COMMITTEE NEURAL NETWORK FORCE PREDICTION MODEL IN MILLING OF FIBER REINFORCED POLYMERS

Transcription

1 COMMITTEE NEURAL NETWORK FORCE PREDICTION MODEL IN MILLING OF FIBER REINFORCED POLYMERS A Dissertation by Devi Kiran Kalla M.S., West Texas A&M University at Canyon, 24 B.E., K.V.G College of Engineering at India, 21 Submitted to the Department of Industrial and Manufacturing Engineering and the faculty of the Graduate School of Wichita State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy August 28

2 Copyright 28 by Devi Kiran Kalla All Rights Reserved

3 COMMITTEE NEURAL NETWORKS FORCE PREDICTION MODEL IN MILLING OF CARBON FIBER REINFORCED POLYMERS The following faculty members have examined the final copy of this Dissertation for form and content, and recommend that it be accepted in partial fulfillment of the requirement for the degree of Doctor of Philosophy with a major in Industrial and Manufacturing Engineering. Janet M. Twomey, Committee Chair Jamal S. Ahmad, Committee Member Krishna Krishnan, Committee Member Ramazan Asmatulu, Committee Member James E. Steck, Committee Member Accepted for the College of Engineering Zulma Toro-Ramos, Dean Accepted for the Graduate School J. David McDonald, Dean iii

4 DEDICATION To My Parents and My Wife Prabhakar Rao Kalla, Farther Rajani Devi Kalla, Mother Radhika Bokka, Wife iv

5 ACKNOWLEDGEMENTS I would like to express my sincerely thanks to my advisors, Dr. Jamal S. Ahmad and Dr. Janet M. Twomey, for their invaluable guidance and support throughout this research work. I acknowledge their valuable suggestions and committed efforts towards the successful completion of my work. The past four years have been a growth experience for me. Dr. Twomey and Dr. Jamal, thanks for all supports in the past years that helped me and lifted big loads off my wife, Radhika Bokka. It is a very special honor to be Ph.D. student under you both. I would like to truly thank to my committee members, Dr. Krishna Krishnan, Dr. Ramazan Asmatulu and Dr. James Steck, for their helpful suggestions and comments. There were times when I wondered of I could have survived graduate school without the love and encouragement of my family. I wish to extend my sincere gratitude to my parents for their continuous support through the years. I would like to dearly thank for all supports from my wife that have brought me this far. Also, I am very sorry for causing her stress and rarely have time for her. I would like to thank Prashanth Lodhia, Amith Deshpande and Sunny Repaka for being good friends during my happy and difficult times. Finally, I am beholden to the almighty God for giving me strength and courage for finishing my dissertation successfully. v

6 ABSTRACT Increasingly, fiber reinforced polymers are being used in aerospace, naval, and automotive industries due to their high specific strength and high stiffness. Some of the damage problems encountered during machining these materials include: delamination, surface roughness and high rate of tool wear. Major factors that affect damage during machining in these materials are cutting forces, tool geometry, feed rate, and spindle speed. The first part of this study aims to develop an approximate mechanistic model to predict the cutting forces in the orthogonal cutting of unidirectional fiber reinforced polymers (FRP) when the fiber orientation varies from to 18. This work utilizes the mechanistic modeling approach for predicting cutting forces and simulating the milling process of fiber reinforced polymers with a straight cutting edge. Specific energy functions were developed by multiple regression analysis (MR) and committee neural network approximation (CN) of milling force data and a cutting model was developed based on these energies and the cutting geometry. Cutting force prediction models were constructed for principal and thrust cutting directions. The models are based on the specific cutting energy principle and account for a wide range of fiber orientations and chip thickness. Results from two forms of non-linear modeling methods, non-linear regression and committee neural networks, were compared. It was found that the committee neural networks provide better prediction capability of smoothing and capturing the inherent non-linearity in the data. The model predictions were found to be in good agreement with experimental results over entire range of fiber orientations from to 18. vi

7 The second part of this study dealt with an improved mechanistic cutting force model for complex tool geometry by sectioning the helical cutting edge into a stacked series of straight edge cutter segments with angular offsets and calculating the forces for each segment, then adding the forces for all segments of the cutting edge. The scope of this work is to establish a three dimensional cutting force prediction model for complex cutting tool geometry using orthogonal machining database developed in first part. The cutting forces predicted have shown a good agreement with experimental results. The third part of this study dealt with building a generalized model to predict cutting forces for any combination of process parameters such as spindle speed (n t ), feed rate (V f ), depth of cut (a e ), rake angle (α i ) and workpiece fiber layup direction ψ. Committee neural network is constructed using machining parameters chip thickness (a c ), fiber orientation angle (θ), spindle speed (n t ) and feed rate (V f ) as input variables and average specific cutting energy values, (K c and K t ) as output variables. Exhaustive experimentation is conducted to develop the model and to validate it. The training of the networks is performed with experimental machining data. Results showed that the model provides good results for unidirectional composites for all fiber orientation. The experimental results show a reasonably good fit to the predicted values, suggesting that the current approach is successful and well suitable for studying the machining of fiber reinforced polymers. Results also showed that the cutting forces are directly dependent on fiber orientation, chip thickness, rake angle, spindle speed, and feed rate. vii

8 TABLE OF CONTENTS Chapter Page 1. INTRODUCTION Fiber-Reinforced Composite Materials Research Rationale Research Scope and Objective Organization of Dissertation BACKGROUND AND LITERATURE REVIEW General Conventional Metal Cutting Theory and Analysis Force Model in Orthogonal Metal Cutting Oblique Machining Theory Review on the Machining of Fiber Reinforced Polymers Experimental Investigation of Chip Formation in Machining Unidirectional FRP s Mechanics of Chip Formation in Multidirectional Laminates Mechanics based modeling of chip formation in machining FRPs Takeyama s Model Bhatnagar et al Model Zhang Model Finite Element Models Mechanistic Models and Specific Cutting Energy Considerations Specific Cutting Energy Considerations Predicting Milling Forces using Specific Cutting Energy Predicting Milling Forces in Metal Machining Predicting Milling Forces in FRP Machining a Puw and Hocheng Model b Yadav Model Artificial Neural Networks Role of neural networks in material modeling ANN Prediction Methods Bootstrapping Train-and-Test Methodology Committee Networks Bias-Variance Dilemma and the Committee Bootstrap Committee Committee Methodology Application of ANNs in composite materials science Concluding Remarks viii

9 TABLE OF CONTENTS (continued) Chapter Page 3. ANN PREDICTION VS MR PREDICTION MODELS General Force Prediction Models Regression Model Bootstrap CN Model Prediction of Cutting Forces Results and Discussions CUTTING FORCE PREDICTION FOR COMPLEX GEOMETRY Introduction Literature on Mechanistic Cutting Force Modeling of Complex Geometry Mechanistic Model Building the Model Rake Angle Effect Experimental Work Workpiece Material Cutting Tool Geometry Cutting Parameters Cutting Forces Results and Discussion Conclusions GENERALIZED ANN PREDICTION MODEL Introduction Experimental Work Workpiece Material Cutting Tool Geometry Milling Configuration Cutting Parameters Fiber Orientation Convention Experimental Results Cutting Forces Specific Cutting Energy Generalized ANN Model Committee Neural Networks Predicting of Cutting Forces Effect of process parameters on cutting forces 132 ix

10 TABLE OF CONTENTS (continued) Chapter Page Effect of uncut chip thickness on cutting forces Effect of fiber orientation on cutting forces Effect of feed rate Effect of cutting speed Analysis of Variables CONCLUSIONS AND FUTURE WORK Conclusions Future Work.147 BIBLIOGRAPHY APPENDIX x

11 LIST OF TABLES Table Page 2.1. Values of Constants in Regression Model for K c Values of Constants in Regression Model for K t Ranges of fiber orientation angle and chip thickness obtainable from milling different laminate orientations Regression Coefficients for Equations (3.1) K c and K t committee member architecture and training parameters Comparison between CN and MR models performance Cutting Angles for Helical Tool Mechanical Properties of Work piece Material Cutting Tool Dimensions Cutting Angles Experimental Matrix for Machining Ranges of fiber orientation angle and chip thickness obtainable from milling different laminate orientations K c and K t committee member architecture and training parameters Model performance for K c and K t Values of constants in regression model for fiber orientation Values of constants in regression model for fiber orientation xi

12 LIST OF FIGURES Figure Page 2.1. Geometries of orthogonal and oblique Cutting Force diagram for orthogonal metal cutting Geometry, force and shear diagrams in oblique cutting Mechanics of chip formation in orthogonal cutting of FRPs Takeyama and Iijima s orthogonal cutting model Experimental results obtained by Takeyama and Iijima Schematic representation of the location of fiber orientation Variations of the in-plane shear strength with fiber angle from orthogonal cutting tests and Iosipescu shear tests Cutting force diagram in Region The contact between the tool node and workpiece material in Region The contact in Region Results by Zang Planing experiment of unidirectional reinforced plastics Milling Experiment for unidirectional composites Experimental and predicted forces Block Diagram of a Committee Network Schematic showing orientation of laminates used in milling experiments Variation of experimental and predicted K c with fiber orientation and chip thickness in up milling unidirectional composite laminates Variation of experimental and predicted K t with fiber orientation and chip thickness in up milling unidirectional composite laminates xii

13 LIST OF FIGURES (continued) Figure Page 3.4. A schematic for Network Committee architecture Comparison between experimental and predicted forces for o laminate at depth of cut of approximately.96 mm Comparison between experimental and predicted forces for 45 o laminate at depth of cut of approximately.76 mm Comparison between experimental and predicted forces for 9 o laminate at depth of cut of approximately.76 mm Comparison between experimental and predicted forces for 135 o laminate at depth of cut of approximately.76 mm Comparison between experimental and predicted forces for a o /45 o /9 o /135 o laminate at depth of cut of approximately.9 mm End mill with chip load elements Schematic view of helical end milling process Schematic representation of the location of effective fiber orientation Flow chart for oblique milling force prediction model procedure Schematic showing lay-up sequence of multidirectional workpiece Cutting Tool Instant force data for milling 6 to fiber direction (ψ =6 ) in the cutting conditions of speed 2 rpm and feed 46.4 mm/min Cutting Forces in milling to fiber direction (ψ = ) in the cutting conditions of speed 2 rpm and feed 46.4 mm/min Cutting Forces in milling 6 to fiber direction (ψ =6 ) in the cutting conditions of speed 2 rpm and feed 46.4 mm/min Cutting Forces in milling 6 o / o /12 o to fiber direction in the cutting conditions of speed 2 rpm and feed 46.4 mm/min 84 xiii

14 LIST OF FIGURES (continued) Figure Page Comparison between experimental and predicted forces F x for laminate at depth of cut of approximately.45mm Comparison between experimental and predicted forces F y for laminate at depth of cut of approximately.45mm Comparison between experimental and predicted forces F z for laminate at depth of cut of approximately.45mm Comparison between experimental and predicted forces F x for 6 laminate at depth of cut of approximately.45mm Comparison between experimental and predicted forces F y for 6 laminate at depth of cut of approximately.45mm Comparison between experimental and predicted forces F z for 6 laminate at depth of cut of approximately.45mm Comparison between experimental and predicted forces F x for 6 o / o /12 o laminate at depth of cut of approximately.45mm Comparison between experimental and predicted forces F y for 6 o / o /12 o laminate at depth of cut of approximately.45mm Comparison between experimental and predicted forces F z for 6 o / o /12 o laminate at depth of cut of approximately.45mm Effect of number of disks on prediction results Schematic showing orientation of laminates used in milling experiments Cutting tool Insert dimensions Up-milling configuration with a single straight edge Machine and Dynamometer setup and axes Fiber orientation angle conventions with respect to instantaneous cutting velocity 16 xiv

15 LIST OF FIGURES (continued) Figure Page 5.7. Instant force data for milling 15 to fiber direction (ψ =15 ) in the cutting conditions of speed 1 rpm and feed 11.6 mm/min F x in milling 3 to fiber direction (ψ =3 ) in the cutting conditions of speed 1 rpm and feed rate 5.8, 11.6 and mm/min F y in milling 3 to fiber direction (ψ =3 ) in the cutting conditions of speed 1 rpm and feed rate 5.8, 11.6 and mm/min F x in milling 12 to fiber direction (ψ =12 ) in the cutting conditions of speed 1 rpm and feed rate 5.8, 11.6 and mm/min F y in milling 12 to fiber direction (ψ =12 ) in the cutting conditions of speed 1 rpm and feed rate 5.8, 11.6 and mm/min F x in milling 3 to fiber direction (ψ =3 ) in the cutting conditions of feed rate 11.6 mm/min and speed 5, 1 and 15 rpm F y in milling 3 to fiber direction (ψ =3 ) in the cutting conditions of feed rate 11.6 mm/min and speed 5, 1 and 15 rpm F x in milling 12 to fiber direction (ψ =12 ) in the cutting conditions of feed rate 11.6 mm/min and speed 5, 1 and 15 rpm F y in milling 12 to fiber direction (ψ =12 ) in the cutting conditions of feed rate 11.6 mm/min and speed 5, 1 and 15 rpm Transformation of F x, F y to F c and F t Principal Forces in milling 3 to fiber direction (ψ =3 ) in the cutting conditions of speed 1 rpm and feed 5.8, 11.6 and mm/min Thrust Forces in milling 3 to fiber direction (ψ =3 ) in the cutting conditions of speed 1 rpm and feed 5.8, 11.6 and mm/min Principal Forces in milling 12 to fiber direction (ψ =12 ) in the cutting conditions of speed 1 rpm and feed 5.8, 11.6 and mm/min Thrust Forces in milling 12 to fiber direction (ψ =12 ) in the cutting conditions of speed 1 rpm and feed 5.8, 11.6 and mm/min.117 xv

16 LIST OF FIGURES (continued) Figure Page Specific Cutting Energy (K c ) in milling 3 to fiber direction (ψ =3 ) in the cutting conditions of speed 1 rpm and feed rate 5.8, 11.6 and mm/min Specific Cutting Energy (K t ) in milling 3 to fiber direction (ψ =3 ) in the cutting conditions of speed 1 rpm and feed rate 5.8, 11.6 and mm/min Specific Cutting Energy (K c ) in milling 12 to fiber direction (ψ =12 ) in the cutting conditions of speed 1 rpm and feed rate 5.8, 11.6 and mm/min Specific Cutting Energy (K t ) in milling 12 to fiber direction (ψ =12 ) in the cutting conditions of speed 1 rpm and feed rate 5.8, 11.6 and mm/min A Schematic Diagram for Network Committee Variation of experimental and predicted K c with fiber orientation in the cutting conditions of speed 1 rpm and feed 11.6 mm/min Variation of experimental and predicted K t with fiber orientation in the cutting conditions of speed 1 rpm and feed 11.6 mm/min Flow chart for calculating cutting forces Comparison between experimental and predicted forces for 3 o laminate at speed 15 rpm and feed rate of approximately 5.8 mm/min Comparison between experimental and predicted forces for 45 o laminate at speed 5 rpm and feed rate of approximately mm/min Comparison between experimental and predicted forces for 135 o laminate at speed 5 rpm and feed rate of approximately mm/min Comparison between experimental and predicted forces for 45 o laminate at speed 1 rpm and feed rate of approximately 11.6 mm/min Comparison between experimental and predicted forces for 135 o laminate at speed 1 rpm and feed rate of approximately 11.6 mm/min xvi

