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UBC Theses and Dissertations

Virtual turning system Zhou, Jing 2005

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V I R T U A L TURNING S Y S T E M  by  JING Z H O U B . S c , Zhejiang University, Hangzhou, China, 1993  . THESIS SUBMITTED IN P A R T I A L F U L F I L L M E N T OF THE REQUIREMENTS FOR THE D E G R E E OF M A S T E R OF APPLIED SCIENCE  in  THE F A C U L T Y OF G R A D U A T E STUDIES (Mechanical Engineering)  THE UNIVERSITY OF BRITISH C O L U M B I A  June 2005  © Jing Zhou, 2005  Abstract The goal of machining industry is to produce the first part correctly and most optimally without resorting costly trials on the shop floor. This thesis presents a Virtual Turning system which predicts the physics of machining rotational parts before actual production on the shop floor. As opposed to measurement of physical dimensions, cutting forces, torque, and power, they are predicted in virtual environment by integrating the laws of metal cutting process and the geometric and solid modeling of the tool-workpiece engagements along the tool path. The proposed Virtual Turning has two fundamental modules. The first module identifies the tool-workpiece engagement geometry along the path, which is used by the second, cutting process simulation engine. The initial workpiece geometry and tool path (Cutter Location) are imported from commercial C A D / C A M systems using industry standard IGES or STEP N C graphics formats. The tool-workpiece intersections along the tool path are identified by applying Boolean intersections of the two parts represented by their Boundaries. In order to expedite the time consuming computations, the in-process machining features along the path are classified, engagement conditions are parametrically modeled, and recalled instead of using Boolean operations recurrently along the tool path. The proposed hybrid model which consist of tool-workpiece engagements modeled by features or solid to solid intersections, can handle turning of a verity of two dimensional, symmetric rotational parts.  n  The contact length between the cutting edge and workpiece, and the chip area removed at each tool position are calculated by applying Green's Theorem to the tool-workpiece engagement boundary. The cutting force coefficients are modeled as a function of chip area, cutting edge contact length, tool geometry, feedrate and cutting speed. The cutting forces, torque, power and static deflections of the tool on the finish surface are predicted along the tool path. The algorithm can handle variety of tool motions which include taper and contour turning operations. The two dimensional Virtual Turning system is experimentally validated in machining a sample shaft with circular and taper features.  iii  Table of Contents  Abstract  ii  Table of Contents  iv  List of Tables  vii  List of Figures  viii xi  Acknowledgment  xii  Nomenclature 1. Introduction  1  2. Literature Review  5  2.1. Introduction  5  2.2. Force Prediction Models  5  2.2.1. Orthogonal to Oblique Transformation for Corner-radius Tools 2.2.2. Mechanistic Force Model for Corner-radius tools  6 10  2.3. Prediction of Chip Geometry  13  2.4. Solid Modeler and Z buffer Methodology  20  2.5. Feature Recognition Technologies  22  2.6. Swept Volume Techniques  23  2.7. Summary  25 26  3. Overview of the Virtual Turning System 3.1. Introduction  26  3.2. Overview of the Virtual Turning System  27  3.3. Tool-Workpiece Engagement Model (TWE Model)  29  3.4. Mechanistic Force Prediction Model (MF Model)  31  3.5. Assumptions of the Virtual Turning System  32  IV  3.6. Summary  36  4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology for Extracting Tool-Workpiece Engagements in Turning  38  4.1. Introduction  38  4.2. Full Solid Modeler-Based Methodology  40  4.3. Tool Swept Area (TSA) Construction  46  4.3.1. Linear Toolpath TSA Construction  46  4.3.2. Circular Toolpath TSA Construction  50  4.4. Green's Theorem-based Analytical Intersection Area Calculation  52  4.5. Feature-Based Methodology  62  4.5.1. In-Process Turning Features  62  4.5.2. Extraction of Material Removal Features  64  4.5.3. Geometric Invariant Machining Feature (giF)  69  4.5.4. Form Invariant Machining Feature (fiF)  71  4.6. Hybrid Analytical, Solid Modeler and Feature-Based Methodology  75  4.7. Implementation and Validation  77  4.8. Summary  82  5. Instantaneous Force Prediction for Contour Turning  83  5.1. Introduction  83  5.2. Mechanistic Model in Simple Turning  84  5.3. Prediction of Cutting Forces in Contouring Turning  89  5.4. Mechanistic Cutting Coefficient Evaluated form the Orthogonal Cutting Database  93  5.5. Experimental Validation for Contour Turning 5.5.1. Cutting Test Design  100 101  v  5.5.2. First Operation  102  5.5.3. Second Operation  106  5.5.4. Third Operation  109  5.6. Conclusion and Future Work  112  6. Conclusions  114  6.1. Conclusions  114  6.2. Future Research Directions  118  Bibliography  119  Appendix A. Circular Toolpath Tool Swept Area Construction  122  A. 1. Critical Position Calculations  122  A. 2. Tool Swept Area of Partial Circular Tool Path  124  Appendix B. Green's Theorem-Based Analytical Area Calculation  127  B. 1. Classes of Generic Tool Engagement Features (teF)  127  B.2. General Area Calculation Algorithm  128  B.3. Area Calculation Derivation for teF4  130  B. 4. Analytical Area Formulations for teFs  135  Appendix C. Engagement Boundary Identification in Geometric and Form Invariant Features  139  C. 1. teF Boundary Identification  139  C.2. Recursive Expression of teF Boundaries  149  vi  List of Tables  Table 4.1: Conditions of Generic Tool Engagement Features (teF) Table 4.2: Simulation Times and Accuracy for Two Solutions Table B . l Green's Theorem-Based Area Formulations for teFs  vn  List of Figures Figure 2.1: Orthogonal to Oblique Transformation for Corner-radius Tool  8  Figure 2.2: Mechanistic Force Model  9  Figure 2.3: Uncut Chip Area Decomposition  10  Figure 2.4: Equivalent Chip Thickness (h )  12  Figure 2.5: Simple Representation of the Corner-radiused Chip Area  12  Figure 2.6: Exact Area Calculation Using Geometric Shapes  13  Figure 2.7: Chip-Area Geometry with a Depth-Direction Variation  14  Figure 2.8: Chip-Area Geometry  15  Figure 2.9: Uncut Chip Area Calculations from Elements  16  Figure 3.1: Virtual Machining Model Proposed by Altintas [CIRP 1991]  20  Figure 3.2: Virtual Turning System  20  Figure 3.3: Dynamically Changing Engagement Geometry  29  Figure 3.4: 2D Cross Section of Turning Process Showing Feed Step Uncut Chip Area  32  Figure 3.5: Typical Cutting Tool Inserts and Generic Cutting Edge Geometry  32  Figure 3.6: Tool Geometry Constructions  33  Figure 3.7: Examples of the Constructed Tools in Virtual Turning System  33  Figure 4.1: Original Solid Modeler-Based Intersection Prototype  41  Figure 4.2: Full Solid Modeler-Based Turning Simulation Methodology  43  Figure 4.3: Three Cases of Tool Swept Area of Linear Toolpath  47  Figure 4.4: Linear Toolpath Tool Swept Area Construction  48  Figure 4.5: Circular Toolpath Tool Swept Area Construction  50  Figure 4.6: Tool Workpiece Engagement (TWE)  53  Figure 4.7: Classes of Generic Tool Engagement Features (teF)  55  Figure 4.8: One Example of teF4 Area Calculation  59  Figure 4.9: Classification of Features Generated from Turning  63  Figure 4.10:Transient Machining Feature (trF)  64  Figure 4.11: Material Removal Features (mrF) Generated during Turning  65  e  viii  Figure 4.12: MRA Decomposition  66  Figure 4.13: Geometry Invariant Features giF  70  Figure 4.14: teF Extraction from giF  70  Figure 4.15: Four Types of Form Invariant Feature fiF  72  Figure 4.16: teF Extraction from fiF  73  Figure 4.17: Hybrid TWE Extraction Methodology  76  Figure 4.18: A n Aerospace Turned Component Model  78  Figure 4.19: Simulation of the Machining for Various Tool Paths on Turning Part  79  Figure 4.20: Extracted Material Removal Features for the Turned Part  79  Figure 5.1: Mechanistic Force Model  84  Figure 5.2: Distribution of Friction Force along Cutting Edge  84  Figure 5.3: Friction Forces and Effective Lead Angle (fa)  86  Figure 5.4: Feed, Radial Forces in Each Region  87  Figure 5.5: General Contour Turning  88  Figure 5.6: Force Prediction of Contouring Turning  89  Figure 5.7: Orthogonal to Oblique Transformation  93  Figure 5.8: Tangential Force Predicted from Different Cutting Coefficient Identifications  95  Figure 5.9: Forces Predicted from Different Cutting Coefficients  96  Figure 5.10: Turning Process Plan of the Test Part  97  Figure 5.11: Tool Paths and Workpiece of First Cut  98  Figure 5.12: Comparisons of the Tangential Forces of First Cut  99  Figure 5.13: Comparisons of the Radial and Feed Forces of First Cut  100  Figure 5.14: The Changes of the Radial Forces with the Depth of Cut  101  Figure 5.15: Tool Paths and Workpiece of the Second Cut  102  Figure 5.16: Comparisons of the Tangential Forces of the Second Cut  103  Figure 5.17: Comparisons of the Radial and Feed Forces of the Second Cut  104  Figure 5.18: Tool Paths and Workpiece of the Third Cut  105  Figure 5.19: Comparisons of the Tangential Forces of the Third Cut  106  ix  Figure 5.20: Comparisons of the Radial and Feed Forces of the Third Cut Figure A . l Circular Toolpath Tool Swept Area Construction Figure A.2 Tool Swept Area of T T Construction e  s  Figure B . l Classes of Generic Tool Engagement Features (teF) Figure B.2: teF4 Area Calculation Figure B.3: The Type of Intersection Point P2 Figure B.4 Area Calculation of Edge e/ Figure B.5 Area Calculation of Edge 62 Figure B.6 Area Calculation of teFs Figure C . l teF Extraction within gift fiF Figure C.2 Circle-Circle intersection of P2 Figure C.3 Circle-Line intersection of P2 Figure C.4 Line-Circle intersection of P2 Figure C.5 Line-Circle Intersection of P3 Figure C.6 Line-Line Intersection of P3 Figure C.7 Workpiece Boudnary Point Q, Figure C.8 Recursive Expression of Boundaries  x  Acknowledgement  I would like to express the deepest appreciation to my research supervisor Dr.Yusuf Altintas for his valuable instruction, guidance, support, understanding and patience, which he has provided throughout my research at University of British Columbia. I would also like to extend my deepest gratitude to my co-supervisor Dr.. Derek Yip-Hoi. He has taken an enormous amount of effort to instruct and help me about the academic and language matters. Without his guidance and persistent help this research would not have been possible. I wish to thank all my colleagues in the Manufacturing Automation Laboratory for sharing with me their knowledge and experience, especially to Fuat and Dimitri, they have given me numerous suggestions when I had problems. And also Fuat helped me finish lots of machining experiments, which are very important to my research. I have learned a lot of them. I would like to thank Xuemei, Joseph, and Xiaobo, they gave me many helps in solid modeling. I also want to thank Yuzhong, a truthfully friend, he helped me in many ways. Finally, words alone cannot express the thanks I owe to Lifeng, my husband, for his persistent encouragement, assistance and patient; to my mother Shumin and my father Rongxian, for their lifelong love and unwavering support; and to my son Ricky, for him I can overcome any difficulty. This thesis and my all previous success are dedicated to them.  xi  Nomenclature  A  uncut chip area of the entire engagement  A,  uncut chip area of region 1  A  uncut chip area of region 2  b  width of cut  Q  tool center position at i feed step on a toolpath  Q-i  tool center position at i - l  CL  cutter location  d  depth of cut (mm)  f  feedrate (mm/rev)  fiF  form invariant feature  F  tangent force  F  radial force  Ff  feed force  Ffr  friction force  Ffrcl  friction force in region 1 of an uncut chip area  Ffrcl  friction force in region 2 of an uncut chip area  F«  global tangent force  F  r  global radial force  F  f  global feed force  F  x  2  t h  t  r  th  feed step on a toolpath  cutting force in X axis direction  Fy  cutting force in Y axis direction  F  cutting force in Z axis direction  z  G,  gravity center of region 1  giF  geometric invariant feature  h  chip thickness  ipF  in-process feature  xii  K,  cutting coefficient of tangent force  K  cutting coefficient of radial force  c  rc  Kf  c  cutting coefficient of feed force edge coefficient of tangent force  K  edge coefficient of radial force  Kfe  edge coefficient of feed force  Kfrcl  cutting coefficient of friction force in region 1  Kf 2  cutting coefficient of friction force in region 2  Kf  edge coefficient of friction force  I  distance between C, and C,-./  I,  the distance from Pi to V  h  distance from P2 to V  L  chip-cutting edge contact length  Lei  chip-cutting edge contact length of region 1  Lc2  chip-cutting edge contact length of region 2  MRA  material removal area  mrF  in-cut material removal features  0  circular toolpath center position  Pa  tool nose arc edge upper tangent point  Pb  tool nose arc edge lower tangent point  Pc  tool end cutting edge lower right point  Pd  tool side cutting edge upper left point  Pi  tool nose curve upper tangent point of a tool  Pi  intersection point between cutting edges of two tools  Ps  Intersection point between P/C, and the tool centered at C,./  Qi  workpiece boundary position intersected with current tool  Qi-i  workpiece boundary position intersected with previous tool  Q  the center position of a circular workpiece boundary edge  re  rc  re  c  xiii  R  circular toolpath radius  Ri.  curve region of an uncut chip area  R  Polygonal region of an uncut chip area  r  tool nose radius  r  i  the radius of a circular workpiece boundary edge  S  the length of a toolpath  teF  tool engagement features  trF  transient feature  TD  boolean difference between two tools  TWE  tool-workpiece engagement  TSA  tool swept area  Toolpath  a tool path  Toolpaths  tool paths  T  start position of a toolpath  2  e  s  T  end position of a toolpath  V  instantaneous feed direction at tool contact point  V  cutting speed (m/s)  a  toolpath angle  Pa  friction angle (degree)  Wr  side cutting edge angle  Wr'  equivalent side cutting edge angle with respect to feed  K  end cutting edge angle  e  r  <f>L + 1  effective lead angle effective lead angle of region 1  </>c  shear angle (degree)  Ts  shear stress (N/mm ) 2  xiv  Chapter 1 Introduction  The manufacturing of shafts, gears, discs and family of all rotational components involves turning operations. The rotational parts typically have varying geometric features along the axis with discontinuities, such as slots, keyways, and grooves. In general, all rotating mechanical parts used in common machinery, such as shafts and gears used in gear boxes, automobile engines, aircraft engine gas turbines, are produced with turning operations. The aim of manufacturing engineers is to optimize the machining cycle time of turning operations while respecting process constraints such as torque and power limits of the machine, breakage of the tool, dimensional tolerance of the part and chatter vibration limits of the machine tool and workpiece structures. The process constraints can be respected by selecting suitable feed, speed, depth of cut and tool geometry. However, the present practice in industry is based on the past experience of process planners only, and the selected cutting conditions may either be too conservative for high productivity machining, or too aggressive which leads to failure and repeated trials until satisfactory performance is achieved. The objective of this thesis is to create foundations of a virtual turning system which is capable of predicting the process behavior before any actual turning test is conducted on real machines. In addition, the virtual turning should lead to optimization of feeds and speeds which lead to minimum machining cycle time, i.e. high productivity, while respecting the physical limits of the process and machine tool.  1  Chapter 1. Introduction  2  The virtual turning can be realized by modeling the tool-workpiece intersection geometry along the toolpath, and modeling the mechanics of turning which leads to realistic prediction of cutting forces, torque, power, deflections and vibrations. Most of the past research has either focused on the modeling of basic cutting mechanics and dynamics of turning process, or geometric modeling of material removal process in solid modeling environment. There has not been much research activity in realizing an integrated virtual machining environment which includes the physics of the process. Virtual turning system has tool-workpiece engagement identification and modeling of process mechanics as a function of tool-workpiece engagement, tool geometry, feed, depth and speed of the cut. Further, optimization of process variables as a function of physical limits of the machine and cutting tool can be achieved by exploiting their results. The thesis presents research conducted in the aforementioned subjects, and their integration to achieve virtual turning system. Henceforth, the thesis is organized as follows. Chapter 2 reviews the relevant previous work in the fields of solid modeling of toolworkpiece intersection and mechanics of turning. The current C A D / C A M systems do not have any built-in algorithm which provides the tool-workpiece intersection. The relevant literature in extracting such geometric information along the toolpath is reviewed. The tool-workpiece engagement geometry may continuously vary along the toolpath, and it strongly affects the uncut chip area, hence the resulting force amplitudes and directions vary at each feedrate increment. The computational cost and accuracy of chip and force calculation methodologies reported in the literature are presented.  Chapter 1.  Introduction  3  Chapter 3 provides architecture of the proposed Virtual Turning system. The system consists of two main modules: The first module identifies tool-workpiece engagement conditions and chip geometry at discrete tool motion intervals, which are used to predict cutting forces, torque, power and deflections in the second module. The inputs and outputs, solid modeling techniques, the influence of the feed motion direction in contour turning, and the assumptions of the system are presented. Chapter 4 describes the algorithms developed to identify tool-workpiece intersection in turning operations. The workpiece geometry and N C Tool Path, i.e. Cutter Location (CL) file are imported from standard C A D / C A M platforms using IGES or STEP N C standards. The intersection of tool and workpiece is identified by two new techniques. The first method is based on the intersection of solid models of workpiece and tool on ACIS solid modeling kernel. The computational cost and robustness of the pure solid modeling approach led to the development of the second algorithm which integrates both solid model and feature-based engagement methods. The chip area is then predicted by applying Green's Theorem to the identified toolworkpiece intersection conditions. The prediction of cutting force, torque and power is presented in Chapter 5. The previously reported mechanistic model of the turning process is adopted by considering changing chip area and orientation of the cutting forces along the cut. The extensions to the algorithm allow handling of contour turning operations at discrete feed increments. The proposed Virtual Turning system is experimentally validated in machining a sample shaft with varying geometry.  Chapter 1. Introduction  4  The thesis is concluded in Chapter 6. The contributions to the literature in Virtual Turning are summarized and the future research directions which lead to the handling of arbitrary tool and workpiece profiles are discussed.  Chapter 2 Literature Review  2.1  Introduction Turning is one of the most commonly used metal cutting operations in industry. Many  research projects have focused on the cutting mechanics, modeling and simulation of turning to understand the physics of the process and increase its efficiency. The modern cutting-process models have stemmed from a fairly good understanding of the metal cutting process gained through the experimental findings of the early years of machining research. Part of this literature review presented in this chapter is concerned with the cutting mechanics, uncut chip area, and chip thickness. The proposed Virtual Turning system, i.e., the geometric and physical simulation of the turning process, is based on the integration within one system of geometric and solid modeling models and static force prediction models for different types of turning process. The system thus combines components in the areas of mechanics of turning, engagement geometry calculation, solid modeler techniques, feature recognition methods, and swept volume generation algorithms. A literature survey related to these aspects is presented in this chapter.  2.2  Force Prediction Models The cutting forces in turning operations are typically represented by the three orthogonal  force components, namely tangential F , radial F , and feed Fj forces. These forces are t  r  proportional to the area of the interference between the tool and the workpiece (uncut chip area)  5  Chapter 2. Literature  6  Review  as well as the length of engagement between the tool edge and the workpiece, and can be calculated as [Altintas, 2000] and [Armarego et al., 1985]: F =K A  +  F =K A  +KL  F =K A  +  t  r  f  lc  rc  fc  KL le  re  c  (2.1)  c  KL fe  c  If the tool rake face has an irregular geometry due to chip breaking grooves and chip tool contact restriction features, the cutting coefficients are identified using mechanistic models. A series of cutting tests are conducted with the specific tool at different speeds, radial depth of cuts, and feedrates. The coefficients are evaluated by curve fitting the force expressions to the measured cutting forces and chip geometry. If the rake face of the tool is smooth and uniform, it is possible to model the cutting edge as an assembly of oblique cutting edges [4,7]. The cutting pressure at each discrete oblique cutting edge element is modeled by applying the orthogonal to oblique transformation method proposed by Armarego [7]. Both approaches will be introduced in this literature review.  2.2.1 Orthogonal to Oblique Transformation for Corner-radius Tools If the insert's rake face is uniformly flat without chip breaking or contact reduction grooves, the turning insert's curve cutting edge can be considered an assembly of oblique cutting edge elements. Oblique cutting mechanics laws lead to the prediction of cutting pressure at each discrete cutting edge element, which depends on the discrete chip area, edge geometry, and orthogonal cutting parameters of the work material (i.e., shear stress, shear angle and friction  Chapter 2. Literature  7  Review  angle) which are mapped using classical mechanics laws proposed by Armarego [7]. The details of the orthogonal to oblique cutting transformation can be found in [1,2,4,7]. The three cutting force components can be expressed as follows: F,=F F =K -b-h lc+  F  f  le  +  lc  =F  +F =K -b-h  fc  fe  fc  F =F F =K -b-h r  rc+  re  K -b le  + K -b +  rc  (2.2)  fe  K -b re  where the oblique cutting coefficients are presented as follows: K  T s  - « „ ) + tan i tan 7sin /?„  <n  cos(  ^«  B  ^cos (fa  sin  2  - « „ ) + tan77 sin B„ 2  n  +  B  s  sin(/J„ - g „ )  T K  2  sin fa cos/ ^/ os (fa + B„ - a ) + tan n sin B  f c  2  2  C  t  cos(0„-a )tani  s  s i n  n  ^  2  n  + t&n7jsmB„  - « J + tan77 sin /J„ 2  ^cos (fa +p 2  n  '  (2 3) n  2  The shear stress (T ), the shear angle (</>), and the friction angle (/?„) are determined from 5  n  the results of the orthogonal cutting tests [1,2]. The uncut chip area is divided into three regions (Figure 2.1) due to the tool nose curve and lead cutting angle.  