\uf8f7\uf8f7 \uf8f8 \uf8f6 \uf8ec\uf8ec \uf8ed \uf8eb \u2212 <\uf8f7\uf8f7 \uf8f8 \uf8f6 \uf8ec\uf8ec \uf8ed \uf8eb \u2212\u2212 = 0,1arctan 2 1 0,1arctan 2 1 )arctan( yfor y yfor yy pi pi Thus the Eq. (5.25) can be written in the following form ( ) 0sin22 =\u2212++ UUUU CBA \u03ba\u03b8 (5.28) where ( )UUU BA \/arctan=\u03ba . Solving Eq. (5.28) for \u03b8 yields ( ) \uf8f4 \uf8f4 \uf8f4 \uf8f3 \uf8f4\uf8f4 \uf8f4 \uf8f2 \uf8f1 \u2212 \uf8f7 \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ec \uf8ed \uf8eb + \u2212 \u2212 \uf8f7 \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ec \uf8ed \uf8eb + = \u2212 \u2212 U UU U U UU U u BA C BA C tc \u03bapi \u03ba \u03b8 22 1 22 1 sin sin , (5.29) When Eq. (5.29) is plugged into Eq. (5.15-a), for a given cutter location point the envelope boundary of the upper-cone surface is obtained. 5.3.2 Envelope Boundary of the Corner-Torus The velocity of arbitrary point IT on the toroidal surface (Figure 5.7) can be expressed in MCS by ( ) \u2192\u00d7+= TTM IFI \u03c9VV FT (5.30) Plugging Eq. (5.14-b) into Eq. (5.30) yields ( ) ( ) ( ) ( ) ( ) ( ) ( )L LLFT sinsin cossincos y\u03c9 x\u03c9z\u03c9VV \u00d7++ \u00d7++\u00d7\u2212+= \u03b8\u03c6 \u03b8\u03c6\u03c6 c cccT M rR rRrhI (5.31) Chapter5. Feasible Contact Surfaces 112 For obtaining the envelope boundary of the corner-torus, Eqs. (5.31) and (5.17-b) are plugged into the following equation ( ) ( ) ( ) 0, TT == TMTMT II,tf N\u00b7V\u03b8\u03c6 (5.32) and considering the vector operations the above equation yields ( ) 0sincos, =\u2212+= TTTT CBA,tf \u03b8\u03b8\u03b8\u03c6 (5.33) where ( ) ( ) ( )LLFL cossinsin y\u00b7\u03c9y\u00b7\u03c9V\u00b7x \u03c6\u03c6\u03c6 ++= cT hA ( )FLsin V\u00b7y\u03c6=TB ( )FLcos V\u00b7z\u03c6=TC Eq. (5.33) can be written in the following form ( ) 0sin22 =\u2212++ TTTT CBA \u03ba\u03b8 (5.34) where ( )TTT BA \/arctan=\u03ba . Solving Eq. (5.34) for \u03b8 yields ( ) \uf8f4 \uf8f4 \uf8f4 \uf8f3 \uf8f4\uf8f4 \uf8f4 \uf8f2 \uf8f1 \u2212 \uf8f7 \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ec \uf8ed \uf8eb + \u2212 \u2212 \uf8f7 \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ec \uf8ed \uf8eb + = \u2212 \u2212 T TT T T TT T BA C BA C t \u03bapi \u03ba \u03c6\u03b8 22 1 22 1 sin sin , (5.35) When Eq. (5.35) is plugged into Eq. (5.15-b), for a given cutter location point the envelope boundary of the corner-torus surface is obtained. Chapter5. Feasible Contact Surfaces 113 5.3.3 Envelope Boundary of the Lower-Cone The velocity of a point IL on the lower-cone surface (Figure 5.7) is given by ( ) \u2192\u00d7+= LLM FII \u03c9VV FL (5.36) The Eq. (5.36) is expanded by substitution of Eq. (5.14-c) as ( ) ( ) ( ) ( )L LLFL sintan costan y\u03c9 x\u03c9z\u03c9VV \u00d7+ \u00d7+\u00d7+= \u03b8\u03b1 \u03b8\u03b1 l llL M c ccI (5.37) Eqs. (5.37) and (5.17-c) are plugged into the following equation ( ) ( ) ( ) 0, LL == LMLMlL IItc,f N\u00b7V\u03b8 (5.38) and after the vector operations the envelope boundary of the lower-cone is obtained in the closed form ( ) 0sincos, =\u2212+= LLLlL CBAtc,f \u03b8\u03b8\u03b8 (5.39) where ( ) ( ) ( )LFLL coscostansin y\u00b7\u03c9V\u00b7xy\u00b7\u03c9 llL ccA \u03b1\u03b1\u03b1\u03b1 ++= ( )FLcos V\u00b7y\u03b1=LB ( )FLsin V\u00b7z\u03b1=LC The Eq. (5.39) is transformed into the following form ( ) 0sin22 =\u2212++ LLLL CBA \u03ba\u03b8 (5.40) where ( )LLL BA \/arctan=\u03ba , and it is solved for \u03b8 as Chapter5. Feasible Contact Surfaces 114 ( ) \uf8f4 \uf8f4 \uf8f4 \uf8f3 \uf8f4\uf8f4 \uf8f4 \uf8f2 \uf8f1 \u2212 \uf8f7 \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ec \uf8ed \uf8eb + \u2212 \u2212 \uf8f7 \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ec \uf8ed \uf8eb + = \u2212 \u2212 L LL T L LL L l BA C BA C tc \u03bapi \u03ba \u03b8 22 1 22 1 sin sin , (5.41) When Eq. (5.41) is plugged into Eq. (5.15-c), for a given cutter location point the envelope boundary of the lower-cone surface is obtained. 