17 LIST OF FIGURES (continued) Figure Page Comparison between experimental and predicted forces for 45 o laminate at speed 15 rpm and feed rate of approximately 5.8 mm/min Comparison between experimental and predicted forces for 135 o laminate at speed 15 rpm and feed rate of approximately 5.8 mm/min Effect of uncut chip thickness on Cutting forces at Fiber orientation (3 ) deg cutting condition of speed 1 rpm and feed rate 11.6 mm/min Effect of uncut chip thickness on Cutting forces at Fiber orientation (12 ) deg cutting conditions of speed 1 rpm and feed rate 11.6 mm/min Effect of fiber orientation on Principal force (F c ) at variable chip thickness with cutting conditions of speed 1 rpm and feed rate mm/min Effect of fiber orientation on Thrust force (F t ) at variable chip thickness with cutting conditions of speed 1 rpm and feed rate mm/min Effect of feed rate on Principal force (F c ) at constant chip thickness (.25mm) and Fiber orientation (3 deg) Effect of feed rate on Thrust force (F t ) at constant chip thickness (.25mm) and Fiber orientation (3 deg) Effect of feed rate on Principal force (F c ) at constant chip thickness (.25mm) and Fiber orientation (12 deg) Effect of feed rate on Thrust force (F t ) at constant chip thickness (.25mm) and Fiber orientation (12 deg) Effect of Cutting speed on Principal force (F c ) at constant chip thickness (.25mm) and Fiber orientation (3 deg) Effect of Cutting speed on Thrust force (F t ) at constant chip thickness (.25mm) and Fiber orientation (3 deg) Effect of Cutting speed on Principal force (F c ) at constant chip thickness (.25mm) and Fiber orientation (12 deg) Effect of Cutting speed on Thrust force (F t ) at constant chip thickness (.25mm) and Fiber orientation (12 deg) xvii

18 CHAPTER 1 INTRODUCTION 1.1 Fiber-Reinforced Composite Materials Emerging from the aerospace industry, lightweight design and engineering has gained an increasing importance. The main objective in many industries is to improve the ratio of performance and weight. Fiber reinforced polymers (FRPs) clearly satisfy this objective. Generally, these composites are classified into three divisions by fiber length and arrangement: continuous/aligned fiber composites, discontinuous/aligned fiber composites, and discontinuous/randomly oriented fiber composites. The physical properties of fiber and matrix together with their combination and orientation determine the performance of fiber reinforced polymers. Due to the availability of a wide spectrum of fiber and matrix materials, FRPs have tremendous growth in aerospace and industrial applications. Generally, most materials, especially brittle ones, exhibit an important characteristic that a small-diameter shape is much stronger than the bulk material. This feature has been used to advantage in FRPs. The fiber, therefore, can provide the key structural properties such as high specific strength and stiffness for FRPs while the polymer matrix primarily transmits the load to fibers and protects them from harsh environment. The most commonly used fibers are carbon, glass, aramid and boron. FRPs are widely used in the transportation, aerospace and chemical industries due to their high specific stiffness and strength. Literature has shown that these materials could meet the requirements of modern technology not met by the conventional materials. Although FRPs are generally fabricated near-net-shape, it is often necessary that additional machining is carried out in order to trim the part to final dimensions and to drill holes required for assembly. 1

19 The increased use of FRP materials in manufacturing in recent years has led to an increased demand for their machining. The finish machining is of more concern than roughing because FRPs are produced almost net-shape which often requires the removal of excess material to control tolerances. This is characterized by high speeds and feed rates, small depths of cut and small uncut chip thickness. However, the weakness of composite materials lies in their susceptibility to machining damage when subjected to improper machining conditions. To minimize damage in machining, it is important to monitor process variables such as the machining forces and temperatures. In most applications, traditional metal cutting machine tools and techniques are still being used. But machining of FRP differs significantly from machining of metals and the theory and experience from metals cannot be directly applied. However, it has been reported that the strong anisotropy and inhomogeneity of FRP introduces many specific problems in machining, such as fiber pullout, delamination, surface damage, burrs and burning. Due to inhomogeneity of FRP composites, they do not machine by plastic deformation in a similar way to ductile metals, producing a uniform chip. They rather machine by fracture of the matrix and the reinforcement fibers in a powdery-like chip. Discontinuous chips are usually observed during the machining due to failure of the fiber and matrix material. Since the chip formation in cutting FRPs is complex, a better understanding of the fiber and fiber-matrix interaction effect on the chip formation is needed. The cutting forces generated during machining are often indicative of the mode by which the chip is formed and the frequency component of these forces may be directly linked to the chip formation mode [1]. Quality of the machined component is also greatly dependent on the mode of chip formation and may be correlated with the cutting forces. Cutting forces 2

20 has a strong influence on tool wear, tool breakage, machining quality and part integrity. The induced forces in machining govern the undesired effects on the workpiece and the cutting tool. A high or low degree of cutting and thrust forces generated can affect the machined surface quality and tool wear, thus indicating that the forces generated are a link between undesired machining effects and parameters of the machining process. The work of various authors [2,3,4], when reporting on milling composite materials, have shown that surface quality and delamination are strongly dependent on cutting parameters, tool geometry and cutting forces. Delamination is directly related to the cutting forces, and hence predicting and controlling the cutting forces would lead to controlling and preventing delamination. Thus a model for the cutting forces is needed for predicting machining quality. The predicted forces can be used to indicate machining quality and can help to determine and reduce the undesired effects on the workpiece as well as the cutting tool. Monitoring techniques in machining FRPs are still limited by the lack of models and sensors for chip formation analysis. The idea in neural network material modeling is to present a neural network with a series of experimental specific cutting energy states and let the neural network learn and generalize the constitutive behavior by using a back propagation type of neural network. In this study committee networks have been considered to effectively treat noisy data associated with machining FRP composites. In order to properly fit the data while training and testing, effort is needed to generalize the network to prevent overtraining of data. Generalization will occur if the network identifies features within the input, which is influenced by size, network architecture, training parameters, degree of nonlinearity and noise in the data. 3

21 1.2 Research Rationale This research effort focuses on Understanding the machinability (as assessed by cutting forces) of carbon fibers reinforced polymers when subjected to orthogonal cutting because no reliable cutting theory is available to guide the machining of fiber reinforced polymers. Since the chip formation mechanism in orthogonal machining of unidirectional FRP is very different from conventional metal machining. Hence, theories used in predicting the chip formation process in machining metals cannot be applied to FRP. A simple approach to treat this problem is by force prediction using the specific cutting energy, or mechanistic modeling. Providing a force prediction model for complex tool geometry such as helical mills and drills (i.e. non-orthogonal cutting) Studying the effect of process parameters such as rake angle, spindle speed and feed rate on the cutting forces of machined CFRP. 1.3 Research Scope and Objective The objective of this research is to study by experiment and analysis the cutting mechanisms in orthogonal machining of CFRP composite materials. The cutting mechanisms are investigated by using orthogonal milling experiments. Orthogonal cutting characteristics such as chip types, cutting forces, and surface morphology, are mainly studied for unidirectional CFRP composites. Knowledge of the cutting mechanisms will lead to better understanding of the cutting process and better designs of material and tooling systems. The specific objectives of this research are: 4

22 1. To study the effects of cutting speed, feed rate and fiber orientation on the cutting forces in orthogonal machining of unidirectional CFRP. 2. To predict, using artificial neural networks, the specific cutting energy for orthogonal machining in a wide range of feed rates, cutting speeds, fiber orientations and chip thickness. 3. To predict the cutting forces in machining unidirectional fiber composites using specific cutting energy (mechanistic modeling). 4. To extend the developed cutting models to include machining multidirectional composites and machining with cutting tools with complex geometry. 1.4 Organization of Dissertation After discussing the need for this research and specific objectives, the chapters to follow will be organized as follows. Chapter 2 describes the background of metal cutting theory and reviews the machining of fiber reinforced polymers. This chapter will present the mechanics and mechanistic modeling of FRP materials. Thorough background of Artificial Neural Network and its prediction methods is also reviewed in this chapter. Chapter 3 compares the ANN prediction model with Multiple Regression Model for cutting forces prediction in orthogonal milling. Chapter 4 discusses applying the extended oblique machining theory to predict the cutting forces in milling with a helical tool. Chapter 5 introduces a generalized ANN model for predicting the cutting forces as a function of process parameters such as cutting speed, feed rate, rake angle, fiber orientation and depth of cut. Lastly, chapter 6 presents the conclusions and recommendations for future study. Because each of chapters 3, 4 and 5 is written as a 5

23 stand alone journal paper, some repetition in the introduction and background sections in these chapters may be necessary. 6

24 CHAPTER 2 BACKGROUND AND LITERATURE REVIEW 2.1 General Although the problems associated with structural composites, physical properties and manufacturing of composites have been investigated in the past four decades, only limited studies on the machining of composite materials have been done. Few theories and investigations have been published on the mechanics of composite cutting. This chapter begins with discussion of important concepts and backgrounds relevant to this research. First, conventional metal cutting theory will be introduced. Then chip formation in orthogonal machining FRP materials will be discussed from a macro mechanics point of view. Problems related with predicting the machining of fiber reinforced composites and the existing cutting models will be discussed briefly. Finally an overview of neural networks and their prediction problems, committee networks and bias-variance dilemma and the committees is provided. 2.2 Conventional Metal Cutting Theory and Analysis Metal cutting processes can be divided into two basic categories: orthogonal and oblique cutting as shown in Figure 2.1. In orthogonal cutting, the cutting edge is straight, perpendicular to the cutting velocity and also normal to the feed direction. But in oblique cutting, cutting edge is inclined at an angle to the cutting velocity and feed direction. 7

25 Figure 2.1 Geometries of orthogonal and oblique cutting [6] a) Orthogonal cutting; (b) Oblique cutting For orthogonal cutting, the mechanics of the cutting process depends in large on the assumption that metal deformation is entirely in the plane containing the velocity and feed direction. It is further assumed that chip formation takes place by shear deformation as the material passes through the shear plane inclined to the cutting velocity vector by the shear angle (see Figure 2.2). The shear angle can be determined by experiments based on the premise that no change in volume of work material takes place before and after cutting, as represented with chip thickness [5]. Research based on chip thickness to predict shear angle is very difficult and predictions are at best good estimates when compared with experimental results. The theory of metal cutting assumes continuous plastic deformation with high strain rate under the compressive stress exerted by a wedge shaped cutting tool. The type of chip formed is dependent on the ductility, brittleness of workpiece material, and friction between the chip and the cutting tool. A continuous chip is common when most ductile materials, such as wrought iron, mild steel, copper, or aluminum, are machined. Discontinuous chips are produced when machining ductile materials at very low speeds and high feeds. 8

26 2.2.1 Force model in orthogonal Metal cutting For the orthogonal cutting process, analytical relationships may be obtained for shear and friction components of force in terms of the horizontal (principal, F p ) and vertical (thrust, F t ) force components. Figure 2.2 shows a graphical description of the cutting forces on the shear plane (F s, N s ) and frictional forces on the tool rake face (F f, N f ). In this Figure α is the tool rake angle, φ is the shear plane angle, and β is the friction angle. The cutting force components horizontal and normal to the shear plane are described by following relations: F s = F p cosφ - F t sinφ N s = F t cosφ + F p sinφ 2.1 Figure 2.2 Force diagram for orthogonal metal cutting [5]. The frictional force components both parallel and perpendicular to the chip tool interface are described by following relations: F f = F p sinα + F t cosα N s = F p cosα + F t sinα 2.2 9

27 Shear and normal stress components on the shear plane are obtained by the following relationships: τ s = σ s = F / A s N s s / A s ( Fp cosφ Ft sinφ) sinφ = 2.3 bt c ( Fp sinφ Ft cosφ) sinφ = 2.4 bt c These shear force components, or stress, depend on the shear angle, φ, width of cut, b, and undeformed chip thickness, t c. Therefore the orthogonal cutting process is determined by the obtaining the shear angle (φ). The workpiece material deforms most near the shear surface; this region is the primary deformation zone. Due to irregularity on the back surface of a chip, it is difficult to obtain the shear angle (φ) experimentally by direct measurement on the chip thickness. But, it can be obtained analytically using the geometric description. Therefore a useful shear angle has to be predicted based on assumptions. The first complete analysis in a so called shear-angle solution was presented by Ernst and Merchant [7]. They assumed the shear stress (τ s ) on the shear plane to be uniformly distributed and that the shear angle (φ) should be the angle of the maximum shear stress. Based on their assumptions, the shear angle can be predicted as: β α φ = Although the equations above do not yield qualitatively accurate shear angle predictions, but they provided an important relationship among the shear angle, the rake angle, which is the most fundamental for tool materials and the friction coefficient between the workpiece and cutting tool materials. 1

28 2.2.2 Oblique Machining Theory Various theories have been developed for analyzing the simple case of orthogonal machining where the tool has zero inclination to the cutting velocity vector. However, three dimensional cutting geometries are widely used in practical machining processes to which orthogonal theories are not directly applicable. In oblique cutting operation the cutting edge is not normal to the cutting velocity. Most analyses assume that the mechanics of oblique cutting in a plane normal to the cutting edge are equivalent to that of orthogonal cutting; hence all velocity and force vectors are projected on the normal plane [6]. The oblique cutting was first addressed by Merchant et al. [7]. In this analysis, the flow in the plane normal to the cutting edge acts as an orthogonal flow. The obliqueness of the cutting process is described by including a sliding process in the direction parallel to the cutting edge. An alternative approach to analyze oblique machining, initiated by Shaw et al. [8], considers the material flow in the plane containing the cutting velocity and chip flow normal to this plane but the normal stresses on the shear plane does not act in a direction parallel to this plane. Figure 2.3 Geometry, force and shear diagrams in oblique cutting [6] 11

29 In Figure 2.3, the angle between the shear and xy planes is called the normal shear angle φ n. The shear velocity lies on the shear plane but makes an oblique shear angle φ i with the vector normal to the cutting edge on the normal plane. The sheared chip flow angle η measured from a vector on the rake face but normal to the cutting edge [6]. Geometrical relation between the shear and the chip flow direction is given by [7]: tanη tan i *cos( φ α ) cos( α ) * tanφ n n n i = 2.6 sinφ n The above relationship defines the geometry of the oblique cutting process. Based on Amarigo et al. [9] oblique model the force components in the directions of cutting force (F p ), the thrust (F t ), and the normal (F r ) are given by the following equations F F t p τ s cos( β n α n ) + tan i tanη sin β n = bh. sinφ n cos ( φ + + n β n α n ) tan η sin β n s n n = bh. τ sin( β α ) sinφn cosi φ cos ( n β n α n ) tan η sin β n F r τ s = bh. sinφn cos( β n α n ) tan i tanη sin β n cos ( φ + + n β n α n ) tan η sin β n 12

30 Hence the corresponding cutting pressure constants (specific cutting energy) are K K pc tc τ s cos( β n α n ) + tan i tanη sin β n =. sinφ n cos ( φ + + n β n α n ) tan η sin β n =. τ s sin( β n α n ) sinφn cosi φ cos ( n β n α n ) tan η sin β n K rc τ s =. sinφn cos( β n α n ) tan i tanη sin β n cos ( φ + + n β n α n ) tan η sin β n 2.3 Review on the Machining of Fiber Reinforced Polymers FRP materials have many advantages over other materials. Major among them is the potential for weight savings as a result of a combination of high strength, high stiffness, and low density. FRP s are inhomogeneous materials consisting of two phases that are deliberately combined to form structures with desired properties. The two phases are the matrix phase and the reinforcement phase. The phases are macroscopically distinguishable with distinct boundaries between them known as interface. The role of the matrix phase is to hold the reinforcement phase, distribute the applied load and protect the composite from hostile environments such as heat, cold, moisture and corrosion. The matrix is therefore responsible for the shape, surface appearance, environmental tolerance and durability [1]. In general, a reinforced fiber provides high stiffness, high strength, and brittleness, and the matrix material provides low density, high thermal expansion, low temperature stability, and low strength [11]. However, the main advantage of FRP materials over conventional metals and alloys are their high specific strength and specific stiffness. FRP can outperform many engineering materials although there are disadvantages too, such as poor temperature performance, high raw material cost, lack of knowledge and know-how, and difficulty in 13