8  Chapter 2. Literature Review  Figure 2.1: Orthogonal to Oblique Transformation for Corner-radius Tool In region 1 the uncut chip area is divided into small differential elements, for each of these elements, the oblique tangential, radial, and feed forces can be determined as:  = K c,i • A r  f,<  F  Where, Ais  =  fc>  K  'U A  u  + K  re  + K  fe'  the chip area of i' element, and L  ci  •L  c4  (2.4)  c,i  L  is the chip-cutting edge contact length of  i element. By summing all the respective force components, the cutting force in region 1 can be t h  determined as equation (2.5), in which 0, is the uniform angular increment of each element:  Chapter 2. Literature Review  xJ  -Y, ti,i i=l  F  F  F ,,=t(F/,,,s™O,-K ,,cos0 ) y  l  (2-5)  i  /=/  = 2Z(F/ij cos0, -F  F  zJ  sm0,)  rlJ  i=l  In region 2 and region 3, the approach angle is assumed to be the side cutting edge angle (y/ ) and half of this angle respectively. The cutting force components can be calculated as: r  x,2  F  F  =  t  2  y,2 = f2 sin(-y ) - F  F  cos(-y/ )  F  r  z,  F  = f  cos(-^ ) - F  F  2  r2  2  r  r  (2.6)  sm(ry/ )  r 2  r  and F 3 = x>  F  l3  F ,3 = F y  sin(- y/ /2) - F  F  f3  z>3  r  =F  cos(- y/ /2)  r3  r  (2.7)  cos(- y/ j2) - F sin(- y/ /2)  f3  r  r3  r  The total forces for the entire uncut chip area in global X , Y and Z directions are found as follows:  F =F  xJ  x  y  = yJ  F  F  F =F z  2J  +F  + F  + y,2  + y,3  x2  F  x3  (2-8)  F  + F, + F, y 2  z 3  The advantages of this model are that it is valid for a range of cutting tool geometries and have been verified for good force prediction from past research. The disadvantages are that the tool nose curve needs to be discretized into small segments at each feed step, because this model is only valid for straight cutting edges. Also the cutting coefficient expressions are complicated,  10  Chapter 2. Literature Review  and they have to be evaluated for each element. Differential cutting forces are summed up to the total X Y Z forces, but these X Y Z forces predicted only represent for one feed step. These disadvantages make the orthogonal to oblique transformation approach less efficient in modeling the contour turning, which likely has large numbers of feed steps. Moreover, this model is valid only for the tools with a flat rake face, which limits the model to be used for general turning processes. Therefore, a mechanistic model is adopted in this research, which is described in the next section.  2.2.2 Mechanistic Force Model for Corner-radius tools In the mechanistic model proposed by Atabey et al. [1,2], cutting forces are represented by a tangential component (F ) and a frictional component (Ff ) as shown in Figure 2.2. F^-is further t  r  resolved into radial (F ) and feed (Ff) forces. Cutting forces at each feed step are modeled as a r  function of the uncut chip area (A) and the chip-cutting edge contact length (L ) as shown in c  equation (2.9): F = F F =K -A lc+  te  lc  +  Ff =Ff +Ffre=Xfr -4 r  C  C  K -L le  +  c  K -L fre  c  Chapter 2. Literature  11  Review  Figure 2.2: Mechanistic Force Model To account for the differences in cutting mechanics between the tool's leading edge and nose regions, an approximated polygonal region is used for force prediction along the leading edge and a region bounded by arcs and lines is used to account for the nose. The continuously changing oblique angle at the nose is the reason for the differences. As a result, the uncut chip area is divided into two zones as illustrated in Figure 2.3.  d  Figure 2.3: Uncut Chip Area Decomposition  Chapter 2. Literature  12  Review  The frictional force in equation (2.9) is modified as follows to account for this change: (2.10)  F =K A +K A +K L fr  frcX  x  frc2  2  fre  c  Where, the areas (Aj and A2) are corresponding to the areas of two regions. K , Kj i, and tc  Kf  rc2  rc  are empirical proportionality coefficients between the corresponding cutting forces and the  uncut chip area, they are modeled as a nonlinear function of cutting speed and uncut chip area or chip-cutting edge contact length; K and K/ are empirical proportionality coefficients between le  re  the corresponding edge forces and the length of the tool cutting edge engaged with the workpiece. For a given tool-workpiece material combination and tool geometry, the cutting coefficients can be identified from a set of cutting tests where the forces are measured for ranges of feeds (/) and depths of cut (d), and the collected data is processed with multivariable regression analysis [Altintas, 2000]. Later, the tangential force and friction force are resolved to X Y Z forces by using an effective lead angle. With all necessary inputs described above, the instantaneous force components can be calculated from the above equations at each feed step of the cutter. Rohi G. Reddy [29,30] used normal force (F„) and the frictional force (Fj) in his mechanistic model for contour turning as shown in equation (2.11). Similarly, these two forces are resolved to the global X , Y, Z direction at the end. F =K -A n  n  c  F =K f  f  (2.11)  A  c  Where K„ and A/are specific cutting energy coefficients, which are functions of equivalent chip thickness, cutting velocity, and normal rake angle. It can be seen that this force model is in  Chapter 2. Literature  Review  13  fact a simplified mechanistic force model analogous to equation (2.9) for the static case, but the uncut chip area^ and cutting coefficients have different definitions. From the review of research up-to-date, it can been seen that the turning static forces can be predicted using the well known existing force models, in which forces are proportional to the area of the interference between tool and workpiece and the cutting coefficients. Therefore, only the mechanistic force model proposed by Atebey [1,2] is adopted in this research. However, it is necessary to develop a new method to identify the continuously changing tool workpiece intersections and the cutting coefficients effectively and accurately. Some developed methods to predict chip geometry are reviewed in the next section.  2.3  Prediction of Chip Geometry Since the early days of metal cutting research, researchers have observed the machining  force to be proportional to the cross-sectional area of the uncut chip being removed. This chip area is defined as that area bounded by the tool edge profiles corresponding to two tool passes. The theoretical analysis of the machining processes, dating back to the early 1940s or before, has considered this fact and modeled the machining force components as proportional to the chip area. However, in the presence of vibration, runout, or special cutter designs such as contour turning applications, establishing an analytical chip-area expression is not a trivial matter when working with processes used by industry, i.e., contour turning, those that exhibit complex tool forms. The most common tool form seen in turning and boring consists of a straight major lead cutting edge, a straight minor cutting edge, and an edge with a corner radius that connects /  14  Chapter 2. Literature Review  blends together the two straight edges. Ozdoganlar [26] termed the ensuing chip-area representation and analysis corner-radiused tools. The chip thickness is a calculated factor based on the chip area. The Equivalent chip thickness is used in calculating cutting forces and tool life, described 1936 by Woxen [9] as shown in Figure 2.4. he  Figure 2.4: Equivalent Chip Thickness (h ) e  The equivalent chip thickness (h ) is the quotient of the approximate chip area, which is the e  product of the depth of cut (a ) and feedrate (j) divided by the active tool edge length (l ). p  saD  K  (2.12)  = SaD  SaD  where a -r -(\-cosK ) p  SaD  e  r  sin*-  [  r  K -r -n r  E  180  |  /  2  (2.13)  Chapter 2. Literature Review  The active tool edge length  15  l$ D a  is considered as a straight line in the equivalent chip area,  as presented in Figure 2.4. However, equations (2.12-2.13) are approximate values for the real chip thickness. Moreover, these geometric and trigonometric methods become complex when tool inclination and rake angle change.  J  d  Figure 2.5: Simple Representation of the Corner-radiused Chip Area A traditional graphical representation of the chip area is shown in Figure 2.5, where r is £  the corner radius and y/ is the lead angle. For the simple situation shown here, where feed r  direction is defined to be parallel to the uncut surface and a depth direction is defined to be perpendicular to the feed direction, the commonly seen chip area expression is  a - fd -a  c  (2.14)  a is the area of the cusp left on the cut surface at the intersection of the current-pass and c  previous-pass profiles. (2.15)  Chapter 2. Literature Review  16  The area model is attractive, since fd is a simple product involving the depth of cut and feedrate, a is a fairly simple function of feed, corner radius and sometimes, in extreme cases, c  the lead and end-cutting edge angles. However, contour turning introduces profile-to-profile variations in the feed direction as well as the depth of cut value and direction. This representation cannot give the correct prediction. A method for computing the exact chip area, in the presence of depth and feed variations, has been developed by Endres [29] based on the addition and subtraction of geometric shapes. Figure 2.6 illustrates the geometric shapes for the large depth case. Subtracting the crosshatched area, a circular segment, from the shaded area composed of three triangles and a circular segment, the exact area is obtained. However, this exact result is computationally complex and fairly algorithmic since it includes many cases with several conditions and requires the coordinates of each of the five points in Figure 2.6 to be computed. Moreover, this primary trigonometric method cannot give general expression of the chip geometry.  Figure 2.6: Exact Area Calculation Using Geometric Shapes Ozdoganlar [26] proposed an analytical representation of the area which is employed for a tool at each time step in time-domain simulations. This analytical representation also opens the  17  Chapter 2. Literature Review  door to analytical machining dynamics, where one seeks analytical solutions for stability limit and vibration level.  Figure 2.7: Chip-Area Geometry with a Depth-Direction Variation Chip-area expressions are derived for "small" and "large" depth cases using a vectorial approach. Large depth of cut case is shown in Figure 2.7. r (1 — sin((// ) Large depth of cut  a, = fd  Small depth of cut  a  Where d  m  =  m  +  ^  )  + r (/, - /) + Ad(^—  + d tm(y/ )) - ac,  X Y r /  e  , = fd  m  m  + r (/. - f) + Ad^d (2r £  m  E  r  - d ) - a, m  c  (2.16)  (2-17)  is the mean depth of cut. /*, a * is the equivalent feed and cusp area. c  Since the presented chip area is an approximation, it involves analytical error compensation and numerical error compensation. Therefore, it is not desirable for static force prediction. Another analytical chip load solution has been developed by Rohit [29,30]. The approach is to divide the intersection region into three zones, depending upon the tool parameters and the  18  Chapter 2. Literature Review  cutting conditions, and to calculate separately the portion of the chip load in each zone, as depicted in Figure 2.8. i  ^ x  Chip width  1  Zone I  Figure 2.8: Chip-Area Geometry  The area of Zone 1  A  d  = area of quadrilateral  OG  f  The area of Zone 2  f  '•ell  The area of Zone 3  d%  (2.18b)  p  p  c  2  R -OcH ^  (  A clll  (2.18a)  c  -OM ^  2  LLEE  2  2  d^  (2.18c)  The total uncut chip area is A - A +A c  cl  cll  + A  clll  (2.19)  Chapter 2. Literature Review  19  This model involves numerical integration, and an equivalent chip thickness is calculated for identifying the cutting coefficients.  A number of calculations are required to separate the  uncut chip area to three zones and get the chip areas, and this method divides the plane of the tool motion into four quadrants, additional identifications are also required to cast the cutting conditions into one of these four quadrants. Instead, a general analytical area representation for all cases is desirable. Atabey [1,2] obtained the uncut chip area by summing up the contributions from a discretization of the intersection into approximate geometric elements. See Figure 2.9.  Figure 2.9: Uncut Chip Area Calculations from Elements The total chip area in Region 1 is evaluated by a discrete summation of all differential elements in the curved region. Region 2 is considered to be a rectangle, although one side of it has a slight curve caused by the corner radius of the previous tool position. Region 3 is a simple triangle. Finally, total uncut chip area is found by adding the areas for each region. This method  Chapter 2. Literature Review  20  also uses numerical integration, and the area is approximate. This method can be used in one constant cutting condition, but it is not efficient for continuously changing geometry cases. Armarego [7,32] identified nine different types of tool-workpiece intersections in turning depending on the depth of cut and feedrate limits, and developed the analytical solutions. But the cutting conditions were originally developed for longitudinal straight cuts. If this method is used on contour turning, it is difficult to cast an uncut chip area to those 9 categories due to the complex calculations of three varying feedrate limits along the contour toolpath. Also it approximates the workpiece boundary as a straight line. Since the feedrate in the turning operation is usually smaller than nose diameter, there are three cases in those nine cases that will not be considered in this research. Other significant research has developed area calculation methods for tools with curved geometry, but most of them are approximations and cannot capture the varying geometries along the arbitrary toolpath and workpiece surface. In this thesis, taking advantage of Green's Theorem-based analytical area calculation, the intersection area and other parameters, which are represented by simple algebraic integration formulae, can be calculated accurately and effectively. Moreover, this analytical solution is generalized for any arbitrary closed area, and the boundary conditions are easy to obtain by calculating few intersection points between the line and arc. The details are given in Chapter 4.  2.4  Solid Modeler and Z buffer Methodology Area calculation methods investigated in the previous section are in fact not capable of  capturing the tool-workpiece engagement along the whole arbitrary toolpath and in-process  Chapter 2. Literature Review  21  workpiece, since the instantaneous intersection geometry varies dynamically and hard to predict. Some research that has been done for contour turning requires the workpiece must be simple so that it cannot be used widely. The biggest difficulty is that the depth of cut varies depending on the relative distance and direction between a toolpath and an in-process workpiece boundary. The accurate depth of cut can be identified by using solid modeling in which the workpiece and the tool are represented as B-rep (Boundary Representation) models, and their in-process geometries and topologies are obtained by applying Boolean operations with the swept area of the tool. In this research a Solid Modeler is used as the basis for extracting the intersections from the continuously changing geometries. Researchers have in the past investigated the potential of solid modelers to support modeling machining processes [6,9,35]. Computational complexity was identified as one of the difficulties in adopting this approach. In addition, research has also demonstrated geometric and solid modeler cutter-workpiece intersection calculations within the context of an integrated virtual machining environment where the modeler provides inputs to the process models [5,6,13,14]. While results are promising, they do not address all the possibilities that can come from the range of geometry, processes, cutting tools, and machine tool axis configurations that are encountered. In particular, most approaches are for 2'/2 D milling operations. There is little research that focuses on turning operations, especially when the initial workpiece is noncylindrical and when multiple turning operations (facing, profiling, grooving) lead to intersecting machining features. The proposed research partly addresses this deficiency. It is also important to mention that other techniques have been studied for finding tool workpiece  Chapter 2. Literature Review  22  engagements. Most notable is the Z-buffer method originally developed for N C verification but adopted to obtain engagement geometry. Examples include Takata [35], Jerard [15,16] and Lazogolu [23]. However, again these researchers target end mill part intersections, but not turning. In this research, A C I S Solid Modeler kernel is used to model the turning process, where 1  the in-process workpiece and tool are constructed as B-rep solid models, tool travels along tool paths, and Boolean operation is used to subtract the intersection area and update the workpiece after each toolpath.  2.5  Feature Recognition Technologies Significant research has investigated the problem of feature recognition. Reviews of this  research can be found in [33,36]. Feature recognition addresses the problem of identifying engineering relevant regions of interest (faces, edges, points) from a C A D model. Typically, two main approaches have been used for recognition of 2D rotational features. One is syntactic pattern recognition, used by Jakubowski(1980), Srinivason (1985), and L i (1988). The 2D boundary of the part is captured as a string of geometric primitives that are then parsed using a grammar to identify feature patterns. This approach does not consider the workpiece boundary, and additional steps are required to generate machining volumes. Another technique is the rule-based feature recognition approach, which was used by Davies et al. (1988) and Joseph and Davies (1990). Features are recognized using decision logic expressed as rules within an expert system as part of a "backward planning" strategy. 1  In this research the ACIS solid modeler product is used to model and manipulate geometry  Chapter 2. Literature Review  23  Feature recognition research has directly supported Computer-Aided Process Planning by targeting features on the final part geometry. The surfaces of these features are used to identify appropriate machining operations to be applied to the initial workpiece. Identifying features for supporting process modeling has not received significant attention. These features differ in that they appear on in-process states of the workpiece. This new type of feature is the focus of the technique described in this thesis. The feature identification approach used is based on a 2D area decomposition algorithm proposed by Cho et al. (1994) and Sakurai and Chin (1994). The difference of in the approach used in this research is the type of decomposition used, and the definitions of in-process machining features that are specified for turning operations. Since there are a small numbers of engagement conditions in turning operations, feature identification method is developed to extract the intersection geometry from three machining features to enhance the computational efficiency.  2.6 Swept Volume Techniques Significant amounts of research have focused on developing swept volume algorithms since swept volumes are used in a variety of applications such as robotic analysis, collision detection, machining verification, and simulation. These methodologies can be classified into mathematical approaches and engineering approaches. Examples of mathematical approaches reported are the Jacobian Rank Deficiency method (JRD) and Sweep Differential Equation (SDE) approach. JRD method has only been demonstrated in parametric and implicit surface sweeping with multiple parameters. The SDE method has been demonstrated for planar parametric curves sweeping. These general methods  Chapter 2. Literature Review  24  have provided some well-established solutions for analytical curves and surfaces sweeping. However, these approaches are not practical in solid model-based applications because analytical expressions of curves or surfaces are not always available. Some engineering methods have been developed to generate swept volumes for N C verification for 5-axis machining. For instance, Sheltami et al. (1998) uses generating curves to get swept volumes of toroidal cutters, Roth et al. (2001) do surface swept by a toroidal cutter during 5-axis machining. Weinert et al. (2003) generates swept volume for the simulation of machining processes. A swept area of a two-dimensional turning tool is the union of the area occupied by the tool at all positions during the motion. The swept area is generated using a boundary representation of the border of the sweeping body, such as line segments and arcs. The boundary of the swept area is developed as the envelope of all plane curves representing the boundary of the body at all positions of the body included in the sweep. Unfortunately, most planar moving bodies cannot be represented by a parametric equation as simple as a circle. The methodology for identifying "envelope points" is described in Ling and Chase [22]. In this research, since the toolpath and tool geometry are a combination of lines and arcs only, a simple swept area algorithm has been developed for the turning process, which is different from [22] in the way of finding extremal points and constructing edges. Details of the algorithms for linear toolpath and circular toolpath are explained in Chapter 4.  Chapter 2. Literature Review  2.7  25  Summary In this chapter, an outline of the literature in mechanics of turning, chip geometry  calculation, solid modeler techniques, feature identification methods, and swept volume algorithms has been presented. It has been shown that well developed turning mechanics models can predict cutting forces accurately, as long as the chip geometry is provided correctly. The tool-workpiece intersection calculation becomes challenging along the contour turning, when the engagement changes dynamically. Solid Modeler techniques are used to model the whole turning process, while the computational complexity is a difficult issue. The contributions intended in this research have been placed in context with the reviewed literature.  Chapter 3 Overview of the Virtual Turning System  3.1  Introduction The modem manufacturing sector requires rapid design, manufacturing, and deployment of  products in small batch sizes. When the batch size is small and the lead period is short, the industry cannot afford to conduct costly test trials on the shop floor. The goal of virtual machining, as proposed by Altintas [CIRP 1991] is shown in Figure 3.1, is to machine the part using a mathematical model of the process in a simulation environment.  CAD M O D E L N C Tool Path Cutter Geometry  MACHINING DATABASE Spindle Dynamics (FRF) Feed Drive F R F and Control Cutter-material Cutting Constants Kinematics of the Machine Volumetric Error Model of the Machine  FINAL PROCESS PLAN Optimized Speed, Feed, Depth, Width, Error Compensation  Cuttcr-part Intersection Geometry'Calculations  Machining Process Simulation  CAD BASED VIRTUAL MACHINING PROCESS SIMULATION SYSTEM  PATH C L File  Improved path planning  Strategy Analysis  simulation results  MONITORING AND CONTROL DATA Peak force, torque, power,tracking error, modal frequencies  Figure 3.1: Virtual Machining Model Proposed by Altintas [CIRP 1991]  26  27  Chapter 3. Overview of the Virtual Turning System  Altintas states that "The part must be produced accurately and most optimally in the shortest cycle period at the first trial on the shop floor, which is possible only i f the mechanics of the metal-cutting process and the dynamic behaviour of the machine tool are modeled accurately using the laws of physics" [Altintas, N S E R C - P & W C Industrial Research Chair Grant Application]. The proposed Virtual Turning system is a component of the CAD-based process simulation module in Virtual Machining Model proposed by Altintas [CIRP 1991] as shown in Figure 3.1, and developed in this thesis. This chapter provides the brief overview of this system, which includes two main modules and their inputs and outputs, the assumptions, and the capability of the system.  