in Flexible End Milling Systems, ASME Journal of Engineering for Industry, Vol.108, pp, 269-279, 1986 .Fu, H.J., R.W.Devor, and S.G Kapoor, A Mechanistic Model for the Prediction of the Force System in Face Milling Operations, Journal of Engineering fo Industry, Transaction ASME, Vol. 106, pp,81-88 1984 Subramani, G, R. Suvada, S.G Kapoor, R. W. Devor, and W. Meingast, A Model for the Prediction of Force System for Cyclinder Boring Process, Proc. XV North American Man-ufacturing Research Conference, pp, 449-446, 1987 J.W. Sutherland, GSubramani, M.J.Kuhl, R.E.Devor, S.GKapoor, An Investigation into the Effect of Tool and Cut Geometry on Cutting Force System Prediction Models, Proc. of XVI North American Manufacturing Research Conference, 1990, pp 264-272 Ph.D Thesis By Zhang G, Dynamic Modeling and Dynamic Analysis of the Boring Machining System, University of Illinois at Urbana-Champain, 1986 G M . Zhang, S.G Kapoor Dynamic Modeling and Analysis of the Boring Machining Sys-tem, Journal of Engineering for Industry, Vol. 109 pp219-226 1987 Frank R. Giordano and Maurice D. Weir, Differential Equations (A Modeling Approach), Addison Wesley Publishing, 1994 Ren H., Mechanics of Machining with Chamfered Tools, Master thesis, University of Brit-ish Columbia, Vancouver, Canada, 1998 E. J. A. Armarego, and Uthaichaya, M., Mechanics of Cutting Approach for Force Predic-tion in Turning Operations, Journal of Engineering Preduction, Vol. 1, No.l, pp. 1-18,1977 E.J.A. Armarego. Material Removal Processes-An intermadiate Course, The University of Melbourn, 1993 Sanjiv G Tewani, Keith E. Rouch, and Bruce L. Walcott, Cutting Process Stability of a Boring Bar with Active Dynamic Absorber, Vibration Analysis-Analytical and Computa-tional ASME 1991, Vol. 37 pp 205-213 D. R. Browning, I. Golioto, N. B. Thompson, Active Chatter Control System for Long-Overhang Boring Bars, Proceedings of SPIE The International Society for Optical Engi-neering, 1997, Vol. 3044, pp 270-280 S. G. Tewani, T. C. Switzer, B. L. Walcott, K. E. Rouch, T. R. Massa, Active Control of Machine Tool Chatter for a Boring Bar: Experimental Results, Vibration and Control of Mechanical Systems, ASME 1993, Vol. 61 ppl03-115 Bibliography 182 [27] S. Jayaram, M. Iyer, An Alalytical Model for Prediction of Chatter Stability in Boring, Transactions of NAMRI/SME, Volume XXVIII, 2000, pp. 203-208 [28] E. W. Parker, Dynamic Stability of a Cantilever Boring Bar With Machined Flats Under Regenerative Cutting Conditions, Journal Mechanical Engineering Science Vol. 12 No 2 1970, pp. 104-115 [29] S.A. Tobias and W. Fishwick A Theory of Regenerative Chatter, The Engineer-London, 1958 [30] S.A. Tobias, Machine Tool Vibration, Blackie, 1965 [31] J. Tlusty and F. Ismail, Basic Nonlinearity in Machining Chatter, Annals of the CIRP, 30:21-25, 1981 [32] J. Tlusty, Analysis of the State of Research in Cutting Dynamics, Annals of the CIRP, 27: 583-589, 1978 [33] J. Tlusty, High-Speed Machining, Annals of the CIRP, 42:733-738, 1993 [34] D. Montgomery, Milling of Flexible Structures, Master's Thesis, University of British Columbia, 1990 [35] J. Tlusty, A Method of Analysis of Machine Tool Stability, Proceedings of the 6th Machine Tool Design and Research Conference, pp5-14, 1965 [36] E.I.Rivin, H. Kang, Enhancement of Dynamic Stability of Cantilever Tooling Structures, Int. J. Mach. Tools Manufact. Vol. 32 pp539-561, 1992 [37] S. Ema, E. Marui, Supression of Chatter Vibration of Boring Tools Using Impact Dampers, International Journal of Machine Tools& Manufacture, Vol. 40 ppll41-1156, 2000 

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