31 machining [12]. During the past, FRPs have been used as substitutes for other traditional materials such as wood, steel, aluminium, plastic and concrete, and new applications are continuously arising. However, new machining techniques are needed in order to expand the successful application of this promising material. As the selection of a cutting process is dependent on the ability of the process to attain the required specifications, selective reviews of published work in the machining of FRPs are highlighted to discuss the advantage and limitations of different processes. Problems in traditional machining of FRPs include fiber pull-out, delamination, fuzzing of the machined surface, and dust like chips that are health hazards. Fiber pill-outs can be reduced with proper choice of tool material and tool geometry. These problems usually occur due to excessive interlaminar stresses and heat generated during machining processes Experimental Investigation of Chip Formation in Machining Unidirectional FRPs The chip formation mechanism in orthogonal machining of unidirectional FRP is very different from conventional metal machining. In the cutting process, metals are sheared and deformed as it passes the shear plane, and then is removed in the form of a chip. However, from the literature researchers indicated that the chip formation of FRPs occurred without plastic deformation producing a uniform chip. They rather machine by shearing and fracture of the matrix and the reinforcement fibers in a powdery-like chip. The cutting process consisted of a series of brittle fractures and hence frictional force generated by the chip sliding up the tool face is negligible in the machining of the FRPs. The chips formed from machining both short and long fiber carbon composites are discontinuous types, in powder and scrap forms. This shows that the chip formation mechanism in machining carbon composites is a serial process of material fracture called brittle fracture, which reveals that 14

32 this cutting mechanism is not observed in common metals [2]. Hence, theories used in predicting the stress distribution in machining metals cannot be applied to FRP. Most of the published studies focused on experimental observation of cutting forces, chip formation, tool wear, and morphology of the machined surface. A few of the studies used theoretical and semi-empirical models to predict the force requirements in machining FRPs. This section will highlight some experimental observations while machining FRPs. The chip formation mechanisms in the orthogonal machining of unidirectional FRPs have been described in great detail by [2,3,13,14]. These works utilized linear orthogonal cutting to study the chip formation process. Their most significant findings are summarized here. While machining FRPs chips are formed by three different modes and the fluctuation patterns of forces produced are highly dependent on them. The modes of chip formation are shown in Figure 2.4 [13]. Koplev and his colleagues [2] investigated the cutting chips, machined surface, and forces in orthogonal machining of machining of unidirectional composites. That is the first effort to find the machining characteristics in the orthogonal cutting of composites. The tests were carried out when the cut direction was both parallel to and perpendicular to the fiber orientation. A quick stop device was used to investigate the chip formation process near the tool tip and in front of it. An innovative aspect of this work was the use of the so called macrochip method to handle and study the many small chips produced by the cutting process. They concluded from this work that the process of machining CFRPs consists of a series of fracture, each creating a chip. Koplev [2] also found the effects of cutting parameters on CFRP material removal. According to this work, increasing the rake angle of the tool slightly reduces the principal 15

33 cutting force, while no definite trend is found on the thrust force. They found the quality of the machined surface to be a function of cutting direction relative to the fiber orientation. It was noted that chip formation in cutting of reinforced plastics is constituted by a series of individual fractures, each of which releases a small discontinuous chip. Chip size was the longest in the orientation and decreased with increasing fiber orientation. In trimming operations where the fibers were inclined towards the cutting directions, a thin layer of redistributed matrix material was found covering the trimmed surface, obliterating details beneath. This behavior is different from the chip formation in ductile metals which occurs through continuous shear. Principal and thrust cutting forces were also monitored in Koplev s [2] study as function of fiber orientation and cutting parameters. Cutting force measurements were correlated to the chip formation and tool wear to examine fundamental relationships. In the study of the CFRP cutting mechanism using scanning electron microscopy, Kaneeda [3] proposed that in cutting laminates with fibers parallel to the cutting direction, the main chip formation mechanism is due to the breaking of the fiber matrix bond. He claimed that the chip forms in two stages as shown Figure 2.4 where the cutting tool has a positive rake angle. In the first stage of the chip formation process, the cutting tool will bend the work material layer. Then in the second stage, fibers will fracture ahead of the tool. When the cutting tool has zero or negative rake angle, fiber buckling was observed in the cutting of laminate with fibers parallel to the cutting direction. The fibers are subjected to extensive compressive stress perpendicular to the fiber ends, and will buckle in front of the tool edge. During the cutting of laminate with fibers inclined away from the cutting direction, the cutting mechanism is primarily due to the cutting of fibers for all rake angles. In fiber cutting 16

34 mode, the authors observed extensive crushed fibers on the machined surface, therefore fiber cutting mode can also be known as fiber crushing mode. Figure 2.4. Mechanics of chip formation in orthogonal cutting of unidirectional FRPs [13]. It was noted in [13] that the chip formation mechanism is highly dependent on the tool geometry and relative angle between the cutting direction and fiber orientation. The mode of chip formation is profoundly affected by the fiber orientation relative to the cutting 17

35 velocity vector and by the cutting edge rake angle. The fluctuation patterns of the cutting forces produced clearly reflect the mode of the chip formation. When cutting parallel to the fibers with a positive rake angle, the fiber-matrix interface is first fractured (peeled) in Mode I fracture and the chip slides over the rake face forming a curved cantilever beam, which later fractures due to bending stress. Mode II loading (sliding) and fracture by buckling occurs when machining parallel to the fibers with a negative rake angle. In machining composites with fiber orientations up to 75 o, and with a positive rake angle, mode II fracture occurs whereby fibers fracture by compression induced shear across the fiber axis. Interfacial sliding fracture along the fiber direction causes the chip to slide upward along the rake face. This mode of chip formation is somewhat similar to machining ductile metals by shearing along a shear plane. A mode III fracture occurs for the composites having 9 and greater fiber orientations. In this case, consecutive planes of fiber-matrix interface are subjected to in and out of plane type of shear [13]. The major forces in orthogonal machining of FRPs are cutting force (principal force) and thrust force. These are dependent on tool geometry, physical properties of the workpiece material, fiber orientation and the type of machining operation. A close relationship exists between the chip forming process at the rake face of the cutting tool and the horizontal force, and between the vertical force and the interaction of the surface of the specimen and the relief of the tool [19]. Machining conditions and cutting tool geometry have less influence on principal and thrust forces in a typical trimming operation. The cutting force increases with fiber orientation up to 75 o, then shows a greater increase while trimming 9 o fiber orientation. The principal force then decreases with further increase in fiber orientation, with sprouting decrease occurring at 15 o and 165 o orientations [13, 14]. As opposed to the 18

36 conventional forces generated in metal cutting, the thrust force is almost always higher than the corresponding principal force, except when trimming o and greater than or equal to 9 o fiber orientation. Generally, thrust forces increase with fiber orientation up to 45 o, and then decrease to 9 o [13, 14]. Higher thrust force generation is the probable effect of elastic energy release or snapping back of the deforming fibers in the contact zone [13, 14]. Cutting speed has virtually no influence on cutting forces; however, depth of cut has a considerable effect on the magnitude of the cutting force, with more aggressive cutting depths resulting in linear increase in both the principal (cutting) and thrust forces. Studies using semi-empirical and theoretical models of the machining were conducted to obtain cutting models to predict the cutting forces. It is thus important at this juncture to review some of these models and to understand their merits and shortcomings Mechanics of Chip Formation in Machining Multidirectional Laminates. Wang et al. [13, 14] studied the chip formation process in the orthogonal cutting of a unidirectional and multidirectional graphite/epoxy composite, respectively, in order to identify the influence of tool geometry and fiber orientation on the mechanisms of chip formation. They found that the chip formation mechanism varied greatly depending on the fiber angle. Due to the cutting mechanisms, the surface quality and cutting forces also dependent on fiber directions. They also noted that varying the tool geometry had an effect on cutting force. They found that an increased rake angle caused a reduction in cutting forces when cutting a material with fiber orientation. At fiber orientations up to 9, an increase in rake angle produced an increase in the thrust force. Higher thrust values were found for tools with smaller relief values, and cutting speed was found to have no effect on the forces involved. An increase in the depth of cut produced a linear increase in the forces involved. In 19

37 a further study [14] they examined the cutting process for a multidirectional laminate of the same material. The lay-up of the panel was [45 /-45 /( /9 /45 /-45 ). Contrary to the results obtained when machining unidirectional laminate, the principal cutting force was found to be greater than the thrust force for all cutting conditions and tool geometries. An empirical cutting force model was developed for machining multidirectional composites. The generalized machining force regression model obtained is based on a Gauss- Newton search and nonlinear regression with the following structure Force = C + Ci X i + Cij X i X j 2.9 i i j Where Force is the machining force, X i are factors, and C, C i, C ij are coefficients. This model predicts forces considering the experimentally proven facts that rake angle α and depth of cut t primarily influence the cutting force, and that clearance angle γ and depths of cut primarily influences thrust force. F c F t 2 = α t +.99α 7.1Vt γt = γ t +.77α 32.54αt γt 2.11 Where Fc is the cutting force, Ft is the trust force, V is the cutting velocity, and t is the depth of cut. For minimal principal cutting force, the optimal rake angle α was determined to be approximately 7 o, and the optimal tool geometry for minimizing the resultant force have 6 o, 7 o rake angles and 17 o clearance angle. They also found that the magnitude of the resultant force obtained during cutting was nearly equivalent to the summation of forces from the independent unidirectional plies as calculated previously on the basis of a rule of mixtures approach. They concluded, therefore, that a multidirectional laminates behaves like an assemblage of independent materials. In order to examine the dynamic effects and the periodicity of facture events in the chip formation process, they used a spectral analysis to 2

38 examine the finished surface profile [15]. For multi directional laminate, spectral analysis showed that no periodicity existed, which agreed with the observation that the material behaved as an assembly of independent materials, each with its own chip formation characteristics. 2.4 Mechanics based modeling of chip formation in machining FRPs The chip formation in unidirectional FRPs can be modeled similarly to the chip formation in metals. In modeling, the material properties of composites were assumed to be homogenous and plastic. A shear plane is considered to form, extending up from the tool cutting edge to the point of intersection of the free surfaces of the work and chip Takeyama s Model [16] Takeyama and Iijima were perhaps the first to develop a model for cutting forces resulting from orthogonal trimming of glass fiber reinforced plastics and using a modified Merchant s minimum energy principle [16]. They suggested that the shearing stress in the shear plane (in the chip formation zone) is a function of fiber direction. Figure 2.5 shows the orthogonal cutting model proposed by the authors. Figure 2.5 Takeyama and Iijima s orthogonal cutting model [16] 21

39 From Merchants model, the power required to the cut the material is given by: b t τ ( θ ) V cos( β α) P = V R cos( β α) = 2.12 sinφ cos( φ + β α) Where P = Cutting power V = cutting speed R = resultant force φ = shear angle α = rake angle β = friction angle on the rake face θ = fiber angle, limited to 9 θ = shear fiber angle, the angle between shear plane and fiber direction τ(θ ) = in-plane shear strength of glass FRP corresponding to the θ w = width of cut t = depth of cut By minimizing the power required to cut the material, the shear angle can be obtained by differentiating P in equation with respect to φ and equating the derivative to zero to obtain: P = φ τ ( θ ') sinφ cos( φ + β α) τ ( θ ') cos(2φ + β α) = φ 2.13 The authors then experimentally determined the in-plane shear strength of GFRP, τ(θ ), as a function of shear angle, θ, using simple shear tests. Using the empirical relationship for τ(θ ) in equation 2.13, they found the shear angle that meets the minimum 22

40 power requirement for different fiber orientation. From the predicted shear angle, the cutting and the thrust force can be obtained from: b t τ ( θ ') cos( β α) F C = 2.14 cos( φ + β α) sinφ and b t τ ( θ ') sin( β α) F t = 2.15 cos( φ + β α) sinφ Where F c is the cutting force and F t is the thrust force. The authors then compared the experimentally obtained F c and F t with that from the Eqn 2.14 and 2.15, and found that the prediction was within experimental errors. The results of [16] are shown in Figure 2.6. Figure 2.6 Experimental results obtained by Takeyama and Iijima [16]. However there are a few problems with the model: a) The model developed is only valid for fiber orientation of less than 9, and no attempt was made by the authors to predict the force requirements in machining laminates with fibers inclined towards the cutting edge. 23

41 b) Chip formation was assumed to be quasi-continuous shear process, however studies have shown that the chip formation is not continuous, thereby creating the dust like chips. Thus merchant s theory cannot be applied to the machining of FRPs. c) There is no standard technique for measuring the shear plane angle FRP materials. As the chips are generally in powder form, it is extremely difficult to measure chip thickness and so calculate the shear plane angle as is done in metal machining. d) There is no evidence whether the mean angle of friction between the chip and tool remains the same for all fiber angles. Therefore, using one mean value for all fiber angles may be incorrect. e) The authors assumed that crack growth along the fiber orientation permitted chip formation but failed to recognize that fiber fracture is required for chip removal Bhatnagar et al Model [17]. Bhatnagar et al. [17] developed the model for the cutting forces in orthogonal trimming of CFRPs, which is also based on Merchant s relationship for the power required for materials removal. They noted that the in-plane shear strength of a material played a key role during machining. Using Merchant s equations, they developed the principal and thrust force models on orthogonal cutting of CFRPs. Based on experimental observations, it was found that the shear plane in trimming negative fiber orientations was defined by the fiber angle. Similar to Koplev et al. [2] observations, Bhatnagar et al. [17] also found fiber fracture during chip generation to be dependent on the fiber orientation. Schematic representation of the location of fiber orientation during orthogonal cutting for [17] is different from other researchers as shown in Figure

42 Figure 2.7 Schematic representation of the location of fiber orientation [17] From simplification this representation has been changed to the regular representation of fiber orientation as shown in Figure 2.5 for description The Iosipescu shear test was used to determine the in-plane shear strength, which is a function of fiber orientation. 2 3 τ ( θ ) = θ +.3 θ.1 θ MPa 2.16 This model is only applicable to positive-angle fiber orientations between 15 o and 9 o, since the chip flows on a plane parallel to the fiber orientation. The cutting and thrust forces are determined from shear plane model as F c τ ( θ ) ac w cos( β γ ne ) = 2.17 sinθ cos( θ + β γ ) ne and, F t τ ( θ ) ac w sin( β γ ne) = 2.18 sinθ cos( θ + β γ ne) The variable friction angle β was calculated using the theory of metal machining, according to the equation µ F f c ne t ne = = = 2.19 tan β F n F F c sinγ + F cosγ cosγ F sinγ ne t ne 25

43 By using the methodology in reverse (i.e. inputting F c or F t, β and θ in equations 2.17 and 2.18), values for in-plane shear strength could be predicted. Figure 2.8 shows a comparison of the predicted and measured in-plane shear strength for carbon fiber composites. Some of the drawbacks of this model are, the requirement of determining in-plane shear strength by actual mechanical testing, varying the friction angle which in turn is dependent on fiber angle, the material of the workpiece material, and the tool geometry. They proposed that there was no single standard test method for the determination of the shear strength at any given fiber angle. However, the proposed approach is not valid for fiber orientations greater than 9 due to change in chip formation mechanisms. The authors assumed that the crack growth along the fiber orientations permitted chip formation but failed to recognize that fiber fracture is required for the chip removal. It is evident from figure 2.8 that the model is not valid for higher positive fiber orientation angles, as the chip formation mechanism differs drastically. Figure 2.8 Variations of the in-plane shear strength with fiber angle from orthogonal cutting tests and Iosipescu shear tests. 26