3.2  Overview of the Virtual Turning System Tool Path (APT C L file) Tool Geometry, In-process Workpiece Model  Conventional C A D / C A M Software (CATIA, Pro/E, UG) or Developed Virtual Turning System  Geometric Solid Modeling j . . . , Engagement Information Tool-Workpiece Engagement Model (TWE Model)  Mechanistic Force Prediction Model (MF Model)  VIRTUAL TURING SYSTEM Optimization based on Maximum Force, Torque, Power. Chip Load, Deflection  Cutting forces, Power, Torque Deflection, Chatter stability  Figure 3.2: Virtual Turning System  Chapter 3. Overview of the Virtual Turning System  28  Figure 3.2 shows a flowchart of the Virtual Turning system. This system is composed of two main modules, i.e., the Geometric and Solid Modeling Tool-Workpiece Engagement Model (TWE model), and the Mechanistic Force Prediction Model (MF model). In the TWE model, an intersection extracting methodology is developed. The outputs of the TWE model are the inputs of the MF model, where the cutting forces, power, and torque are calculated by using a mechanistic force prediction approach. The inputs and outputs of these two modules, along with the brief introduction of these modules are presented in the follows. Three inputs are required to the TWE model. First, an A P T C L file (Cutter Location file) generated from C A D / C A M software, which describes the toolpaths, is read and saved to a cutter location array. Each pair of nodes represents the start position and end position of one N C block in the APT file. If the toolpath is an arc, the center position of this arc, the radius, or the tangent direction of the start position are also stored in the data structure. In the simulation, the-tool moves along the tool path, the instantaneous intersections and force calculations rely on the tool position and the feed direction at each machining step. Second, a 3D workpiece STEP file (Standard for Product Model Data file), which is also exported from C A D / C A M software, is translated to an A O S B-rep (Boundary Representation) model by a solid modeler translator, 1  which is ready to be manipulated. Additionally, an in-process workpiece model, which is the updated workpiece or the final part after the intersection calculations in TWE model, can also be inputted as an initial workpiece for the next process simulation. Third, tool geometry, which is described by a nose radius, a side and an end cutting edge angles, and a side and a back rake  1  In this research the ACIS solid modeler product is used to model and manipulate geometry  Chapter 3. Overview of the Virtual Turning System  29  angles, are imported separately, since a turning C L file does not provide entire tool geometry information. The intersections of the tool and the workpiece at every feed step along the toolpaths are identified using the TWE model. The outputs of the TWE model are uncut chip area (A, A/, A2 (mm )), chip-cutting edge contact length (L , L , L 2 (mm)), feedrate (f (mm/rev)), depth of cut 2  c  c/  C  (d (mm)), cutting speed (F(m/min)), workpiece radius (r (mm)), effective lead angle (fa (rad)), and machining time (T (s)). These outputs are needed for force calculation in MF model. The instantaneous cutting forces, power, and torque, which are the outputs of the MF model, and also the workpiece deflection and chatter stability, which can be predicted from the forces easily, will be used to optimize the process at desired feed increments by selecting feeds and spindle speeds based on a set of machining constrains, such as machine tool maximum / minimum speeds and feeds, maximum power and torque, and stability limits.  3.3  Tool-Workpiece Engagement Model (TWE Model) As described in the literature review, tool-workpiece engagement geometry can be  predicted when the workpiece is simple, and when the tool trajectory, depth of cut and feedrate are known. However, for complex workpiece geometry and toolpath, the geometry of cut varies at every machining step and is hard to predict, one example is shown in Figure 3.3.  Chapter 3. Overview of the Virtual Turning System  30  Figure 3.3: Dynamically Changing Engagement Geometry One of the solutions to the problem of capturing and manipulating the realistic, complex geometry dynamically in the C A D / C A M environment is the use of solid modelers. The ACIS Solid Modeler is one of the most commonly used solid modeling kernels, thus the proposed TWE model in Virtual Turning system is based on the ACIS solid modeling kernel. ACIS, the 3D Geometric Modeler, is an integrated software library of geometric and solid modeling algorithms which can be used in the development of any application requiring the representation and manipulation of 3D geometry. ACIS represents the exact shape of an object because it creates and records the equations of the curves and surfaces. A boundary representation (B-rep) is use to define complex 3D shapes in terms of the geometry of faces and edges and the topology (the relationships between these faces and edges) that define the physical boundary of the object. In the machining process, the tool solid model moves along the toolpaths, and intersects the workpiece continuously. Boolean operations, which include union, subtraction, and intersection, are used to obtain the tool-workpiece intersection and update the in-process workpiece. Within these operators, first, all the intersections between the two bodies are identified. Second, the intersection graph is imprinted onto both bodies. This splits faces with these intersection curves  Chapter 3. Overview of the Virtual Turning System  31  into new faces. Third, a decision is made to determine which of the new faces are to be kept and which should be discarded. Finally, the new B-rep model after the Boolean operation is reorganized to ensure a valid topology. Every Boolean operation leads to complex computations on the B-rep model, and the computation is not fully reliable. Due to the computational complexity and robustness problems surrounding Boolean operations, analytical and feature-based methodologies are developed in this research to increase the efficiency and robustness of the process simulation, which will be described in Chapter 4.  3.4  Mechanistic Force Prediction M o d e l  In the force prediction model, a mechanistic approach, with the tool having a nose radius, is used to predict the cutting forces [1,2], which will be described in Chapter 5 in detail. As shown in the literature review, the cutting forces are represented as a function of the intersection geometries as shown in the follows: F = K-(A,L ) C  (3.1)  The intersection geometries come from TWE model, which is described in the previous section. Since the proposed Virtual Turning system is aiming to simulate the whole turning process, the MF model has the ability to predict forces, power and torque continuously, i.e., the MF model uses intersection geometries at every machining step to calculate the instantaneous forces along all cutting steps.  Chapter 3. Overview of the Virtual Turning System  32  In contour turning, tool feed direction changes along the toolpath, while the forces are predicted with respect to the feed direction, i.e. the predicted feed force at each step is in the same direction with the instantaneous tool feed direction at that step, and the predicted radial force is in the direction that is perpendicular with the feed direction. These two local forces need to be resolved and summed in the global X Y Z directions for further simulation and optimization. Another challenge of this force model is to predict forces correctly and continuously for different types of turning. Since the mechanics of these different turning operations are regarded as the same cutting principles, one force model is applied for many types of turning operations. Therefore, no matter what type of a turning operation is, the predicted forces are carried out in the local coordinate system with respect to the cutting mechanics, and then these forces are projected onto the global X Y Z directions for predicting the cutting forces, power and torque along the whole machining process. After the cutting forces, powers and torques of the whole process are predicted, they can be presented together to find the critical process parameters, such as the maximum cutting forces, the maximum power and torque, and the maximum chip load (engagement area), compared with the machining constrains, such as the machine-allowed forces, power, torque, and chip load, to increase the material removal rate (increase feed or speed) i f the process parameters are too low, or decrease the feed or speed if they are too high. And also the chatter stability is predicted by using depth of cuts (also form TWE model) and cutting speed.  Chapter 3. Overview of the Virtual Turning System  3.5  33  Assumptions of the Virtual Turning System In this section the assumptions that are made in this research are outlined. These are based  on the limitations of the force prediction model adopted and simplifications to the cutting tool geometry. •  Rigid workpiece and cutting tool: As such deflections due to flexure and dynamics are neglected. This constraint is based solely on the limitations of the force prediction model that is currently used. A more sophisticated model that calculates deflections can easily be incorporated when available. The location of the cutting tool can be adjusted accordingly to account for this during the engagement calculations.  •  2D modeling of workpiece geometry: In turning operations, as the workpiece rotates, the tool moves longitudinally along the rotational axis, hence the tool actually sweeps out a 3D helical volume. Since as discussed in the previous section, the cutting forces are related to the uncut chip area, little accuracy is lost in reducing the problem to manipulating 2D cross sections of the workpiece and swept volume. Again, this assumes that dynamics are not considered. If this were not the case then the true impact of vibrations and chatter from process instability on form and surface finish can only be accurately modeled in 3D. A consequence of this simplification is that area calculations need only be made at feed step intervals, i.e., the distance moved per revolution of the workpiece as illustrated in Figure 3.4. While static deflections can be modeled in 2D this is not done in this research.  34  Chapter 3. Overview of the Virtual Turning System  positions of the tool  Figure 3.4: 2D Cross Section of Turning Process Showing Feed Step Uncut Chip Area •  2D modeling of tool geometry: For the tool workpiece intersection calculations, the region of interest on the cutting tool that defines the uncut chip areas is in the region of the tool nose. As can be seen from Figure 3.5, for a wide range of different tool geometries, this defaults to either a circular edge or two straight edges with an interconnecting circular edge. As is also shown in the figure, these three pieces of geometry can be defined by the tool nose radius (r ), e  the side cutting edge angle  (y/ ), r  and the end cutting edge angle  (K ). r  Figure 3.5: Typical Cutting Tool Inserts and Generic Cutting Edge Geometry  Chapter 3. Overview of the Virtual Turning System  35  Therefore, the tool is constructed by using its major and minor straight cutting edges, nose curve, side cutting edge angle, and end cutting edge angle. Figure 3.6 illustrates the geometric construction of the generic tool shape that is used. r : tool nose radius MV side culling edge angle K,: end cutting edge angle /.: tool height W: tool width c  ••.  1. Tool center position OYX, \„) 2. fool nose are edge e upper tangent point P X„ "X, - r^-osiy,) Y = J'„ + r .v/n(i|i ) 3. Tool nose arc edge e, lower tangent point l\ X .V„ • r,iiniK ) 2  t  a  a  r  t  r  4. Tool straight edge e, upper left point P  d  Xj  .V„ [- r cos(y,) * [I. - rjl • ,v/«(v))]to«(H/,)J r  t  ft" K t-r, 5. tool straight edge e, upper right point P  L  [W- r ( I< .V//J(K,)1 /C<W(Vi " r )  II  K  c  X = UcOS(K ) • Xf, e  }„  r  II i'(>l{KV) I i'l,  Figure 3.6: Tool Geometry Constructions This construction method provides sufficient tool geometry information for intersection calculations. Further, this tool model is general enough to be applied to the different types of insert geometries, such as facing tools, contour turning tools, grooving tools, and boring tools as described in case (a), (b), (c), and (d) as shown in Figure 3.7.  • (a) Facing  (b) Contour Turing  (c) Grooving  (d) Boring  Figure 3.7: Examples of the Constructed Tools in Virtual Turning System  Chapter 3. Overview of the Virtual Turning System  36  Since the rake angle of the cutting tool face where the uncut chip area is calculated typically does not lie in the plane of the workpiece cross-section, the face geometry is projected onto this plane. Straight lines project to lines, while the circular edge defining the tool nose radius projects to an ellipse. It is assumed for rake angles typically encountered in practice that this ellipse can be reasonably approximated with a circle with radius equal to the tool nose radius. •  Tool path geometry: The toolpaths in this research are followed by the nose centre of the cutting tool. Toolpaths consist of linear and circular components only. Spline toolpaths are assumed to be discretized into small linear segments.  •  Force modeling for contour turning: The adopted mechanistic cutting force model is verified from the cutting tests in the past only on the longitudinal straight cutting. In that condition, the feed is in the direction of the spindle axis and the depth of cut is constant during machining. In this research, the mechanics of contour turning, along with different types of turning operations, is assumed to apply the same cutting principles. Therefore, when the feed has a machining axis component and a radial component (in contour turning or taper turning), the force model is still assumed to be valid. More cutting tests should be done in the further to verify this force modeling approach.  •  Tool feed direction on contour turning: Tool feed direction varies along the contour toolpaths, the two feed directions at tool successive positions separated by a feed interval are the tangent vectors along the tool path curve. Due to the feed step is considerably small and usually the tool nose radius is significantly smaller than tool path curve, the difference between two feed vectors is fair small. Therefore, in this research the feed at each feed step is in the direction of the toolpath tangent vector of the tool at current feed step.  Chapter 3. Overview of the Virtual Turning System  •  37  Depth of cut of contour turning: The depth of cut of contour turning is defined as a distance from the workpiece boundary, which intersects with the cutting edges of the tool at current step, to the instantaneous feed vector at the tool-part contact point, which is the offset of the feed vector from the tool nose center to the tool curve edge.  3.6  Summary In this chapter, the Virtual Turning System is outlined. This system is capable of capturing  tool-workpiece intersections along the whole toolpaths, and predicting cutting forces, torque and power for the majority of common turning process, such as regular turning, facing, grooving, and boring operations. The system is based on solid modeling technology, and other techniques and algorithms, such as analytical and feature-based methodology, tool construction method, process orientation methodology, are used for speeding up the computation and augmenting the capability of the system. Several basic assumptions of this system are presented for future improvement. In the following chapters the details of the intersection methodology and force prediction in contour turning will be presented.  Chapter 4 A Hybrid Analytical, Solid Modeler and Feature-Based Methodology for Extracting Tool-Workpiece Engagements in Turning  4.1  Introduction As described in the previous chapter, in the Virtual Turning system, the simulation of  turning processes is used to optimize cutting conditions so as to minimize machining cycle time while facilitating production of correctly machined parts from the very first component. However, this process requires an accurate calculation of Tool-Workpiece Engagement (TWE) geometry to give chip area characteristics used in predicting instantaneous cutting forces, power, and torque at positions along the tool path. This becomes challenging when the initial workpiece geometry has a shape history (e.g., castings or forgings), when the tool path is complex, or when the tool edge is complex, such as during contour turning with formed tools and groove turning. Solid modelers are increasingly becoming an option for performing these calculations due to the increased robustness and efficiency that is evolving in this technology. These modelers are used to perform Boolean intersections between 2D representations of the cutting tool and the in-process workpiece to extract the engagement geometry. For complicated turned components, particularly those machined from non-cylindrical workpieces, these intersections must be performed at feed increments corresponding to each rotation of the workpiece to guarantee that changes are properly identified. This requirement can easily lead to several thousand Boolean intersections that must be performed to simulate a part. These Boolean operations greatly increase simulation time and the likelihood of modeler errors when intersections between  38  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  39  marginal geometry are attempted. Thus, there is a motivation to integrate intelligence into the TWE calculations to increase efficiency and improve robustness. This chapter describes research that combines analytical and feature-based methodologies to augment the use of a solid modeler. Exploiting the cutting tool insert similarities, where they engage the workpiece, and the 2D turning process simplifications make it possible to identify a limited number of engagement conditions that occur over significant regions of each tool pass. In this research these regions are formalized as In-Cut Material Removal Features (mrF). The use of the term "in-cut" to define these features is to emphasize that they are regions of interest in the in-process workpiece during material removal as opposed to the traditional definition of features that refers to the geometry of the final part. One consequence of the use of these features is the motivation to develop methodologies for extraction and parametrization. Further, within each region it is possible to characterize the engagement geometry at each feed step of the tool (i.e., at each workpiece revolution) as a small set of Tool Engagement Features (teF). The parameters of each type of teF derive from the machining process parameters. These can be combined into the appropriate formulations and solved analytically using Green's Theorem to find uncut chip area characteristics that are used in modeling the cutting forces. While Green's Theorem is not new and is used to find general 2D areas in the solid modeler, the need in these cases for a generic solution necessitates that the calculations be performed numerically. Due to the limited number of teF types that have been identified, these can be directly formulated and solved without numerics to reduce computations. The rest of this chapter is organized as follows. Since a pure solid modeler-based methodology is also an option for engagement, the development of such an approach is  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  40  described in Section 4.2. This approach serves as a basis with which to extract the TWEs and compare the efficiency of the hybrid methodology. As part of this, details about the construction of tool swept areas are given in Section 4.3. Following this methodology, an analytical approach (based on the teF classification given above and Green's Theorem) for extracting tool workpiece intersection parameters are presented in section 4.4. Section 4.5 develops the method for extracting material removal features. This is essentially an area decomposition procedure that divides the material removal area into three feature types. Section 4.6 outlines the overall hybrid analytical, solid modeler, and Feature based methodology, which is the combination of the methods described in above sections. A discussion of the implementation details along with results from validation on an industrial aerospace component follows in Section 4.7. The chapter ends with a general discussion of this approach and some directions for future tool-workpiece engagement research.  4.2  Full Solid Modeler-Based Methodology As described in the previous section, TWE is a key issue for modeling the turning process.  A Solid Modeler-based methodology is one of the solutions for this problem. The prototype of this methodology originally developed in author's laboratory is shown in Figure 4.1. After being significantly improved for computational efficiency with added functionality for different types of turning in this research, a complete Solid Modeler-based solution is shown in Figure 4.2, which serves as a basis for TWEs extraction and is compared to the hybrid solution later.  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  41  The A C I S solid modeling kernel is used to modeling and capture the geometry of the 1  workpiece, its in-process state, toolpaths, and the cutting tool, and to perform Boolean operations and other geometric operations in extracting the TWEs and their parameters.  Toolpath  Tool Construction ^ at Each Feed Step -N  Initial Block  -N  Boolean Intersection  Intersectioin Area from A C I S Function  In-process Workpiece Update Workpiece  Figure 4.1: Original Solid Modeler-Based Intersection Prototype Figure 4.1 presents the basic procedures of this methodology. The input requirements are mostly as discussed in Chapter 3. However, the initial workpiece is created as a 2D block by using the bounding box of the workpiece, later the workpiece geometry is extended to the arbitrary model created in the C A D environment in the improved full Solid Modeler solution, which will be described in the following paragraph. Toolpaths are provided as cutter location data generated from a C A M application. In addition to the path geometry, process parameters such as spindle speeds (rpm) or surface cutting speeds (m/min) and feeds (mm/rev) are also contained in this data. Tool geometry is constructed by constant depth of cut, side and end cutting edge angle and tool nose radius. The toolpath is discretized based on the feedrate. The smallest step that can be taken is the feed per revolution (referred to as a feed step). Once the positions along a toolpath have been evaluated, an intersection between the tool geometry and  1  In this research the ACIS solid modeler product is used to model and manipulate geometry  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  42  the in-process workpiece geometry is performed by using the intersection operator in the ACIS kernel, and the intersection area is obtained using ACIS kernel functions. The in-process workpiece is updated by using the Boolean subtraction operator for the intersection at the next feed step. The main problem of this methodology is that the geometry of the solid in-process workpiece model becomes increasingly complicated as Boolean subtractions remove the tool shape at each feed step. Much of the topological and geometric information stored in the solid model does not contribute to the intersection at a given step. This makes the localization effort for the Boolean operations between the tool and the in-process workpiece solid time consuming and inefficient. To solve this problem, the Tool Swept Area (TSA), which is the swept region of a tool along a toolpath, is constructed and intersected with the in-process workpiece. Since there is an order of magnitude less toolpaths than total feed steps, the in-process workpiece is less complex than in the original method. Consequently the localization effort is reduced for the Boolean operations and the computational time is improved. Other improvements are also applied to increase the efficiency further for the complex toolpath and workpiece. Details are given in the full Solid Modeler methodology presented in the following paragraphs. Figure 4.2 (shown in the next page) gives an overview of the full Solid Modeler-based TWE extraction methodology. The inputs required are the toolpaths, the initial workpiece geometry, and tool geometry information. In the first step of this solution, the initial workpiece is represented as a 3D ACIS solid model obtained either directly or through STEP translation from the C A D system where the model was originally created. A 2D cross section of the initial workpiece for the TWE calculations is obtained by slicing the model with a plane through the  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  43  machining axis. For convenience and consistency with the axis configuration on lathes, the X Z plane is used, and the tool is projected onto the same plane. (  Start End  2.Construct C u t t i n g Tool G e o m e t r y  1 . C o n s t r u c t Initial  no  Workpiece Geometry i  yes 3.Creat T o o l Swept Area (TSA)  -'' N e w ^ Toolpath?.  >  4.Generate M a t e r i a l Removal Area (MRA)  9,Update In-Process Geometry yes  6.Step t h r o u g h N e x t F e e d r a t e Interval  5.Discretize T o o l p a t h at F e e d r a t e Interval  7.Translate/Construct  .Intersect T D with M R A  Tool Difference (TD)  no  'End o f \ ^ Jbolpath?