Machining Feature Based Geometric Modeling of. Twist Drills

Transcription

1 Machinin Feature Based Geometric Modelin of Twist Drills Jian Zhu A Thesis in The Department of Mechanical & Industrial Enineerin Presented in Partial Fulfillment of the Requirements for the Deree of Master of Applied Science (Mechanical Enineerin) at Concordia University Montreal, Quebec, Canada Auust 2011 Jian Zhu, 2011

2 CONCORDIA UNIVERSITY School of Graduate Studies This is to certify that the thesis prepared By: Entitled: Jian Zhu Machinin Feature-based Geometric Modelin of Twist Drills and submitted in partial fulfillment of the requirements for the deree of Master of Applied Science complies with the reulations of the University and meets the accepted standards with respect to oriinality and quality. Sined by the final examinin committee: Dr. A. K. W. AHMED Chair Dr. A. AGUNDUZ Examiner Dr. W. TIAN Examiner Dr. C. Z. CHEN Supervisor Approved by Dr. MARTIN D. PUGH Chair of Department or Graduate Proram Director Dr. ROBIN DREW Dean of Faculty Date SEPTEMBER 15, 2011

3 iii Abstract Machinin Feature Based Geometric Modelin of Twist Drills Jian Zhu To pursue hih accuracy, efficiency and reliability in the machinin industry, hih quality cuttin tools play an important role. It is always in hih demand in industry for new cuttin tools in order to achieve better cuttin performance and lower machinin costs. To develop a new tool, a new approach is first to establish the accurate solid model of the tool, second to predict the tool performance in machinin in order to optimize the tool, and third to rind the tool on a 5-axis rindin machine. To ensure the eometric model of a twist drill in ood areement with the machined one, a new approach to eometric modelin of the twist drill based on parametric machinin features is proposed. In this work, the solid model of a twist drill includes four machinin features, such as the flutes, the first flank, the drill split (ash), and the land. These features are parameterized based on the rindin wheel eometry and the 5-axis rindin wheel path. The effective rindin ede is calculated, and the eometric model is established. This approach is implemented in the CATIA V5 R20 to build the solid model of the twist drill. This model is enuine in terms of the actual machined twist drills. Therefore, it is essential to predict its machinin performance of the actual cutters in machinin simulation.

4 iv Acknowledements I wish to express sincere ratitude to my supervisor, Dr. Chevy Chen, for ettin the thesis started in the positive and riht direction. It was Dr. Chen who instilled the desire in my heart to develop an accurate model of twist drill of such level, which is enuine in accordance to the actually machined drill. I also wish to extend a measure of ratitude to my fellows, Shuanxi Xie, Chun Du, Maqsood Khan, and Muhammad Wasif, in CAD/CAM lab office. These entlemen were tireless of answerin my questions. When I encountered a problem, they helped me and inspired me. Especially, I would like to ive my special thanks to my wife and my two kids whose love enabled me to complete this work.

5 v Table of Contents List of Fiures... viii List of Tables... xii Chapter 1 Introduction Manufacturin-based Desin Feature-based Modelin Definition Parametric feature based part desin and modelin Literature Review Objective of the Thesis Outline of the Thesis Chapter 2 Body Modelin Introduction Drill Body Parameters The procedure of modelin the drill body Chapter 3 Flute Modelin Introduction The Direct Method of Machinin Flutes Parametric Model of the Standard Grindin Wheels Mathematical Model of the Multi-axis CNC Grindin Process Parametric equation of effective rindin ede Flute Cross-Sectional Profile Application of the Direct Method Inverse Method of Machinin Flutes... 33

6 vi Formula of the Flute Surface and its Normal Conjuate Relationship between the Flute and the Wheel Grindin Wheel Profile Grindin Wheel Profile Applications of the Inverse Method Example of Geometrically Modelin of the Flutes Effective rindin ede obtained by MATLAB prorammin Input to CATIA Chapter 4 Flank Modelin Mathematical models of the flank Mathematical model for the quadratic flanks Mathematical model for the planar flanks Parametric modelin of the conical flanks of the drill tip Parametric modelin of the planar flanks Chapter 5 Drill Split Modelin Introduction Parametric Modelin of the Drill Split Construction of the Drill Split Mathematical model of the ash Final model of manufacturin feature-based twist drill Chapter 6 Conclusion & Future Work Conclusion Future Work References Appendix A Code for Direct Method to find EGE by a standard parallel wheel... 85

7 vii Appendix B Code for special functions ) Function 'Rx' ) Function 'Ry' ) Function 'Rz' ) Function 'T' ) Function ' findsolution_enhanced ' ) Function xh_zh_xph_zph_anle... 97

8 viii List of Fiures Fiure 1-1 Standard eometry of a two flute twist drill [1]... 1 Fiure 1-2 Conical point... 2 Fiure 1-3 Planar point... 2 Fiure 1-4 Split point... 2 Fiure 1-5 Radial (split) point... 3 Fiure 1-6 Corrected cuttin ede point... 3 Fiure 1-7 Conical point with internal coolin hole... 3 Fiure 1-8 Curved cuttin ede point with S-split and internal coolin hole... 4 Fiure 2-1 Drill body and its key dimensions Fiure 2-2 Dimension of drill radius Fiure 2-3 Parameter of drill diameter Fiure 2-4 Dimension and parameter of drill back taper anle Fiure 2-5 Dimension and parameter of drill body lenth Fiure 2-6 Dimension and parameter of drill overall lenth Fiure 2-7 Dimension and parameter of drill shank diameter Fiure 3-1 The profile of a standard anled rindin wheel Fiure 3-2 The parameters of the rindin wheel profile Fiure 3-3 Illustration of the kinematics of multi-axis rindin of flutes Fiure 3-4 The parameters of the rindin wheel profile Fiure 3-5 The three-dimensional flute and its cross section of the drill machined viii

9 ix usin the No. 1 rindin wheel Fiure 3-6 The three-dimensional flute and its cross section of the drill machined usin the No. 2 rindin wheel Fiure 3-7 The three-dimensional flute and its cross section of the drill machined usin the No. 3 rindin wheel Fiure 3-8 A contact curve of the flute surface shown with the rindin wheel not shown Fiure 3-9 The profile of a non-standard rindin wheel Fiure 3-10 The effective rindin ede of the virtual rindin wheel Fiure 3-11 A contact curve between the flute surface shown and the rindin wheel not shown in the first example Fiure 3-12 The effective rindin ede of the wheel in the first example Fiure 3-13 The cross sectional profile of the wheel in the first example Fiure 3-14 A contact curve between the flute surface shown and the rindin wheel not shown in the second example Fiure 3-15 The effective rindin ede on the wheel in the second example Fiure 3-16 The cross sectional profile of the wheel in the second example Fiure 3-17 A contact curve between the flute surface shown and the rindin wheel not shown in the third example Fiure 3-18 The effective rindin ede on the wheel in the third example Fiure 3-19 The cross sectional profile of the wheel in the third example Fiure 3-20 Effective rindin ede enerated by a parallel rindin wheel ix

10 x Fiure 3-21 EGE in CATIA Fiure 3-22 EGE and helical curve Fiure 3-23 The envelope surface swept by EGE Fiure 3-24 Flute modelin: Step Fiure 3-25 Flute modelin: Step Fiure 3-26 Flute modelin: Step Fiure 3-27 Flute modelin: Step Fiure 3-28 Flute modelin: Step Fiure 3-29 Flute modelin: Step 6, addin wheel surface at final position Fiure 3-30 Flute modelin: Step 7, create final flute shape Fiure 4-1 Conical flank and rindin cone Fiure 4-2 Conical flank modelin: Step Fiure 4-3 Conical flank modelin: Step Fiure 4-4 Conical flank modelin: Step Fiure 4-5 Conical flank modelin: Step Fiure 4-6 Conical flank modelin: Step Fiure 4-7 Calculation of d y Fiure 4-8 Conical flank modelin: Step Fiure 4-9 The cutter coordinate system is translated alon its positive X axis direction by a half of the web thickness Fiure 4-10 The cutter coordinate system is rotated around its X axis by a half of the drill point anle x