5.4 Analyzing the Distribution of Feasible Contact Surfaces In the previous section the generic cutter surfaces have been partitioned into three boundary point sets by utilizing the tangency function. Then for each surface patch the envelope boundary which defines the limits of the feasible contact surfaces (FCS) has been formulated by calculating the parameter \u03b8. For a given cutter geometry, any change in the tool tip velocity VF and the angular velocity \u03c9 will effect the range of \u03b8 and in turn the location of the point set CWEK which represent the feasible contact surfaces. Therefore the cutter motion type has direct effect in the construction process of FCS. In 5-axis machining the cutter rotation axis can be tilted in any direction. In general during machining the side face of the cutter has instant contact with the in-process workpiece and the engagement angle covers the angular range of [00, 1800]. For example in Figure 5.8(a-b) Taper-End and Ball-End mills are performing 5-axis motions and only the side faces of these cutters contain the point set CWEK. But on the other hand in 3-axis milling the tool rotation axis is fixed and with respect to the tool tip velocity, more cutter surfaces may be involved in machining. This is illustrated for 3-axis plunge motion in Figures (5.8-c) for a Flat-End mill and ((5.8-d) for a Ball-End mill respectively. Chapter5. Feasible Contact Surfaces 115 (a) (b) (c) (d) Figure 5.8: Instantaneous cutter contact surfaces in (a,b) 5-axis, and in (c,d) 3-axis plunge motions Because 3-axis motion can cover more contact surfaces, in this section the distribution of the FCS on the cutter will be analyzed with respect to this motion type. For defining the limits of the FCS two boundaries are used: the surface boundaries and the envelope boundary. The surface boundaries are fixed for a given tool motion and they can be calculated from section (5.2). For calculating the envelope boundary, the formulae developed in section (5.3) will be utilized by considering that the angular velocity is zero. Also the tool tip velocity VF will be denoted by f which is short for the feed. In Figure (5.9-a) the generic cutter profile and the feed vector f are shown. Initially the feed vector is coincident with the zL axis of the TCS. From this figure the feed angle \u03c8 which is measured from the zL axis in the clock wise direction can be obtained as \uf8f7 \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ec \uf8ed \uf8eb = \u2212 L L1 \u00b7cos zf zf\u03c8 (5.42) Any change in the direction of feed vector will effect the distribution of the FCS on the cutter. For clearly seeing this, the generic cutter is partitioned into two sections (Figure 5.9- b): Front face (F) and Back face (B). An unbounded plane which passes through yL and zL axes of TCS separates these two faces. Chapter5. Feasible Contact Surfaces 116 (a) (b) Figure 5.9: Feed angle ranges Figure (5.9-a) illustrates that the feed angle\u03c8 takes different