^--"  9.Obtain A r e a s a n d Centroids from Properties of i n t e r s e c t i o n  Cutting F o r c e s Prediction  Figure 4.2: Full Solid Modeler-Based Turning Simulation Methodology As described in Chapter 3, the cutting tool is originally constructed in step 2 by making an edge loop composed of a side cutting edge, an end cutting edge, and a tool nose curve, then converting to a solid model. When the tool does facing, grooving, and boring, a machining setup angle and a machining direction are given to transform the tool to the correct orientation with respect to the specified machining operation.  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  44  The TSA is constructed in step 3 based on the tool geometry and the toolpath. A swept area is the union of the area occupied by the tool at all positions during motion over that path. The TSA is generated by constructing its boundary (consisting of line and arc segments) as a sequence of edges to which a face is added by the modeler to give a closed 2D shape. Since the TSA is very important for correct workpiece updating and TWE calculation, its construction algorithm will be described in detail in the next section. The Material Removal Areas (MRA) can be obtained by performing a Boolean intersection between the TSA and the in-process workpiece (step 4). In Step 5 the toolpath is discretized based on the feedrate. To speed up computation times, intersections at only a few steps need to be calculated when the engagement is not changing. Determining when these invariant engagement conditions occur is part of this research in another solution. The TWE of each feed step can be viewed as the Boolean difference between the tools at successive positions separated by the feed step intersected with the MRA. Therefore, two consecutive positions of the tool along the toolpath need be identified in Step 6, and the Boolean difference between these two tools (TD) is generated in Step 7. For linear toolpaths, the TD is constant and only a transformation is needed to locate it for different feed steps. But for a curved toolpath, TD needs to be generated at each feed step. Step 8 performs the engagement calculation by intersecting the TD with the MRA. Compared to the original prototype shown in Figure 4.1, the intersection between the TD and MRA at each feed step is another important improvement made in this research. This is because the complexity of the MRA does not increase continuously as the tool goes through more toolpaths.  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  45  To match the input requirements of the force prediction model adopted in this research, the TWE is decomposed into sub-regions due to the difference in cutting mechanics over these regions. Areas and centers of gravity for each of these regions are then calculated by extracting face properties using ACIS kernel functions. The tool position is incremented (Step 6), and the calculations performed in Steps 6, 7, 8, and 9 are repeated until the chip areas at all positions for a given tool path have been evaluated. Step 3 is repeated until all toolpaths have been processed. Areas and centers of gravity are saved to a file that is subsequently used by the force prediction model. This approach is valid only under the assumption that the workpiece, tool, and surrounding machine tool structure are rigid. If compensation of the tool location due to flexure and dynamics are to be considered then the process model needs to be evaluated at each position of the cutting tool and deflection information feedback to adjust the location of the tool at the next step. The advantages of this methodology are that it is simple to implement, and it is a generic solution that works regardless of the complexity of the workpiece geometry. The disadvantages are that performing Boolean operations at each step are computationally expensive, and surfacesurface intersections in solid modelers are not fully stable, particularly when marginal overlaps between the tool and workpiece occur. Boolean operations may fail in these cases. By reducing the number of operations that need to be performed, computational efficiency can be increased and the likelihood of modeling errors reduced (though not completely eliminated). Towards this goal, analytical area and centroidal calculations and the use of features will be described in Section 4.4 and 4.5.  46  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  4.3  Tool Swept Area (TSA) Construction The TSA, analogous to the Swept Volume in 3D is the total area that a tool occupies over  one toolpath. It is constructed by a series of ordered edges, which include tool boundary edges and new envelope edges. The construction of the TSA for each toolpath requires finding the outer or/and inner envelope points of these edges, then connecting all together to form an edge loop. Using ACIS functions to convert the edge loop to a solid body, the TSA is represented as a solid model for Boolean operations to be performed with the in-process workpiece solid model. For different types of toolpaths, i.e., linear toolpaths and circular toolpaths, the TSA construction methods are different. These are described in section 4.3.1 and section 4.3.2.  4.3.1 Linear Toolpath TSA Construction Linear toolpath TSA has three cases with respect to edge loop connection as shown in Figure 4.3. y/ is the side cutting edge angle, K is the end cutting edge angle, a is the toolpath r  r  angle, and P is the complementary angle of y/ , /? r  =  /2~ l r  n  x /  • T , T are the start and end s  e  positions of the toolpath, P P is the common tangent line offset from T T , and P , P are the s  e  s  e  s  e  corresponding tangent points of the tool at T , T positions. Pj, P/ are the side cutting edge s  e  upper points of the tool at T , T positions (refer to Tool Geometry Construction as shown in e  s  Figure 3.6). Similarly, P , P ' are the end cutting edge upper points of the tool at T , T positions. c  c  e  s  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  case 1  a <  P, K  case 2 K  r  Tool boundary edge  r  case 3  < a < (3  Tool path  > R, K  a  r  New envelope edge  Figure 4.3: Three Cases of Tool Swept Area of Linear Toolpath In case 1, when a < P, K , two new envelope edges are formed by P P r  s  e  and P / / , other  TSA edges are constructed by tool edges at T and T positions. In case 2, when K < a < P, new s  e  r  envelope edges are PdPd and P P '. Finally, in case 3, when a > p and K , new envelope edges C  are P P s  e  C  r  and P P '. Since a, /3 and K are known before the construction, it is straightforward to C  C  r  classify a linear toolpath TSA into one of the three cases. In each case, the TSA is constructed by connecting the corresponding new envelope edges and tool edges that have been identified. This method is specific to modeling swept areas for turning operations based on the generic tool  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  48  geometry of figure 3.6. However, a general linear toolpath TSA algorithm is desirable for all there cases, and it is in fact simple and straightforward as described in the follows. It can be seen that a TSA of a linear toolpath is the convex hull enclosing the tool shape at the start and end toolpath positions. To achieve this, the upper furthest point and lower furthest point of the tool, with respect to the toolpath, need to be identified and connected to form the new envelope edges, as illustrated in Figure 4.4 . The procedure is presented in the following algorithm.  Figure 4.4: Linear Toolpath Tool Swept Area Construction Algorithm Linear Toolpath TSA Construction INPUT: Tooltiy/,., K , r P , P , P , P ), i = 1 to 2, toolpath T T r  a  a  b  c  d  s  e  OUTPUT: TSA boundary edge loop STEP: 1. P , P <— toolpath T T offset r distance s  e  s  e  £  2. D (i = a to e)±- signed distance between P, (z = a to e) to T T . h  s  e  49  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  3. Find Upper furthest point P  u  where D < 0 and A = max(\Di <0\) u  4. Find Lower furthest point Pi where A > 0 and A = max(\Di > 0\) 5. New upper envelope edge e  u  <— connect P of the tool at toolpath start and end positions u  6. New lower envelope edge ei «— connect Pi of the tool at toolpath start and end positions 7. 7X4 boundary edges <—{{e } i, e {e } i , ej e  PuP  h  s  P Pu  In step 1, the offset tangent point P - T +r -h , where h is the unit vector e  e  £  2  2  perpendicular to toolpath unit vector h, (P is obtained similar to P ). Let L denote the toolpath s  T -T T -T length |7"7^S, « = ~f r =— e  t  e  h = (-n ,n ). 2  ly  Step 2 finds the distance 79, from each  lx  tool point Pi (i from a to e) to the toolpath T T . D - \(P -T ) x hj . For a 2D case, this equation e  reduces to 79, =  s  i  t  e  [(x, - x )(y - y ) - (v, - y )(x - x )], where Pfa yd, T (x , y ), and T (x , y ) v  e  s  s  e  s  h  e  e  e  s  s  s  give the x and y coordinate notations. It must be noted that, A is a signed distance. In the upper region with respect to the toolpath, the distance A is negative, and the furthest point P  u  corresponds to the largest negative A in Step 3. Similarly, the distance A is positive in the lower region, and the furthest position Pi corresponds to the maximum positive A in Step 4. These furthest points at the start tool position and end tool position are connected to form the new envelope edges. In the last step, the 7X4 boundary edges are composted of three parts, two envelope edges e and a, tool boundary edges {£«>}/>/> from P to Pi at the end toolpath position, u  u  50  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  and the tool boundary edges {e } s  PP  from Pi to P at the start toolpath position. In the example u  shown in Figure 4.4, the TSA is {eye^e/e^e^}.  4.3.2 Circular Toolpath TSA Construction Outer Sweep Env  Inner Sweep Envelope  •Toolpath  Toolpath  (a) R > r  £  (b)R< r  s  Figure 4.5: Circular Toolpath Tool Swept Area Construction The above figure illustrates the swept area generated by the tool sweeping along a circular path. In Case (a) the toolpath radius R is bigger than the tool nose radius r . In Case (b) R is E  smaller than r . The difference between these two cases is that there is no inner sweep envelope e  in the TSA of (b) since the entire interior area is swept out. The Swept Area of (a) is the region between two edge rings, the Outer Sweep Envelope, and the Inner Sweep Envelope. The Outer  51  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  Sweep Envelope consists of three circular edges ey, e , e and three straight edges e , e , e^. The 3  5  2  4  first three edges are generated from sweeps of the tool nose curve, the tool upper right point P , c  and the tool upper left point Pd (See Figure 3.6). The latter three edges come from the tool boundary edges that are tangent to the first three circular edges respectively. Similarly, the Inner Sweep Envelope consists of three circular edges ej, e&, and eg, which are generated from the sweeps of the tool upper left point Pd, the tool nose curve, and the tool upper right point P  c  respectively. To construct the TSA of the circular toolpath, the outer and inner sweep envelope edges need to be generated. The corresponding critical points Py to P9 need to be calculated. Note that Ps is a transient position, below which the inner envelope edge ej is formed by Pd of the tool. This is because the angle of the tangent vector of the sweeping envelope formed by P at P# is d  just the tool side cutting edge angle (y/ ). When the^ angle of the tangent vector is bigger than y/ r  r  (below), the tool motion is covered by this sweeping envelope e?. Similarly, Pg is the critical position below which the inner envelope edge eg is formed. The angle of the tangent vector of the sweeping envelop formed by P below P? is bigger than the tool end cutting edge angle (K ). c  r  According to the properties of common tangents and planar rigid motion, these points can be obtained easily. For example, Py and P can be formulated as follows: 2  Pi:  * i =X  -(R + r )-cos(y/ )  m  e  Y^Y +(R  + r )-sm(y, )  X =X  +(R + r )-sin(K )  0]  P: 2  r  2  0X  Y =Y -(R 2  m  e  (4.1)  r  c  r  + r )-cos(K ) c  r  (4.2)  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  52  Where Oi (Xoi, Yoi) is the center of the toolpath. Other position calculations are presented in Appendix A . According to these positions and the tool geometry relationship, the corresponding critical toolpath positions A to F can be identified. Six sections of toolpath are constructed, i.e., A B , BC, CD, DE, EF and F A . In real turning operations, it is common that only a portion of a circular tool path be encountered. To determine the tool swept area of a given toolpath it is first necessary to find which section or which combined sections the path belongs to. Then the tool swept area is constructed by generating the outer and inner edge loops based on the boundary conditions of those sections. Details of this circular toolpath TSA construction are presented in Appendix A .  4.4  Green's Theorem-based Analytical Intersection Area Calculation As briefly discussed in the introduction, the disadvantages of the full Solid Modeler  methodology are that performing Boolean operations at each step is time consuming and that surface-surface intersections in solid modelers may fail in particular cases. Also the generic nature of area calculations performed by a solid modeller requires a general-purpose numerical solver. As computational speed is a critical concern, efficiency can be increased by applying an analytical solution directly to calculate the uncut chip area. This improvement means that Boolean operations need not be preformed at each step to obtain the chip shape, i f the boundary information is provided. A Feature recognition methodology can first be used to identify the boundary conditions as described in next section.  Chapter 4. A Hybrid Analytical,  Solid Modeler and Feature-Based  Methodology  It is possible to characterize the engagement geometry as a small set of Tool Engagement Features (teF). The parameters of each type of teF, such as area, chip-cutting edge contact length, and gravity center, derive from the machining process parameters, i.e., these teF parameters can be expressed as functions of feedrate, cutting speed and tool geometry. These teF expressions can be combined into several appropriate formulations and solved analytically using Green's Theorem to find the uncut chip area characteristics that are used in modeling the cutting forces. The formulations of this solution using Green's Theorem will be the subject of this section.  Figure 4.6: Tool Workpiece Engagement (TWE) Green's Theorem is widely applied in the study of mathematics; it can convert the double integral to a line integral over its boundary.  (4.3)  The area and centroid calculation are two of the main applications of Green's Theorem. Figure 4.6 illustrates the Tool Workpiece Engagement (TWE). The boundary edges consist only  54  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  of lines and arcs. After the equations of lines and arcs are specified as parametric equations (x(t), y(t)), area equations are expressed as follows. The three formats are equivalent. In this research the middle format is used in TWE calculations for better geometric understanding. A = | xy' dt = | - yx' dt = ^ 2  (xy'-yx' )dt  2  (  (4.4)  Mathematically, the TWE can be expressed as follows: M ^ ^ P . ^ . i ' i W . r / i )  (4.5)  where the boundary is a set of connected edges of arc or line type: f (1 - u)Pj + uP P(u) = <  0<u<1  j+l  [i c  r  + Vc ° O ) c  s  c  r  s i n  O)]  -,  Line segment  9 <u< 6 t  M  Arc  v  (4.6) '  and e,:  Edge on the boundary of TWE  Pj, P :  End points of edge e  C:  Center of tool nose along the tool path  r:  Tool nose radius  6:  Angle of vector P C in world coordinate system.  j+1  i  c  i  i  i  i  In general, two or more edges define the boundary of a teF. These edges are either portions of the in-process workpiece boundary before the machining of the current material removal area (MRA) or portions of the cutting tool boundary. These groups are referred as the sets W{eJ and T{eJ respectively. This is sufficient for identifying the various combinations of engagement that are possible for any turning part under consideration.  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  55  Figure 4.7 illustrates six commonly occurring teFs that have been identified for generic turning inserts such as those described in the previous chapter that cover a wide range of machining conditions. , J..  —/  "U " V V Ql, Case J. d < h  Case 2. h<d< h. f < rz  dl  Case 3. h < d < h. n<f < 2n  /W*7  1q,  Case 4.d>li.f<  n  /  y  1\ I  C 'i  vCase 5. d> h. rz<f < 2r&  Case 6. Grooving  Figure 4.7: Classes of Generic Tool Engagement Features (teF)  56  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  Parameter Depth of cut (d) mm  Constraints  teFl  d<h  V  teF2 teF3 teF4 teF5  V  l <d < /, 2  V  V 'V  d>h  V  0<f<r  Feedrate (mm/rev)  c  y/ >0  V  V  V  V V  V  V V  r  Vr( g)  V  V  V  c  V  V  V  r <f<2r c  V  V  V  V  V  V V  de  ¥r<0  Ri V V R1+R2 Table 4.1: Conditions of Generic Tool Engagement Features (teF)  Regions of teF  Each teF is defined parametrically corresponding to the depth of cut (d), the feedrate (/), and the insert lead angle (y/ )- As indicated in the accompanying table (Table 4.1), feature types r  are differentiated by constraints applied to these parameters. Depth of cut (d), which is the distance from the workpiece boundary to tool contact point measured perpendicular to the instantaneous feed direction, determines the number of sub-regions that makeup the TWE. Feedrate ( / ) indicates how far the tool moves during one revolution of the workpiece. Its direction is an instantaneous tangent vector of the tool feed motion. The tool nose radius r and £  the side cutting edge angle y/ are geometric properties of the turning tool. In Figure 4.7, / is the r  distance between the successive two tool positions C;C,.;. If the tool path is a straight line, / is equal to feedrate/ If the tool path is a circular edge,/is the arc length and / is the chord length. Pi is the tool nose curve upper tangent point of the tool, and P is the intersection point between 2  the two tools. V denotes the instantaneous feed direction that is tangent to the tool nose curve at  57  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  C„ and // and l are the distances fromP,, P to V respectively. Appropriate analytical equations 2  2  can be formulated for each case by giving the boundary conditions of the engagement, such as intersecting positions and angles. For case (1), the type of teF], d is smaller than l (P V), 2  2  there is only one curve region R/,  the boundary of which is composed of two edges. One is tool nose curve, and the other is workpiece boundary edge. Since d is fairly small, the workpiece boundary edge within one feed interval can be regarded as a straight line. Geometric equations for the uncut chip area A/, chiptool contact length Lc/, and gravity center 67 of the region are as follows:  L =r y>-sm{</>)) cl  GC, =y  (4.8)  E  (sin {</>)) {tx - sin(0))  (4.9)  3  £  r.-d^  f  where ^ = 2 cos"  V  e  T  J  For teF2 and teF3 as shown in case (2) and case (3), d is lies between // and l , i.e., only 2  the curve region A1 lies inside the engagement, while / is smaller than tool nose radius r for e  teF2 and bigger in teF3. The difference between the two cases is the possible type of P (the 2  intersection point of cutting edges of two tools): it can be nose arc-nose arc intersection in both cases, or nose arc-side cutting edge intersection in teF2, but end cutting edge- nose arc intersection in teF3. Both cases need to identify the equivalent side cutting edge angle y/r with respect to the instantaneous feed direction at the successive locations, i.e., y/ '-y/ r  r  + a , where  58  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  a is the toolpath angle. For teF4 and teF5 as shown in cases (4) and case (5), two regions Ri and R2 are included in the engagement. teF6 is a special case for straight groove cutting. To describe all the cases illustrated above, a common area calculation algorithm is expressed in this section, which is based on a general Green's Theorem-based line integral. Details of the developed equations for each case are listed in Appendix B. Assuming the area integrated by a straight line e, is A,, and that integrated by an arc e is A , a  a  the parametric equations of e, and e are, a  (4.10)  where the end points are P (X , Y ), P (X , j  i  t  M  M  Y) M  (4.11)  where the centre point is C(X ,Y ) C  C  A = ^-yx'du t  , 61,62 are the parametric bounds for the circular edge.  (y  i+yiX i i- i) x  i +  x  +  2  (4.12)  where P  t  (4.13)  where 61 and 62 are the parametric bounds for the circular edge.  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  59  Since Ai and A are signed areas, the total area A of a region is the sum of all sub-areas a  integrated by all edges, where n is the total number of edges of A, m is the number of arc edges, and n-m is the number of line edges. n  m  n-m  ^ =I 4 = I X + I X * i=0  j=\  (4-14)  k=\  Centroidal positions are calculated in a similar manner and will not be discussed in this thesis. A general Green's Theorem-based area calculation algorithm is illustrated with the example (teF4) as shown in Figure 4.8.  Figure 4.8: One Example of teF4 Area Calculation Algorithm Area_Calculation INPUT: C„€,_,, {Pi}, ({Pi}:  the set of end points, n: number of edges, m: number of circular edges, k:  number of zones) OUTPUT:  e„n,m,kj  A„A ,A 2  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  60  STEP: For region RjJ = 1 to 2 From i = 1 to n C A S E geometry_type (e,)=LINE A = \ -yxdu i  )  where P , (x , y ), i+  i+l  i+l  =  ,  P (x , y ) t  t  t  C A S E geometry_type(e/) ARC s  If center is C, A =  ^-yx'du-^-~-rY cos(u)-^-s'm(2u)  t  e2  c  01 where 0, = x + arctan(^ , lx ) / C(  0<Q, <2n  PiCi  0 =2n + arctan 2(y . ., x 2  P +/C  )  0 <9 <2k  Pi+/C/  2  If center is C,./ Aj = |^-j>x'd« =  Y ^ 14  rY cos(w) c  where y/, = 2;r + arctan 2{y  PC  y/ = x + arctm(y 2  Y^  PCi  ,x )  Wi  1  sin(2w)  Wi  0<y/, <2n  PjCj  Ix )  0 <y/ <2n  PiCj  2  Aj <— Af + Aj End End A = A +A }  2  End In Figure 4.8, points C;,C,_,are the consequence tool nose center positions along the tool path. The interval is / = | C C _ | . TWE has two regions R ,R /  /  /  X  2  for cutting force calculation. At  61  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  first, the intersection point P needs to be identified, if the equivalent side cutting edge angle, 2  f  y/ '=y/ +a , r  f  r  , ^  , P2 is an intersection point  sin  is smaller than a critical angle  K  2  \  2  r  e J J  between two curves. If y/ ' is positive as shown in the above figure, curve region R is known to r  }  be bounded by edges {e ,e ,e } , and the close to polygonal region R is bounded by l  2  3  2  edges{e ,e ,e ,e }. e,,e are a portion of the tool nose curves, and e is the line segment PyC, 3  4  5  6  2  3  truncated by e at P . e , e are tool straight cutting edges, and e is a portion of the workpiece 2  3  4  6  5  boundary edge. The signed areas covered by all the edges are formulated and summed up to give the total area equations. The formulations are given by the closed form equations (4.15) and (4.16). For any teF of this class appropriate boundary conditions are applied to these standard equations to get the results. A, =—— (4[7c,(cos# -cos0 ) + Yc _ (cosy/ 2  4  J  i  ]  -cosy/,)]  2  + r [sin(26' ) - sin(26>) + sin(2y ) - sin(2(//,)] E  2  -2r {6 -6, E  i  =^{ sy4  A  x  + y ,  2  -  ;  x 4  y3  + x  4ys  + ^(Y + Y,)(X  2  (4.15)  2  3  - sy x  + x 4  -X,)  3  s y i ~ iy5 iy3 x  +x  - syi)  (- )  x  4  16  All the boundary conditions (point coordinates and angles) are derivable analytically. P , P are the tool nose curve upper tangent points at C , , C,_,. P is the intersection point between x  4  2  Cj and C,-_|. Q , Q._ are the workpiece boundary positions intersected with the tools at the two f  x  positions. The coordinates of these boundary points are P (x„ y,), Q\(x% ys), and Qi-i(x , y ), and ;  4  4  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  62  can be calculated using f, d, the tool geometry and the workpiece boundary in the MRA. 0j and 02 are the angles of vector P/C7, P2C,, and y/i, y/ are the angles of vector P2C/, PjC,./. 2  Thus each region can be parametrically defined in terms of area A , chip-cutting edge contact length L , the gravity center G , tool side cutting edge angle if/ , feedrate / , depth of c  r  cutd . A parametric form for any of the TWEs shown in Figure 4.6 can be expressed as, teF(R (A L , G,), / , d, y/ i  L  cj  r  ,K ),i r  =  lto2  The formulations of the other teF cases are listed in Appendix B. A methodology for the identification and expression of the boundary conditions is presented in the following section.  