44 2.4.3 Zhang model [18] An analytical model by zhang et al. [18] predicts forces generated in cutting unidirectional CFRPs by considering chip formation caused by fractures and plastic deformation similar to metal cutting. The model holds good for fiber orientation θ ranging from to 9 in orthogonal machining. To predict forces, this model assumes the application of the principal of superposition. This model divided the cutting region, based on the deformation mechanism, into three regions, namely, chipping, pressing and bouncing and calculated the cutting forces by adding the forces from the three regions. The workpiece and cutting tool system is divided among three distinct deformation Regions, as shown in Figures 2.9, 2.1, and Figure 2.9. Cutting force diagram in Region 1. Figure 2.1. The contact between the tool nose and workpiece material in Region 2. Figure The contact in Region 3. By modeling Region 1 and referring to Figure 2.9, sinφ.tan( φ + β γ ) + cosφ F z1 = τ1hac 2.2 ( τ / τ )cos( θ φ)sinθ sin( θ φ)cosθ

45 cosφ tan( φ + β γ ) sinφ F y1 = τ 1hac 2.21 ( τ / τ ) cos( θ φ)sinθ sin( θ φ)cosθ 1 2 where θ is the fiber orientation varying from to 9, φ is the shear plane angle, γ is the rake angle of the tool, τ 1 and τ 2 are shear strengths of the work material in AC and BC directions, a c is the real depth of cut, and h is the thickness of the workpiece in yz plane. 2 are By modeling Region 2 and referring to Figure 2.1, the final cutting forces in Region F y2 = P real *(cos θ µ sinθ ) 2.22 F z2 = P real *(sin θ + µ cosθ ) 2.23 where P real is the real resultant force in Region 2, and is given by: P real = K * P 2.24 where the coefficient K is a function of fiber orientation. That is, K = f (θ), which is dependant on experimental values and µ is the coefficient of friction, so friction force is f real = P real *µ 2.25 By modeling Region 3 and referring to Figure 2.11, the cutting forces in Region 3 can be summed as 1 F y3 = r e E3 h (1 µ cosα sinα) F z3 = r 2 e E3 h cos α where α is the clearance angle of tool, µ is the coefficient of friction between the clearance face of the cutting tool and the workpiece material, E is the effective elastic modulus, and r e is thickness of Region 2. 28

46 The total cutting forces are F = F + F + F 2.28 y y1 y2 y3 F = F + F + F 2.29 z z1 z 2 z3 In this mechanics model, the parameters to be determined by experiment for a given workpiece are τ 1, τ 2, β, Ε, ν, µ, Ε 3, and Κ. The comparison of predicted and experimental cutting forces is shown in Figure Comparison between model predictions and experimental measurements when the depth of cut and fiber orientation change Comparison between model predictions and experimental measurements when the rake angle and fiber orientation change Figure 2.12 Results by Zang [18]. Figure 2.12 [18] indicate that regardless of assumptions, the prediction of variation of cutting forces with the parameters such as depth of cut, fiber-orientation and rake angle is good and has captured the major deformation mechanisms in cutting FRPs. The trends of the predicted cutting forces and thrust forces matched the experiments although the actual values varied by 27% and 37% respectively, from the experimental values. Drawbacks of this model are the requirement of properties of the material, and the high error in prediction of cutting 29

47 and thrust force. Also, the validation range for the model in terms of fiber orientation is limited to less than Finite Element Models Arola et al. [22] observed from experiments that both cutting and thrust forces registered a minimum in the 15-3 fiber angle range and increased thereafter up to 9. They have conducted a two-dimensional finite element analysis of the chip formation process for unidirectional composites. Chip separation was modeled using a critical stress criterion based on the strength of the components of the composite. The cutting forces obtained from the experiments agreed well with numerical analysis but variation of thrust force were incorrect owing to difficulties in defining the fracture plane properly on account of the simplifying assumptions made in the model and did not match with other researchers. Finite element method was used to simulate the chip formation process and predict the extent of machining damage. A two-dimensional cutting model to predict the cutting forces in relation to fiber orientations and developed an adaptive three- dimensional finite element algorithm. Nayak et al. [23] have done analysis, where the cutting forces, contact pressure and frictional shear at the tool-fiber interface in simulation of machining of UD-GFRP composite were calculated by simulation and validated with the experimental results. It was seen that both the thrust and cutting forces matched reasonably well with the experimental results. There are some analytical as well as empirical models proposed over the years to predict the cutting force. However, all the mechanisms are shown as schematic views without a detailed discussion or simulations to support the suggested chip formation mechanism. Though all the models listed here performed well within the prescribed range, none of them is universally accepted. 3

48 2.5 Mechanistic Models and Specific Cutting Energy Considerations Specific Cutting Energy Considerations Specific cutting energy is the machining power per unit volume of material removed by machining. In mathematical terms it can be written as F v P F p = c m c s = = 2.3 vac Z w Ac The specific cutting energy is defined as a quantity of energy consumed in removing a unit volume of material. Thus, it is a quantity that gives adequately accurate relationship to describe machining in energy form. The specific cutting energy in a process can be broken down into three broad elements. The specific cutting energy can vary considerably, based on the given material, its properties, and changes in the cutting speed, depth of cut, feed speed, and tool rake angle. In general, it is affected by machining conditions. However, specific cutting energy is constant at higher spindle speeds and feed rates for a given tool rake angle geometry. In machining CFRPs, specific cutting energy is higher at small depth of cut. This can be explained by the fact that small piece of workpiece material have greater strength than larger pieces. To consider the effect of machining parameters, generally constants are used to complete the relation between specific cutting energy. It can also be observed in two-dimensional, orthogonal cutting termed as size effect, which depicts the sensitivity of material to micro defects. The higher the sensitivity less is the energy required to remove thicker chip of material workpiece. In machining CFRPS, the specific cutting energy is related to the physical property and structure of the composite workpiece. High-density fiber composites require high energy. Carbon/Epoxy on the other hand is brittle and more sensitive to material defects and requires low energy [4, 2]. 31

49 2.5.2 Predicting Milling Forces using Specific Cutting Energy Predicting milling forces in metal machining The mechanistic modeling method is developed from the relationship of chip load and cutting force. The mechanistic modeling approach has been applied extensively to milling processes. Model for milling forces using specific cutting energy represents the forces as the product of the cutting pressure coefficients and cross-sectional area of uncut chips. Smith and Tlusty [26] classify these models of the milling process according to increasing levels of sophistication and accuracy. These are: 1. average rigid force static deflection (ARFSD) 2. instantaneous rigid force (IRF) 3. instantaneous rigid force, static deflection (IRFSD) 4. instantaneous force with static deflection feedback (IRFSDF) 5. regenerative force, dynamic deflection (RFDD) More details of these models have been documented in the literature [26]. The present work is based on the IRF so the review is focused on this model. The cutting force coefficients are determined by using empirical power-law relationships for cutting force coefficients in an analytical force model. Thus, accuracy of the extracted specific cutting pressure coefficients is closely related to the accuracy of the measured cutting forces. Mechanistic models are based on a theory that the machining forces are proportional to the uncut chip area. The relationship can be expressed as follows F = A ( K cosα + K sinα) x c n f F = A ( K cosα + K sinα) 2.31 y c n f 32

50 Where A c is the uncut chip area, K n is the specific normal pressure, and K t is the specific friction pressure. K n and K f are functions of process parameters, and workpiece and tool materials. They are determined for a given material from a series of machining experiments covering a specific range of process parameters. The end milling process was modeled using the mechanistic approach by Kline et al. [27]. A large number of cutting tests with different combinations of cutting conditions are required in order to determine the average coefficients in the models. To reduce the number of cutting tests Yucesan et al. [28] represented the cutting pressure coefficients as a function of tool rotation angle, and later Yucesan and Altinas [29] improved the representation as a function of both the rotational angle and the position of a cutting edge point. Shin and Waters [3] introduced instantaneous pressure coefficients, which is a function of effective chip thickness, and also developed a systematic model that can extract a range of the coefficients from a single cutting test to minimize the experimental cost of obtaining the coefficients. Feng and Meng [31] introduced a two-component mechanistic force model that considered local cutting mechanics of differential cutting edges by introducing the force coefficients K t and K r as polynomials of the axial depth of cut. The authors also comment on the size effect in metal cutting. That is, an increase in specific cutting pressure with a decrease in the undeformed chip thickness, which the authors mainly attribute to tool flank friction. Due to the empirical nature of the model, a separate a set of experiments was required to identify the numerical values of the empirical model parameters for the particular work piece cutter combination. Budak et al. [32] proposed another approach to predict cutting forces predicting the cutting pressure coefficients for different geometries based on the orthogonal cutting experiments. They suggested the mechanistic model and the relationship between force 33

51 coefficients depending on the milling experiments for cutter geometry. They showed that the force coefficients and cutter geometry can be predicted by cutting data base. Oxley [33] made a notable attempt to analytically predict cutting conditions during orthogonal machining. The foundation of his theory is the use of plane normal to the cutting edge. Using this plane, Oxley developed a model of chip formation based on the slipline field analysis of experimental flow fields. Furthermore, he used a triangular shear zone for making reliable predictions. In this theory, the shear plane angle is determined first, which is subsequently followed by prediction of forces and temperature. Lin et al. [34] extended Oxley s orthogonal cutting model [33] to oblique machining assuming stabler s flow rule and independence of cutting force and thrust force upon inclination angle. They proposed that the cutting force components in the cutting velocity and feed directions would be independent of the inclination angle and obtained these components using Oxley s orthogonal theory, considering only the normal rake angle of the tool. The third component of the cutting force was found from the constraint that the projection of the resultant cutting force on the rake face should not have any component perpendicular to the chip flow direction. The approach proposed by Lin et al. [34] also has a limited range of applicability due to using Stabler s flow rule, valid only or small rake angles and zero friction, and ignoring the effect of inclination angle in the cutting force Predicting milling forces in FRP machining a Puw and Hocheng Model Puw and Hocheng [4, 2] proposed a mathematical model for the prediction of the cutting forces in milling of FRPs using single insert end mills. This approach extends the concept of chip load to deal with the anisotropic and directional properties of composite 34

52 materials. A planing test was used to obtain the specific cutting forces under various cutting conditions. Figure 2.13 Planing experiment of unidirectionally reinforced plastics Figure 2.13 shows the planing operation, which can be considered as milling with a cutter of indefinite radius. K, K, K, K, were experimentally determined. Experiments showed cii tii c t that the specific cutting energy was dependent on chip load and cutting speed. The specific cutting energy for cutting force, K c, and for thrust force, K t were K c a b c = C V w ae 2.32 K t a' b' c' = C' V w ae 2.33 Where a e is the depth of cut (shown as t c in Figure 2.13), w is the width of cut, and V is the cutting velocity. The above equations were used together with concept of chip loading to predict cutting forces in planing of composites. The instantaneous chip thickness in milling or edge trimming was calculated by using equation 2.34, and specific cutting pressures were calculated from equations 2.32 and 2.33 a = sinθ 2.34 c a f Where a f the feed per tooth and θ is the immersion angle. 35

53 Figure Milling experiment for unidirectional composites. As shown in Figure 2.14, the local milling forces can be calculated by using local chip load composed of F X and F Y for planing. Edge trimming parallel to fiber direction: F x = FxII y cosθ + F sinθ 2.35 F F cosθ F sinθ 2.36 y = yii x Edge trimming perpendicular to fiber direction is determined by F F x y = Fx yii cosθ + F sinθ = Fy xii cosθ F sinθ Where F F xii yii = K = K cii tii w a w a c c F x = Kc w ac 2.41 F y = Kt w ac 2.42 Some of the prediction results by Puw and Hocheng are shown in the Figure

54 Figure Experimental and predicted forces [4, 2]. The experimental results agreed with the proposed model for thermoplastic carbon/peek, while further modifications are required for the thermoset carbon/epoxy. A modified model considering more directional force response can be used, since the chip formation in brittle thermoset composites is more sensitive to the variations in orientation. The prediction of cutting forces also should account for the generation of damage caused by machining. The limitation in this approach, however, was that it is only valid for the two fiber orientations tested and thus may not be extended for force prediction for other fiber orientations or multidirectional laminates b Yadav model Yadav [21] extended the work of Puw and Hocheng [4] to encompass all possible fiber orientations from to 18 o by introducing specific cutting energy as a continuous function of fiber orientation and chip thickness. He conducted the experiments with constant 37

55 feed and speed on unidirectional and multi-directional CFRP laminates in order to collect cutting force data for developing the specific cutting energy functions and verifying the force prediction model. Four fiber orientations used were, 45, 9, and 135, measured clockwise from the feed direction. The K c and K t values were predicted for instantaneous values of the immersion angle and the fiber orientation angle θ. Nonlinear multiple regression (MR) was used to fit a parametric mathematical function to the K c and K t. The regression equation used to predict the K c value is: K c (Ө,a c ) = [a c^n]*[a* sin θ +b*( sin θ )^2+c*( sin θ )^3+d*(sin θ )^4+e*( sin θ )^5] 2.43 TABLE 2.1 VALUES OF CONSTANTS IN REGRESSION MODEL FOR K c Coefficient Value a b c d e n.22 The equation was set to be the same for predicting the Kt values as: K t (Ө,a c ) = [a c^n]*[a* sin θ +b*( sin θ )^2+c*( sin θ )^3+d*(sin θ )^4+e*( sin θ )^5] TABLE 2.2 VALUES OF CONSTANTS IN REGRESSION MODEL for K t Coefficient Value a b c d e n.1 38

56 The model gave better results for various cutting conditions and milling parameters. Radial depth of cut was found to have no or little effect on the specific cutting energy. Therefore, while predicting cutting and thrust forces, it is important to take into consideration the effect of radial depth of cut and has little relevance in predicting specific cutting energy. It was found that forces are affected by both the fiber orientation angle θ and the end mill cutter immersion angle φ which in turn determine the instantaneous chip thickness. It was also found that the increments fiber orientation angle θ and immersion angle φ increment influence the magnitude of predicted specific cutting energy and therefore need to be adjusted for better results. In his work he has not considered the effect of speed and feed rate, this would result in more influencing parameters and widen the application range of the model. Since fiber orientations, 45, 9 and 135 used were limited, the present work will include small increments of fiber orientations in order to study the effects of fiber orientation in detail. Instead of MR, artificial neural work is introduced in present study. The following section will introduce neural networks and shows how well neural network performed for material modeling. 2.6 Artificial Neural Networks: A neural network model consists of an input layer, one or more hidden layers of computing neurons and an output layer of computation neurons. Artificial neural networks have been inspired by the neural architecture and operation of the brain [35]. The commonly used neural networks in material modeling are backpropagation type of neural networks. In the backpropagation, each presentation of data sets and the input values are compared with the desired output values and adaptive weights with in the network and are incrementally adjusted to minimize the output error. ANN is proven to be a promising field of research in 39