11 xi Fiure 4-11 The cutter coordinate system is rotated about its Y axis by the relief anle Fiure 4-12 The final rindin wheel position Fiure 4-13 The solid model of a twist drill with a flank face enerated Fiure 5-1 The helix uide is drawn in order to machine the ash Fiure 5-2 The rindin wheel is located at the startin point of the drill split Fiure 5-3 The second flank is enerated by removin the drill stock with the surface swept by the effective rindin ede alon the uide curve Fiure 5-4 A drill with a blue ash and the orane second flank are shown Fiure 5-5 Drill point with ashin done Fiure 5-6 Final machinin feature-based eometric model of twist drill xi

12 xii List of Tables Table 2-1 Drill body parameters Table 3-1 The parameter values of the rindin wheels Table 3-2 Parameters in flute modelin and prorammin Table 4-1 Parameters for conical flank modelin Table 5-1 Parameters for drill split modelin xii

13 1 Chapter 1 Introduction Of all machinin processes, drillin is very important because nearly 25% of the metal cuttin processes are drillin operations and approximately 40% of workpiece materials are removed by drills [1]. Throuhout the past century, twist drills have been drastically improved and have played an important role in the metal cuttin industry. Up to now, more than 200 types of twist drill points have been shown in the literature and market. Fiure 1-1 shows a standard twist drill, and Fiures 1-2 to 1-8 show some common drill points in the market. Fiure 1-1 Standard eometry of a two flute twist drill [1]

14 2 Fiure 1-2 Conical point Fiure 1-3 Planar point Fiure 1-4 Split point

15 3 Fiure 1-5 Radial (split) point Fiure 1-6 Corrected cuttin ede point Fiure 1-7 Conical point with internal coolin hole

16 4 Curved cuttin ede S split Drill axis Internal coolin hole Fiure 1-8 Curved cuttin ede point with S-split and internal coolin hole Althouh drillin seems a relatively simple process, it is really a complex and difficult process. Since drillin occurs inside the workpiece, heat is accumulated, and the cuttin temperature could be hih, especially, the cuttin process contains a lare portion of chisel cuttin (a cuttin process by chisel ede which has more than 45º neative rake anle) or the chips are not small enouh. Another problem is lubrication and coolin is difficult to carried out because the chips block the coolant in the flute. To overcome these difficulties, more complex drills have been desined, for example, a Racon point with a round flank, a helical point with a raised tip, a wavy point with a curved cuttin lip, and a four-facet split point, etc. It is very important to accurately model drills in 3D for drill analysis. Generally, advanced drill modelin should have two aspects, manufacturin-based desin and feature-based modelin.

17 5 1.1 Manufacturin-based Desin Manufacturin drill modelin means that the drill modelin should be accurately based on its manufacturin process. In a new drill desin, drill modelin should be always based on what kind of manufacturin process is used to produce a drill. Most drills are machined with rinders. The drill features, for example, the flutes, are part of the envelope surfaces formed by the rindin wheel throuhout the rindin route. Thus, drill modelin should always follow the envelope theory, which represents the relationship of the drill shape and its manufacturin process. In conventional and standard drill analysis, drill modelin should also consider how a rinder machines a drill. Today, the complexity of the drill points and the hih labour costs require that the rindin machines must be accurate, versatile, and automated. This requirement leads the appearance of 4-axis, 5-axis and even 6-axis CNC tool-rindin machines. Nowadays most of drills are produced in these multi-axis rinders. The multi-axis rindin machines manufactured by different companies have their own software, which is based on the same or different mathematical models. This means different rinders may use different manufacturin features to machine a drill. So, to accurately built a 3-D solid model, it is very important to know the manufacturin features and the related mathematical models.

18 6 1.2 Feature-based Modelin Definition A mechanical part consists of several eometric features. Features in eometric feature-based modelin are defined to be parametric shapes associated with eometric parameters (such as lenth, width, radius etc), positional parameters (such as offset distance, positional anle etc), and orientational parameters (such as orientational anle, riht hand of a helix etc). Now, parametric feature-based part desin and modelin becomes one of the kernel techniques of the new computer-aided desin. It is a dispensable advanced technique since the parametric CAD models of the part features can be easily modified in the part desin optimization process. By definition, the parametric feature-based desin and modelin is to determine the key feature dimensions as the parameters and to specify the relationships (or constraints) amon the parameters and other part dimensions. Fortunately, the functions of definin parameters and constraints are provided in some major CAD/CAM software. Applyin these functions, all part dimensions can be calculated by assinin data to the feature parameters and the solid model of the part can be chaned accordinly and updated in seconds Parametric feature based part desin and modelin

19 7 Mechanical part desin includes a number of decision-makin processes and activities. Generally, a mechanical part desin has four phases: conceptual solutions, desin exploration, desin refinement, and final CAD models and enineerin drawins. In contrast with the conventional part modelin method which all part dimensions have to be defined independently, parametric feature-based modelin allows the feature parameters and the eometric, positional, and orientational constraints to be specified or related. This can reatly reduce the leadin time in part desin because the solid 3D CAD model of a part can be easily attained and modified. Generally, four steps are necessary to implement the parametric feature-based part desin and modelin, which are to define the dimensions of key features as the parameters, to define the relationships or constraints between the parameters and the dimensions, to establish the 3D solid model with CAD/CAM software, and to input the parameters and the constraints in the part model. Once the parametric feature-based model of a part is built, the part can be chaned automatically by assinin different parameter values. 1.3 Literature Review Many technical articles have discussed about the eneralized models of cuttin tools, includin the mathematical and manufacturin models [2-22]. Enin and Altintas [2] described a mathematical model of eneral end-mills often used in the industry. For twist drills, the first accurate eometry model was developed by Galloway [3] in However,

20 8 the rindin cone was not unique. In 1973, Armareo et al. [4-5] studied drill point sharpenin by the straiht lip conical rindin method and developed an analysis of the straiht lip conical rindin concept. He discovered the conical rindin processes of flank are determined by four dependent factors, whereas the drill point was specified by three parameters. He ave the relief anle on the flank as the fourth parameter to et the unique solution. A few years later, Tsai and Wu [6] developed a mathematical model that describes drill flank eometry includin the conical, hyperbolical, and ellipsoidal drills and the flank of the drill is represented with coincide. This study ives an accurate method to represent the quadratic drill eometry which enables the flank to be analyzed accurately and conveniently by computer. In 1983, Radhakrishnan [7] first derived the mathematical model of the planar split drill point. Fuelso [8] in 1990 improved the standard straiht-cuttin-ede model by rotatin the drill about its axes by anle ω before sharpenin. This improvement solved the problem that the clearance anle is too small near the chisel ede. The new method led to a curved cuttin ede, and, from then on, the curve cuttin ede was introduced and applied widely in some drillin processes. Lin et al. [9] first developed the helical drill point in 1995 to alleviate the disadvantaes of existin planar micro drill point. Ren and Ni [10] developed a new mathematical model for an arbitrary drill flute face by sweepin the polynomial representation of the flute cross-sectional alon the helix drill axis with helical movement. The typical mathematical models for multi-facet drills were presented usin anle-solid-block method in Ger-Chwan Wan s works [11]. However, the Boolean operation used here can not apply to helical multi-facet point.