4.5  Feature-Based Methodology Green's theorem based analytical equations described in the previous section require the  boundary positions, i.e., the end points and angles of the edges of TWEs. This translates into finding the coordinates of the points Pi, P , P3, P4, Qi, and Q .i, as shown in Figure 4.8. To 2  t  identify these positions effectively, feature concepts are introduced in this section. A feature identification methodology is developed in this research to achieve this goal.  4.5.1 In-Process Turning Features A feature-based methodology developed in this research is based on the decomposition of a material removal area (MRA) generated during turning into in-process features. Figure 4.9 shows the taxonomy for features generated during turning process, similar to the one proposed by [YipHoi and Huang, 2004] for 2 V2 D milling. Of interest are the in-cut features.  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based  Methodology  63  Figure 4.9: Classification of Features Generated from Turning Tool Engagement Features (teF) define the shape of the engagement over a single revolution of the workpiece. Material Removal Features (mrF), on the other hand, are regions in the removal volume that correlate with specific types of engagement changes over a complete tool pass (a toolpath). Essentially, for a particular mrF the teF is of one type. The extraction and parametric expression of a teF are fully discussed in the previous section. There are three types of mrF. A Geometry Invariant Feature (giF) is a region within a tool pass where the geometry of the teF at each rotation of the workpiece remains unchanged. This is the case along a linear tool path where the tool cuts at a constant depth of cut. A Form Invariant feature (fiF) is one where the class of the teF, as defined in Figure 4.7, remains unchanged over the corresponding region.  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based  Methodology  64  Hence the shape or topology of the teF boundary is fixed, though its geometry varies. In contrast to a giF, a fiF occurs when the tool is fully engaged with the workpiece and the depth of cut varies continuously over the region defined by the feature. The third type of mrF is a Transient Feature (trF). These features occur when the tool breaks into or out of the workpiece at the start or end of a pass or when the tool transitions between adjacent regions, as shown in Figure 4.10. Machining Direction  trF  Tool enters workpiece boundary  —  S  tool transitions between adjacent regions  Figure 4.10:Transient Machining Feature (trF) trFs have unpredictable boundaries and consequently their constituent teFs do not fit any of the classes presented in Figure 4.7. The parameters of trFs will be extracted by using solid modeler functions, the same as the methodology used in the full Solid Modeler solution. The rest of this section focuses on the parametrization and extraction of mrFs.  4.5.2 Extraction of Material Removal Features Examples of mrFs for a single pass are illustrated in Figure 4.11. The giF occurs in regions 1,3, and the fiFs correspond to regions 2, 4, and 5. The start region and the regions between  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based  Methodology  giFs I fiFs belong to the trF class. The details of the decomposition one mrF into giFs,fiFs,  65  or  trFs are presented in this section.  Figure 4.11: Material Removal Features (mrF) Generated during Turning The approach for MRA decomposition starts by identifying the edges of the MRA boundary that correspond to the workpiece boundary, by traversing MRA boundary edges and comparing them with TSA and tool boundary edges. The edges that are different from the TSA and tool edges are workpiece boundary edges. The start and end positions of these edges are indicated by  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based  66  Methodology  the points Q in Figure 4.12. Figure 4.12 shows a MRA which is generated by Boolean t  intersection between the TSA of a circular toolpath T T and the in-process workpiece. s  trF-*!  !  fiF  >l  e  htrF  Figure 4.12: MRA Decomposition The toolpath corresponding to the MRA is then discretized at the feed step from the start position T to its end T . These points indicate the tool center positions at each step along this s  e  toolpath. These are denoted by the round points T, in Figure 4.12. Additional strip points C are t  added, corresponding to the tool center positions when the tool leading edge passes though an end point Q . If C, is not coincident with any T this means that between the previous tool t  h  location (T .,) and the next tool location ( Z ^ the tool leading edge passes through a eJ  discontinuity between two workpiece boundary edges, resulting in a transient engagement feature (trF) in this region. giF and fiF type features occur between adjacent locations of these transition points when the tool leading edge continuously intersects the same workpiece  67  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  boundary edge. As shown in the example of Figure 4.12, the workpiece boundary edge end point Qi results in a transition point C, on the tool path. A trF is defined between the feed step points T .i and T \ that bracket C,. A fiF is present between feed step point T and T , which define eJ  Si  sj  eJ  the start and end tool center positions of the fiF on the workpiece boundary edge Qt+iQt. Identification of feed step points along the tool path as well as the bracketing of different types of engagement features are needed in extracting these features. A n algorithm for doing this follows:  Algorithm M/L4_Decomposition INPUT: MRA, Toolpath edge (T T ), TSA, e  s  Tool geometry TG(r , y , K ), feedrate (/). c  r  r  OUTPUT: giF I fiF, and trF STEP: 1. e/...<?„, <— FindWorkpieceBoundarvEdges (MRA, TSA, Tool) 2. Tj =T + j-f S  <— DiscritizeToobathAtFeedratelnterval (T„T*, f)  where j = 1 to n,n is the number of feed steps, n = floor (SI f),  S is the length of  toolpath T T . e  s  3. From i = 1 to m (m is the number of workpiece boundary edges) for each e„ Qt+h Qi *~ Cj+i,Ci  Get the end points of e,  <— FindCorrespondingToolPositionfrWB (Oi+i, Qi)  68  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  T j, T ,i e  FindTransitionEn^  s  Qi, C,+/, C,)  Qej, Qs,t *— FindWorkpieceBoundaryPositionfromToolPositioti (T j,  T )  e  S]i  giFj or fiF, <— IdentifyMRFforWorkpieceBoundarv&Toolpath (QejQsj T j, T ) e  trFt  SJ  <— FindtrFBetweenTwoEdges (T j, T j.i) s  Output  e  giFj/fiFj, trFj  End To find the tool position from the workpiece boundary or vice versa, the geometric relations between workpiece boundary and tool center are used, and these are also used in teF boundary position identification within giF I fiF, which will be introduced in the following sections and presented in Appendix C in detail. As shown in Figure 4.12, Qj.; to  Qi+2  are the workpiece boundary positions, and C,-./ to C,+2  are the corresponding tool center positions on the toolpath. If the toolpath is a circular line, R denotes the radius of the toolpath, and O is the center of the toolpath. The Transition Engagement Identification algorithm is presented as follows. Algorithm FindTransitionEngagementPositionsOnToolpath INPUT : C  C T , (R, O if toolpath is an arc)  i+h  h  OUTPUT : T , eJ  s  T  sj  STEP: 1. Get the length between C, and T (j = i and i+1) s  Linear Toolpath  CjT,  S=  cos  1  'CjO-CjT^  v  \R Circulaar Toolpath  R  2  2. If Q corresponds to the start position of workpiece boundary edge,  69  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  n=  ceilingyy/j-j,  T =T +n-f SJ  s  If Cj corresponds to the end position of the workpiece boundary edge,  T =T,+n.f eJ  End Note that while this algorithm identifies the transition positions, these are also equivalent to the end positions of giFs and fiFs along the toolpath. In the following sections, the giF and fiF identification and parametrization are presented.  4.5.3 Geometric Invariant Machining Feature (giF) To recognize of a giF from any non-transient mrF, it is easy to see that it is sufficient to determine that the workpiece boundary and toolpath are parallel straight lines. As shown in Figure 4.13, there are two cases for this feature, one is when the workpiece boundary Q Q s  e  intersects the straight cutting edge of the tool, and the other when the intersection occurs on the tool nose curved edge. teF at /"' feed step  5  (a) QeQx intersects straight edge of cutting Tool  teF at / " ' step  (b) QeQs intersects curve edge of cutting Tool  Figure 4.13: Geometry Invariant Features giF  70  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  Since in a giF all teFs (at each feed step) have the same geometry, the extraction of the teF is performed once for the entire giF. Figure 4.14 illustrates an example. With the tool position at Ci and the previous position at C,./, the corresponding workpiece boundary positions Q , Q .i, the t  t  depth of cut (d), and the intersection boundary points Pi, P2, and P3 are calculated using 2D linear and circular components intersections. These positions can be expressed as a function of the tool nose center position C . Since C, is on the known toolpath T T , it can be easily calculated ;  s  e  from the number of feed steps.  Figure 4.14: teF Extraction from giF The boundary points are calculated as follows: h is the unit vector of P/Cz, h is the unit vector of T T ,. h ,h art the unit vectors s  3  e  s  2  4  perpendicular to n,, h respectively. Then, 3  h = [-cosy/,, sin y ]  (4.17)  h =[smy/ , cosi/zj  (4.18)  x  r  2  r  71  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  (T  -T)  *>--fcf:e  ( 4  s\  n4 = [-h ,h ] 3y  t  P  2  2  =  C  '  l  - ^ 2  -h,  s  +  l  1  2f  (4.23)  2  a =C r -nj+ l+  (4-22)  3  -r/)-fh,n j  2  r  (4.21)  ^ ^ - n  P =C +n \4f\n n Y-{f' 3  \(Q -C )xhj\-n  B  t  1 9 )  (4-20)  3x  P,=C +r  -  t  (4.24)  2  ^ =|(a-C )x"3| +^  (4.25)  )  After these boundary positions are expressed as a function of tool position C , feedrate, and t  tool geometries, they can be calculated easily and fast for any position of the tool on the toolpath. Based on this, the Green's theorem based analytical equations can be applied to extract the parameters of each teF.  4.5.4 Form Invariant Machining Feature (fiF) AfiF is defined when the geometry of the teF changes in a predictable manner. There are four cases obtained from different combinations of line/arc workpiece boundary edges and line/arc toolpaths. The workpiece boundary Q Q and toolpath T T can be of types line-line (not e  s  e  s  parallel), line-arc, arc-line and arc-arc which are shown in Figure 4.15. Each case has two subcases where (a) illustrates Q Q intersecting the leading cutting edge of the tool, and (b) e  s  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  72  illustrates Q Q intersecting the tool's circular nose. In the case of one workpiece boundary edge e  s  intersecting both the straight tool edge and the circular tool edge along a toolpath, it can be separated into two single cases (a) and (b), and then dealt with separately. — - Machining Direction  Case 1. &a-ttne nn-line  Case-2 line feTv-arc  Case 3 &0,-arc T T> - line e  Case. 4 &&-arc T^-arc  Figure 4.15: Four Types of Form Invariant Feature fiF Similar analytical formulae have been developed that capture the intersection boundary points of the cutter, which are used to define the boundary of the teF at that location along the toolpath for a fiF. When the toolpath is a line segment, these formulae are calculated for each of the cases in the same way as identified above for the giF, i.e., the equations are the same as (4.17 ~ 4.25), except Q is different in equation (4.24) if the workpiece boundary is a circular t  edge. When the toolpath is an arc segment, since the feed direction varies at successive feed steps, these analytical expressions must be applied at each location along the tool path for the teF to be extracted. The formulae for arc toolpath as shown in Figure 4.16 are as follows.  73  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  o —  R  Machining Direction  (4.26)  (4.27)  P = C + v • n, 3  t  (4-28) v = -*/(/•»; -n ) -I 2  2  5  +r  2 E  -(/•«;  Q^Cj+r^hj+liQ^C^xhjln, d^Qi-CJxfijl  -n ) 5  (4.29) + r,  (4.30)  In these formulations, O is the circular toolpath center, and R is the radius of the toolpath.  74  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  Toolpath can be a line segment or an arc, for any position C, of a tool on a toolpath it can be expressed as follows: C,,=•  (l-t)T +tT s  0<t<l  e  | 0 + [i?cos(O  i?sin(0]  (4.31)  a,<t<a  2  where T ,T are the known start and end positions of the tool path, and a,,a are the s  e  2  corresponding parametric angles of T and T . Successive expressions for C, using C/.y can be s  e  obtained from the above equation, f C,_, + At(T„ - T,), line toolpath C =< r ' [0 + [V cos(At)-V sm(At) V cos(At)-V sin(At)\ x  y  y  1  x  arc toolpath  v  (4 32) ' '  where V = [V V ] = C,_, - O , Ois the circular tool path center. x  v  The tool increment is defined by At =  f/ /  L  line !  L / R'  , where L = \T T \ is the line tool path e  a  r  s  c  length, and R is the radius of tool path. Using the previous expression, the tool position C, is easy to calculate from the previous position C,./. From this, boundary positions P can also be expressed using P,_/. It can be seen t  that these positions of a teF do not need to be calculated from beginning at each feed step, further speeding up computation. In conclusion, from the depth of cut and feedrate, the teF type is identified from the classification as shown in Figure 4.7. After all the boundaries are obtained as described above, the areas and centroids can be calculated using a Green's Theorem-based analytical formulation.  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  75  Furthermore, by using the boundary expressions in terms of C* and by rearranging the Green's Theorem-based area equations, the areas can be expressed as a function of the toolpath parameter t (see equation (4.31)), A = F(t), where t - i- At, / is the feed step number and At is a tool position increment. W ith this expression, the area can be directly predicted with respect to (  the feed step along the toolpath. The details for deriving the boundary positions are shown in Appendix C.  4.6 Hybrid Analytical, Solid Modeler and Feature-Based Methodology The hybrid methodology developed in this research merges the Green's Theorem-based analytical formulation, the Feature-based boundary identification methodology, and the Solid Modeler solutions together, to provide an efficient and complete mechanism for extracting teF parameters during the machining of each tool pass. A flowchart of this hybrid methodology is presented in Figure 4.17. Details of each step have already been given in the above sections. The steps of this Hybrid methodology are briefly described here. Results from the Full Solid Modeler Methodology and this Hybrid approach will be generated and compared using an industrial example in the next section.  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  Q  StartJ  1. C o n s t r u c t Initial Workpiece Geometry  2. C o n s t r u c t C u t t i n g Tool Geometry  3. C r e a t e T o o l S w e p t A r e a (TSA) for T o o l P a t h  4. C r e a t e M a t e r i a l R e m o v a l A r e a (MRA)  5. D e c o m p o s e M R A into Material R e m o v a l F e a t u r e s (mrF)  yes  no  6. A p p l y S o l i d M o d e l e r for iel< E x t r a c t i o n  8. Identify teF T y p e  7. C a l c u l a t e P a r a m e t e r s With Solid Modeler  9. A n a l y t i c a l l y Calculate Parameters for teF T y p e  •mm 10.  Update In-Process Geometry  yes  Figure 4.17 Hybrid TWE Extraction Methodology  Cutting Force Prediction  11  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  As with the full Solid Modeler Solution, after the initial two steps where 2D models of the workpiece and the cutting tool geometry are created in the modeling environment , the TSA for 2  the first tool path is generated. This is used to create a material removal area (MRA) for the tool path by performing a regularized Boolean intersection between the TSA and the workpiece (Step 4). In Step 5, this MRA is decomposed into removal features each belonging to one of the three types described in the previous section. Transient material removal features are processed to extract their constituent teFs using generic functions in the solid modeler (Step 6). teF parameters (areas and centroids) to be used in cutting force prediction are also extracted using the generic property evaluation functions of the modeler (Step 7). Non transient features are differentiated into giF and fiF types (Step 8) and analytical techniques based on the teF classification in Figure 4.7 are applied to calculate the area and centroidal parameters (Step 9). At this point a single toolpath has been processed for its engagement geometry. Before proceeding to the next toolpath the in-process workpiece must first be updated. This is accomplished in Step 10 by performing a Boolean subtraction between the current in-process workpiece and the TSA for the just completed tool path. These steps are repeated until all tool paths have been processed. This hybrid methodology combines the generality of solid modeler-based functionality for handling transient engagement conditions with analytical solutions that enhance efficiency for regions where the engagement changes in a predictable and continuous way. Features are used to help in the formalization of the methodology.  2  In this research the ACIS solid modeler a Dessault Systemes product is used to model and manipulate geometry  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based  Methodology  78  4.7 Implementation and Validation The implementation and validation of the hybrid methodology is described in this section. A n aerospace turned component shown in Figure 4.18 is used for this purpose. Two solutions (full solid modeler and hybrid) for extracting the teFs and their parameters are implemented separately in Visual C++ using the ACIS 3D modeling kernel and toolkit on a Windows Pentium4, 2.6GHz/512Mb, X P Workstation. Display and interaction with the in-process model utilizes the HOOPS 3dGS computer graphics database. Parametrized teFs are extracted at each feed step along a tool path. In addition, it is necessary to generate the tool swept area for each tool path and to subtract this from the in-process workpiece for toolpath / to prepare the workpiece for toolpath  These areas are accordingly subtracted using Boolean function calls.  Simulations of the machining for various tools and tool paths on the aerospace component are shown in Figure 4.19. Figure 4.20 illustrates examples of giFs ,fiF, and trFs generated for one tool pass as part of the solution.  Figure 4.18: A n Aerospace Turned Component Model  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  1_  • Figure 4.19: Simulation of the Machining for Various Tool Paths on Turning Part  Figure 4.20: Extracted Material Removal Features for the Turned Part  79  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  Full Solid Modeler Solution Total Simulation Times (sees.) Total Number of Intersection Area Calculation Number of Intersection Area Calculation in giFs One Example  Hybrid, Feature Based, Analytical Solution  172.094  14.984  4613  1503  3133  23  5.360  0.058  Times  Toolpath  (sees.)  Single Intersection  0.020  0.00017  One Example of A (mm )  0.064444  0.06444  L (mm)  1.208196  1.208196  Effective Angle (red)  0.777640  0.777640  2  Accuracy  80  Table 4.2: Simulation Times and accuracy for Two Solutions The computation times and accuracy for both solutions are listed in the above table. The feedrate used in this simulation is 0.126 mm/rev. From the comparison of total simulation times it can be seen that the pure solid modeler solution is about an order of magnitude slower than the hybrid solution. This is because Boolean operations need to be performed at each feed step. This can also be seen from the total number of intersection area calculations (4613) that must be performed. Moreover, the generic nature of area calculations performed by a solid modeler requires the use of a general purpose numerical solver. It is to be expected that the calculation time for this would be longer than an equivalent analytical solution, which involves exact integration over a small number of edges. Feature identification further speed up the calculations, especially for the geometry invariant features where only one intersection calculation is needed. Only 23 area calculations are needed for all the giFs identified in the test part. On the other hand,  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  81  3133 area calculations are required with the pure solid modeler solution. Furthermore, from table 2 it can be seen that the chip area, length and effective lead angle of both solutions are calculated as the same values. This verified the correctness of Green's Theorem-based analytical equations. And in conclusion, the both solutions can achieve the same the accuracy. It should be noted that the total simulation times indicated in Table 2 are actually greater than the sum of the intersection times or the sum of single toolpath times in the case of the hybrid solution. The reason for this is that the computation times for tool swept area calculations, material removal area subtraction, in-process workpiece updates, and other implementations for visualization are all included as the same as the full Solid Modeler solution. However, the time for solid modeling of both solutions is small compared to the total intersection and area calculation times, this makes the hybrid solution is still an order of magnitude faster than the full solid modeler solution. The problem of robustness must be considered in full solid modeler solution, when applying large numbers of Boolean operations during simulation. In particular, Boolean operation errors can show up when boundary entities on two faces undergoing a Boolean operation overlap in a marginal way. Unlike design applications which are user interactive, these errors when they occur must be handled automatically. One approach to circumventing this problem is to represent the 2D workpiece and tool representation as thin 3D extrusions of slightly different thicknesses. Other strategies are incorporated as described in Yip-Hoi [1]. One of these is to perturb the position of the tool along the toolpath when an intersection operation fails. Contrarily, the hybrid approach is better in robustness since fewer intersections as indicated in the table translate into less opportunity for Boolean problems. From robustness  Chapter 4. A Hybrid Analytical, Solid Modeler and Feature-Based Methodology  82  point of view, the hybrid approach also made a big improvement compared to the full solid modeler solution.  4.8  Summary A pure solid modeler solution and a hybrid analytical, solid modeler and feature based  methodology for tool/workpiece engagement calculations in general turning processes are described in this research. The accuracy and computational efficiency are compared as shown in Table 2. It can be seen that the hybrid solution has significantly better computational efficiency than the pure solid modeler solution, while achieving the same accuracy. This is because Boolean operations which are applied at each feed step consume significant computational time. Whereas the latter hybrid solution which employees algebraic calculations has a significantly smaller processing time. Moreover, the identification and parametrization of geometric invariant features and form invariant features, eliminates large amounts of repetitive calculations, therefore leading to further improvements in the computation time. As discussed in the assumptions section, one area for future work will consider deflection and dynamics which result in process induced variations in the geometry of the intersection area. The 2D model will need to be extended to a 3D model for capturing this effect.  A 3D  methodology is also necessary for capturing engagement conditions for non-symmetric parts due to the initial workpiece having previously machined non-turned features such as holes and slots. The feature identification needs to be extended to 3D volume features.  