57 predicting experimental trends and has become increasingly popular as they can often solve problems much faster compared to other approaches with the additionally ability to learn [36]. These models are versatile and have the capacity to continuously learn as additional material response data becomes available. Backpropagation networks require multiple layers, which can solve complex problems. In addition, the backpropagation method uses the delta rule for training, which reduces the error rapidly [36] Role of neural networks in material modeling A neural network has several features, which make it suitable for material modeling. It has learning capability. It can acquire, store and retrieve knowledge. The learning capabilities allow neural networks to be directly trained with the results of experiments. Neural networks, which can be defined as massively parallel computational models for knowledge representation and information processing, have unique learning capabilities that can be used in learning complex nonlinear relationships, and they are also noise and fault tolerant and truly adaptive systems. These types of characteristics are found to be suitable for the material modeling task, which often deals with material behavior that contains nonlinearity, path dependency, and uncertainty. A new method to develop a constitutive material model using NN has been proposed by Ghaboussi and his co-workers [37]. Unlike the conventional modeling method, which uses mathematical formulations to represent approximately the experimentally observed behavior of materials, NN-based material modeling method offers a fundamentally different approach to modeling the constitutive behavior of materials. The NN learning capability is a very special characteristic that allow NN to be trained directly with the experimental data. During training, the NN learns the underlying constitutive material behavior present in the material data and then stores the 4

58 obtained knowledge in its connection weights. A method to extract the constitutive behavior of material from a non-uniform material test has been introduced and applied to modeling of laminated composite material [35]. A multi-layered feed-forward neural network has been proven to be suitable and effective as material models in computational mechanics. The multi-layered neural network is the most widely applied neural network, which has been utilized in most of the research related to the composites [38]. A back propagation algorithm can be used to train these multilayer feed-forward networks with differentiable transfer functions to perform function approximation, pattern association and pattern classification [39]. Neural network is proven to be a promising field of research in predicting experimental trends and has become increasingly popular as they can often solve problems much faster compared to other approaches with the additional ability to learn ANN Prediction Methods This dissertation is limited to ANN models that are used for prediction purposes only. The class of ANN employed in this dissertation and considered in this Chapter is restricted to multilayer, feedforward-sigmoidal architecture, trained by backpropagation. Prediction models are models that try to capture the functional relationship between a set of input (predictor) variables and a set of output (response) variables. ANN models develop a mapping from the input variables to the out put variables through an iterative learning process with little or no prior assumption of model form. Like nonparametric regression; multilayer, feed forward networks are a new class of prediction models. Actually, they belong to the same category with nonparametric estimators of unknown mappings, where a particular form of the function f: x y describes the 41

59 relationship between x and y, thus y can be determined from any given value of x. Twomey and Smith [4] have shown that the use of sigmoid logistic function (for multilayer, feedforward networks) can map any continuous function, given a sufficient number of hidden nodes and examples Typically, statistical (nonparametric regression) prediction and neural network prediction are measured in terms of a mean squared error (MSE) of a the targeted function, y=g(x), and of the ANN approximation function, fˆ ( Tn, x), respectively. The goal is to minimize this MSE, which consists of the sum of two terms, bias 2 and variance. Since the dependent function g(x) is usually unknown, g(x) is often defined as the expected value of the output y corresponding to any input x, E[y x]. Therefore, the mean squared distance or error between fˆ ( Tn, x) and E[y x] is given by: Mean squared error bias 2 - the squared error between the expected E[(E[y x] fˆ ( Tn, x) ) 2 ] = fˆ ( Tn, x) and the targeted function: ˆ (E[ f ( Tn, x) ] E[y x]) 2 + variance - variance of approximating function ˆ( T, x) error: E[ fˆ ( Tn, x) E[ fˆ ( Tn, x) ] 2 ]. From preceding expression, we can notice that the dilemma has been formed according to: first, nonparametric models require large amount of data to insure convergence and to control variance. The problem with large amount of data is slow convergence rate, and large sample sizes are rarely found in the real world. Leaving us to deal with small sample sizes, in which the estimators may become too dependent on the particular sample resulting in high variance of the estimators. Second, since real world models are unknown and difficult to identify, a good model that fits data well and causes significant reduction in f n 42

60 bias is hard to come up. However, a highly parameterizing model may fit data too close and result in variance increase. In order to reduce both bias and variance while guarantee convergence, a statistical nonparametric estimator (including ANN estimator) requires good model structure and large numbers of observations. The following methodologies try to reduce both bias and variance Bootstrapping: The bootstrap method of error estimation was originated by Efron [41]. It gives the nonparametric maximum likelihood estimate of the excess error of a prediction model that corrects for the bias of the apparent error. Unlike cross-validation data sets, bootstrap data sets are generated by resampling Fˆ with replacement, whereas cross-validation does it without replacement. Bootstrap samples are produced as follows. Let Fˆ be the empirical distribution function where random sample T n is drawn with equal probability mass n 1 on t 1, t 2,, t n ; t i =(x i,y i ). And, let * T n be a random sample of size n drawn iid with replacement from Fˆ, where t i * is a single random observation. If an observation is taken more than once, probability mass n 1 times the number of being selected is assigned to that observation. The true error estimation is obtained through the independent bootstrap training sets T* 1, T* 2,, T* B ; where B is the total number of bootstrap samples, each created as previously ˆ * described. The prediction model f ( T b, x ) is constructed according to each T* b. The first i term of the bootstrap estimate of true error is the resubstitution error of the application model. The second term is the difference of the average error over the original sample and the average error over the bootstrap sample (mean apparent error), summed over all bootstrap 43

61 validation models divided by B. The bootstrap error estimate is given as following (Twomey and Smith [4]): n B n n b b Err ˆ * 1 * * * BOOT = L[ yi, fˆ( Tn, xi )] + L[ yi, fˆ( T, xi )] n B n n L[ yi, fˆ( T, xi )] i= 1 b= 1 i= 1 i= For ordinary bootstrap (described in this section), B validation model plus one application model (trained on all original n observations) are constructed; where B value traditionally starts from a minimum of Train-and-Test Methodology The train-and-test method is the most commonly employed method of neural network evaluation. It obtains an estimate of true error from an independent set of data not used to construct the prediction model. This method is often referred to as cross validating the network ; where the test set error represents the true error of the prediction model. Subdividing a given sample of size n into two sub-samples such that n 1 +n 2 =n. One sample of size n 1 is used to build the prediction model fˆ( Tn 1, x), and another sample of size n 2 is used to validate the trained network. The train-and-test estimate of true error is the error over the validation set of size n 2 (Twomey and Smith [4]): Eˆ rr n 1 = L[ y, fˆ( T, x )] 2.46 T / T n2 j n1 j j= n1 + 1 Train-and-test method, unlike the resubstitution and resampling, does not use all available n data for constructing the application network. It constructs the network from a reduced data set of size n 1 ; where only one model is built Committee Networks It is believed that a combination of many different predictors may improve overall predictions. In recent years, a committee network has been one of the focused subjects in the 44

62 ANN community. A committee network (CN) is composed of several individually trained neural networks. These individual networks can be trained either on the same data set or different subsets of data. The outputs of the individual networks are usually combined according to some rules to produce the final output of the CN. Common approaches such as simple averaging or weighted averaging have been used. The networks in the committee can be either trained using the same data set or different subsets of data. The outputs of these committee members are normally combined using a fuser, which is established from a certain rule such as the simple average and weighted average. Figure 2.16 demonstrates a typical committee network. Network 1 f 1 (X) Out put f (X) Network 2 f 2 (X) Input Vector X Fuser (Σ) Network B f B (X) Figure 2.16 Block Diagram of a Committee Network The committee s output is the weighted sum of the committee member s outputs, which is written as B f ˆ ( X) = α f ( X), 2.47 i= 1 i i 45

63 where f i (X) is the output from committee member i, and α i is the corresponding combinationweight, i = 1,2,.,B. The terms α i are chosen such that the error of the committee is minimized. The simplest way to determine α i is to use the simple average method (SAVG), which assumes that all the committee members are equally important. The committee s output based on the SAGV is written as B f ˆ ( X) = 1/ B( f i ( X)) 2.48 i= 1 Some researchers have demonstrated that CN can perform better than a single network. Twomey and Smith [4] considered CN s derived from the resampling validation networks. They showed that using a CN approach by fusing all the resampling validation networks outperformed the single network that was developed using reduced data set Bias-Variance Dilemma and the Committee The concept behind building an accurate predictor is to be able to optimally balance the tradeoff between bias and variance (that contributing to the final prediction) to reduce the total error. According to Geman, Bienenstock, and Doursat, [42] the mean squared error of a statistical or ANN estimator can be broken down into two parts: bias 2 and variance, given as following E[( fˆ( Tn, x) E[ y x]) ] = ( E[ fˆ( Tn, x)] E[ y x]) + E[( fˆ( Tn, x) E[ fˆ( Tn, x)]) ], " bias " "variance" where fˆ ( Tn, x) is the empirical estimator built from the training set T n ={(x 1, y 1 ),, (x n, y n )}; and E[y x] is the true estimator that we try to emulate. An empirical estimator fˆ ( Tn, x) is believed to be a biased estimator of the true estimator E[y x], if (based on the entirely possible population of T n ) the mean of fˆ ( Tn, x) is different from the mean of E[y x]. 46

64 47 However, the empirical estimator ), ( ˆ x T f n is considered to yield high variance, if (based on giving a different training set of T n ) an output of ), ( ˆ x T f n significantly differs from other outputs of ), ( ˆ x T f n. From equation (2.49), we can notice that both bias and variance can give the empirical estimator poor performance. Typically, decreasing the prediction error is usually achieved by reducing variance caused by an excessive number of parameters in the ANN predictor [43]. [43] Viewed the committee predictor as a linear combination of different ANN network predictors; where bias 2 and variance of the committee can be formulated as following: Bias 2 [ ), ( ˆ x T f com ] = 2 1 ] [ )], ( ˆ [ = x y E x T f E K k k k k α. 2.5 And the variance: Var[ ), ( ˆ x T f com ] = = = )], ( ˆ [ ), ( ˆ K k k k K k k k k k x T f E x T f E α α = ( ) = K k k k k k k x T f E x T f E )]), ( ˆ [ ( )], ( ˆ [ α + ( ) ' ' ' ' ' ' )]), ( ˆ [ )], ( ˆ [ ( )], ( ˆ ), ( ˆ [ k k k k k k k k k k k k x T f E x T f E x T f x T f E α α 2.51 [43] indicates that if the simple average decision combining is used to form the committee, then the first term of (2.49) becomes: = = = = K k K k k k K k k k x T f E x T f E x T f E )], ˆ( [ )], ( ˆ [ )]), ( ˆ [ ( α, which exactly makes (2.49) equal to the bias 2 term in (2.5). As the result, the bias 2 term of the committee predictor is equal to the bias 2 term of a single predictor; Bias 2 [ ), ( ˆ x T f com ] =

65 Bias 2 [ f ˆ( T, x) ]. For the weighted average decision combining under the constraint of K k = 1 α = 1 without the constant term, the same result is still applied [43]. k Bootstrap Committee Let B be the total number of bootstrap samples. In bootstrap error estimation procedure [4]: n items are randomly sampled with replacement from the original data set of n points for each resampling process to create a (bootstrap) resampled set of data. The leftout data points are the items that have not been selected. One item or data point can be randomly selected more once. Repeat the resampling process B times. Therefore, B resampled sets of training data (size of n) can be used to construct B network members or predictors of the committee based on a particular decision-making method (simple averaging, voting, or weighted averaging), and then test on the original data set (size of n). Efron [44] suggested that the number of bootstrap replicates (B) should be between 2 and 2 in order to obtain a good, stable error prediction for the parameter estimation problem. Since the number of networks in the committee may become very large, to reduce the computational time it is necessary to control the training time and the number of networks for the optimal overall performance [43]. [43] Parmanto stated that training a large committee network may result in too much expense in terms of time, unless the networks are trained on parallel computers. From Twomey s [3] study on the estimation of the ANN prediction error using bootstrap method, the optimal number of bootstrap samples is between 2 and 3. There is no significant improvement in the generalization performance after Committee Methodology In any particular prediction problem, if the accessible data can be used to truly represent the entire population, a single predictor with the minimum error (or the best 48

66 performance) on unseen cases will be selected as the best forecasting/prediction model used to emulate the actual model or system. This is the most typical case of the model selection that selects one of many available predictors and discards the rest. In the real world, the amount of data is usually limited which may or may not represent the entire population of interest. This circumstance increases the risk of getting poor prediction performance or misestimating the true performance of the prediction model. Parmanto[43] states that the single best predictor does not always guarantee to be the right model since typically the original data must be split into two parts: training set and testing set; in which only the testing set (not used to construct the model) is used to predict the model performance. It is the better idea to use all available data to construct and evaluate the predictor under limited data [3]. Parmanto [43] indicates that neural network models can inherently suffer through high variance, where two possibilities can occur. First, some neural network models may perform better than others on different unseen data or cases. As it is known that the network model changes significantly with a slightly change in training set; as the result, different models will perform differently on different unseen data. Second, with excessive complexity in neural network models, different models may obtain different useful information about the actual system. From the two preceding reasons of Parmanto [43], one best (selected) model may not perform well on all varieties of unseen data; or it may not contain all useful information about the true model. In this research, it is impractical to use only one best model to predict the future outputs since the experimented data is sparse. Therefore, the committee methodology is used to combine the various performances and information of different prediction models together. The two studies by [36] and [45] demonstrated the 49

67 application of ANN to the critical issue of delamination in drilling FRPs. The advantages to a neural network approach are well documented the ability to model complex non-linear, multi-dimensional functional relationships without any prior assumptions about the nature of the relationships Application of ANNs in composite materials science For materials research, a certain amount of experimental results is always needed first to develop a well performing neural network, including architecture, training functions, training algorithms and other parameters, followed by training process and evaluation method. After a network has learned to solve the problems based on these datasets, new data from the same knowledge domain can then be put into the trained network, in order to output realistic solutions. The process of creating ANNs for materials research can, therefore be summarized in terms of the following stages: 1. Database collection: analysis and preprocessing of the data. 2. Training of the neural network: this includes the choice of it architecture, training functions, training algorithms and parameters of the network. 3. Test of the trained network: to evaluate the network performance. 4. Use of the trained ANNs for simulation and prediction. The greatest advantage of ANNs is its ability to model complex non-linear, multidimensional functional relationships without any prior assumptions about the nature of the relationships, and the network is built directly from the experimental data. However, the limitations of the ANN method are as follows: 1. Training data of the database should have close relationship with the predicting parameters. 5

68 2. Sufficient training data for complex ANNs are necessary. 2.7 Concluding Remarks Because of the obvious limitations of the shear plane theory approach and the wide range of possible mechanisms by which the chip formation may take place in machining FRPs, a mechanics approach to the solution of this problem over the entire range of fiber orientations is unrealistic. It has been shown that the mechanics models previously developed could only give reasonable accuracy in predicting the cutting forces for a limited range of fiber orientations (15 < θ < 75). This is because the material in this range of fiber orientations behaves during somewhat similar to homogeneous materials during cutting. A simpler approach to treat this problem is by force prediction using the specific cutting energy, or mechanistic modeling. In a mechanistic model, the cutting forces are determined as the product of the uncut chip area and the specific cutting energy. The specific cutting energy depends on the cutting conditions, workpiece properties and fiber orientation. A limited study of Yadav [21] has shown that this method predicts with reasonable accuracy the cutting forces for all fiber orientations. The prediction method could be further improved by better approximation of the specific energy function over the fiber orientation range of concern. The proposed research in this dissertation would extend the work of Yadav [21] to include more experimental data for supporting the prediction model and by using ANN approximation instead of multiple regressions. The following chapter s present results from efforts aimed at these two areas. 51