21 9 In 1988, the direct and the inverse problems related to the flute and rindin wheel were first discussed [12] and then mathematically solved by many researchers. K.F. Ehmann [13] developed a proram and presented a well solution for the inverse problem. J.F. Hsieh [14] proposed a eneral mathematical model of the tool profile and helical drill flank and solved the two problems by usin conjuate surface theory. Alon with the development of CAD/CAM technoloy, CAD approach becomes an attractive way to simulate the eometrical and rindin features of twist drill. Thanks to the objective of the thesis, which is feature-based modelin of the twist drills, eiht more related papers are reviewed [15-22]. Based on the Galloway's models, Fujii et al. [15-16] first presented an analysis about the drill point eometry by usin computer aided desin system. However, the proposed cone parameters were difficult to measure and set. Fuh [17] applied the computer aided desin to analyze the quadratic surface model for the twist drill point. Sheth and Malkin [18] reviewed commercial CAD/CAM software for the desin and manufacture of components with helical flutes. enineers desin the profile of the tool and the helical flute. The CAD system could help Kaldor et al. [19] dealt with eometrical analysis and development for desinin the cutter and the rindin wheel profile. The direct and the inverse methods allow prediction of the helical flute profile and the cutter profile, respectively. Kan et al. [20] proposed an analytical solution to helical flute machinin throuh a CAD approach, and a eneralized helical flute machinin model usin the principles of differential eometry and kinematics, was formulated. Vijayarahavan [21] etc. developed an automated 3D model software based on eometry and manufacturin parameters, and it can be output with solid eometry format which

22 10 can be meshed and analyzed in FEA software efficiently. Li e al. [22] presented a method to automatically measure the relief anle and rake anle of the standard twist drill based on 3-D model created by PRO/E. However, the Boolean operation used in [21] and [22] is not feasible in some tanential area. 1.4 Objective of the Thesis The objectives of the thesis include (a) findin the relationship between the mathematical, manufacturin and eometric models of a twist drill and (b) buildin a parametric solid model of a twist drill with the CATIA V5 R20 software, based on its machinin features. The machinin features of the twist drill include the drill body, two flutes, the drill flank, the split (or the ash), and the land. In this work, all the machinin features will be parameterized and their solid models will be constructed with the CATIA V5 R20 software. 1.5 Outline of the Thesis Basically, the document comprises of seven chapters. Chapter one introduces the parametric desin and modelin and reviews literature on this topic. Chapter two eometrically models the drill body. Chapter three proposes the parametric modelin for flute rindin with standard wheels in the direct method and renders a new method of determinin the rindin wheel profiles of non-standard wheels for machinin desined

23 11 flutes in the inverse method. Chapter four defines the machinin features accordin their mathematical models and builds the parametric modelin of the flank. Since the main cuttin lip is the intersection of the flank face and the flute face, the flank modelin and the flute modelin are very important. Chapter 3 and chapter 4 will describe them in detail. Chapter five constructs the drill split, and in Chapter six, conclusions are drawn for the thesis.

24 12 CHAPTER 2 BODY MODELING 2.1 Introduction Body modelin of a twist drill defines the drill's body profile. Only eometrical model is presented for the body modelin. 2.2 Drill Body Parameters There are five parameters of the drill body, such as the drill diameter, shank diameter, overall lenth, body lenth, and backtaper anle. Table 2-1 Drill body parameters Parameters Definition value D Drill diameter 12 mm D shank Drill shank diameter 14 mm L Drill overall lenth 120 mm L body lenth) Drill body lenth (drill diameter 80 mm

25 13 DiameterBackTaper drill body Taper anle of a tapered twist 0.1 o 2.3 The procedure of modelin the drill body All five parameters are used in the twist drill body modelin. These parameters define the correspondinly eometric features. The 3-D model and its key dimensions are shown in Fi 2.1. BackTaper Body lenth (Lbody) Drill radius Overall lenth (L) Shank radius Fiure 2-1 Drill body and its key dimensions Fiure 2-2 to 2-7 show the dimensions and correspondin parameters of the twist

26 14 drill. Fiure 2-2 Dimension of drill radius Fiure 2-3 Parameter of drill diameter

27 15 Fiure 2-4 Dimension and parameter of drill back taper anle Fiure 2-5 Dimension and parameter of drill body lenth

28 16 Fiure 2-6 Dimension and parameter of drill overall lenth Fiure 2-7 Dimension and parameter of drill shank diameter

29 17 Chapter 3 Flute Modelin 3.1 Introduction The flute is one of the most important features of the twist drill. The flute determines the cuttin forces and the core size that is very important to the cutter riidity; and, at the same time, it provides accommodation for chips and evacuates them durin machinin. For a twist drill, its flutes are vital to the tool riidity and chip evacuation. However, these two characteristics are contradictory with each other. To attain a tool with hih riidity, the cross section area of the flute and the flute depth should be small so that the core radius is lare. On the contrary, to quickly evacuate the chips, the larer the flute space, the quicker the chip flow. It is difficult to optimize the flute shape for hihest tool riidity and fastest chip evacuation. So the flute model that accurately reflects its manufacturin process is very important. To machine the flutes of the twist drill in practice, there are two different methods, i.e., the direct and the inverse methods. The major difference between these methods is what type of wheel is selected and how the rindin wheel is determined. In the direct method, a standard rindin wheel is first selected; while, in the inverse method, the rindin wheel is nonstandard, and its profile is determined based on a prescribed flute profile. The two methods share the same steps in the flute machinin process. In the second step, the cuttin ede of the flute is specified as the wheel path. Then, in rindin,

30 18 when the wheel moves alon the path, it sweeps a volume, in which the stock material is removed, eneratin the flute. Mathematically, the flute can be represented as part of the outside surface of the wheel swept volume, which is the envelope of the wheel durin machinin. In this section, a parametric model of the flute is rendered accordin to the flute manufacturin process. 3.2 The Direct Method of Machinin Flutes The direct method employs standard rindin wheels to rind the flutes. To represent the flutes, the envelope theory should be applied to the standard wheel movin alon the pre-determined wheel path, which is the helical cuttin ede. As a result, the effective rindin ede can be found, and a mathematical model of the flute can be formulated. In this thesis, two types of standard rindin wheel, which are often used in drill manufacturin, are adopted. A eneral model of these rindin wheels is built in the followin Parametric Model of the Standard Grindin Wheels Two standard rindin wheels used in this work are straiht (or parallel) and anled (or bevel) wheels. For the straiht wheel, the flanks of both sides of the wheel are straiht and normal to the wheel revolvin circumference; and, for the anled wheel, one flank of the wheel is straiht and the other is inclined with an anle to the wheel circumference.

31 19 Usually, these types of rindin wheels are popular and economic, and they are easy to dress after the wheel worn out. Fiure 3.1 illustrates the two rindin wheels with the parameters. Of the two wheels, the anled wheel is more eneric in shape, and the parametric equation of the anled wheel can represent the straiht wheel by settin the anle to 90 o. Z X Y Fiure 3-1 The profile of a standard anled rindin wheel To derive the parametric equation of the anled wheel, a rindin wheel coordinate system x y z is established in such a way that the oriin is at the center of the straiht flank face, the z axis is alined with the wheel axis and points inside the wheel, and the x and y are perpendicular to each other ad on the straiht flank face (see Fiure 3.1). The profile of the rindin wheel is a polyon H 0 H 1 H 2 H 3 H 4 and is shown on plane xz in Fiure 3.2.