Chapter 5 Instantaneous Force Prediction for Contour Turning  5.1 Introduction A substantial amount of research has been done to predict static turning forces for given depth of cut, feedrate, cutting speed, and tool geometry. A mechanistic model has emerged as a successful approach for cutting force prediction. The mechanistic model proposed by Atabey [1,2] presents the cutting forces in tangential and friction directions. Friction force is perpendicular to the cutting edge and passes through the gravity center of the uncut chip area. This model gives good prediction of force direction to simplify the force calculation. Particularly for the force prediction in dynamics of turning when the uncut chip area becomes irregular, the force model remains the same. However, the aforementioned model is capable of computing mechanics parameters for simple workpiece geometry at one feed step, where feed is in the direction of the spindle axis and the depth of cut is constant. In contour turning, the feed direction varies along the toolpath. Therefore, the forces are predicted in the local coordinates with respect to feed direction. Local tangential force (F,) is in the same direction with global tangential force ( F t ) ; local feed Force (F/) and local radial force (F ) need to be resolved and summed in the global Feed F f (spindle r  axis, called Z) and global Radial F  r  (X axis) directions, which will be discussed in Section 5.3,  for power, torque calculation, and machine constrain-based optimization of the turning process. Deflection is predicted based on local feed and radial forces in local coordinates.  83  Chapter 5. Instantaneous Force Prediction for Contour Turning  84  In addition to feed direction, instantaneous depth of cut, uncut chip area, chip-cutting edge contact length and effective lead angle change along the contour path or when the workpiece geometry varies. The cutting force prediction requires dynamic identification of removed chip shape at each feed increment to simulate part turning. The tool-workpiece intersection is identified through a geometric and solid modeling system, which was presented in Chapter 4. The rest of this chapter is organized as follows. Section 5.2 presents details of the adopted mechanistic force model, which is slightly modified to improve force prediction efficiency for virtual turning. The mechanistic model of contour turning is presented in section 5.3. Section 5.4 introduces a new method to identify the mechanistic cutting coefficients from the orthogonal cutting database. This is followed by experimental validation on an Aluminum test part in Section 5.5. The chapter ends with conclusions and recommendations for future research.  5.2  Mechanistic Model in Simple Turning In the mechanistic model proposed by Atabey [1,2], the turning tool has a nose radius, and  the cutting forces are represented by the tangential force (F ) and friction force (Fp), shown in t  Figure 5.1.  Chapter 5. Instantaneous Force Prediction for Contour  85  Turning  Figure 5.1: Mechanistic Force Model Since the chip thickness distribution at each point along the cutting edge contact point is different and dependent on the tool nose radius (r ), side and end cutting edge angle, feedrate (f) £  and radial depth of cut (d), the distribution of the force along the cutting edge-chip contact zone also varies. At any contact point, the differential cutting forces are modeled as a function of local chip load (dA) and chip-cutting edge contact length (dL ), See Figure 5.2. c  dF = dF + dF = K t  tc  te  tc  • dA + K  te  • dl  c  Figure 5.2: Distribution of Friction Force along Cutting Edge  86  Chapter 5. Instantaneous Force Prediction for Contour Turning  The direction of each differential tangential force is perpendicular to the 2D cross-section of the workpiece and the tool, as shown in Figure 5.1. However, the direction of differential friction force varies in different regions of the uncut chip. In the tool straight cutting edge region (Region 2 shown in Figure 5.3), the chip thickness does not change and the effective lead angle is the same as the side cutting edge angle. The direction of each differential force remains the same, i.e., perpendicular to the same straight cutting edge, as well as the magnitude. While in the tool nose curve region (Region 1 shown in Figure 5.3), the differential chip area changes continuously, and the friction force acts perpendicular to the cutting edge segment for each differential element, it can be predicted by assuming that each component of the friction force passes through the gravity center of each related region (Figure 5.3). The friction force component of each region is added up vectorially to find the total friction force (Ff ). r  Due to the different mechanics at the tool nose curve and straight cutting edge regions, the total tangential force (F ) and friction force (F/ ) are modeled as follows: t  r  F = F + F„ =K -A t  lc  F =F  lc  fr  m  +F  frc2  +  K, • L e  +F  =K  fn  frc]  The cutting coefficients K ,K ,K ,K ,K tc  frcl  frc2  c  le  fre  • A, + K  frc2  •A +K 2  jm  •L  c  are obtained from cutting tests and  curve-fitting technics. They are non-linear functions of chip load, chip-cutting edge contact length and cutting speed. Friction force Ff is considered to consist of two cutting force components Ff \ and Ff 2 r  rc  rC  corresponding to the cutting forces in region 1 and region 2, which are associated with uncut  Chapter 5. Instantaneous Force Prediction for Contour  chip areas j4/ and A . Later, F 2  87  Turning  is resolved into the feed (Ff) and radial directions (F ) with  fr  r  respect to the resultant effective lead angle<f) . See Figure 5.3. L  F  =F -sm(<P )  r  F  f  fr  =F  fr  L  -cos(^)  (5.3)  Region 2  Region 1  Figure 5.3: Friction Forces and Effective Lead Angle (<f>i) In the original mechanistic force model, the calculation of effective lead angle (tpi) is  tp  L  =  +  V '^ /  2  . For relatively large radial depth of cut, the effective lead angle tends to  A,+A  2  approach the side cutting edge angle (y/ ). However, since the effective lead angle defines the r  direction of total friction force (Ffri,  Ff ), r2  Ff i r  and  Ff  r2  (Ff ), r  which is the resultant force evaluated from the two regions  in the tool nose region and straight cutting edge region are assumed to  contribute to the direction of Ff . Therefore, the fa, which was considered to be dependent only r  on the geometric information in the original model, showed some discrepancies between  88  Chapter 5. Instantaneous Force Prediction for Contour Turning  measured and predicted effective lead angles. Atabey introduced modification factor K , which m  is a linear function of chip length (Z ) and cutting speed (V), to correct the effective lead angle c  calculation. The modified effective lead angle i s ^ = K (L ,V)-$ m  c  L  . Later, the radial force and  the feed force are calculated using this lead angle. F =F .sm(fa r  fr  F =F f  • cos(^)  fr  To eliminate using the modification factor and to minimize the discrepancy caused by fa, a slight modification to the original model is presented as follows. Fj  rl  and Fj  are calculated in  r2  each region separately instead of calculating the resultant friction force F/ . The forces are r  resolved by using the effective lead angle of each region and summed up to form resultant F , Ff r  as shown in Figure 5.4. From equation (5.2), F ^ a n d F  fr2  Ffrl  Kfrcl ' l  =  A  Ffr2 - K fre  re  •2 A  frc2  The edge coefficient K  Kf  +  +  are obtained as  ' L,  (5.5)  c  Kfre ' L  c2  is assumed to be constant in both regions, and the radial force and  feed force in each region are calculated as follows. F  =F  rl  -sm(fa)  frI  F i =F f  -cos(fa)  fr!  (5.6)  where fa is the gravity vector angle shown in Figure 5.4 F  =F  r2  -sm(y/ )  fr2  Ff  r  =F  2  -cos(^)  fr2  (5.7)  wherey/ is the side cutting edge angle. Final radial and feed force is: r  F =F -F r  rl  r2  l  F =F F f  fl+  f2  (5.8)  89  Chapter 5. Instantaneous Force Prediction for Contour Turning  Region 2  Region 1  Figure 5.4: Feed, Radial Forces in Each Region As a result, the modification factor is not used and the predicted radial and feed forces match well with the measured data as shown in Section 5.5.  5.3  Prediction of Cutting Forces in Contouring Turning  Workpiece  F  r  boundary  Toolpath Initial Workpiece X  Final  Part  Machining  o  Figure 5.5: Contour Turning  Axis  90  Chapter 5. Instantaneous Force Prediction for Contour Turning  As shown in Figure 5.5, the forces predicted using the mechanistic force model described in the previous section are the local forces with respect to instantaneous feed direction at each feed step. To apply the mechanistic model to contour turning, the local forces F , F , and F/need t  r  to be projected to global X Y Z directions. Tangential force F, is in the same direction as the global tangential force F t . Radial and Friction forces F , F/at each feed step need to be resolved r  in global machine axes (X, Z). The projected forces in global axes are called global radial ( F ) r  and feed ( F f ) forces, respectively. The global forces F t , F , and F f at each machining step are r  shown in Figure 5.6.  Z ( F ) f  Figure 5.6: Force Prediction of Contouring Turning The global cutting forces at any feedrate step are:  F  r  =F  f  • sin(a) + F • cos(a) r  F f = Ff • cos(a) + F • sin(a) r  (5.9)  91  Chapter 5. Instantaneous Force Prediction for Contour Turning  Where, a is the angle between the instantaneous feed direction and the machining axis shown in Figure 5.6. Using equation (5.9) the instantaneous global F t , F , and F f are predicted at r  each machining feed step. When all these three forces are calculated at all feed steps along the toolpath, the whole cutting process is simulated. Based on forces, powers, torques, and the deflections, the machining process can be optimized. However, it can be seen that the global force calculation method of contour turning (equation (5.9)) is not efficient enough in terms of projection twice. In the local coordinates with respect to the instantaneous feed direction, the local side cutting edge angle (y/') varies with a, r  i.e., y/ '=y/ r  r  +a, where y/ is the fixed side cutting edge angle of the tool along machining. r  Local friction forces (Ff i, Ff ) need to be projected to the local radial (F ) and feed (Ff) r  r2  r  (equations (5.4-5.7)) and then be projected again to global radial ( F ) and feed ( F f ) forces r  (equation (5.9)). To increase computational efficiency, mathematical simplification  is achieved by  manipulating the above formulations. From equations (5.5-5.7), the local F and Ff are r  expressed as the combination of the friction forces in two regions, F =F r  rl  - F  r2  Ff =F ,+F f  = F , • sin(fa') - F fr  =F ,  f2  • sin(t// ')  fr2  fr  -cos(fa') + F  r  fr2  -cos(y/ ') r  where fa' and y/ ' are the local gravity angle and local side cutting edge angle at any feed r  step. Similar to y/ ' = y/ + a, the global gravity angle is fa - <j>, '+a . r  r  Substituting F , Ff, fa' and y/ ' in equation (5.10), the global forces in equation (5.9) are r  obtained as  t  Chapter 5. Instantaneous Force Prediction for Contour Turning  92  F  (5.11)  F  r  =F  sm(<p,) - F  =F  cos(fa) + F  frl  f  frl  fr2  fr2  sm(y/ ) r  cos(y/ ) r  Where fa is the angle of a vector that goes through the gravity center of tool nose region and points to the tool nose center in global coordinates. The global force formulation shown in equation (5.11) can significantly simplify the force calculation during arbitrary contour turning, because the side cutting edge angle (y/) remains unchanged, and only the global gravity center r  angle (fa) is calculated directly in global coordinates without projection twice at each feed step. However, it must be noted that the force expressions for contour turning are based on the assumption that the changes in side cutting edge angle along the contour toolpath do not affect cutting coefficients significantly. In this research this assumption is used to simplify the force calculation based on the experimental results shown in Section 5.5. needed at different contour conditions to generalize the comment.  More cutting tests are Moreover, the force  expression (equation (5.11)) is merely the mathematical equations without physical meaning. As presented in the previous section, the cutting force coefficients used in the equations (5.2-5.4) are identified from the cutting tests or from the orthogonal database (which will be discussed in the next section). Instantaneous tool-workpiece intersection information (uncut chip area (A), contact chip length (L ), global gravity center of the tool nose region (fa), and c  tool geometry information (y/, K , r ) are generated from the tool-workpiece engagement model r  r  £  described in Chapter 4. The instantaneous global forces F t , F , F f , power, and torque of all r  machining steps are predicted using the extended mechanistic force model. The comparison  93  Chapter 5. Instantaneous Force Prediction for Contour Turning  between the predicted forces and the measured forces of a test part will be presented in Section 5.5.  5.4  Mechanistic Cutting Coefficient Evaluated form the Orthogonal Cutting Database The mechanistic cutting coefficients K, , Kf i, Kf 2, K , and K/ are identified from turning c  rc  rC  re  re  tests for each tool/workpiece combination, and they are modeled as non-linear functions of the A, Lei, L and V: c2  K =b A 'V > b  le  b  0  K =m L <V > m  frc!  0  m  cl  [N/mm ]  (5.12)  2  K =n L 'V > n  fic2  0  n  c2  where bo, b\, b , mo, m\, m2, no, «/, and n are empirical constants that are evaluated from 2  2  the experimentally measured force data using the least-square method. There are three advantages to the mechanistic cutting coefficient identification method. First, they are expressed as a function of a few geometric variables with simple expressions. Second, the empirical constants of mechanistic coefficients are valid for all cutting conditions. Third, K, expresses the tangential coefficient of the whole uncut chip area, and Kp i and K/ c  c  rc2  express the friction coefficients in two regions of the uncut chip area. These expressions make the force prediction simple because the uncut chip area is only separated into two regions at each feed step. The simplicity and coverage of multiple cutting conditions lead to efficient cutting force computation in contour toolpaths where the chip geometry changes continuously. Contrarily, the orthogonal to oblique transformation force prediction method digitizes the  Chapter 5. Instantaneous Force Prediction for Contour Turning  94  cutting edge into small micro-elements, which requires at least an order of magnitude more computation at each feed step. On the other hand, the disadvantage of the mechanistic method is that it is only valid for one tool geometry and workpiece combination. Hence, each tool geometry must be calibrated through cutting tests. The orthogonal to oblique transformation force prediction method [7] uses fundamental material properties, such as shear angle, shear stress, and friction angle to determine the oblique cutting coefficients.  The advantage of this method is that it is valid for a range of cutting tool  geometries. Therefore, once an orthogonal cutting database is established, the oblique cutting coefficients of any tool geometry of a material can be identified. However, when a tool has a nose radius, the cutting edge must be considered as an assembly of the straight oblique cutting edge elements. This force method is less efficient in contour turning, because there are a large number of different tool-workpiece intersections along the toolpaths and in each instantaneous uncut chip area, the cutting edge has to be discretized to small segments, and all the differential cutting coefficients need to be calculated through orthogonal to oblique transformation [7]. Moreover, the method requires that the rake face of the cutting tool is uniformly flat (i.e., without any chip breaking or contact reduction grooves). As an alternative to the pure mechanistic or the pure orthogonal to oblique transformation at each tool-workpiece engagement, the mechanistic cutting coefficients are directly evaluated from the orthogonal cutting database. An oblique tool with curved nose and smooth rake face are considered, and the cutting forces are predicted in sets of given cutting conditions using orthogonal to oblique  95  Chapter 5. Instantaneous Force Prediction for Contour Turning  transformation by digitizing the cutting edge. The predicted forces are used like the measured forces in fitting the mechanistic cutting force coefficients. The following example is given to illustrate the above identification method. The existing orthogonal cutting database of Aluminum 6061-T6 and the tool geometry used are shown as follows. •  Existing Orthogonal Cutting Database of Aluminum 6061-T6 K  Edge cutting coefficients: (N/mm)  te  = 24.416 -1.9907e~ • V + 0.045502 • V - 0.16499 • RN  f  = 9.8695 - 6.0891'e' • V + 0.13283 • V - 0.00162 • RN  re  =0  5  K  2  5  K  2  Shear stress: T = 244.49 + 0.336 • RN (N/mm ) Shear angle: fa = 72.755 + 53.650f + 0.0073698 • V + 0.2972 • RN (degree) Friction angle: J3 =53.473 - 4.2403f + 2.5759e~ • V - 0.069330 • V + 0.297 • RN (degree) f - feedrate (mm/rev), R N - Side rake angle (deg), V - Cutting speed (m/min) s  2  5  2  a  •  Tool Geometry Tool nose radius: (mm)  r = 0.7874  Side cutting edge angle: (degree)  1//7 =22.5  End cutting edge angle: (degree)  K = 32.5  Side and back rake angle (degree)  .  e  r  - GLf— a  p  =0  Force Prediction Based on Orthogonal to Oblique Transformation As introduced in the literature review, three differential cutting force components can be  expressed as follows:  96  Chapter 5. Instantaneous Force Prediction for Contour Turning  dF = dF + dF = dK -b-h + K -b t  tc  le  tc  le  dF = dF + dF = dK f  fc  fe  dF =dF +dF r  rc  fc  -b-h + K -b  =dK -b-h  re  (5.12)  fe  rc  + K -b re  Where the oblique cutting coefficients are given in equation (5.13) refer to [4,7]. T , B„, and s  fa come from the orthogonal database, rake angle a , oblique angle i, and TJ come from the tool n  geometry. cos(0„ - a„) + tan/tan?7sin fi  n  dK„. =  sin^  J cos (fa+/3 - « „ ) + tan 77 sin /?„ 2  2  r,  dK  sin(^„-a„)  sin^„ cosi  r r  ^ (fa+/3 2  cos  - « „ ) + tan ?7sin 2  n  2  (5.13)  B  n  cos(/3 - a„) tan / + tan TJ sin /?„  s  dK  2  n  n  sin fa ylcos (fa +p - « „ ) + tan ?7sin 0„ 2  2  2  n  The differential forces for the regions 1 of each uncut chip area in global X , Y and Z directions are identified in equation (5.14), region 2 and 3 are shown in equation (2.6-2.7), and the total integrated forces F , F , F are expressed in equation (5.15). x  y  z  Region 1 F  Dynamometer axes Y  Figure 5.7: Orthogonal to Oblique Transformation  97  Chapter 5. Instantaneous Force Prediction for Contour Turning  n i=l  F =t  (F/i t  vI  0, -F cose ) rli  (5.13)  t  i=l n  zj = H( /u ^  F  F  x  = x,I  F  -Ku ^)  cos  F  sin  + x,2  +  F  x,3  F  y = y,l + y,2 + y,3  F  F  F  (5.14)  F  K=PzJ+Py,2+P ,3 z  For given cutting speed (m/min) 250, 375, 500, feedrate (mm/rev) 0.05, 0.075, 0.1, 0.125, 0.15, 0.175, 0.2, 0.25, and the depth of cut (mm) 0.2, 0.4, 0.7, 1.0, 1.5, 2.0, 2.5, 3.0, the forces in total 216 cutting conditions are predicted. The predicted cutting forces (F , F , F ) are regarded x  y  z  as the measured forces in the 216 different sets of cutting conditions. After Least-square curve fitting [1,2], the final mechanistic cutting coefficients are shown as follows.  •  Mechanistic Cutting Coefficients: 4(N/mm)  £«(N/mm)  A/ (N/mm)  Ajre(N/mm)  48.873815  20.025611  43.391623  47.789728  e  tc = 1863 82-A  .v ~  {00675)  {  K  = 375.382 Kfrcl ~~ Kfrc2 ~  241519 4 • L  -Lj-°-  442)  ( 2  -  0  1 2 8 9 9 7 )  _ y (-0.0511434)  0 3 9 ? m )  .  y{-i-i20065)  (N/mm ) 2  ( N/mm ) 2  ( N/mm ) 2  Where A (mm ), L , L (mm), V (m/min) 2  c!  c2  From the previous cutting test validations, the pure mechanistic cutting coefficient identification method (least square techniques from the measured data) gives less than 10%  Chapter 5. Instantaneous Force Prediction for Contour Turning  98  errors when enough measured forces are conducted, and the pure orthogonal to oblique transformation method also gives good prediction when discretized cutting edge elements are small enough. Therefore, the error of the predicted force, in which the mechanistic cutting coefficients  are evaluated from the orthogonal database, remains the same order as the  orthogonal to oblique transformation method. The comparison between the forces whose cutting coefficients are evaluated from orthogonal to oblique transformation and orthogonal database are shown as the following figures. F o r c e s C o m p a r i s o n (V=30m/min)  d e p t h of c u t  0  0 0 5  feedrate  Figure 5.8: Tangential Force Predicted from Different Cutting Coefficient Identifications Figure 5.8 shows forces predicted from different cutting coefficient identification methods. Dots represent the forces that are calculated using the orthogonal to oblique transformation method, and the stars represent the forces that are calculated using the mechanistic method, in which the mechanistic cutting coefficients are evaluated from the orthogonal cutting database.  99  Chapter 5. Instantaneous Force Prediction for Contour Turning  The plot shows that the forces predicted from the two methods match well. The following figures also show the forces predicted using the two methods in different cutting conditions. F o r c e s C o m p a r i s o n (d=0.4mm  V=40m/min)  — 300  C u t t i n g C o e f f i c i e n t s evaluated from O r t h o g o n a l t o O b l i q u e T r a n s f o r m a t i o n M e c h a n i s t i c C u t t i n g C o e f f i c i e n t s e v a l u a t e d from O r t h o g o n a l D a t a b a s e  0.25  0.05  F o r c e s C o m p a r i s o n (d=1.25mm  u- 100 0.05  0.25  0.15  feedrate (mm) V=50m/min)  0.15  feedrate (mm)  Figure 5.9: Forces Predicted from Different Cutting Coefficients The above comparisons demonstrate that the mechanistic cutting coefficients evaluated from the orthogonal database provide good force prediction results. The forces predicted using  Chapter 5. Instantaneous Force Prediction for Contour Turning  100  these mechanistic coefficients are very close to the forces predicted from the orthogonal to oblique method. More cutting tests will be done to compare the forces that are calculated from different cutting coefficients. Depending on the accuracy of the well-known orthogonal to oblique approach and the mechanistic cutting coefficient identification method, it can be concluded that mechanistic cutting coefficients evaluated from the orthogonal database method is feasible and practical.  5.5  Experimental Validation for Contour Turning To test the tool-workpiece intersection and mechanistic force prediction model, an  Aluminum 6061-T6 test part was machined on the Cincinnati Falcon 300 C N C turning center. A turning tool, P052.1 - Holder PT 135789, Insert PC 157838 with a nose radius of 0.7874mm, was used in the experiments. Kistler 9257B dynamometer and MalDaq 6.0 software were used to measure the cutting forces. The rake face of the tool was flat, hence it was possible to use the orthogonal to oblique transformation theory. The predicted forces are based on the mechanistic force model of contour turning, in which the cutting force coefficients are predicted from the orthogonal database. Tool-workpiece intersection geometry was generated from the tool-workpiece engagement model. The cutting tests are designed as follows.  Chapter 5. Instantaneous Force Prediction for Contour Turning  5.5.1 Cutting Test Design  Initial Workpiece  Second Cut  First Cut  Third Cut  Figure 5.10: Turning Process Plan of the Test Part The turning operations to produce the final test part include three series of cuts. The first cut tests the correctness of the intersection geometries and the force results on the initial cylindrical workpiece with the contour toolpaths. The second cut tests those on the contour inprocess workpiece, which is the resulting workpiece after the first cut, with the contour toolpaths. And the third cut tests the contour workpiece with the straight toolpaths. The comparisons  Chapter 5. Instantaneous Force Prediction for Contour Turning  102  between the measured and predicted cutting forces along the contour turning are presented in the following sections.  5.5.2 First Operation  Initial workpiece and tool paths  Figure 5.11: Tool Paths and Workpiece of First Cut Figure 5.11 shows the designed initial workpiece and toolpaths in a C A D environment and the simulation result of the turning process in the developed tool workpiece intersection model of this research. The tool moves along the toolpaths, and tool-workpiece engagement at each machining step is captured and calculated. The material removal area (Boolean intersection between the tool swept area and the in-process Workpiece) is obtained and subtracted from the workpiece. Since the forces are proportional with the intersections as shown in equations (5.2-5.4), the material removal area of each toolpath and the corresponding predicted forces are shown in the following figures to verify the intersections and predicted forces.  