values in five different regions (shaded area). The angles which separate these regions are defined by the geometric parameters \u03b1 and \u03b2 of the generic cutter from section (5.2). Also the second and the fourth columns of Table (5.1) show the feed angle ranges corresponding to these five regions. In this table the first and the third columns show alpha numeric symbols i.e. F1, B2 etc. for different ranges of the feed angle. Letters F and B denote the front and the back faces of the cutter (Figure 5.9-b) respectively. Table 5.1: The angular ranges of the feed angle According to the feed angle ranges given in Table (5.1) the cutter can have three types of motion: Ascending type (F1 to F4), Horizontal type (F5) and Descending type (B1 to B4). With respect to these tool motions the following propositions can be given Chapter5. Feasible Contact Surfaces 117 Proposition 5.1: When the cutter has ascending and horizontal type motions which cover the range of [ ]090,\u03b2\u03c8 \u2208 , the point set CWEK is located on the front face (F) of the cutter. Therefore only the front face can contain the feasible contact surfaces during the machining Proposition 5.2: When the cutter has descending motion which covers the range of ( ]\u03b2\u03c8 \u2212\u2208 00 180,90 , the point set CWEK is located on both the front and the back faces of the cutter. Therefore both faces can contain the feasible contact surfaces during the machining. Note that feed angles ( \u03b2\u03c8 < ) and ( \u03b2\u03c8 \u2212> 0180 ) are excluded from these propositions. because in the former one the top circle of the upper-cone touches the workpiece and in the latter one all cutter surfaces totally plunge into the workpiece. These propositions will be proven for the corner-torus surface of the generic cutter by considering the feed angle in the ranges of F1, F3, B2 and B4 from Table (5.1). The results for the other surface patches can be found in Table (5.2). The results of these propositions will be used in the next chapter during the cutter work piece engagement identifications. Proof: It has been shown in section (5.2) that two of the geometric parameters for describing the corner-torus are \u03b8 and \u03c6 . Figures (5.10-a, b) illustrates that \u03c6 is measured in CCW direction from the tool rotation axis and \u03b8 is measured in CCW direction from the xL axis of TCS. It can be seen from these figures that the\u03c6 angle of an arbitrary point P must lie within the range of ]90,90[ 00 \u03b2\u03b1 \u2212\u2212 i.e. )90()|()90( 00 \u03b2\u03c6\u03b1 \u2212\u2264\u2208\u2264\u2212 TGPP (5.43) When the lower ( )\u03b1\u03c6 \u2212= 090L and the upper ( )\u03b2\u03c6 \u2212= 090U limits of \u03c6 from Eq. (5.43) are plugged into Eq. (5.35), the following expressions are obtained: ( ) ( ) \uf8f7\uf8f7\uf8f8 \uf8f6 \uf8ec\uf8ec \uf8ed \uf8eb = \u2212 L LL x\u00b7f z\u00b7f \u03b1\u03b8 tancos 12,1 m (5.44-a) Chapter5. Feasible Contact Surfaces 118 ( ) ( ) \uf8f7\uf8f7\uf8f8 \uf8f6 \uf8ec\uf8ec \uf8ed \uf8eb = \u2212 L LU x\u00b7f z\u00b7f \u03b2\u03b8 tancos 12,1 m (5.44-b) where superscripts L and U represent the lower and the upper limit respectively. Note that the angular velocity in Eq. (5.35) becomes zero in 3-axis machining. (a) (b) Figure 5.10: The corner-torus with upper and lower surface boundaries. \u0001 Cutter has a feed angle ( \u03b2\u03c8 = ) In this case the angle between the feed vector f and zL axis equals to \u03b2 and