32 20 Fiure 3-2 The parameters of the rindin wheel profile. The radius of the wheel circumference is R w, the thickness of the wheel is W w, and the inclined anle of the ede is. The lenths startin from H 0 to the five vertexes in the polyon are denoted as L 0, L 1, L 2, L 3, and L 4, where L 0 is zero. The coordinates of H 0, H 1, H 2, H 3, and H 4 are 0, 0, T W, L W, L1 L2 L1 Ww L2 L1 w, 0, T 1 w L, and, 0, 0 3 T cos, 0, sin, 0, 0, 0 T, respectively. To find the parametric equation of the polyon edes, a parameter, h, the lenth on the polyon startin from H 0 is used. Then the equations of the polyon edes with parameter, h, are derived as T h 0 For ede HH, h L1 q 1 Ww ; 1 For ede HH 1 2, L1 h L1 cos 0 L1 h L2 q2 Ww h L1 sin ; 1

33 21 For ede 2 3 For ede HH 3 4, Rw 0 L h L q L3 h ; 1 HH, L4 h 0 L3 h L4 q 4 0 ; 1 The wheel is constructed by rotatin the profile polyon about z by 360 derees. Suppose the rotation anle v starts from axis x and is a parameter of the wheel surface, the rotation matrix and the parametric equations of the wheel surface are formulated as The eneral surface equation is r cos sin 0 0 x v v sin v cos v 0 0 ry r r z x h cosv x h sin v z h 1 q T (3.1) Thus, For surface S1 h, v enerated with HH 0 1, where 0h L1 and 0 v 360 ; hcos v hsin v S 1 hv, (3.2) Ww 1 For surface S2 h, v enerated with HH 1 2,

34 22 where L1h L2and 0 v 360 ; 1 1 S 2 hv, (3.3) L1 h L1 cos cos v L h L cos sin v Ww h L1 sin 1 For surface S3 h, v enerated with HH, 2 3 where L2h L3 and 0 v 360 ; Rw cos v Rw sin v S 3 hv,.(3.4) H3 h 1 For surface S4 h, v enerated with HH 3 4, where L3h L4 and 0 v 360. L4 h cos v L h sin v S 4 hv, (3.5) Mathematical Model of the Multi-axis CNC Grindin Process To truly represent the eometry of a machined flute, a mathematical model of the multi-axis CNC rindin process for the flute is necessary, which is established here accordin to the kinematics of the ANCA CNC tool rindin machine used to cut the flute.

35 23 In the actual flute rindin process, the wheel axis could be in a skew orientation in terms of the cutter axis, which is usually in horizontal, and the cutter is simultaneously rotated and fed alon its axis. Althouh the rindin kinematics is that both the wheel and the cutter move at the same time, it can be converted to an equivalent kinematics that the cutter is stationary and the wheel moves and rotates for the same flute eometry. Thus, the cutter flute can be modeled by representin the wheel in the flute machinin process in the cutter coordinate system. In the equivalent rindin kinematics, first, the cutter coordinate system x y z d d d is fixed, and the rindin wheel coordinate system x y z is coincided with it before rindin. Durin machinin, the rotation anle of the cutter chanes, and the wheel location and orientation chane accordinly, which can be decomposed in the followin steps. The wheel coordinate system is translated alon its axis z by z, which is equal to kz. k z is the coefficient related with the helix anle of the cuttin ede. The T translation matrix is 0,0, z M z The wheel coordinate system is rotated about its z axis by. The rotation matrix cos sin 0 0 sin cos 0 0 M R is z, The wheel coordinate system is translated alon its x axis by x, which is equal to Rw c x r k. k x is related with the taper anle of the cutter, if it is a tapered twist

36 24 T drill. The translation matrix is x,0, x M The wheel coordinate system is rotated about its axis R matrix is M x, cos sin 0. 0 sin cos x by anle. The rotation The wheel coordinate system is rotated about its y axis by anle. The rotation R matrix is M y, cos 0 sin sin 0 cos Fiure 3-3 illustrates the first three steps of the kinematics when is equal to 360 derees. Usin Euler rule, the equivalent matrix of the five transformation matrices can be derived as M M (0,0, z ) M ( z, ) M ( x,0,0) M ( x, ) M ( y, ) d T R T R R cos cos sin sin sin sin cos cos sin sin sin cos x cos sin cos cos sin sin cos cos sin sin cos sin cos x sin (3.6) cos sin sin cos cos z

37 25 Fiure 3-3 Illustration of the kinematics of multi-axis rindin of flutes Applyin this equivalent matrix of the machinin kinematics, the wheel surfaces can be represented in the cutter coordinate system while the wheel cuts the flute. The eneral equation is h, v,,, k, k h, v S M S (3.7) d d x z d In detail, the wheel surfaces can be found in the cutter coordinate system as The equation of surface hv S S in the cutter coordinate system is 1, x1 h, v hcos cosv hsin cos sinv Ww sin sin xcos h, v y h, v h sin cosv h cos cos sinv Ww cos sin x sin z 1 h, v h sin sinv Ww cos kz 1 1 (3.8)

38 26 where 0h L1 and 0 v 360 ; d S The equation of surface hv S in the cutter coordinate system is 2, x2 h, v h, v y h, v z2 h, v 1 1 w w 1 L h L cos sinsinv W h L 2 2 L h L cos cos cosv sin cos sinv W h L sin sin sin x cos L h L cos sin cosv cos cos sinv W h L sin cos sin xsin 1 1 w 1 sin cos kz (3.9) where L1h L2 and 0 v 360 ; The equation of surface hv S in the cutter coordinate system is 3, d S x3 h, v z3 h, v 3 3 sin sin L cos h, v y h, v 3 3 Rw cos cosv sin cos sinv L h sin sin xcos Rw sin cosv cos cos sinv L h cos sin xsin Rw v 3 h kz (3.10) where L2h L3 and 0 v 360 ; The equation of surface S hv d S 4, in the cutter coordinate system is x4 h, v L4 h cos cosv sin cos sinv x cos h, v y h, v L h sin cosv cos cos sinv x sin (3.11) z4 h, v L4 h sin sinv kz where L3h L4 and 0 v Parametric equation of effective rindin ede In the multi-axis CNC rindin of a flute, a complex volume swept by the wheel while

39 27 movin and rotatin alone the cutter, and the stock material within the volume is removed. Eventually, a flute is formed. Geometrically, the volume outside surface is the envelope of all the eometries of the wheel at different locations and in different orientations. In this work, the volume is called wheel swept volume, and its surface is called wheel swept surface. Usin the well established envelope theory, the wheel swept surface can be formulated. Specifically, at a moment of the machinin, there are a roup of wheel points, at each of which the wheel surface normal is perpendicular to the instantaneous wheel feedin direction. These wheel points define a curve on the wheel that is called effective rindin ede in this work. The effective rindin ede is the curve of the wheel swept surface at that moment. Thus, in the cutter coordinate system, the eneral equation of effective rindin ede is h, v h, v h, v d d d S S S 0 h v (3.12) Since the wheel includes four surfaces, the effective rindin ede consists of four pieces, and their equations are For 0h L1, the effective rind ede equation is f1 h, v hsin cosv xsin k zcos 0. (3.13) For L1h L2, the effective rindin ede is, cos cos sin sin sin L cos sin L cos cos sin f h v k k v v v 2 z z 1 1 W sin cosv sin hsin cosv x cos sin sinv x cos sin 0 w (3.14). For L2h L3, the effective rindin ede is

40 28 f h, v L sin cosv k sin sinv hsin cosv x cos sinv 0.(3.15) 3 3 z For L3h L4, the effective rindin ede is 2 2, L L L cos cos f h v x sin k cos v sin h v sin 4 4 z 4 4 k hcos x hsin 2L hcosv sin 0 z 4. (3.16) With the four equations, the four pieces of an effective rindin ede at a machinin time can be found. By repeatin this step at all the machine times, the wheel swept surface can be found, thus, the flute is represented. In eneral, in the multi-axis CNC rindin of flutes, the wheel orientation remains the same in the process, so the effective rindin ede keeps the same shape alon the flute. The effective rindin ede has to be found once, instead of at every machinin time. Therefore, the machined flute can be enerated by sweepin the effective rindin ede alon the cuttin ede Flute Cross-Sectional Profile In this work, an actual flute cross-sectional profile refers to the intersection between the machined flute model and a plane perpendicular to the cutter axis. An effective rindin ede is a 3-dimensional curve, and the actual flute cross-sectional profile is a 2-dimensional curve. In practice, the cutter sometimes is desined, in which the flute cross-sectional profile is iven. To check the machinin accuracy, the desin and the actual flute cross-sectional profiles are compared to find their maximum deviation. With the equation of the effective rindin ede of the wheel at a machinin time, the mathematical model the flute profile at the d z equal to zero can be derived. Similarly, the profile consists of four sements due to the four wheel surfaces.