103  Chapter 5. Instantaneous Force Prediction for Contour Turning  • Material Removal Area  Toolpath  • • • I n - p r o c e s s Workpiece  ~" 24'""  M a c h i n i n g Direction  M e a s u r e d v s Predicted Tangential Force  100  150  200  250  300  350  400  Time (s)  Figure 5.12: Comparisons of the Tangential Forces of First Cut  104  Chapter 5. Instantaneous Force Prediction for Contour Turning  26  25  OA  2  J? . 2  3  M e a s u r e d v s Predicted Radial Force  1 Measured Fr Predicted Fr  10,21,22 ,23,24,25  1  w  10,11,12 •  7-  f  —-  \ tit 06  f L i  •  1 t 50  100  150  200  250  300  350  40  Time (s)  Measured v s Predicted Feed Force  50  100  150  200  250  300  350  400  Time <s)  Figure 5.13: Comparisons of the Radial and Feed Forces of First Cut  2  1  20  Chapter 5. Instantaneous Force Prediction for Contour Turning  105  It must be noted that the unit of the X axis is the machining time as shown in Figures 5.12 and 5.13. Since the machining direction in turning simulation is leftwards, the forces shown in the figures are left and right reversed with respect to the real machining direction. From the force comparison shown in the above figures, the measured forces and the predicted force are in good agreement. Tangential forces and feed forces are proportional to the tool-workpiece intersection area; i.e., the forces are big when the uncut chip areas are big. This trend is consistent with the adopted mechanistic force model, and demonstrates that the intersection model developed in this research provides correct geometric information along the contour toolpaths. The radial force is not always proportional to the uncut chip area and changes the direction. The reason is that the sign of the radial force in the two regions (tool nose region and straight cutting edge region) is opposite if the side cutting edge angle is positive.  Figure 5.14: The Changes of the Radial Forces with the Depth of Cut  106  Chapter 5. Instantaneous Force Prediction for Contour Turning  Illustrated in Figure 5.14, the final radial force (F ) is positive when the uncut chip area r  only has the tool nose region corresponding to the small depth of cut. When the depth of cut increases, the radial forces in the two regions cancel each other, and F drops to zero in certain r  depth of cut (shown as the middle case in the above figure). And then F becomes negative and r  the magnitude increases as the depth of cut increases. This theoretical force analysis is verified by the measured radial data (Figure 5.13). As the result, the predicted radial forces match the measured radial forces very well. The intersections and forces are shown to be accurate and correct in the first cut.  5.5.3 Second Operation  Initial contoured workpiece and tool paths of the second cut  Figure 5.15: Tool Paths and Workpiece of the Second Cut  Chapter 5. Instantaneous Force Prediction for Contour Turning  Figure 5.16: Comparisons of the Tangential Forces of the Second Cut  107  108  Chapter 5. Instantaneous Force Prediction for Contour Turning  28  4-«*«*'^^2a«iBS!^M«JL2_J ^  M e a s u r e d v s Predicted Radial Force  202;23 6.7  1,2  !1  ]  J§  Time (s)  Measured v s Predicted F e e d Force M e a s u r e d Ff Predicted Ff  2i 18,19,201  6.7  1  i  m  A  Ifi )i A)  5  £ 200  lAii fv  1  •  I  I,, .„.,  100  150 Time (s)  Figure 5.17: Comparisons of the Radial and Feed Forces of the Second Cut  Chapter 5. Instantaneous Force Prediction for Contour Turning  109  The comparisons of the second cut also present that the predicted forces match the measured forces very well. The good agreement again demonstrates that the proposed mechanistic force model and the intersection methodology are capable of contour turning, in which the workpiece has waved surface and the toolpaths are non-parallel with the workpiece. The challenge here is that the uncut chip area is hard to predict since the workpiece is not simple cylindrical block. By using the proposed hybrid solid modeler, analytical and feature-based method, the geometric information of the tool-workpiece intersections is obtained correctly and effectively. These correct intersections lead to correct predictions of the cutting forces.  5.5.4 Third Operation  Figure 5.18: Tool Paths and Workpiece of the Third Cut  110  Chapter 5. Instantaneous Force Prediction for Contour Turning  • Material Removal Area  Toolpath  WBMIn-process  Workpiece  Figure 5.19: Comparisons of the Tangential Forces of the Third Cut  Chapter 5. Instantaneous Force Prediction for Contour Turning  M e a s u r e d v s Predicted Radial Force  Measured v s Predicted Feed Force  Figure 5.20: Comparisons of the Radial and Feed Forces of the Third Cut  Chapter 5. Instantaneous Force Prediction for Contour Turning  112  As with the previous two cuts, the predicted forces match well with the measured forces in the third cut. This consistency verifies that the proposed force model and the intersection model are valid for straight toolpaths with contour workpiece. However, the predicted tangential forces are a little bigger than the measured tangential forces, but the errors remain less than 15%. This deviation may come from the predicted cutting coefficients or from the noise of experiments. There are big discrepancies in the radial forces of Figure 5.20, this may be because the radial forces are very small, less than 50 N . Due to the effects of chips and noise, the measured forces are not completely reliable.  5.6  Conclusion and Future Work From the force comparisons it can be seen that the measured forces and predicted forces  are in good agreement, especially the magnitude and direction of the radial forces. The predicted tangential force is a little bigger than the measured force, but the error is less than 15%. The conclusions can be summarized as follows. The tool-workpiece intersection methodology works well and captures the correct instantaneous uncut chip areas and in-process geometries for force prediction. The mechanistic cutting coefficients evaluated from the orthogonal database are accurate enough. The assumption that the changes of mechanistic cutting coefficients due to the changes in side cutting edge angle along the contour toolpath (Section 5.4) are neglectable, is acceptable because the forces accurately predicted.  Chapter 5. Instantaneous Force Prediction for Contour Turning  113  The mechanistic force model accurately predicts of the instantaneous cutting forces along the contour turning. To further verify the mechanistic force model, more experiments should be conducted to fully identify the following factors. The exact mechanistic cutting coefficients need to be identified from the cutting tests. The effect of a change in the side cutting edge angle along the contour toolpath to the mechanistic cutting coefficients, and also the effect on the predicted cutting forces, needs to be investigated. More cutting tests on different materials and with different tools. Overall, the proposed two main models of the Virtual Turning system are verified from the experimental results. The forces at each machining step along the arbitrary contoured toolpath and workpiece are predicted. Later the forces will be used to optimize the turning process by changing the feedrate and cutting speed. This force model can also be easily extended to dynamics of turning.  Chapter 6 Conclusions  6.1  Conclusions A prototype Virtual Turning system, which can predict the cutting forces, torque, power  and deflections along the toolpath, is developed in this thesis. The system has two integrated components: Tool-workpiece engagement identification based on C A D techniques, and process simulation based on the laws of metal cutting mechanics. The tool-workpiece intersection is identified from tool geometry, imported workpiece geometry and tool motion information from standard C A D / C A M software systems. Two fundamental approaches are developed to identify the tool engagement conditions. The first method is based on Boolean intersection of tool and workpiece by using their Boundary Representation models in ACIS solid modeling kernel. Since the computational cost is quite high with the first method, a hybrid analytical, feature-based solid modeling approach is developed as a viable alternative. The engagement conditions are grouped as a class of geometric features, and as they are encountered along the toolpath, they are retrieved as opposed to repetitive computation of recurring engagement conditions. Green's theorem is then used to evaluate the chip area at each tool engagement feature. The hybrid model improved the computational efficiency of tool-workpiece intersection by significantly reducing Boolean operations and numerical area calculations in solid modeler. The process is simulated by using the tool-workpiece intersection and previously developed Mechanistic Model of the turning process. The transformation of orthogonal cutting  114  Chapter 6. Conclusions  ll5  to discrete, oblique cutting edge elements along the tool engagement zone takes significant computational time which hinders the practicality of Virtual Turning Simulation system. In order to reduce the computational complexity and time, the cutting coefficients are evaluated from the orthogonal cutting database by considering the classified chip features and areas. As a result, the cutting force is predicted as just function of total chip area and cutting edge engagement length estimated from the tool-workpiece intersection engine. The overall prototype Virtual Turning system is experimentally validated in machining a sample Aluminum workpiece on a C N C lathe. The predicted and measured cutting forces are shown to have sufficient agreement for practical use of the system in basic turning operations. The contributions of the thesis can be summarized as follows: •  A n experimentally validated, prototype Virtual Turning Process simulation system is developed. The system is one of the first reported in the literature.  •  A solid modeler-based tool-workpiece intersection algorithm is developed by applying the Boolean intersections of their boundary representation models at each toolpath. The proposed modeling approach reduces the computational cost by using Toolpath Swept Area intersecting with the workpiece at each toolpath in comparison to the tool intersecting with the workpiece at each feed increment. The reduced solid model complexity and number of Boolean operations decreases the computational cost significantly, since the number of toolpaths are used as opposed to the number of feedrate increments which are typically an order of magnitude bigger.  •  A Tool Swept Area (TSA) algorithm is developed for toolpaths containing line and arc segments based on the tool geometry and a toolpath. The TSA is generated by  Chapter 6. Conclusions  116  identifying envelope edges of the path and connecting them with the tool edges. A general convex hull algorithm is used for the linear toolpath TSA construction and identified critical points of the tool swept envelopes used for circular toolpath TSA construction. The proposed simple algorithm is used to represent tool swept area in two dimensional turning paths in a computationally efficient manner, although it is not applicable to more generic turning operations. •  A hybrid algorithm to evaluate the tool engagement and chip area is developed based on the combination of the solid modeling method, a feature identification algorithm, and an analytical Green's Theorem based method for calculating chip areas . The use of features and analytical formulations for the majority of toolpaths during machining to extract the engagement parameters increases the computational efficiency. The solid modeler is used to construct the workpiece, the tool, and the toolpath, generate material removal areas, and extract tool-workpiece engagements when situations which cannot be handled analytically are encountered. In short, the proposed hybrid technique can handle a variety of cutting tool engagement conditions.  •  Tool Workpiece Engagement (TWE) geometry has been grouped into a small set of classes the areas and centroids of which are expressed in appropriate formulations that can be solved analytically using Green's Theorem. This method increases the computational efficiency due to two reasons: first, generic numerical solvers in the solid modeler are not used; second, Boolean intersections are not required to obtain the  Chapter 6. Conclusions  117  intersection solid for extracting the required parameters, which is computationally expensive. •  A novel use for in-process machining features has been developed along with feature recognition algorithms. These features are classified as geometric invariant, form invariant and transient features. A n area decomposition algorithm is applied to the material removal area along a toolpath segment to generate these features. For a geometric invariant feature, all TWE geometry within the feature is the same, thus the boundary position calculations are performed only once. For form invariant features parametric expressions of lines and arcs are used in finding the intersections between linear and circular components. As a result all the boundary positions of a TWE are calculated as a function of machining parameters (feedrate, depth of cut), tool geometry (y/ , K, r ), and tool center positions along the toolpath. In short, with the exception of r  £  the transient features, the boundary conditions at each step along the toolpath are determined analytically enhancing computational efficiency. •  A mechanistic force model previously developed at U B C is adopted with slight modifications for improved computational efficiency and force prediction in radial direction. A n algorithm is developed to predict cutting forces in contour turning operations, where the tool engagement conditions and the directions of the cutting forces continuously vary. The predicted forces at each feed step are projected to the global X Y Z directions of the toolpath, and used in evaluating power, torque, and deflection in contour turning operations.  Chapter 6. Conclusions  118  6.2 Future Research Directions The proposed Virtual Turning system does not consider the structural dynamics of the system, hence the forced and chatter vibrations are not included in simulations. An accurate prediction of chatter stability and dimensional form errors left on the finish surface are still unresolved research topics, and need to be further investigated before including them in Virtual Turning Simulations. The thesis dealt only with two dimensional tool-workpiece intersections. In order handle a variety of turning operations, three dimensional workpiece and multi-axis tool motions need to be studied. Parts having slots, holes and other non-symmetric features require three dimensional modeling of tool-part intersection algorithms in order to simulate their turning process in virtual environment.  Bibliography [1]  Atabey, F., "Modeling of Mechanics and Dynamics of Boring", M.A.Sc. Thesis, University of British Columbia, 2001  [2]  Atabey, F., Lazoglu, I., Altintas, Y . , "Mechanics of Boring Process Part I", International Journal of Machine Tools and Manufacture, Design, Research and Application Vol. 43,Issue 5, pp. 463-476, 2003  [3]  Atabey, F., Lazoglu, I., Altintas, Y . , "Mechanics of boring processes - Part II: Multi-insert boring heads", International Journal of Machine Tools and Manufacture, Design, Research and Application 43 (2003) 477 - 484  [4]  Altintas, Y . , "Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design ", Cambridge University Press, 2000.  [5]  Altintas, Y.,Spence, A . D . , "End milling force algorithms for C A D systems", Manufacturing Technology CIRP Annals, Vol. 40, pp. 31-34, 1991.  [6]  Altintas, Y.,Spence, A . D . , " A Solid Modeller based milling process simulation and planning system", Transactions of the ASME, Journal of Engineering for Industry, Vol. 116, pp. 61-69, 1994.  [7]  Armarego, E.J.A. and Uthaichaya, M . , "Mechanics of Cutting Approach for Force Prediction in Turning Operations", J. of Engineering Production, Vol. 1, pp. 118, 1977.  [8]  Armarego, E.J.A., Whitfield, R.C., 1985, "Computer Based Modelling of Popular Machining Operations for Force and Power Predictions", Annals of CIRP, Vol. 34, pp.65-69, 1985.  [9]  Yip-Hoi, D., Dutta, D., Huang, Z., " A Customizable Machining Feature Extraction Methodology for Turned Components", Journal of Manufacturing Systems, Vol.22 /No.2, 2003  [10] Yip-Hoi, D., Huang, X . , "Cutter Engagement Feature Extraction from Solid Models for End Milling", Computer Aided Design, 2004 [11] Carlsson, T., Stiernstoft, T., " A Model for Calculation of the Geometrical Shape of the Cutting Tool-Workpiece Interface", Annals of the CIRP Vol.50/1/2001, 41-44 [12] Floriani, L., "Feature Extraction from Boundary Models of Three-Dimensional Objects", IEEE Transactions on Pattern and Machine Intelligence. V o l . 11, No.8, August 1989  119  Bibliography  120  [13] Fussell, B.K., Hemmett, J.G.,Jerard, R.B. "Geometric and mechanistic modeling integration for five-axis milling force prediction", Proceedings of the Japan-USA Symposium on Flexible Automation. Kobe, Japan. Vol. 2. pp. 747-750, 1994. [14] Fussell, B.K., Jerard, R.B.,Hemmett, J.G., "Modeling of cutting geometry and forces for 5-axis sculptured surface machining", Computer Aided Design, Vol. 35, pp. 333-346, 2003. [15] Fussell, B . K., Jerard, R. B., and Hemmett, J. G., "Robust Feedrate Selection for 3Axis N C Machining Using Discrete Models", Journal of Manufacturing Science and Engineering — May 2001 ~ Volume 123, Issue 2, pp. 214-224 [16] Jerard, R.B., Fussell, B . K . , Erean, M.T., and HemmeR, J.G., "Integration of Geometric and Mechanistic Models of N C Machining into an Open-Architecture Machine Tool Controller", Proc. IMECE, Symp. on Dynamics and Control of Material Removal Processes, DSC-Vol. 2, (Nov. 2000), A S M E , 675-682. [17] Jimenez, P., Torras, C , " A n Orientation-Based Pruning Tool to Speed Up Contact Determination between Translating Polyhedral Models", International Journal of Robotics Research Vol. 20, No.6, June 2001, 466-483 [18] Abdel-Malek, K., Yeh, Harn-Jou, "Geometric Representation of the Swept Volume Using Jacobian Rank-deficiency Conditions", Computer-Aided Design, V o l 29, No.6, 457-468, 1997 [19] Kim, Y J . , Varadhan, G., Lin, M . , Manocha, D., "Fast swept volume approximation of complex polyhedral models", Computer-Aided Design 36 ( 2004) 1013-10287. [20] Lee, J.Y., Kim, K . , " A feature-based approach to extracting machining features", Computer-Aided Design, Vol.30, No. 13, 1019-1035, 1998. [21] Lee, J., Sung J., Kim, M . , "Polygonal boundary approximation for a 2D general sweep based on envelope and Boolean operations" The Visual Computer (2000) 16:208-240 c, 2000 [22] Ling, Z K and Chase, T 1996, "Generating the swept area of a body undergoing planar motion," ASME J. Mech.Design, Vol 118, pp221-233. [23] Lazoglu, I., and Liang, S. Y . , 1996, "Feedrate Optimization on Complex Workpieces for C N C Milling Machines," ASME 1996 IMECE Symposium on the Physics of Machining Process-Ill, Atlanta G A , pp. 129-138. [24] Mounayri, H . E l , Spence, A . D., Elbestawi, M . A . , "Milling process simulation - a generic solid modeller based paradigm", ASME Journal of Manufacturing Science and Engineering, 120(2), pp. 213-221, May 1998.  Bibliography  121  [25] Montgomery, D. and Altintas, Y . , "Mechanism of Cutting Force and Surface Generation in Dynamic Milling", Transaction ofASME, Journal of Engineering for Industry, pp. 160-168, vol. 113, 1991 [26] Ozdoganlar, B., Endres, W., " A n Analytical Representation of Chip Area for Corner-Radiused Tools Under Both Depth-of-Cut and Feed Variations", Journal of Manufacturing Science and Engineering, Vol. 122, Nov. 2000. [27] Ozdoganlar, O. B., Endres, W. J., 1998, "An Analytical Stability Solution for the Turning Process with Depth-Direction Dynamics and Corner-Radiused Tooliling", Proceedings of ASME Dynamic Systems and Control Division, Vol. 64, 511-518 [28] Rao, P.N., Rao,J.S., "Towards improved Design of Boring Bars Part 1 :Dynamic Cutting Force Model with Continuous System Analysis for the Boring Bar Performance", Int. J. Mack Tools Manufacture., Vol. 28, No. 1, pp.33-44, 1988. [29] Reddy, R.G., DeVor, R.E., Kapoor, S., " A mechanistic force model for combined axial-radial contour turning", International Journal of Machine Tools & Manufactured (2001) 1551-1572 [30] Reddy, R.G., Kappor, S., Devor, R.E., " A Mechanistic Force Model for Contour Turning", Journal of Manufacturing Science and Engineering, August2000, Vol.122, 398-405. [31] Spence, A . D . , L i , Z. "Parallel processing for 2-1/2 D machining simulation", Proceedings of the 6th A C M Symposium on Solid Modeling and Applications. Ann Arbor, MI. Vol.pp. 140-148, 2001. [32] Samaranayake, P. Armarego, E.J.A. "Technological Performance Prediction Models For Turning with Rounded Corner Tools. I - Theoretical Development", Machining Science and Technology, Vol. 2, December, 1999. [33] Subrahmanyam, S., Wozny, M . , " A n overview of automatic feature recognition techniques for computer-aided process planning", computers in Industry 26 (1995) 1-21 [34] Saturley, P.V., Spence, A . D., "Integration of Milling Process Simulation with OnLine Monitoring and Control", IntJAdv ManufTechnol (2000) 16:92-99 [35] Takata, S., " A Cutting Simulation System for Machinability Evaluation Using a Workpiece Model", CIRP Annals, Vol. 38/1, pp. 417-420, 1989 [36] Tseng, Y . - J , Joshi, S.B., "Recognition of interacting rotational and prismatic machining features form 3D mill-turn parts", INT. J. PROD. RES., 1998, Vol. 36, N o . l l , 3147-3165 [37] Yang, Z., Abdel-Malek, K . , "Approximate swept volumes of N U R B S surfaces or solids", Computer Aided Geometric Design 22 (2005) 1-26.  Appendix A Circular Toolpath Tool Swept Area Construction  A . l Critical Position Calculations To construct the TSA of the circular toolpath, the outer and inner sweep envelope edges need to be generated. The corresponding critical points P/ to Pg need to be calculated.  Figure A. 1 Circular Toolpath Tool Swept Area Construction As illustrated in Figure A . l , the Swept Area is the region between the Outer Sweep Envelope and the Inner Sweep Envelope. The Outer Sweep Envelope consists of three circular edges a, e$, e$ and three straight edges e , e , e . The first three edges are generated from the 2  4  6  portion of the sweeping envelope of the tool nose curve, tool upper right point P , and tool upper c  left point Pd (See Figure 3.6). The centers of these three arcs are Oj, 0 and O3, and the toolpath 2  122  123  Appendix A. 1 Circular Toolpath Swept Area Construction  radius is R. The latter three edges come from the tool boundary edges that are tangent to the first three circular edges respectively. Similarly, the Inner Sweep Envelope consists of three circular edges ej, eg, and ep, which are generated from another portion of the sweeping envelope of the tool upper left point Pd, the tool nose curve, and the tool upper right point P . c  Given the tool geometry (r , y/ , K ), the toolpath geometry (R, O/), and according to the E  r  r  properties of common tangent and planar rigid motion, the following formulas can be established: \ o , = o,+h x  °  x  j Y  2  =Y  0j  \X  0  =X  Q  s  r  2  3  X, =X ;  0  "> 'y v  ( ->  }  A  e  r  s  x  £  2  x  i  =o X  t  Y =Y 3  0j  r  e  + A • cos(/c ) + (r +R)r  r  (  r  )  r  cos(K )  e  r  (A.6)  3  where /, =  (W-r (\  + smy/ ))  s  r  cos(y/ +K ) r  4  5^ '  sin(/r )  £  + /, • sim> )-(r +R)-  D  (A  P:  p.'  3  (A.4)  +(i? + r ) - s i n ( ^ )  2 = o , +(R + r )-sin(K ) ' Y =Y -(R + r )-cos(K )  x  p .  }  r  -(R + r )-cos(if/ )  0i  y =r  r  = (y  3  '  + sim> )))tanf> )  £  ° ~° /o o \ 3  ( A  g  • cos(y/ ) + (L-r (l  £  v.  (^r) + r simX)  +1, sinfK,) - r cos(^ )  0)  -r  c o s  X, = X  0  Y, = Y  Qi  r  + (R + /,)• cos(/c ) + r • sin(/c ) r  £  r  +(/? + /,)• sin<y ) - r • cos(/c ) r  £  r  (A.7)  124  Appendix A.J Circular Toolpath Swept Area Construction  X =X 5  -r  Qi  Y =Y 5  + (L-r (\  r  e  6  6  r  + sin (//,.))• tmy/  e  r  +L-r +R-sin(K )  Qt  X =X P:  -cosy/ +R-cosK  e  Q  - (r + R) • cosi// +(L-r (l E  r  + sin^)) • t a n ^  E  Y =Y +L-r +R-sm(iy ) 6  0j  (A.8)  r  s  r  (A.9)  r  o +o _ , 2  3  ^  v -d,  +  2  (A. 10)  P: 7  where v = (-v , n 2  d = R-, s  2  v) Ix  3  +R-cos(y/ )  X =X  +i?