69 CHAPTER 3 ANN PREDICTION VS MR PREDICTION MODELS General In the present research, we improve the work of Yadav [21] by improving the prediction capabilities of the specific cutting energy models. The specific cutting energy model for predicting cutting forces is generated using two empirically based nonlinear fitting and approximation techniques. The first is the traditional approach of nonlinear ordinary least squares multiple regression (MR). The second is a special form of artificial neural networks, committee neural networks (CN). Prediction models from both techniques are constructed using two machining parameters - chip thickness (a c ) and fiber orientation angle (θ) as input variables and average specific cutting energies in the direction of cutting speed, K c, and normal to the cutting speed, K t, as output variables. The continuous specific energy functions are then used, in combination with the cutting geometry, to predict cutting forces in milling of unidirectional and multidirectional FRPs. Table 3.1 lists the ranges of fiber orientation angle and chip thickness that are obtained when milling four laminate configurations ψ = o, 45 o, 9 o and 135 o. This Table demonstrates that using a milling test in association with these four laminates cutting data is obtained for fiber orientations encompassing the entire possible range and for a wide range of chip thickness. The four laminate orientations were ψ = o, 45 o, 9 o and 135 o, measured clockwise from the y-axis as shown in Figure 3.1. It is noted here that a o laminate is the same as a 18 o laminate. The FRP blank is modeled here as semi-infinite plane and thus all possible laminate orientations are within the range o ψ 18 o. 1 This chapter was adapted from Sheikh-Ahmad, J., Twomey, J., Kalla, D, Lodhia, P., Multiple Regression and Committee Neural Network Force Predication Models in Milling FRP. Machining Science and Technology, 27, 11:

70 y ψ 45 ο 9 ο 135 ο Figure 3.1. Schematic showing orientation of laminates used in milling experiments Table 3.1 Ranges of fiber orientation angle and chip thickness obtainable from milling different laminate orientations. Depth of cut a e = 1.8 mm ψ θ (degrees) a c (mm) 45 o o o o (18) ο ( ο ) The goal of this work at this point is to obtain a continuous specific cutting energy function for the given material-cutting tool combination. The independent variables in this function are fiber orientation and uncut chip thickness. Therefore, all experiments were conducted at a constant spindle speed of 38 rpm, a constant feed speed of 217 mm/min, and with one cutting edge mounted to the tool. 3.2 Force Prediction Models In order to facilitate the force prediction, the specific cutting energies shown above have to be fitted to a mathematical function. In this work we use multiple regression and committee neural networks for function fitting and approximation. For simplicity, the following assumptions were made in deriving the force prediction models: - The cutting tool is sharp and straight and has a zero lead angle. 53

71 - The width of the cutting tool is greater than the thickness of the composite laminate and the chip slides perfectly on the rake face with no side spread (no lateral forces). - The heat generated from cutting does not affect the material properties of the workpiece Regression Model Nonlinear multiple regression (MR) is used to fit a parametric mathematical function to the data in Figures 3.2 and 3.3. The form of the mathematical function used was based on the physical behavior of the composite material in cutting and consisted of two product terms; a power term to account for the dependence of the specific cutting energy on uncut chip thickness (as demonstrated in Figures. 3.2 and 3.3) and a polynomial to account for the dependence on fiber orientation. A reasonable form of this function was found by trial and n error: K c, t = ac [ b sinθ + c sinθ + d sinθ + e sinθ + f sinθ ] 3.1 where b, c, d, e, f and n are regression coefficients and are given in Table 3.2. Also shown in the table are values for the correlation coefficient R 2, indicating reasonable adequacy of the regression models. Table 3.2 Regression coefficients for Eq. (3.1) K c Coefficients for K t n b c d e f R

72 Comparisons of the regression models and experimental data are shown in Figures 3.2 and 3.3 for K c and K t, respectively. It is apparent from this comparison that the regression models for K c and K t are reasonably accurate only for a narrow range of fiber orientations in each laminate. The models break away from the experimental data at the beginning of tool engagement and at exit. This is apparently influenced by the sinusoidal form of the regression model, which is profoundly depicted by model results, and has no relationship to the physical phenomena being investigated. More appropriate choices of the mathematical function would improve the prediction capability of the MR model. But this also would require more exhaustive trails. The poor behavior of the MR model in these cases is purely mathematical in nature and clearly demonstrates the limited flexibility of parametric modeling in capturing the entire trend of the experimental data. Such limitation is a known weakness in regression analysis using parametric models and may be avoided by utilizing a more capable data fitting tool such as committee neural networks Bootstrap CN Model A subset of the data used to construct the regression model was used to build and test the CN (n = 338). The total sample was randomly divided into CN training data (7%, n= 238) and test data (3%, n=1). Based on the results of [3, 33], B = 2 bootstrap samples were generated and used to build the CN. Separate CN were built for predicting K c and K t. Bootstrap CN were built according to the description provided under Committee Networks under chapter 2. Network architecture, training parameters and stopping criteria were selected through experimentation and examination of networks trained on twenty independent samples T n, for each problem, at each level of n. The network architecture used was a multi-layered fully connected perceptron with one input neuron (x), two hidden layers with ten neurons in each and one output neuron (y). A depiction of the CN used in this work 55

73 is shown in Figure 3.4. The transfer function was the Sine, and a traditional backpropagation learning algorithm was used with the learning rate of.15 and the momentum (smoothing) factor of.9. All networks were initialized to the same set of random weights between -.5 and.5. Network training ceased when the mean weight change was below a given threshold (.1) for 1 generations. The multi-layer perceptron trained by backpropagation was the basis of the committee members. Using one bootstrap sample, network architecture and training parameters were determined as shown in Table 3.3. For both models for K t and K c all committee members were trained with the identical architecture and training parameters. The results of the CN prediction of K c and K t are shown in Figures 3.2 and 3.3, respectively. The accuracy measures for both MR and CN prediction models of K c and K t are also provided in Table 3.4. For comparison purposes the MR model results are based on data that was used in building and testing the CN. The superior fitting of the experimental data by the CN prediction model is clearly evident in Figures 3.2 and 3.3, where the CN curves closely follow the variation of the experimental data in the variables space shown. Based upon the root mean square error (RMSE) and coefficient of variance (COV) of the training and testing where, the MR model RMSE is 2.5 to 8. times higher than the CN model in predicting K c and K t. The CN model predictions of K c are also better than those for K t. Table 3.3 K c and K t committee member architecture and training parameters Input layer 2 input nodes (a c,θ) Hidden layers 2 layers, 1 nodes each Output layer 1 output node (K t or K c ) Learning rule Delta Leaning rate.15 Transfer function Sine Number of epochs 3 56

74 a c (m m ) o L a m in a te 9 o L a m in a te o L a m in a te 1 8 o L a m in a te 2 K c M R K c C N K c K c (N/mm 2 ) θ i (D e g re e s) Figure 3.2. Variation of experimental and predicted K c with fiber orientation and chip thickness in up milling unidirectional composite laminates a c (m m ) K t M R K t CN K t o Lam inate 9 o Lam inate 135 o Lam inate 18 o Lam inate K t (N/mm 2 ) θ i (D egrees) Figure 3.3. Variation of experimental and predicted K t with fiber orientation and chip thickness in up milling unidirectional composite laminates 57

75 a c θ a c θ a c θ j=1 2 B Fuser B f ˆ CN[ Tn, x] = = α j f j[ Tn, x] j 1 Output (K c or K t ) Figure 3.4. A Schematic for network committee architecture Table 3.4: Comparison between CN and MR models performance Training Data (n = 238) CN MR K c K t K c K t RMSE COV (%) Testing Data 1 (n = 1) CN MR K c K t K c K t RMSE COV (%) Testing Data 2 (n = 22) CN MR K c K t K c K t RMSE COV (%)

76 Those terms defined and used in Table 3.4. to monitor the performance of both MR and CN analysis are the root-mean-square error (RMSE), coefficient of variation (COV%) and absolute fraction of variation (R 2 ), which are as follows: 1. RMSE: Root-Square-Error defined as RMSE = n m = 1 ( y n x 2 predicted, m t arg et, m) 2. Cov : Coefficient of variation in percent defined as RMSE Cov = 1 xt arg et 3.3 Prediction of Cutting Forces Once the specific cutting energy function is determined, force prediction for any laminate structure can be made. For force prediction in the milling operation of a specific unidirectional laminate structure, ψ, the cutting geometry is first analyzed and the instantaneous chip thickness, a c, and fiber orientation angle, θ i, for a given immersion angle, φ i, are determined from ψ φi φi < ψ θ i = 18 ( φi ψ ) φi > ψ φi ψ = a c = a sinφ 3.3 f i a f V f = 3.4 N n t spindle speed. Where V f is the feed speed, N is the number of cutting edges in the cutter and n t is the 59

77 The specific cutting energies are then predicted using the appropriate prediction model. The forces F c and F t are calculated from Eq The forces along the machine coordinates F x and F y are calculated from the transformation Eq Force predictions with the CN model were obtained from testing data set 2. F ( φ ) = K ( a, θ ) a ( φ) a, and 3.5 a c t t c c c c c t t F ( φ ) = K ( a, θ ) a ( φ) a 3.5 b F = F sinφ + F cosφ, and 3.6 a x c t F = F cosφ + F sinφ 3.6 b y c t For a multidirectional laminate structure, the cutting forces are determined by superposition of the cutting forces required to independently machine each unidirectional ply in the laminate structure. That is: F F m c ( i ) = [ K c( ac, θi ) ac( φi ) at ] j j= 1 φ, and 3.7 m t ( i ) = [ K t ( ac, θ i ) ac( φi ) at ] j j= 1 φ 3.8 Where a t here is the ply thickness and m is the number of plies in the laminate structure. The principle of superposition is used here under the condition that the adhesive strength between the different plies plays an insignificant role in the machining behaviour of the individual plies and that these plies will behave under the condition of milling as if they were single separate plies. The validity of such condition has been supported by the findings of [13] and [5], and is attributed to the inferior strength of the epoxy polymer as compared to the fiber material. 3.4 Results and Discussion Figures 3.5 to 3.8 show comparisons between the experimental and predicted cutting forces for the various laminate configurations used in this study. These results show that the 6

78 present force prediction models provide reasonable agreement with experimental results for all unidirectional laminates used in the present study. Because the specific cutting energy functions obtained by regression analysis do not capture the force oscillation associated with cutting fibres and machine dynamics as exhibited in Figures 3.2 and 3.3, the regression model predictions smooth the force signal, but generally follow the main trend of the force dependence on engagement angle. However, the better prediction capability of the CN model of the highly nonlinear cutting force signals is obvious. The CN models tend to more closely follow the inherent nonlinearity of the data while the regression model tends to deviate at entry and exit points. Reasons for this could be attributed to the rigidity of the mathematical regression model as it is apparent that the polynomial function is always profoundly depicted. The regression models for K c and K t exhibited great deviation from the experimental data at tool entry and tool exit, and this in turn is translated into deviations in the predicted forces in the same tool engagement positions. Such mathematical rigidity is not found in the CN models and thus their predictions are better. The cutting forces for a multidirectional laminate having a layup sequence [ /45 /9 /135 ] were calculated using Equations 3.7 and 3.8, the results are shown in Figure 3.9. Here, the number of plies m = 4, the ply thickness a t = 2.5 mm and the depth of cut is approximately.9 mm. This figure clearly shows that the CN force prediction model and the principle of superposition adequately simulate the cutting forces when milling the multidirectional laminate. The force prediction is simply additive and is independent of layup sequence, and hence it would similarly work for any layup sequence. 61

79 6 5 Forces (N) o Laminate Cutting F x F y MR CN φ i (Deg) Figure 3.5. Comparison between experimental and predicted forces for o laminate at depth of cut of approximately.96 mm 6 5 Cutting Forces (N) F x F y MR CN 45 o Laminate φ i (Deg) Figure 3.6. Comparison between experimental and predicted forces for 45 o laminate at depth of cut of approximately.96 mm 62

80 Cutting Forces (N) F x F y MR CN 9 o Laminate φ i (Deg) Figure 3.7. Comparison between experimental and predicted forces for 9 o laminate at depth of cut of approximately.76 mm 6 Cutting Forces (N) F x F y MR CN 135 o Laminate φ i (Deg) Figure 3.8. Comparison between experimental and predicted forces for 135 o laminate at depth of cut of approximately.76 mm 63

81 14 12 Multiply Laminate Forces (N) Cutting Fx Fy MR CN φ i (Deg) Figure 3.9. Comparison between experimental and predicted forces for a o /45 o /9 o /135 o laminate at depth of cut of approximately.9 mm A mechanistic force prediction model based on CN and regression models of specific cutting energy functions was developed. The specific cutting energy functions utilized in this model encompass the entire range of fiber orientations possible in a composite laminate, and hence the model can be widely applicable. The CN model was shown to outperform the standard form of regression modeling by smoothing the noisy data and capturing the inherent non-linearity in the experimental data. The CN model accuracy was 2.5 to 8 times better than that of the MR model. The force prediction model is capable of predicting forces in milling unidirectional composites. The model could be further extended to other machining processes such as turning, drilling and milling with a helical tool. Modeling in this work was conducted at the workpiece material-tool property level (i.e., specific cutting energy) rather than for process specific output (i.e., cutting forces, 64

82 delamination). This lends the current methodology the flexibility to be applied to other cutting processes involving the workpiece-tool pair. A required condition is that analysis of the cutting geometry can be made in terms of the uncut chip thickness and fiber orientation angle for each ply in the laminate structure. This methodology is rather generic and straightforward for straight edge tools such as in drilling, turning, planning and milling with a helix angle cutters. The procedure can be extended to more complex geometries such as non-zero helical cutters by sectioning the helical cutting edge into a stacked series of straight edge cutter segments with angular offsets and calculating the forces for each segment, then adding the forces for all segments of the cutting edge. One obvious limitation of the current model is that it is valid only for one rake angle and one cutting speed. The model can be easily extended to include the effect of rake angle and cutting speed by including cutting data for different levels of these parameters. 65

83 CHAPTER 4 CUTTING FORCE PREDICTION FOR COMPLEX GEOMETRY 4.1 Introduction Advances in the aerospace industry have created several engineering challenges which include the introduction of complex geometry and difficult to machine composites. Machining is a major manufacturing process in the fiber reinforced polymer industries. Because of the increasing complexity of machined part geometry; use of complex milling tools has become widespread. These cutting tools are used to make components to the desired size and shape and they include helical mills, burr tools, compound helical mills and various types of drills. Sawing, planning, milling, drilling and turning etc. are some of the most frequently used machining processes. All of the processes have a common aspect that they all involve a single or multiple teeth. For a straight tooth cutter, with no lead angle, there is likely to be no axial force component. However, a force normal to the cutting edge is present. Oblique cutting is the basis for understanding many machining operations such as milling with a helical cutter. In this chapter orthogonal force model developed in the previous chapter is used to obtain the cutting forces during helical milling. Mechanistic neural network cutting force prediction model for straight edge milling was derived in the previous chapter 3. Prediction of cutting forces for machining processes has gained increasing attention. This is because of the fact that cutting forces are key factors that determine the machining parameters that affect the stability of machining operations, influence tool-workpiece vibrations, cutting tool failure, dimensional accuracy and finish of the workpiece and stability of the machine tool itself. Cutting forces acting on the milling cutter are one of the most 66

84 important physical output variables that give significant machining process information. In milling, cutting forces directly affect the product quality and the process efficiency. Excessive cutting forces result in low product quality, serious damage such as surface ply delamination and through-thickness edge cracking can occur at the machined edges. While small cutting forces often indicate low machining efficiency. Therefore it is important that the cutting force be maintained close to optimum value during the machining process. The cutting forces are directly influenced by several machining parameters, such as depth of cut, feed rate, as well as the cutting tool design and the workpiece material. Based on the surface finish requirements of the machined part, the cutting force needs to be optimized to maintain the process efficiency by selecting optimum machining parameters for a certain cutter-part combination. Hence, developing a reliable cutting force predictive model is critical in for process planning and optimization of process parameters in milling operations. The scope of this chapter is to establish a three dimensional cutting force prediction model for complex cutting tool geometry using orthogonal machining database developed in previous chapters. The results are compared with results obtained from other oblique cutting forces and available experimental data. The methodology used for straight edge cutting tool is extended for complex geometries such as helical cutters. This involves sectioning the helical cutting edge into a stacked series of oblique edge cutter segments with angular offsets and calculating the forces for each segment, then adding the forces for all segments of the cutting edge. This methodology could be applied to other cutting tool geometries such as compound-helix tools and inserted tools. 67