41 29 For 0h L1, and, the parameters of the points of the flute profile can found by solvin the system of equations, Eq. 3.16, f1 h, v 0. (3.17) h sin sinv Ww cos kz 0 For L1h L2, and, the parameters of the points of the flute profile can be found by solvin the system of equations, Eq. 3.17, f2h, v w 1. (3.18) L h L cos sin sinv W h L sin cos kz 0 For L2h L3, and, the parameters of the points of the flute profile can be found by solvin the system of equations, Eq. 3.18, 3 f h, v 0. (3.19) Rw sin sinv L3 h cos kz 0 For L3h L4, and, the parameters of the points of the flute profile can be found by solvin the system of equations, Eq. 3.19, f4 h, v 0. (3.20) L4 hsinsinvkz 0 After solvin the systems of equations, the parameters of the profile points can be attained, and the coordinates of the profile points in the cutter coordinate system can be calculated by substitutin the parameter values to Eqs Application of the Direct Method The direct method of rindin flutes is rendered with all theoretical formula in the

42 30 above sections. To demonstrate its validity, the direct method is applied to several practical examples. In these examples, a solid carbide twist drill with the diameter of 20 mm and the cuttin ede helix anle of 30 derees is adopted. Three rindin wheels with different parameter values are used to machine the flutes of the twist drill. The parameter values of these wheels are listed in Table 3.1, and the illustrative diaram of the wheel is plotted in Fi Table 3-1 The parameter values of the rindin wheels used to cut the drill flutes Grindin wheel Wheel radius Wheel width Wheel outer width Wheel anle No. R w (mm) W w (mm) W o (mm) (de.) Z R w H 0 H 1 Outer width W w Ф H 2 X H 4 H 3 Fiure 3-4 The parameters of the rindin wheel profile.

43 z 31 In the first example, the No. 1 rindin wheel is used to machine the drill flute. The enerated flute and the cross section of the flute are shown in Fi Cross section of helical drill at r iz = 0 Wheel anle = 45 o y Y Z X x Cross section of helical drill at r iz = 0 Wheel anle = 45 o 10 5 Y X 0 Fiure 3-5 The three-dimensional flute and its cross section of the drill machined usin the No. 1 rindin wheel. In the second example, the No. 2 rindin wheel is used to machine the drill flute. The enerated flute and the cross section of the flute are shown in Fi. 3-6.

44 z 32 Cross section of helical drill at r iz = 0 Wheel anle = 55 o y Y Z X x Cross section of helical drill at r iz = 0 Wheel anle = 55 o 10 5 Y X 0 Fiure 3-6 The three-dimensional flute and its cross section of the drill machined usin the No. 2 rindin wheel. In the second example, the No. 3 rindin wheel is used to machine the drill flute. The enerated flute and the cross section of the flute are shown in Fi. 3-7.

45 z 33 Cross section of helical drill at r iz = 0 Wheel anle = 65 o y Y Z X x Cross section of helical drill at r iz = 0 Wheel anle = 65 o 10 5 Y X 0 Fiure 3-7 The three-dimensional flute and its cross section of the drill machined usin the No. 3 rindin wheel. 3.3 Inverse Method of Machinin Flutes In the tool manufacturin industry, sometimes cutters are desined with the flute

46 34 cross-sectional profiles, and the manufacturin tolerances of the flutes are hih. To accurately make the flutes, the direct method of machinin flutes usin standard rindin wheels cannot realize this oal; thus, the inverse method is necessary. The inverse method of machinin flutes is to compute the rindin wheel profile based on the flute desin, make a non-standard wheel with the calculated profile, and rind the flute with a set of appropriate cuttin parameters, in order to achieve hih flute accuracy. The kernel technique of findin the wheel profile is the conjuate theory between the virtual rindin wheel and the desined flute. This method is introduced in the followin Formula of the Flute Surface and its Normal In eneral, a flute surface is a curve (or multiple curves) on the cutter cross section sweepin alon the cuttin ede. A flute surface is shown in Fi In this work, a d d d cutter coordinate system x y z is established in a way that the z d axis is alon the cutter axis from the bottom to the top and the x d and the yd axes are on the cross section. Here, a eneric mathematical representation of the desired flute surface F rv, in the cutter coordinate system is adopted as d Fx d rv, F dr, v Fy d r, v, (3.21) Fz d rv, where the parameter r represents the radial variable and the parameter v represents the rotatin variable. So the equation of the normal vector N d rv, of the flute surface is

47 35 i j k Nx d rv, Nd Nd Fxd Fyd Fzd N dr, v Ny dr, v. (3.22) r v r r r Nz d rv, Fxd Fyd Fzd v v v Fiure 3-8 A contact curve of the flute surface shown with the rindin wheel not shown Conjuate Relationship between the Flute and the Wheel In the reverse method of machinin the flutes, the wheel profile is to be determined (see Fi. 3-9), while, at beinnin, the wheel coordinate system xyz is defined with the x axis in line with the wheel axis. Accordin to the kinematics of the multi-axis CNC rindin of flutes, the wheel, toether with its coordinate system, can be represented d d d in the cutter coordinate system x y z. For this purpose, it is assumed that the rindin wheel coordinate system xyz is coincided with the cutter coordinate d d d system x y z before rindin. Durin machinin, the rotation anle of the cutter chanes, and the wheel location and orientation chane accordinly. The specific

48 36 steps are provided in the followin. Z Generative Curve X(h) Z(h) X Y Fiure 3-9 The profile of a non-standard rindin wheel. The wheel coordinate system is rotated about its z axis by. The rotation cos sin 0 0 sin cos 0 0 M R matrix is z, The wheel coordinate system is translated alon its x axis by x, which is equal to Rw c x r k. k x is related with the taper anle of the cutter. The translation x M T matrix is x,0,0 The wheel coordinate system is rotated about its R rotation matrix is x, cos sin 0 0 M. sin cos The wheel coordinate system is rotated about its x axis by anle. y axis by anle. The The

49 37 R rotation matrix is y, cos 0 sin M. sin 0 cos Usin the Euler rule, the equivalent matrix of the four transformation matrices can be derived as R T R R,,, k, k z, x,0,0 x, y, d M x z M M M M cos cos sin sin sin sin cos cos sin sin sin cos x cos sin cos cos sin sin cos cos sin sin cos sin cos x sin. cos sin sin cos cos (3.23) A point on the z axis is represented as T z in the wheel coordinate system, and it can be represented in the cutter coordinate system as 1 b1 c1 d b2 c z d z 2 M,,, kx, kz, (3.24) b3 c 3 1 where b1 cos sin sin sin cos, b2 sin sin cos sin cos, b3 cos cos, c1 x cos, c2 x sin, and c3 0. Durin machinin, the wheel contacts with the flute at an effective rindin ede, thus, they are conjuate with each other. Accordin to the conjuate theory, at the points where the wheel contacts with the flute surface durin machinin, the normals to