-sin(^ )  x  3  9  9  7  =o  x  P:  oo  Y =Y  0i  9  Qi  r  r  -R-cosOr,)  (A.ll)  (A.12)  A.2 Tool Swept Area of Partial Circular Tool path In the real turning operation, it is common that a portion of the circle tool path is encountered. To determine the tool swept area of a given tool path, the location of the given tool path needs to be identified at first, i.e., to find the given tool path belongs to which section or which combined sections with respect to the whole circle toolpath. According to the critical positions P; to Pg, and the tool geometry relationship, the corresponding critical toolpath positions A to F are calculated. Therefore, six sections of toolpath are constructed, i.e., A B , B C , CD, DE, EF and FA. Then the tool swept area is constructed by generating the outer and inner edge loops based on the boundary conditions of those sections.  Appendix A. 1 Circular  Toolpath Swept Area  a = o,  x  125  Construction  -R-cos(iy )  x  r  (A. 13)  Y =Y +R-sm( ) B  0i  ¥r  X =X +l,  cos(K ) + r sm{K )  r  B  B  7  ^Y =Y +l, B  c  r  (A.14)  sin(/c ) - r cos(/c,.)  7  X  e  r  r  =X  e  + R • sin(K-,.)  0  (A.15)  Y =Y -R-cos{K ) c  D  \X  B  =X 7  0/  r  r • cos(y/ ) + (L - r (1 + sim>,))) e  r  {  £  r  Y =Y +L-r B  X  E  =X  7  +R-cos(y/ )  0j  E  X  F  Y  F  (A. 16)  £  r  (A. 17)  Y =Y -R-sm(y, )  E  F  (y/ )  tan  0i  =X  r  +R-sm(^ )  0i  r  (A. 18)  =Y -R-cos(w ) 0>  r  One example of a circular toolpath TSA construction is presented in the follows.  Figure A.2 Tool Swept Area of T T Construction e  s  126  Appendix A. 1 Circular Toolpath Swept Area Construction  As shown in Figure A.2, the given toolpath T T is inside A B and B C sections after section e  s  identifications. It is known that outer envelope edge is ei and the inner envelop edge is e and e 6  5  in section A B and B C respectively. Therefore the swept area is the combination of the envelope edges ei, es and e<j, and tool boundary edges at toolpath start and end positions. Assume the tool points at Te are {P , Pb, P , Pd), tool points at Ts are {P \ Pi,', P ', Pd'}, they can all be a  c  a  c  calculated by using T , T , and tool construction equations (see Figure 3.6). Outer envelope e/ e  s  can be constructed as follows: Unit direction vector  « = e  ei end point P :  p -Q / yjp O \  P =T +n •r  e  e  Similarly,  e  e  =^ ~  ei start point P :  s  s  £  s  1  .  (A.20)  \  Q  P =T +n -  s  (A- 9)  e  (A.21)  r  (A.22)  £  From equations (A. 19 ~ A.22), the end positions of ei are calculated, along with (9;and R, e\ is constructed. As shown in Figure A . 2 , P is the connected point between es and e<$ and calculated in t  equation (A. 10). es is constructed by center O3, end points Pd and P,; e^ is constructed by center O2, end points P, and P . c  Finally, the TSA of T T is consisted of a list of counter clockwise edges: {e/, e , e^, e , e , e , e  s  2  4  5  6  e?, e& e }. Where e , £3 and e are the tool edges at the start position and ey, e§ and eg are the tool 9  2  4  edges at the end position, ei and e , e^ are the outer envelope and inner envelope edges. TSA of 5  other sections is constructed in the similar manner.  Appendix B Green's Theorem-Based Analytical Area Calculation  B.l Classes of Generic Tool Engagement Features (teF) Figure B . l illustrates six commonly occurring teFs that have been identified for TWE calculations. — / V:;  I  Case 2. h<d<h. f<n  Case 1. d< h  If .. QJ  W  - -  / T V .a,  / I  Case 3. h<d< h. n<f< 2rz  Case 4.d>l.f< n  / •2,  ''•C, f d  Case 5.d> I,. n<f < 2n  Case 6. Grooving  Figure B . l Classes of Generic Tool Engagement Features (teF)  127  128  Appendix B Green's Theorem-based Analytical Area Calculation  In Figure B . l , / is the distance between the successive two tool positions C,C,./. If the tool path is a straight line, / is equal to feedrate/ If the tool path is a circular edge,/is the arc length and / is the chord length. Pi is the tool nose curve upper tangent point of the tool, and P is the 2  intersection point between the two tools. Note that P can be the intersection between two tool 2  nose curves or between tool nose curve and major straight cutting edge (when / is small) or minor straight cutting edge and tool nose curve (when / is big). Identifying the types of P is 2  included in the detailed algorithm. V denotes the instantaneous feed direction that is tangent to the tool nose curve at C„ and // and l are the distances fromP,, P to V respectively. The depth 2  2  of cut d is the distance from the workpiece boundary to V . In case 1, the teF only has one region Ri and is covered by two edges. In case 2 and 4, depth of cut d is smaller than // but bigger than l , there are only R/ region in both cases. In case 3 and case 5, d is bigger than 2  hence there are  two regions in each case.  B.2 General Area Calculation Algorithm After identifying the six types of teFs, appropriated analytical equations can be formulated for each case by giving the boundary conditions of the engagement, such as the numbers of enclosed edges, properties of edges (linear or circular). All the intersecting positions and angles, which are required by these formulations and treated as the inputs to get the final results, are also analytically derived from Feature identification algorithm and will be described in Appendix C in detail.  129  Appendix B Green's Theorem-based Analytical Area Calculation  A general Green's Theorem-based area calculation algorithm is expressed in the following algorithm. teF of each case presented in Figure B . l follows this algorithm, and it is integrated and rearranged to form some fixed formulations which will be shown in the next section. Algorithm A r e a C a l c u l a t i o n INPUT:  C„C _ ,{P,},e ,n,m,kJ l  l  i  ({P,}: the set of end points, n: number of edges, m: number of circular edges, k: number of zones) OUTPUT:  A ,A ,A }  2  STEP: For region Rj,j = 1 to 2 From i = 1 to n C A S E geometry_type (e,) = LfNE Aj = I- yx'du where P  i+I  2  (x , y ), P (x , y ) i+l  i+1  t  t  i  C A S E geometry_type(e,) = A R C If center is C  t  where 0 =K + arctan(.y > . /x l  /  6 =2n 2  c  + arctan 2(y  PjCi  )  0<0, <2n  , x  Pj+iCi  )  0<0 <2n  Pj+iCi  2  If center is C,.y  where y/, =2K + arctan 2(y  PjCj  y/ = n + arctan(^p. 2  +;C/  i  ,x  PjCj  Ix  ) )  0<y/ <2n 7  0 <y/ <2K 2  130  Appendix B Green's Theorem-based Analytical Area Calculation  Aj <— Aj + Aj End End A=A +A x  2  End  B.3 Area Calculation Derivation for teF4 Detailed derivation of formulations is given to teF4, because it is the most commonly encountered engagement type. The final expressions of other types of teF are listed in the next section.  Figure B.2: teF4 Area Calculation In B.2, points C,(Xc,, Yc,) , C , _ , ( X c , Yc _ ) are the consequence tool nose center M  j  l  positions along the tool path. The interval is / = [Cj-C^y |. TWE has two regions i?,, R for cutting 2  force calculation. P ,P are the tool nose curve upper tangent points at C X  A  / 5  C  M  . P is the 2  intersection point between C, and C,-_, . Q , Q _ are the workpiece boundary positions t  t  t  131  Appendix B Green's Theorem-based Analytical Area Calculation  intersected with the tools at the two positions. The coordinates of these boundary points are PJ(XJ, yd, Qi( 5: ys), and Qi-i(x4, y ), 6/ and 62 are the angles of vector PjCi, P2C1, and \p\, y/ are the x  4  angles of vector  P 2 C / , PiG-/-  2  All the boundary conditions (point coordinates and angles) are  derivable analytically as shown in Appendix C. At first, the type of intersection point P2 needs to be identified, if the absolute value of equivalent side cutting edge angle, y/ '=y/ +a r  I ^  f  y/ ' < cos"  r  (y/ '<0) , is smaller than a critical angle r  , P is an intersection point between two curves, Figure B.2 presents this 2  r  \ eJ 2r  l ^  f  intersection type. Figure B.3 shows the critical angle y/ ' =cos  1  where P  2  r  and P4  \ sJ 2r  overlap.  Figure B.3: The Type of Intersection Point P2 If y/ ' is positive as shown in Figure B.3, curve region i?, is known to be bounded by r  edges{<?,,e ,e,}, and the close to polygonal regionR is bounded by edges{e ,e ,e ,e }. 2  2  3  4  s  6  e ,e are ]  2  132  Appendix B Green's Theorem-based Analytical Area Calculation  a portion of the tool nose curves, and e is the line segment P/C, truncated bye atP . e ,e are 3  2  3  4  6  tool straight cutting edges, and e is a portion of the workpiece boundary edge. The signed areas 5  covered by all the edges are formulated and summed up to give the total area equations. The edge e, is an arc from P to P , li  2i  which is corresponding to the angles 6 ,9 l  2  ,  illustrated in Figure B.4. / \  io.  Pi Figure B.4 Area Calculation of Edge e  x  The parametric equation of ei is:  (B.l)  133  Appendix B Green's Theorem-based Analytical Area Calculation  The area Ayy that is covered by ey is: 2  A  2  = j j - YXdu = -^Y -r Yc  cos(w) - ^ - s i n ( 2 « )  r  n  s  t  c?  2  (B.2) - r„ {4Y {cos(0 ) Cj  The edge e  2  2  -  008(69,) + re (sin(26?2 ) - sin(26?y)) - 2re (92 - 9,))  is an arc from P ,to P , , which is corresponding to the angles 2  3  y/,,y/ , 2  illustrated in Figure B.5.  Figure B.5 Area Calculation of Edge e  2  The parametric equation of e is: 2  \X = Xc _j + r cos(w) , Y = 7c,-_ + r sin(u) t  '2 •  £  7  £  y/j<u<y/  2  (B.3)  134  Appendix B Green's Theorem-based Analytical Area Calculation  The area A12 that is covered by e2 is: 2 , ¥2 ^Tc,-; cos(w) -- -sin(2w)j ¥1  2 A  = j j - yx'du = — 2  l2  4  £  {4Yc _,{cos(y/ )-cos(y/j) i  (B.4)  + r (sin(2^ ) -sin(2y/ ))-2r (y/  2  e  2  7  E  2  -y/,))  The edge es is a line from P3 to Pi. The parametric equation of e3 is:  e,=\ 3  [X = (l-u)X  3  3  1 Y = (l-u)Y  3  The area A13 covered by  +uX, , 0<u<J +uY,  (B.5)  is: c< ,j I- yx' du = —  A= 13  (Y +Y,)(X -Xj) '-^y^ — 3  (B.6)  3  The total area of region Rj is: A,  =AJJ+A +A 12  4  13  -(4\Yc (cos0 - cos 6j) + Yc _j (cosy/ i  2  t  cosy/j)]  2  (B.7)  + r [sin(26> ) - sin(2<9 ) + sm(2y/ ) - sin(2^ )] E  2  7  2  - 2r (0 -0 +Y -¥ )) £  2  I  2  y  + ^ <X + Y, )(X  1  3  3  -  X,)  For region R , since/is fairly small (f < r ), P3 and P are very close, the small arc segment 2  £  4  between them can be approximated as a line segment. The accuracy lost here is neglectable. Therefore, one straight edge e is used to represent the connection between P3 to £?<_/• As a result, 4  if QiQi-i is a linear component or its radius is fairly big compared to the feedrate, region R is 2  close to a polygonal region, and according to Green's Theorem, the area A2 is formulated as follows:  Appendix B Green's Theorem-based Analytical Area Calculation  J  n-l  *  i=0  (B.8)  -~ {X Y 2  3  135  -X Y  4  4  +X Y -X Y +X Y  3  4  5  5  4  5  -X)Y  J  5  + X,Y  -X Y,)  3  3  where P,(X,, Y,), P (X , Y ), Q (X , Y ) and Q,{X , Y ). 3  3  3  hl  4  4  5  5  B.4 Analytical Area Formulations for teFs The area calculations are derived in the same manner as shown in the previous section. The coordinates of boundary conditions are shown in Figure B.6. Boundary positions. Tool centers on the toolpath: C^Xc^Yc^) C.  (Xc YCi) h  Tool nose radius:  r  t  Intersection points:  P, (X Y,) h  P (X , Y ) 3  3  3  Workpiece boundary: Q,.,  (X4Y4)  £ (X ,Y ) ;  5  5  If Qi-iQi is a circular edge: Center: Q (Xq, Yd), radius: r  q  Figure B.6 Area Calculation of teFs  Appendix B Green's Theorem-based Analytical Area Calculation  136  The final results that are used directly in Virtual Machining system are listed in Table B . l . Other engagement characteristics, such as gravity centers, chip-side cutting edge contact length, are pre-formulated in the same method and are used the developed system, they are not listed here due to the space limitation. Also the extreme cases of the type of P , i.e., the intersection 2  between tool nose curve and straight cutting edge, is not commonly encountered if / is small, hence, only curve-curve intersection is considered in this table when/< r . As shown in Figure e  B.5, workpiece boundary segment QtQi-i can be linear or circular edge, and it may intersect with the tool nose edge or the tool side cutting edge.  Table B . l Green's Theorem-based Area Formulations for all teFs  A =  1  }  r ($  sin(^)),0 = 2cos ( ;  B  r —d )  6  (B.9)  teFl A =0 2  If QtQi-i is a linear component Aj =—— (4\Yc (cos6 4 i  teF2  2  - cos 0j) + Yc (cos^ - cosy,)] { y  2  + r [sin(2# ) - sm(26, ) + sm(2ip ) - sin(2^ )] £  2  - 2r e - e E {  2  2  i + ¥ 2  -  ¥ l  7  ) ) + - (Y + Y )(X - X )  If QjQi-i is a circular component  2  5  4  5  4  (B.10)  Appendix B Green's Theorem-based Analytical Area Calculation  A, = -~(4[Yc (cos9 i  ~cos9,) + Yc _ (cosy/ - cost//;)]  2  i  1  2  + r [sin(2c? ) - sin(26> ) + sm(2y/ ) - sm(2y/,)) E  2  y  2  —r - 2r (0 e  9, + y/  2  y/,)) +  2  * (4Y {cos((p ) q  cos(<p )  2  y  + r (sin(2(p ) - sin(2<p )) - 2r (<p - q> )) q  2  y  q  2  l  where 9i and 9 are the angles of vector 0,-C,-, P C 2  2  y/j, y/ are the  h  2  angles of vector P C,.i, Qi-iCj./, and cp,, cp are the angles of vector 2  2  Qt-iQ, QtQ. r is the radius of the circular workpiece boundary. A =0 q  2  If P? is the intersection between two curves, same as teF2 If P is the intersection between tool straight cutting edge and tool 2  nose curve, A, = -~(4[Yc (cos9 i  - cos9 ) + Yc _ (cosy/ - c o s ^ ) ]  2  1  i  1  2  y  + r [sin(26> ) - sin(2(9 ) + sm(2y/ ) - s'm(2y/,)]  teF3  £  2  y  - 2r (6 -6 +y/ -y/ y + -  (Y + Y )(X -X )  1  E  2  1  2  1  (B.ll)  2  )  2  b  2  +  b  A  Q/Qii  where Pb(Xb, Yb) is the tool nose curve right bound. AQtQi-i is the area covered by edge QtQt-i.  A =0 2  Aj =~ (4[Yc (cos0  -cos 9) + Yc _ (cosy/ -cosy/,)]  L  i  2  l  i  1  2  + r [sin(2r? ) - sm(29,) + sm(2y/ ) - sin(2^ )] E  teF4  2  -2r (9 -9, E  2  A = {X Y -X Y L  2  2  3  4  4  3  2  +y/ - )) 2  + l(Y +Y )(X  y/l  3  +X Y -X Y 4  5  5  y  4  !  + X Y,-XJ  3  5  5  (B.12)  -X,)  +  X,Y -X Y,) 3  3  Appendix B Green's Theorem-based Analytical Area Calculation  A, =—— (4[yc (cos# - cos 0/) + Yc j (cosy/ ;  4  2  i  -cos^,)]  2  + r [s'm(20 ) - six\(20 ) + s\n(2y/ ) - sm(2y/,)] E  teF5  2  -2r (9 E  l  -0,+yy -y/,))  2  L  X  3  4  - XY - XY )+ 3  3  4  Aj =——(4\Yc (cos0 i  4  teF6  5  +X Y,+X Y  Y  2  1  + | ( F + Y,)(X  2  A = (is 2  2  +X Y -X Yj  4  4  3  -X Y  5  5  QiQii  - cos 9 ) + Yc ,(cosy/ -cos (//-,)]  2  I  i  2  ,  2  -2r (0 E  2  -0  l  6  A  + r [sin(2<9 ) - sin^f?,) + sin(2y/ ) - sin(2y/ )] E  (B.13)  -X,)  3  2  +y/ -y/,)) 2  A =0 2  /  (B.14)  Appendix C Engagement Boundary Identification in Geometric and Form Invariant Features  C.l  teF Boundary Identification As described in Appendix B, for any teF shown in Figure B . l appropriate boundary  conditions have to be applied to the established equations to get the results. All the boundary conditions (point coordinates and angles) are derivable analytically and presented in this section.  (a) teF Extraction from giF  o ^,9  R,^-''  —  Machining Direction  (b) teF Extraction from fiF Figure C. 1 teF Extraction within gifl fiF  139  Appendix C. Engagement Boundary Identification  140  Figure C . l shows teF is extracted from geometric invariant feature (giF) and one example of form invariant feature (fiF). Since toolpath, workpiece boundary, and tool boundary edges can all be expressed as implicit or parametric equations, the intersections between tool successive positions, or in other words, boundary positions of a teF, can be derived analytically, and expressed as a function of C„ i.e., tool nose center position along the toolpath. For teF2 and teF3, P , Qi, Qi-i need to be calculated, for teF4 and teF5, Pi, P , P3, Qi, and 2  2  Qi.i need to be calculated. Besides, C„ C,./, d, //, and l need to be obtained for all the cases. In 2  the rest of this section, deviation of these boundary points is presented. For better understanding, the terminologies used are listed as follows: T , T : the start and end positions of a toolpath; s  e  O, R: the center position and radius of a circular toolpath; S: the length of a toolpath (curve length for circular toolpath). C Ci-i: tool successive center positions on a toolpath; h  Q , Q : the start and end positions of a workpiece boundary edge; s  e  /: distance between C and C,./; t  Q, r : the center position and radius of a circular workpiece boundary edge; q  Q Qj-f. workpiece boundary positions at one feed step; h  C.l.l  Parametric Expression of Toolpath TsTe and Workpiece Boundary QsQe. Toolpath can be a linear segment or an arc, expressed as follows,  f (l-t)T+tT 0<t<l C -< ' ]tf + [i?cos(0 #sin(7)] a,<t<a e  (C 1) 2  141  Appendix C. Engagement Boundary Identification  a, ,ct are start and end angles of the toolpath, and are calculated from T O and T O. 2  s  The parameter t is  i-At  t  e  i = 1 to n  f /  (C.2)  a, + i- Aa = a, + iWhere n is the total feed steps in a toolpath, n = ^/^, and i is the i step. t h  Any workpiece boundary position Q can be expressed as follows; t  \ Q i  (l-u)Q +uQ s  0<u<l  e  [Q + [r cos(u) r sin(w)] q  (C.3)  B <u<B  q  l  2  It must be notes that the parameter u is calculated with respect to the toolpath parameter t, and it may not be uniformly incremented with feedrate.  C.1.2  Unit Vector Expression  As shown in Figure C . l , n is the unit vector of P]C « is the unit vector of toolpath x  h  3  (instantaneous feed direction in circular toolpath). n is the unit vector of C,C,./. h ,n , and h 5  2  A  are the unit vectors perpendicular to n , n and n respectively. They are expressed as follows. x  3  5  «, = [ - c o s ^ , s i n ^ J  (C.4)  n = [smy/ , c o s ^ J  (C.5)  r  2  r  T  -T  \T. -T. n = i C,.(0 3  T  -T e  L  [-i?sin(Q, i?cos(p]  \c,(t) ~  R  s  (C.6) = [- sin(f), cos(0]  6  Appendix C. Engagement Boundary  n = 4  142  Identification  (C.7)  [-cos(7), -sin(Y)]  r -r n =•  f f \ - sin(r - — ) , cos(r - —)] 2i? 2R  5  (C.8)  where |C,C,. ,1 = / = 2R s i n ( ^ )  «6 = i  [-cos(f-—) 2R  (C.9)  •sin(f-^-)] 2R  C.1.3 Pi Calculation Pi is the tangent point between tool side cutting edge and tool nose curve at tool current position (C) on the toolpath. This position is invariant with respect to C due to the rigid tool t  geometry. It can be expressed for all the cases: P^Ct+r^n,  (CIO)  C.1.4 P Calculation 2  P is the intersection point between cutting edges of two tools at C, and C,-.y. There are 2  three cases: circle-circle intersection, circle-line intersection and line-circle intersection.  Appendix C. Engagement Boundary Identification  143  Figure C.2 Circle-Circle Intersection of P  2  For circle-circle intersection as shown in Figure C.2, P can be expressed as: 2  2 = C, - --h  p  l  5  (C.ll)  +l -n d  6  where Id is the distance from P to C,C,./, substitute Id and L , P is: 2  '  c c  2  3  r —  2  (C.ll)  C, - R s i n ( ^ - ) - n . + . r J - R sin (-^-) •w. 2  2  Figure C.3 Circle-Line Intersection of P  2  Appendix C. Engagement Boundary Identification  144  For circle-line intersection as shown in Figure C.3, P2 can be expressed as: 2 = ij-i  p  +d -n  p  2  2  = P,-l-n +d -h 5  2  =  (C.12)  2  C +r -n,-l-n +d -n j  E  5  2  2  where d is unknown. On the other hand, P is on the circle centered at C„ 2  2  \P -C \ =r 2  2  t  (C.13)  2 e  Substitute equation (C.12) to equation (C.13), it becomes: K  ~' 5  + 2 -^2\ = e  l H  d  r  (- )  2  C  14  Solving equation (C.14) to obtain two d , small one is required, it is: 2  d =-n 2  . -J(n -V) -{\V\ -r ) 2  2  V  2  2  (C.15)  2  £  Where, V = r - rij - I -n . Therefore, P is expressed as: E  5  2  P =Ci+r -n -l-n +d -n 2  E  l  5  2  (C.16)  2  Similarly, line-circle intersection of P is calculated as follows and shown in Figure C.4. In 2  which, Pb is the tangent point between tool nose curve and end cutting edge, n is the unit vector b  of PbCj, n ' is the unit vector perpendicular to n . They are expressed as follows. b  b  P =C r -n b  i+  E  n = [sin(K ), r  b  (C.17)  b  - cos(«:,.)]  n '=[cos(K ),sin(K )] b  r  r  (C. 18) (C.19)  Appendix C. Engagement Boundary Identification  145  Figure C.4 Line-Circle Intersection of P  2  P is formulated as: 2  p  2 = i+ s c  +l-n +d -n '  r  5  3  b  where  d  3  = -n ' # -  ]{n 'W) -( 2  b  b  w  W = r • n +1 • n £  C.1.5  b  5  Calculation  P is the intersection point of P/C, and the circle centered at C,./. It also can be circle-line 3  intersection or line-line intersection as shown in Figure C.5 and Figure C.6. For the first case, it can be expressed in equation (C.21). After substituting Pj and rearrange the equation, P  3  becomes equation (C.22). At the same time, P is on the circle whose center is at C/.y as shown 3  in equation (C.23).  146  Appendix C. Engagement Boundary Identification  Pl\  Figure C.5 Line-Circle Intersection of P3 P =(l-v)C +vP, 3  0<v<l  i  (C.21)  After substituting Pf. (C.22) P =C +v-r -n i  i  £  P -C„ 3  1  2 7  =r/  (C.23)  Substituting (C.22) into (C.23), the equation becomes:  (C.24) =>  \l-n +vr -n,\  = r  2  5  s  2 £  Solving equation (C.24) gives two values of parameter v, since the left side intersection with circle C,./ is required in this case, bigger v is taken to achieve this requirement as shown in equation (C.25). P =C;  +vn,  3  (C.25) v = J(hn n )  2  r  5  - L  2 c  +r  2 £  -(l-n,-n ) 5  Appendix C. Engagement Boundary Identification  147  Figure C.6 Line-line Intersection of P3 In this case, P3 is the intersection point between line P,d and Pi _iP . ti  J p , = (7-*,)<:,+*,/>,  1  2  (C.26)  Pi=Pi,i-,+hn  2  Substitute P, and Pij.i, I3 and tj can be calculated. c  t i- e +t  =C +r  r  i  l(n  -rij-l-n  s  - s i n ^ , +n  5x  +l -n  5  5y  cos y/  3  -cosy,)  2  (C.27)  -smy/  r  r  Therefore, P3 is calculated as: 3 = C +r -n  p  i  _ l(h  5x  s  ~ -n  + l -n  l  I  5  -siny/,. + n  3  5y  -cosif/ )  11 —  cos y/ -siny/,. r  2  r  (C.28)  148  Appendix C. Engagement Boundary Identification  C.1.6 Workpiece boundary Q Calculation t  Figure C.7 Workpiece Boundary Point of Q  t  From Figure C.6, workpiece boundary position Q may intersects with the straight cutting t  edge of C,, or tool nose curve of C,. The relations with C, in these two cases as shown in equation (C.26): If Qj intersects with the side cutting edge of the tool  Q =P +lq-h i  1  2  = Cj +r -Hj +lq-n s  2  If Qi intersects with the tool nose curve: Qi-Ci  =r  E  (C.26)  149  Appendix C. Engagement Boundary Identification  Together with equation (C.3), two equations have two unkowns, l and u can be solved. q  Consequently, Q, can be solved, and also expressed as a function of C,.  C.1.7 Depth of Cut d Calculation d is defined as the distance from Q to instantaneous feed direction (n ): t  3  d = \Q -C \xH +r i  l  3  £  (C.27)  Similarly, the distance between Pi, P and n are: 2  3  l =\P -C \xH +r  £  (C.28)  l =\P -C \xn +r  £  (C.29)  l  2  l  2  i  3  i  3  C.2 Recursive expression of teF boundaries  Figure C.8 Recursively Expression of Boundaries  150  Appendix C. Engagement Boundary Identification  A successive express of C, can be obtained from equation (C.l) as shown in equation (C.30), and a simplified expression is in (C.31).  [ C i  =  Cj_, + At(T - T ), line toolpath e  s  \0 + [v cos(At) - V sin(At) V cos(At) - V sin(At)\ x  y  =>  G =  where, V = [V V ] = C x  y  y  M  x  arc toolpath  \C _! + f -h , line toolpath t  I  3  0 + Nj,  (C.31)  arc toolpath  - O , tool increment At  (C.30)  f/^,  line  f/  arc  From equation (C.31), all boundary positions can be expressed as the recursive equations with respect to their previous positions.  Pij=Ci+  r e  \Cj_, + r -h; + f -h , line toolpath E  1  3  0 + r -h, + N , E  arc toolpath  t  Therefore, Pj =P _  Pij and Pu-i  ]i  ]  + f • h line toolpath 3  (C.32)  X  Pu  Y  Pj  = E, Xp,„ -cosif/j-Yp^  .sin(g)  +  i=E 2  X _,  • sin(%) + Y  Pli  _j  •  PlJ  E,=(l-cos( / )(O  cos(%)  + r -n ) + sm( / ){O  f  R  ^  ;  +r • n )  f  x  E  lx  R  y  E  ly  where,  E =(l-cos(// ){O 2  R  y  + r -H ) + sm(f/ )(O E  ly  R  x  +r • n ) E  Ix  151  Appendix C. Engagement Boundary Identification  For linear toolpath, P  2j  For circular toolpath, P  P j and Pli-l  2  =  2i  2  = P j_, + f • n  3  function{P _ ) 2i  l  (C.33)  Due to the complicity of the equation and the limitation of the space, the detailed expression will not be given in this table. For linear toolpath,  For circular toolpath P  3J  and \Xp j=F +Xp _ -cos(y^-Yp _ sm(y ) i ' y[Yp =F -Xp ^ • sm(y ) + Yp _j • 3  P3,i-1  l  3i  3J  2  1  hi  3  r  R  R  3i  Fj=(l-  cos{ / )(O  F =(I-  cos(y )(O  f  R  x  where, 2  R  y  +v-n )  circu  ]y  ar  toolpath  COS(J/ ) R  + sm(f/ )(O . +v-n ) + sm(y )(O lx  (C.34)  . s / > >-  R  y  +vn )  R  x  +v-n )  ly  lx  Using the successive expression, especially for linear toolpath, boundary positions P can t  be calculated using P .,. The computational speed is significantly increased. t  

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