85 4.2 Literature on Mechanistic Cutting Force Modeling of Complex Geometry Of the different force modeling techniques in developing the cutting force models, mechanistic modeling method is the most robust, simple and efficient technique that is widely used in industry and current research studies. Mechanistic modeling has gained popularity due to its simplicity involved in building a force model with this approach. Such models attempt to correlate machining cutting forces to the in-process chip geometry by way of experimentally determined cutting constants. The underlying assumption behind the mechanistic methods is that the cutting forces are proportional to the uncut chip area. However, there are numerous problems yet to be overcome in creating such a force modeling system. One such difficulty is the representation of complex tool geometry. Some of these geometries are flat, ball nose, helical end mill etc. A mechanistic model for the prediction of cutting forces in end milling was introduced by Kline et al. [27]. This model was used to study problems of cornering and forging cuts. The mechanistic model was based upon the relationship between the cutting forces and the chip load. In their model, the cutter is divided into a stack of thin disk elements along the axis. Each disk element has a finite thickness. The cutting action of each disk is treated as orthogonal element machining to calculate the forces. It was stated that the forces are proportional to the chip load area on the end milling. The authors determine the overall cutter forces by computing the vector sum of instantaneous forces in incremental slices of a helical cylindrical end-mill. Figure 4.1 shows the end mill with the cutting edge divided into chip load elements. They used average pressure coefficients, which are functions of feed, radial and axial immersion. Based on Kline et al. [27], elemental cutting forces for 68

86 the i th axial disk element, the j th angular position of the cutter, and the k th flute, acting on cutting edge element can be expressed as df T = K T D z f sin[ β ( i, j, k)] df = K df 4.1 R R T where f is the feed per tooth, D z is thickness of the elemental disk, β(i,j,k) is the angular position of this particular disk in the cut, and K T and K R are the empirical cutting parameters (cutting pressure coefficients) which relate the elemental cutting forces to the undeformed chip geometry on the elemental cutting edge in tangential and radial directions, respectively. K T and K R values are obtained from the experimental measurements of cutting forces. The tangential and radial elemental forces are resolved into the external x, y coordinate system is by following equations df ( i, j) = ( df x N f k= 1 R sin( β ) + df T cos β ) N f dfy ( i, j) = ( df k = 1 R cos( β ) + df T sin β ) 4.2 The average forces acting along x and y directions are given by F x = N f N z N θ k = 1 i= 1 j= 1 K T D z f cos[ β ( i, j, k)]sin[ β ( i, j, k)] K N θ R K T D z f sin 2 β ( i, j, k) 4.3 F y = N f N z N θ k= 1 i= 1 j= 1 K R K T D z f cos[ β ( i, j, k)]sin[ β ( i, j, k)] + K N θ T D z f sin 2 [ β ( i, j, k)] 4.4 where N θ is the total number of angular increment for a rotation of the tool equal to γ, N D is the number of cutting edges and N Z is the number of elementary plates in the axial direction. It is noted here that due to the assumption of orthogonal machining elements, this model is not capable of predicting the axial force component. 69

87 Figure 4.1 End mill with chip load elements [27] Lin et al. [34] proposed a simple semi-empirical method for predicting the cutting forces during oblique cutting based on Stablers s rule for chip flow direction and by ignoring the effects of the inclination angle on the force components F c and F t. The cutting force components F c and F t are assumed to be independent of inclination angle, and thus can be determined from the orthogonal machining data for the same cutting geometry (i.e. undeformed chip thickness and rake angle). Then, by using these two force components, the third component F a can be found for the given inclination angle, i by assuming that the resultant cutting force must lie in the plane normal to the cutting face and containing the resultant frictional force acting in the chip flow direction. The model for predicting cutting forces is based on the assumptions: (1) the cutting and feed forces can be determined from orthogonal machining by making (i = ), irrespective of its actual value and by making (α = α n ) (2) the chip flow direction can be determined using Stabler s flow rule (η = i). 7

88 The velocity radial rake angleα, and the normal rake angle, α are related by: r tanα = n tanα r cosη 4.5 where η is the chip flow angle and α n is the normal rake angle. By applying vector analysis, the final result is n F a = F (sin i cosisinα tanη) df c n sin i sinα tanη + cosi n t cosα tanη n 4.6 Armarego and Whitfield [9] developed a cutting force model for end-milling based in the undeformed chip thickness concept and circular tool path by using classical oblique cutting analysis, which incorporates the tool geometry to calculate the cutting force in the cutting, thrust, an lateral directions, based on the orthogonal machining experimental data. Budak et al. [32] used the oblique cutting model for milling force prediction using a unified mechanics approach. This predictive approach for milling force relies on an experimentally determined orthogonal database, incorporating the tool geometrical variables, and milling models based on a generic oblique cutting analysis. This method allowed determining the milling force coefficients without additional calibration milling tests, and therefore it is very practical for optimal tool design and process planning. This method eliminates the need for the experimental calibration of each milling cutter geometry which is needed for the mechanistic approach to force prediction, and can be applied to more complex geometry. This would minimize the cost of prediction models. In the force analysis of the milling operation, starting from the basic force model, the analysis can be successively extended for geometric features such as multi tooth, axial and radial rake angles, helical cutters etc. 71

89 Li et al. [53] developed a theoretical model for cutting forces in helical end milling based on Oxley s predictive machining theory [33] and using the approach by Lin et al. [34]. The model predicted the milling forces from input data of workpiece material properties, tool geometry and cutting conditions. This was done by discretizing the helical end milling cutter into a number of slices along its axis to account for the helix angle effect. The cutting action of an individual tooth within each slice was modelled as oblique cutting with a cutting edge having an inclination angle that is equal to the helix angle of the cutter. The cutting forces in the oblique cutting are predicted in a similar way to that discussed in [34] and equation 4.6. The article presents an implementation of the theory in a computer program and the simulation results substantially agree with experimental results. 4.3 Mechanistic Model In the present study, a model for cutting forces in helical milling has been developed based on methods of Lin et al. [34] and Li et al. [53]. This is done by dividing the helical end milling cutter into a number of disks along its axis to account for the helix angle effect. The cutting action of an individual tooth within each slice is modeled as an oblique cutting edge with a cutting edge having an inclination angle equal to the helix angle. In the present approach, the cutting force coefficients K c and K t are assumed to be independent of inclination angle and thus can be directly identified from the orthogonal milling tests in chapter 3. Once the specific cutting energy coefficients are calculated from the orthogonal machining data base, the elemental cutting forces df c and df t in the oblique cutting of each disk are calculated by equations 4.1 and The elemental axial force df z is calculated using equation (4.6). The elemental forces are then resolved into normal df x and feed df y directions by equations 4.14 and The total forces in the end milling are then calculated 72

90 from the sum of the forces on each tooth segment at every slice. Experimental milling tests have been conducted to verify the prediction model. Details of the predictive model are described below. For simplicity, the following assumptions were made in deriving the force prediction model: The workpiece is rigidly clamped on the machine table. The deflection of the tool caused by the cutting force is neglected. The cutting tool is sharp. Ignoring the effect of inclination angle on the cutting forces. Zero friction Buidling the Model An end mill with helix angle of i, diameter of D, and N number of flutes is assumed. Figure 4.2 shows a schematic of the helical mill sliced into a number of disks. The force directions are clearly indicated in the figure. The two cutting force components df c and df t are determined from the orthogonal machining database as explained earlier. Then, by using these values of F c and F t, the third component F a can be found for the given value of i and η as described by equation 4.6. Assuming that the bottom end of one flute engaged in the workpiece is designated as the reference immersion angle φ, the bottom end points of the remaining flutes are offset at angles φ n () = φ + n φ p ; n =, 1 (N-1) 4.7 where φ p = 2π/N At an axial depth of cut z the lag angle for a particular disk j is given by 73

91 ψ j = (z tan i)/r 4.8 The immersion angle for disk j on flute n at axial depth of cut z therefore is given by φ j () = φ + n φ p - ψ j 4.9 Z j th Disc i w first disc dw dfz (a) Front View dfx Feed df t a e φ j df c dfy (b) Top view of any given disk j Figure 4.2 Schematic view of helical end milling process 74

92 The elemental cutting df c, thrust, df t, and axial df a forces acting on disk j of an end mill are given by, df c (φ j, z) = [K ce + K c {a j (φ j, z)}] dw 4.1 df t (φ j, z) = [K te + K c {a j (φ j, z)}] dw 4.11 df a (φ j, z) = df a (φ j, z) 4.12 where dw is the height (width) of the each disk, a j (φ j, z) = a f sin φ j is the uncut chip thickness and a f is the feed rate per tooth, and is given by a f = V f N n t 4.13 where V f is feed speed, N is number of cutting flutes on the cutter and n t is the cutter rotational speed. K ce and K te are edge force components which are neglected in this study. The elemental forces are resolved into feed df y and normal df x directions using the relationships: df x = -df c * sin (φ) + df t * cos (φ) 4.14 df y = df c * cos (φ) + df t * sin (φ) 4.15 df z = - df a 4.16 The total forces acting on the cutter are the sum of the forces on each tooth segment and are given by. N F x = df x j= N F y = df y j= N F z = df z j=

93 4.3.2 Rake Angle Effect The specific cutting energy data base for orthogonal milling was generated for rake angle of 15 o and a specific value of cutting speed and a specific value of feed rate. To apply this data base to tools with different rake angles, some adjustment to the specific cutting energy coefficients is required. To adjust the specific cutting energy coefficients corresponding to a specific normal rake angle α n, the effective fiber orientation concept is utilized by equating the specific cutting energy coefficients for the same effective fiber orientation angle θ e for both cases [1]. The effective fiber orientation angle is defined as the relative angle between the tool rake face and the fiber orientation, θ e = 9 + (α n θ), as suggested by Ramulu et al. [1], where α n is the normal rake angle. θ e is defined purely from geometrical considerations. Schematic representation of the location of effective fiber orientation is shown in Figure 4.3. Adjusting the specific cutting energy coefficients for rake angle is carried out as follows. Let α ο = 15 o be the reference rake angle for the orthogonal cutting database and α n be the normal rake angle for the helical tool. If α ο = α n, then no adjustment is necessary and the database is used directly. In cases the two angles are different, then the database is queried for specific cutting energy coefficients that correspond to the same θ e for the helical tool. For example, for a fiber orientation θ = 3 ο and normal rake angle α n the effective fiber orientation angle for the helical tool is given by θ e = 9 o + (α n θ) = 6 o + α n The effective fiber orientation angle for the orthogonal cutting case is given by θ e = 9 o + (α o θ 2 ) = 15 o - θ 2 76

94 Making the effect of the two angles equal, we determine a fiber orientation θ 2 in the orthogonal database that will result in the same θ e, i.e. 6 o + α n = 15 - θ 2 Then, θ 2 = 45 - α n That is, cutting fiber orientation θ = 3 o with α n degrees normal rake angle tool is equivalent to cutting fiber orientation θ 2 = 45 - α n with a 15 o rake tool. It is obvious from this treatment that when α n = 15 o then θ 2 = 3 o. Figure 4.3. Schematic representation of the location of effective fiber orientation 4.4. The flow chart for calculating the milling forces for a helical tool is shown in Figure 77

95 Start Input Cutting forces, Tool Geometry, Cutting Parameters, Work materials, Number of discs Calculate feed per tooth Set initial immersion angle(φ i ) to zero φ i cos -1 (1-2a c /d t ) No Stop Yes Calc. instantaneous Chip Thickness a c F x = F y = F z =, j 1 = Calc. immersion angle and lag angle Yes j 1 j No φ i = φ i + (N*2π)/6* Samples per sec If immersion angle = No α n = α o Yes No Yes j 1 = j Adjust rake angle from eff. fiber orientation θ e Print F x, F y, F z Recall K c & K f in NN Find Instantaneous cutting forces Calculate Axial forces Compute the forces in X, Y & Z axis Find the cumulative force j 1 = j Figure 4.4. Flow chart for oblique milling force prediction model procedure. 78

96 4.4 Experimental Work To verify the predictive cutting force model, a series of edge milling operations on the CFRP material were performed under dry cutting conditions on a Fadal VMC2 CNC milling machine with a helical tool in an up-milling configuration. Two-flute end milling cutter with 3 helix angle was used to machine the CFRP workpiece. A three-component platform dynamometer was mounted between the work and the machining table to measure cutting forces Workpiece Material The work piece material used in the experiment is a unidirectional continuous carbon fiber reinforced composite with 5.6 mm thickness (approximate constitution 6% fibers, 4% epoxy resin). The laminate orientations ψ = and 6 were used to generate the specific cutting energy data for training and testing the neural networks. Multidirectional workpiece is also made. Multidirectional workpiece contained three unidirectional workpieces each is 2.8 mm thick. These were laid up so that the upper workpiece has 6 fiber direction followed by and 12 ply. The overall thickness was 8.4 mm with length 5.8 mm and width 5.8 mm. Figure 4.5 illustrates the construction of the workpieces. Figure 4.5 Schematic showing lay-up sequence of multidirectional workpiece. 79

97 4.4.2 Cutting Tool Geometry The composite cutting process used was end milling. The tool used was an end mill with two cutting edges, as shown in Figure 4.6. Helical end mills are used to dampen the sharp variations in the oscillatory components of the milling forces, and they are used when the depth of cut is large but the width of cut is small. Diameter of the tool is 9.5 mm, which half of the straight edge tool used in previous chapters. Figure 4.6. Cutting tool. The cutting angles are shown in Table 4.1. TABLE 4.1 CUTTING ANGLES FOR HELICAL TOOL Parameter Dimensions (Degrees) Normal Rake Angle 14 Helix angle 3 Radial Rake Angle Cutting Parameters In the mechanics of milling approach, the cutting force coefficients are predicted using the oblique cutting model and the orthogonal cutting database. [4] presented the method for orthogonal to oblique transformation. When a cylindrical end mill with helical flutes is sliced into a stack of very thin disks, the cutting action of each disk is an oblique 8

98 cutting process because of the helix angle, which is illustrated in schematic view of helical end milling process Figure 4.2. In order to account for the helix angle i, the helical end milling cutter was divided into s number of small discs along its axis in the z direction. In this case the work piece thickness is 5.6mm so it is divided in to 2 discs, where s = 1, The thickness of each disk, dw =.19 mm. All experiments with helical tool were conducted at a constant spindle speed of 2 rpm and a constant feed speed of 46 mm/min. Reason for having speed and feed constant is to compare the helical forces with single straight edge tool forces obtained at speed of 1 rpm and feed of 23 mm/min. Since the diameter of the helical tool is half of the straight edge tool and it has 2 teeth, the speed is doubled and feed is quadrupled in order to obtain similar chip per tooth. Depth of cut is half of the straight edge tool experiments, which is.5 mm Cutting Forces A sample of the signal acquired during milling one of the unidirectional two-ply composites is shown in Figure 4.7. The directions of forces F x, F y and F z are along the normal, feed and axial directions respectively as shown in Figure 4.2. The signal represents approximately three engagements of the cutter. The graph shows three downward half - sine functions. The data collected during the milling process is repetitive and shows that spindle revolution is a major signal component and is the smallest frequency component. Other components are impurities or undesirable noise because of system flexibility and material inhomogeniety. The force signal clearly shows 6 tool engagements taking place (two engagements per revolution) and dynamic resonance after engagement. The fluctuation in the force signal is likely due to excitation of the force measuring system. The rise of the cutting 81