50 38 the two surfaces are in line. Since the wheel surface is a revolvin surface, a normal to the wheel surface passes throuh the wheel axis. Therefore, for the flute points, the normals to the desined flute surface at these points pass throuh the wheel axis. The conjuate relationship can be formulated in the cutter coordinate system so that the contact curve on the flute surface or the effective rindin ede can be found. The intersection point between a normal to the desined flute surface at the contact point and the wheel axis can be represented as r v 2 dr v r v r v Fx d, Nx, Fd r, v 2 N d r, v Fy d r, v 2Ny d r, v, (3.25) Fz d, 2 Nz d, Where 2 is distance between the contact point and the intersection point. Then, the equation of the conjuate relationship is d r, v r, v z F N. (3.26) d 2 d be derived. This equation can be represented in the scale form as d d 1 b1 c1 Fx d r, v 2 Nx r, v 1 b2 c 2 Fy d r, v 2 Ny r, v. (3.27) b c Fz r, v Nz r, v d 2 d Solvin the above equations by eliminatin 1 and 2, the followin equation can c c b c b c Nx Fz Nz Fy Fz Fy b c Fx Nx d d 3 d d 2 2 d 3 3 d d b3 Nyd b2 Nzd b3 Nyd b2 Nzd d. (3.28) With this equation, the relationship between the two parameters, r and v, is

51 39 attained, and the contact curve on the flute surface can be found, which can be represented as Fx d r, v r C d Fy d r, v r. (3.29) Fz d r, v r Grindin Wheel Profile The main objective of the inverse method is to find the rindin wheel profile so that a non-standard rindin wheel can be made for machinin the desined flute. The contact curve has been found in the above section in the cutter coordinate system. Since the contact curve on the flute surface and the effective rindin ede on the wheel surface are the same, the equation of the effective rindin ede in the wheel coordinate system can be found by usin the inverse kinematics of the flute machinin. Based on Eq of the kinematics of the flute machinin, the equation of the inverse kinematics is R R T R,,, k, k y, x, x,0,0 z, d M x z M M M M cos cos sin sin sin sin cos cos sin sin cos sin x cos sin cos cos cos sin 0. cos sin sin sin cos sin sin cos sin cos cos cos x sin (3.30) Therefore, the effective rindin ede can be found as r r r Ex E Ey dmc d. (3.31) Ez

52 40 The wheel surface can be expressed as r r 2 2 Ex Ey cos Wx r, 2 2 W r, Wy r, Ex r Ey r sin. (3.32) Wz r, Ez r Fiure 3-10 The effective rindin ede of the virtual rindin wheel Grindin Wheel Profile Based on the representation of the wheel surface in the wheel coordinate system, the wheel profile is the intersection curve between the wheel surface and the principle plane xz. Thus, the rindin wheel profile can be formulated as Exr Eyr 2 2 Px r P r Pyr 0. (3.33) Pzr Ezr

53 Applications of the Inverse Method The inverse method of rindin flutes is provided in the above sections. To show its validity, the inverse method is now applied to three examples. In the first example, a drill is desined; its diameter is 10 mm, its web thickness is 2 mm (or the core radius is 1 mm), the point anle is 140 derees, the helix anle of the flute is 30 derees. The maximum rindin wheel radius is specified as 29 mm. The cross sectional profile of the flute is provided. By usin the inverse method, the contact curve between the wheel and the iven flute is first found; the curve actually is the effective rindin ede of the wheel. Then, the cross sectional profile of the wheel is determined. As results, they are plotted in the followin diarams. Fiure 3-11 A contact curve between the flute surface shown and the rindin wheel not shown in the first example.

54 42 Fiure 3-12 The effective rindin ede of the wheel in the first example. Fiure 3-13 The cross sectional profile of the wheel in the first example. In the second example, the same drill as that in the first example, except the point

55 43 anle is 120 derees. in the first example. The rindin wheel radius is 39 mm, which is larer than the wheel Similarly, the inverse method is applied, and the contact curve between the wheel and the iven flute (or the effective rindin ede of the wheel) and the cross sectional profile of the wheel are found. Here, they are shown in the followin diarams. Fiure 3-14 A contact curve between the flute surface shown and the rindin wheel not shown in the second example.

56 44 Fiure 3-15 The effective rindin ede on the wheel in the second example. Fiure 3-16 The cross sectional profile of the wheel in the second example. In the third example, the drill is the same as that in the first example, except the point anle is 140 derees in this example. The rindin wheel used in this example is 59 mm in radius, which is twice as lare as that in the first example. After the inverse method is applied, and the contact curve between the wheel and the iven flute (or

57 45 effective rindin ede of the wheel) and the cross sectional profile of the wheel are found. Here, they are shown in the followin diarams. Fiure 3-17 A contact curve between the flute surface shown and the rindin wheel not shown in the third example. Fiure 3-18 The effective rindin ede on the wheel in the third example.

58 46 Fiure 3-19 The cross sectional profile of the wheel in the third example. With the profile of the rindin wheel determined, a correspondin rindin wheel can be made. Then it can be used to rind the desined flute with hih accuracy. The next steps are the same as the steps in the direct method, and they are not elaborated here. 3.4 Example of Geometrically Modelin of the Flutes Based on the above analysis, several MATLAB prorams are desined for findin the points on the effective rindin ede by a standard straiht wheel. The data of the points is then input to the CATIA. Usin B-spline curves to connect these points, the effective rindin edes are then built in CATIA. Sweepin these EGEs, alon with the rindin wheel position, the flute surface is finally created in CATIA accurately.

59 47 The parameters used in prorammin and eometrically modelin are list below. Table 3-2 Parameters in flute modelin and prorammin Drill diameter Core diameter Grindin wheel Helix anle Wheel (mm) (web diameter (mm) ( ) inclination thickness) anle ( ) (mm) Effective rindin ede obtained by MATLAB prorammin Fiure 3-20 shows the effective rindin edes (EGEs) enerated by a parallel rindin wheel. The red curves are enerated by the corner and the cylindrical surface of the rindin wheel. The straiht blue line is by the wheel side surface.

60 48 Fiure 3-20 Effective rindin ede enerated by a parallel rindin wheel Input to CATIA Fiure 3-21 shows the EGE (red color) in 3-D CATIA model. The white points are input from the data calculated by prorammin. The red EGE sweeps alon the drill's helical curve, whose color is yellow, and forms the main flute, which is also an envelope surface.

61 49 EGE point EGE Fiure 3-21 EGE in CATIA The yellow curve in Fiure 3-22 is a helical curve with the lead same as the drill helical flute. Fiure 3-22 EGE and helical curve

62 50 Fiure 3-23 shows the EGEs at different positions. The envelope surface is formed by sweepin the EGE alon the helix curve of the drill. Sweepin surface Fiure 3-23 The envelope surface swept by EGE To et the final flute, we need to consider the surface of the rindin wheel at final position. Here, the kinematics of the rindin process is also built in 3-D model. The movement of the rindin wheel coordinate system is shown in Fiure 3-24 to Fiure First step, as shown in Fiure 3-24, the wheel coordinate system coincides with the drill coordinated system (the orane color) and then translates alon its axis z to a new position (blue color) by the flute lenth z (here = 60 mm).

63 51 Fiure 3-24 Flute modelin: Step 1 Second, the blue coordinate system rotates about its z-axis by θ as shown in Fi The new wheel coordinate system is shown in red. The relationship between the flute lenth and the anle is z D tan drill 360 Substitutin the values of the flute lenth (deree). z, the helix anle, and the drill diameter D drill, the value of θ is obtained as 60mmtan mm

64 52 Fiure 3-25 Flute modelin: Step 2 Third, the red coordinate system rotates about its x-axis by anle λ as shown in Fiure The new coordinate system is shown in white. Normally, the anle λ is set to around 90 so that the wheel orientation corresponds the helix direction. Fiure 3-26 Flute modelin: Step 3

65 53 Fourth, the wheel coordinate system (white) rotates about its y-axis by α. The new wheel coordinate system is shown in yellow (Fiure 3-27). Fiure 3-27 Flute modelin: Step 4 Finally, the yellow coordinate system translates alon its x-axis by x. the new one is shown in red (Fiure 3-28). This final coordinate system (red) defines the wheel's final position and orientation. The x is calculated accordin to the followin formula Dwheel Dcore 1 x 2 2 cos Substitutin the values of the rindin wheel diameter D wheel, the core diameter D, and the anle, the value of x is core

66 54 150mm 3mm 1 x 2 2 cos mm Fiure 3-28 Flute modelin: Step 5 Since the connection area of the end of the flute and the drill body is formed by a pure wheel surface, we need to draw a wheel at its final position. Fiure 3-29 draws the rindin wheel at its final position.

67 55 Wheel surface Fiure 3-29 Flute modelin: Step 6, addin wheel surface at final position Addin the final wheel surface, as shown in Fiure 3-29, to the envelope surface obtained before, and doin a split operation (splittin the volume from the drill body by the envelope surface and the final rindin wheel surface), the final flute (reen) is then obtained in Fiure 3-30.

68 56 Fiure 3-30 Flute modelin: Step 7, create final flute shape

69 57 Chapter 4 Flank Modelin In eneral, the eometry of a twist drill tip includes the flank faces and the chisel ede, which is the intersection curve between the flank faces. Since some drills have ashes and some do not, the ash is rearded as a different feature, which will be discussed in the followin section. The drill tip is a feature that sinificantly affects the drill performance and life, so it is crucial to correctly construct the drill tip eometry in the solid model of the cutter. Currently, the drill tip flanks are classified into four main types accordin to their shapes, planar, quadratic, helical, and multi-facet flanks. The planar flanks include sinle plane flanks and multi-plane flanks; the quadratic flanks include conical, cylindrical, ellipsoidal (Racon), and hyperboloidal flanks; the helical flanks include constant and non-constant helix anle flanks; and the multi-facet flanks are special flanks which combines multi-type flanks in one point. Different flanks are made with different manufacturin methods. Since the flank faces determine the cuttin ede shape, the drill point anle, and the relief anle, it is very important to model the manufacturin methods in order to build the solid flank models in accordance to the actual flanks. Eventually, the chisel ede is found automatically as the intersection between the flanks. Therefore, parametric modelin of the twist drill flank is vital to drill simulation. Since the planar model is simple, this work will discuss the mathematical models mainly for a popular quadratic flank, conical flank. The eometric models will be built for conical and planar flank.

70 Mathematical models of the flank Mathematical model for the quadratic flanks The eneral mathematical model of the quadratic flanks is x y z 0 (4.1) a a c where, for conical flank, 1, a 0, c 0 a and set tan ; c for hyperboloidal flank, 1; for ellipsoidal flank, 1; and for cylindrical flank, 0. Specially, for conical flank the points in cone coordinate system has the followin relationship x y z tan 0 (4.2) The point on conical flank in drill coordinate system has the followin relationship

71 59 x cos y sin S d d 2 xd sin yd cos cos zd sin d tan S tan xd sin yd cos sin zd cos d 0 2 (4.3) The detailed processes to derive this equation will be shown in section 4.3, 'Parametric modelin of the conical flanks of the drill tip'. Fiure 4-1 Conical flank and rindin cone Mathematical model for the planar flanks Another popular drill point has planar flank. The rindin wheel coordinate system coincides with the drill system at the beinnin. Followin a series translation and rotations, the wheel end face ets to the flank position and be able to work out the flank

72 60 face. The transformation matrix is d T Trans( s,0,0) Rot( x, / 2 ) Rot( y, ). The transformation and rotations are shown in the parametric modelin section. From the transformation matrix, the mathematical model of planar flank is S( u, v) T S u, v d d cos 0 cos s u cos sin sin cos cos 0 v sin sin cos sin cos s ucos ucos sin vsin usin sin vcos 1 (4.4) 4.2 Parametric modelin of the conical flanks of the drill tip To build the parametric model of the conical flanks of the twist drill tip, the 'half point anel', 'half cone anle', 'distance from the cone vertex to drill tip', 'skew distance from drill axis to cone axis', and 'chisel ede anle' are taken as the parameters. Table 4-1 Parameters for conical flank modelin Parameters Definition value Drill half point anle 70 o

73 61 Half cone anle 45 o Distance from the cone vertex d to drill tip measurin alon the cone axis 10 mm S Skew distance between the drill axis and the cone axis 3 mm Chisel ede anle 100 o After the rindin wheel is properly oriented, a conical flank can be rinded with the side of the wheel in a path. First, the cone coordinate system coincides with the drill coordinate system. z y x Fiure 4-2 Conical flank modelin: Step 1 Second, the cone and its coordinate system rotates about its x-axis about.

74 62 ϕ x Fiure 4-3 Conical flank modelin: Step 2 Third, the cone and its coordinate system translates alon its z-axis with d. z d Fiure 4-4 Conical flank modelin: Step 3 Fourth, the cone and its coordinate system translates alon its x-axis with S.

75 63 S x Fiure 4-5 Conical flank modelin: Step 4 Fifth, the cone and its coordinate system translates alon its y-axis with d y. dy z y x Fiure 4-6 Conical flank modelin: Step 5

76 64 O1 O2 dy O3 A θ B C E d dy D S Fiure 4-7 Calculation of d y It is important to define the tip of the drill point as the oriin of the drill system. However, the tip is formed by the two cone surfaces after rindin the two flanks. So, the cone surfaces should pass throuh the oriin, say, point A, of the drill system. To ensure this requirement, the translation d y is calculated as below. AD and BD are the radius of the white circle. DE is the stew distance S. Thus, AD BD d tan DE S

77 65 y 2 2 d tan S 2 2 d CD AE AD DE (4.5) Substitutin the values of d, and S, d y 9.539mm tan Finally, to et better distribution of relief anles alon the cuttin lip, the cone surface and it coordinate system rotate about z d with anle.as shown in Fiure 4-8. The red, reen, and blue squares represent xz, xy, and yzplanes, respectively. ω Fiure 4-8 Conical flank modelin: Step 6 Accordin to the above process modelin, the transformation matrix from the drill coordinate system to the final cone coordinate system is

78 66 M M ( S,0,0) M (0, d,0) M (0,0, d) M ( x, ) M ( z, ) T T T R R d y cos sin 0 S cos sin cos cos sin tan sin sin sin cos cos d d S (4.5) S M * S d d x cos sin 0 S xd 2 2 y cos sin cos cos sin dtan S y d z sin sin sin cos cos d z d x cos y sin S d d sin d coscos d sin tan x sin y cossin cosd d d d 2 2 x y z d S 1 zd (4.6) Callin Equation (4.2), the mathematical model for the conical flank is x cos y sin S d d 2 xd sin yd cos cos zd sin d tan S tan xd sin yd cos sin zd cos d 0 2 (4.7) 4.3 Parametric modelin of the planar flanks To build a parametric model of the planar flanks of the twist drill tip, the relief and the point anles are taken as the parameters. After the rindin wheel is properly oriented,

79 67 a flat flank can be rinded with the side of the wheel in a path. Since the manufacturin process is quite simple, its model can be easily established. The main steps of constructin a planar flank of the drill tip include four steps, which are described in the followin. Assumin the cutter coordinate system initially is set up at the center of the bottom plane of the cutter and its Z axis is alined with the cutter axis and points towards the tool shank. First step, this coordinate system is translated alon its positive X axis direction by a half of the web thickness. This coordinate system at the two locations is plotted in the followin diaram. Y Y X X Z Fiure 4-9 The cutter coordinate system is translated alon its positive X axis direction by a half of the web thickness. Second, rotate the cutter coordinate system about its X axis by a half of the drill point anle, shown in the followin diaram.