99 forces in each cycle is due to the increase in undeformed chip thickness from zero at the cutting edge entry to a maximum at exit. The sample rate was 2, samples/sec # of Samples 12 One Spindle Revolution (645) One Spindle Revolution (645) One Spindle Revolution (645) 1 8 Forces (N) 6 4 Fx Fy Fz 2-2 Figure 4.7. Instant force data for milling 6 to fiber direction (ψ =6 ) in the cutting conditions of speed 2 rpm and feed 46.4 mm/min. The interpretations were developed from signals of cutting forces during a signal tooth rotation. Figure 4.8 to 4.1 show forces at constant speed, feed rate, and various fiber orientations as a function of the immersion angle φ along the positive axis of the dynamometer. The feed rate is 46.4 mm/min, depth of cut is.45mm, and the spindle speed is 2 rpm. The lead angle for end milling is and the radial rake angle is 16. The acquired force data along the X, Y and Z axes after filtering for one tooth engagement when machining different laminate lay-ups is shown in these graphs. The forces generated are dependent on the instantaneous immersion angle φ and instantaneous fiber orientation θ. 82

100 Fx Fy Fz Cutting Forces (N) Immersion Angle (φ), (Deg) Figure 4.8.Cutting Forces in milling to fiber direction (ψ = ) in the cutting conditions of speed 2 rpm and feed 46.4 mm/min. 8 6 F x F y F z Cutting Forces (N) Im m ersion A ngle (φ), (D eg) Figure 4.9.Cutting Forces in milling 6 to fiber direction (ψ =6 ) in the cutting Conditions of speed 2 rpm and feed 46.4 mm/min. 83

101 7 6 5 Fx Fy Fz Cutting Forces (N) Imm ersion Angle (φ), (Deg) Figure 4.1.Cutting Forces in milling 6 o / o /12 o to fiber direction in the cutting conditions of speed 2 rpm and feed 46.4 mm/min. 4.5 Results and Discussion In all cases, cutting forces start from zero where the leading tooth of the cutter starts cutting the workpiece. In this study 2 flute cutter is employed. Forces that were analyzed during the trimming operation of CFRP composite materials were normal force along the X- direction, feed force along the Y-direction, and axial force along the Z-direction. The cutting forces are also predicted by using Committee Neural Network analysis model for orthogonal milling as discussed earlier and in chapter 3. Figures show comparisons between experimental and predicted cutting forces for the, 6 and 6 / /12 laminate configurations machined at a depth of cut of approximately.45 mm. In these figures, the dotted line represents the measured cutting forces and the solid line represents the calculated cutting forces from the mechanistic model. The good agreement between the modeled and 84

102 the measured cutting forces clearly demonstrates the accuracy of the prediction model. These results show that the present force prediction model provides reasonable agreement with experimental results of complex tool for, 6 unidirectional laminate composites and multidirectional laminate composites (6 o / o /12 o ). The ability to predict the cutting forces for complex tool geometry allows the user to gain valuable insight into the cutting process. This information can be used for reducing or eliminating the need for expensive process trials P re d ic te d F x E x p e rim e n ta l F x 2 5 Cutting Force (N) Im m e rs io n A n g le φ i (D e g re e s ) Figure 4.11 Comparison between experimental and predicted forces F x for laminate at depth of cut of approximately.45mm. 85

103 3 P re d ic te d F y E x p e rim e n ta l F y Cutting Force (N) Im m e rs io n A n g le φ i (D e g re e s ) Figure 4.12 Comparison between experimental and predicted forces F y for laminate at depth of cut of approximately.45 mm 3 Predicted F z Experim ental F z Cutting force (N) Im m ersion Angle φ i (Degrees) Figure 4.13 Comparison between experimental and predicted forces F z for laminate at depth of cut of approximately.45 mm 86

104 8 P redicted F x E xperim ental F x Cutting Force (N) Im m ersion A ngle φ i (D egrees) Figure 4.14 Comparison between experimental and predicted forces F x for 6 laminate at depth of cut of approximately.45mm. 8 P redicte d F y E xperim enta l F y Cutting Force (N) Im m ersion A ngle φ i (D egrees) Figure 4.15 Comparison between experimental and predicted forces F y for 6 laminate at depth of cut of approximately.45mm 87

105 8 P redicted F z E xp erim e ntal F z Cutting force (N) Im m e rsio n A ngle φ i (D e gre e s) Figure 4.16 Comparison between experimental and predicted forces F z for 6 laminate at depth of cut of approximately.45mm P re d ic te d F x E x p e rim e n ta l F x Cutting Force (N) Im m e rs io n A n g le φ i (D e g re e s ) Figure Comparison between experimental and predicted forces F x for 6 o / o /12 o laminate at depth of cut of approximately.45 mm. 88

106 12 1 Predicted F y Experim ental F y Cutting Force (N) Im m ersion Angle φ i (Degrees) Figure Comparison between experimental and predicted forces F y for 6 o / o /12 o laminate at depth of cut of approximately.45 mm Predicted F z Experimental F z Cutting force (N) Immersion Angle φ i (Degrees) Figure Comparison between experimental and predicted forces F z for 6 o / o /12 o laminate at depth of cut of approximately.45 mm. 89

107 There are apparent differences between the predicted and experimental forces as in the case of 6 laminate and for multidirectional laminate. The errors between the calculation and measured forces could be due to simple orthogonal to oblique transformation and difference in the depth of cut between orthogonal data base and oblique cutting. The trends of the predicted normal, feed and axial forces matched the experiments although the actual values varied by 15%, 38% and 14% respectively, from the experimental values for 6 laminate and 27%, 24% and 25% respectively for multidirectional laminate. Effect of edge force component due to rubbing is neglected in this study. Considering this effect might improve the results. It is clearly shown from Figure 4.2 that if number of disks increases the prediction result is much better. Overall, the results indicate that the proposed model is effective and accurate. 5 4 Predicted Fx for 2 disks Experim ental Fx Predicted Fx for 1 disks Cutting Force (N) Im m ersion Angle φ i (D egrees) Figure 4.2 Effect of number of disks on prediction results 9

108 4.6 Conclusions An improved mechanistic cutting force model is presented in this work to deal with complex tool geometry by sectioning the helical cutting edge into a stacked series of straight edge cutter segments with angular offsets and calculating the forces for each segment, then adding the forces for all segments of the cutting edge. The cutting forces generated by each element on the portion of the engaged cutting edge are evaluated by applying oblique cutting transformation to the orthogonal cutting data, which allows the model to be extended to any cutter geometry without milling calibration test. This work is limited to flat-end milling cutters. For ball-end milling, the geometric structure of the milling cutter is quite complex compared to that of the flat-end cutter. In flat-end milling, the chip thickness direction is always horizontal and normal to the cutter surface at the cutting element position so it is easy to calculate chip thickness, where as in ball end milling the chip thickness direction on the cylindrical part can be horizontal as well as surface normal. This creates the problem in defining the chip thickness direction for ball-end milling. Once the chip thickness is found same approach can be used to predict the cutting forces in ball-end milling. 91

109 CHAPTER 5 GENERALIZED ANN PREDICTION MODEL 5.1 Introduction Today, fiber reinforced polymers are being used extensively in the aerospace, transportation, automotive as well as other types of industries due to their high specific strength and high specific stiffness. Although near-net shape can be obtained in the production of components made of FRP laminates, machining processes are still required for dimensional accuracy, surface finishing and assembly with other parts. Currently used CFRP material often needs to be edge trimmed to their final shapes or dimensions after full cure of the laminates. But due to its inhomogeneous nature, machining FRP s encounters numerous problems such as delamination, burning and accelerated tool wear. To minimize damage in machining, it is important to monitor process variables such as the cutting forces. If extreme care is not taken to avoid excessive cutting forces during milling process, serious damage such as surface ply delamination and through thickness edge cracking can occur at the machined edges. There are several parameters that influence the cutting forces acting on the cutter. These parameters include instantaneous fiber orientation angle, uncut chip thickness, cutting speed, feed rate and other parameters associated with cutting tool geometry. Because of the shear magnitude of these parameters, prediction of the milling forces may become a challenging task.. Prediction of milling forces is critically important in milling operations, because milling forces correlate strongly with cutting performance such as tool wear, surface accuracy, tool breakage etc. Koplev and his colleagues [2] investigated the cutting chips, machined surface, and forces in orthogonal machining of unidirectional composites. That was the first effort to find 92

110 the machining characteristics in the orthogonal cutting of composites. They concluded that the process of machining CFRPs consists of a series of fracture, each creating a chip. Koplev et al. [2] also found the effects of cutting parameters on CFRP material removal. According to their work, increasing the rake angle of the tool slightly reduces the principal cutting force, while no definite trend is found on the thrust force. The machining characteristics were considered only for parallel and perpendicular fiber orientations. They also found that the quality of the machined surface is a function of cutting direction relative to the fiber orientation. It was concluded from this work and others that the principal cutting mechanisms correlate strongly with the fiber orientation and tool geometry [2,3, 2]. Arola et al. [48] investigated the delamination in surface plies of graphite/epoxy laminates caused by edge trimming using polycrystalline diamond and carbide cutters. They found that the formation of delamination is affected by several machining variables such as cutter geometry, feed rate, and rotation direction. They documented that feed rate plays a dominant role. They concluded that an increasing of the cutting speed leads to a better surface finish. Arola et al. [15] studied the chip formation mechanism in the edge trimming of graphite/epoxy laminates. They concluded that the characteristics of chip formation are primarily dependent on the fiber orientation, with only secondary effects from tool geometry, cutting speed and feed rates. Some works have been published, aiming to highlight the modes of chip formation [13,16,17] and the dependence of cutting forces on machining parameters [1,13,15,16]. Many researchers have attempted to develop models that would predict cutting forces based on the geometry and physical characteristics of the process. Such prediction could then be used to optimize the process. Takeyama and Iijima [16] were the first to develop a model 93

111 for cutting forces resulting from orthogonal trimming of FRPs using modified Merchant s minimum energy principle. The model developed is only valid for fiber orientation of less than 9, and no attempt was made by the authors to predict the force requirements in machining laminates with fibers inclined towards the cutting edge. Bhatnagar et al. [17] developed a model for the cutting forces in orthogonal trimming of CFRPs, which is also based on Merchant s minimum energy principle for materials removal. Some of the drawbacks of this model are, the requirement of determining in-plane shear strength by actual machining, varying the friction angle which in turn is dependent on fiber angle, the material of the workpiece material, and the tool geometry. An analytical model by Zhang et al. [18] divided the cutting region based on the deformation mechanism, into three regions, namely, chipping, pressing and bouncing and calculated the cutting forces by adding the forces from the three regions. The trends of the predicted cutting forces and thrust forces matched the experiments although the actual values varied by 27% and 37% respectively, from the experimental values. Drawbacks of this model are the requirement of properties of the material, and the high error in prediction of cutting and thrust force. Also, the validation range for the model in terms of fiber orientation is limited to less than 9. Hocheng et al. [4] proposed a mechanistic model for the prediction of the cutting forces in milling of FRPs using single insert end mills. The limitation in this approach, however, was that it is only valid for the two fiber orientations tested and thus may not be extended for force prediction for other fiber orientations or multidirectional laminates. Sheikh-Ahmad and Yadav [5] extended the work of Hocheng et al. [4] to encompass all possible fiber orientations from to 18 o by introducing specific cutting energy as a continuous function of fiber orientation and chip thickness. In this work, specific cutting 94

112 energy data was obtained from end milling tests of, 45, 9 and 135 laminates. Multiple regression was used to approximate continuous specific energy functions. The work of Sheikh-Ahmad et al. [52] has overcome the some limitations of the Sheikh-Ahmad and Yadav [5] by considering wide range of laminate orientations and by using committee neural networks for function approximation. These modifications have significantly improved the prediction capability of the new model. The limitation of the model, however, is that it did not consider the effects of cutting speed and feed rate on cutting forces. The purpose of this work is to extend the experimental database for the CNN prediction model described in chapter 3 to include more fiber orientation angles and the effects of cutting speed and feed rate on cutting forces. The cutting speed and the feed rate constitute the basic independent parameters and govern the productivity through their direct effect on material removal rate. The model is chapter 3 can be easily extended to include the effect of rake angle, feed rate and cutting speed by including cutting data for different levels of these parameters. Examination of Table 3.1 shows that milling of the four laminates, 45, 9 and 135 o did not provide data for the entire fiber orientation range < θ < 18 o and specific cutting data is missing for fiber orientations in between those provided by the four laminates. Furthermore, the specific cutting data of the previous model did not take into account the effects of cutting speed and feed rate. A generalized prediction model should be readily able to account for all process parameters such as cutting speed, depth of cut, and feed rate. This chapter will highlight the experimental approach used to collect data, CNN structure and performance and the prediction of cutting forces based on the specific cutting energy function over a wide range of spindle speeds and feed rates. This generalized model should be able to predict cutting forces for any combination of process parameters (n t, V f, a e, 95

113 α, i) and workpiece fiber layup direction ψ, where n t is the spindle speed, V f is the feed rate, a e depth of cut, α rake angle and i is the cutting edge inclination angle. The effects of rake angle and inclination angle on cutting forces have been discussed in chapter 4. The effects of cutting speed and feed rate are discussed in this chapter. 5.2 Experimental Work Workpiece Material The workpiece material used in the experiment is a unidirectional continuous IM6/epoxy carbon fiber reinforced composite laminate with 2.8mm thickness (approximate constitution 6% fibers, 4% epoxy resin). The workpiece was cut from a large panel measuring 762 X 712 X 2.8 mm (length X width X thickness). These were cut into a similar geometry with different layup directions and each set had, 3, 45, 6, 9, 12, 135 and 15 fibers layup with respect to the feed direction. The specimens used for milling experiments featuring 5.8 X 5.8 X 2.8 mm (length X width X thickness) were cut from the laminate with required fiber orientations. Figure 5.1 is a schematic showing the orientation of laminates used in milling experiments. It is noted here that a o laminate is the same as 18 o laminate. The laminates ψ =, 3, 6, 9, 12, and 15 were used to generate cutting forces and specific cutting energy data for training and testing the neural networks. The laminates ψ = 45 and 135 were used to verify the force model prediction. The mechanical properties of the work piece material (IM6/Epoxy) are listed in Table

114 Figure 5.1. Schematic showing orientation of laminates used in milling experiments. Table 5.1 Mechanical Properties of Workpiece material (IM6/EPOXY) Typical Fiber Properties U.S. Units SI Units Tensile Strength 827, psi 5,7 MPa Tensile Modulus Chord x 1 6 psi 279 GPa Ultimate elongation 1.9% 1.9% Carbon content 94.% 94.% Density.636 lb/in g/cm 3 Specific heat At 167 F (75 C) At 347 F (175 C).22 Btu/lb, F.27 Btu/lb, F.22 cal/g, C.27 cal/g, C Electrical resistance, 12K 15.5 ohms/ft.53 ohm/cm Electrical receptivity, 12K 4.5 x 1-5 ohm-ft 1.4 x 1-3 ohm-cm 97

115 5.2.2 Cutting Tool Geometry The composite cutting process used was end milling or routing. The tool used was an end mill with a single cutting edge, as shown in Figure 5.2. The cutter dimensions are found in the Table 5.2. The cutting insert dimensions are shown in Figure 5.3. The cutting angles are shown in Table 5.3. Insert Figure 5.2. Cutting tool. TABLE 5.2 CUTTING TOOL DIMENSIONS Height Diameter Number of inserts 7 mm 19.4 mm 1 98