80 68 Y Y Z X Z X Fiure 4-10 The cutter coordinate system is rotated around its X axis by a half of the drill point anle. Third, the cutter coordinate system is rotated around its Y axis by the relief anle. X Y Y Z X Z Fiure 4-11 The cutter coordinate system is rotated about its Y axis by the relief anle. The plane passin throuh the X and the Y axes represent the flank face of the twist drill tip. The plane is shown in red in the followin diaram, and, by applyin the Boolean

81 69 operation - removin, a flank face is finally enerated in the solid model of the drill. Fiure 4-12 The final rindin wheel position Fiure 4-13 The solid model of a twist drill with a flank face enerated.

82 70 Chapter 5 Drill Split Modelin 5.1 Introduction The twist drill performance is often measured with the thrust force and the torque durin machinin. Amon the features that affect the drill performance, the chisel ede is particular sinificant. Since the rake anle of the chisel ede is neative, durin drillin, the chisel ede locates the drill at a position on the part surface and enerates a lare amount of thrust force. Therefore, the lenth of the chisel ede should be optimized for a drill with a lon chisel ede enerates a lare thrust force and a drill without the chisel ede cannot be located while machinin. To address this problem, the drill split (also called the drill ash) is widely adopted in the drill manufacturin industry. The main advantae of drill split is that the chisel ede is shortened and a secondary cuttin ede is enerated with positive rake anles. Thus, the drill split can effectively solve the problems of conventional twist drills. In this thesis, the parametric model of the drill split is established; the detailed procedure will be introduced in the followin section; and, based on the machinin features, a mathematical model of the drill split is firstly proposed. By optimizin the drill split parameters, the best drill performance can be achieved. 5.2 Parametric Modelin of the Drill Split

83 71 The parametric model of the drill split includes some parameters, such as the ash anle to XY plane, the ash axis, the location of the startin point, the rake anles at the startin and the endin points of the ash (axial rake at tip and axial rake at center), and the exit anle (walk anle) etc. these parameters defines the ash features. For example, the ash axis is determined by the 'S ash offset', 'S ash radius', and 'Gash anle in XY plane'. Based on the drill split parameters, the drill split can be modeled with CATIA V5. Table 5-1 Parameters for drill split modelin Parameters Value used in this model 1 Basic ash anle 55 Gash anle in XY plane Axial rake at tip Axial rake at center S ash radius S ash offset Walk anle Walk lenth Gashin wheel diameter mm 0.1 mm mm 120 mm

84 Construction of the Drill Split At the startin point of the drill ash, draw a helix uide with the helix anle equal to the ash anle and around the ash axis. The helix stops at the endin point of the ash. The helix uide is shown in white in the followin diaram. Fiure 5-1 The helix uide is drawn in order to machine the ash. Accordin to the rake anle at the startin point of the ash, draw the rindin wheel. Based on the rake anle at the endin point of the ash, the rake face can be enerated by sweepin the effective rindin ede alon the helix uide. The rake face is shown in the followin diaram in blue.

85 73 Fiure 5-2 The rindin wheel is located at the startin point of the drill split. Accordin to the exit anle of the ash, a uide curve is defined shown in the followin diaram. Based on the uide, the effective rindin ede of the wheel is found, and the surface is swept alon the uide curve is enerated. Usin the Split operation, remove, the ash and the second flank face are enerated, which are shown in the followin diaram.

86 74 Fiure 5-3 The second flank is enerated by removin the drill stock with the surface swept by the effective rindin ede alon the uide curve. The drill split (or ash) and its second flank are enerated, and a drill with a ash is shown in the followin diaram. Fiure 5-4 A drill with a blue ash and the orane second flank are shown

87 Mathematical model of the ash Assumin the ashin axis is a unit vector k k k position P p p p axis of the helical uide line is T K at the 1 2 3, the transformation matrix for helically rotatin around the 1 2 3, R Trans p p p Rot K Trans K Trans p p p K cos sin 1 cos s k k1 1 cos cos k1k2 1 cos k3 sin k1k3 1 cos k2 sin p k1 p1 k1k2 1 cos k3 sin k2 1 cos cos k2k3 1 cos k1 sin p k 2 2 p 2 k1k3 k2 k2k3 k in 3 1 cos cos p k 3 3 p Let then 1 2 3, Trans p p p Rot K a11 a12 a13 p1 a a a p a31 a32 a33 p T R K a11 a12 a13 p k1 p1 a21 a22 a23 p k2 p 2 a31 a32 a33 p k3 p p a11 a12 a13 a11 k1 p1 a12 k2 p2 a13 k3 p3 p1 a21 a22 a23 a21 k1 p1 a22 k2 p2 a23 k3 3 2 p a31 a32 a33 a31 k1 p1 a32 k2 p2 a33 k3 p3 p (5.1) The formula (5.1) is the eneral transformation matrix for ashin process. Assumin the axis of the helical uide line (denoted by Z ash ) is parallel to the end surface of the rindin wheel (denoted by S end ), the rake surface of the ash then can be simplified as a swept surface created by a straiht-line profile, which is parallel to the Z ash and on the S end, sweepin alon a helical uide curve.

88 76 S then The straiht-line profile is: L q mk staiht q mk q mk q mk The swept surface or the rake face, S, of the ashin is: r, r, R L K straiht a a a a k p a k p a k p p a a a a k p a k p a k p p a a a a k p a k p a k p p It is a surface with two variables For example, if S r, Sr,, m K P p T 1 p2 0 T, q q q m k mk 2 m k (5.2) 1 2 3, R Trans p p p Rot K Trans K Trans p p p K cos sin 0 p p1 sin cos 0 p p cos sin 0 p1 1 cos p2sin sin cos 0 p2 1 cos p1 sin ,

89 77 L q mk staiht q1 q2 q3 m, and, S r, R L K straiht cos sin 0 p1 1 cos p2sin q1 sin cos 0 p2 1 cos p1 sin q q3 m Thus, S r, R L K straiht q1 cos q2 sin p1 1 cos p2 sin xr, q1 sin q2 cos p2 1 cos p1 sin yr, q3 m zr, m, 1 1 (5.3) It can be seen that r r , 1, x p y p q p q p const (5.4) The above equation shows that the S, 2 2 radius equal to q p q p r is a part of a cylinder surface with the 5.3 Final model of manufacturin feature-based twist drill In this thesis, the other features of twist drill like the lands, the internal coolin holes, and etc., are relatively simple; and they are not shown in details althouh they were completed in final model. Fiure 5-5 shows the drill point with ashin done.

90 78 Fiure 5-5 Drill point with ashin done Fiure 5-6 shows the final parametric model for machinin feature-based twist drill. The eometric features, such as the body, the flutes, the flanks, the radius splits, the internal coolin holes, and the land, are well completed. This model exactly reflects the rindin features if the rindin errors, such as vibration errors, are inore.

91 79 Fiure 5-6 Final machinin feature-based eometric model of twist drill