the cutter has an ascending motion. For this feed angle the dot products in Eqs. (5.44,a-b) are evaluated as follows \u03b2cos|| LL z||fz\u00b7f = )90cos(|| 0LL \u03b2\u2212= x||fx\u00b7f Plugging the above equations back into Eqs. (5.44-a, b) yields \uf8f7\uf8f7 \uf8f8 \uf8f6 \uf8ec\uf8ec \uf8ed \uf8eb = \u2212 \u03b2 \u03b1\u03b8 tan tan cos 1L2,1 m (5.45-a) 0U 2,1 0=\u03b8 (5.45-b) Chapter5. Feasible Contact Surfaces 119 Note that when the lower and the upper limits of \u03c6 are equal to each other i.e. )90()90( 00 \u03b2\u03b1 \u2212=\u2212 , the corner-torus disappears from the generic cutter. For the corner-torus geometry to be present, the inequality )( \u03b1\u03b2 < must hold and thus under this consideration )tan(tan \u03b1\u03b2 < . In calculations these inequalities will be used frequently. The ratio inside the parenthesis of Eq. (5.45-a) is greater than one and thus the inverse cosine function does not have a real solution. In Eq. (5.45-b) \u03b8 has repeated roots of zeros. From the above results it can be concluded that when the generic cutter moves with the feed angle equal to \u03b2, the envelope boundary of the corner-torus contains only one point which corresponds to angles ( )\u03b2\u03c6 \u2212= 090U and 0U2,1 0=\u03b8 . In Figure (5.11-a), F1 represents this point which is located on the front face of the corner-torus. Thus this proves the proposition (5.1) for this motion type. (a) (b) Figure 5.11: Envelope boundary sets on the (a) front, and the (b) back faces of the corner-torus. \u0001 Cutter has a feed angle ( \u03b1\u03c8 = ) In this case the cutter is performing an ascending motion with a feed angle equals to \u03b1. After evaluating Eqs. (5.44-a, b) with \u03b1, the following equations are obtained 0L 2,1 0=\u03b8 (5.46-a) \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ed \uf8eb = \u2212 \u03b1 \u03b2\u03b8 tan tan cos 1U2,1 m (5.46-b) Chapter5. Feasible Contact Surfaces 120 The ratio inside the parenthesis of Eq. (5.46-b) is positive and less than one. Therefore for a given parameter set ( )\u03b2\u03c6 \u2212= 090U and U2,1\u03b8 the upper surface boundary of the corner-torus contains two points which are members of the envelope boundary. These are two symmetric points with respect to the feed vector and they are located on the front face of the corner- torus. The repeated roots in Eq. (5.46-a) and the lower limit of\u03c6 , ( )\u03b1\u03c6 \u2212= 090L generate a point on the lower surface boundary of the corner-torus. This single point which is also a member of the envelope boundary is located on the front face. From these results it can be seen that the upper and the lower surface boundaries of the corner\u2013torus contain envelope boundary points on the front face. Therefore it can be concluded that changing \u03c6 between ( )\u03b1\u03c6 \u2212= 090L and ( )\u03b2\u03c6 \u2212= 090U generates new envelope boundary points on the front face. F3 in Figure (5.11-a) shows an example of the envelope boundary for this motion. Thus for this motion type the envelope boundary on the front face and the portion of the upper surface boundary on the front face draw the limits of the point set CWEK. Thus this proves the proposition (5.1) for this motion type. F3 in Figure (5.12) illustrates the point set CWEK with shaded area. Chapter5. Feasible Contact Surfaces 121 Figure 5.12: Feasible contact surfaces of the toroidal part with respect to the cutter feed angle. \u0001 Cutter has a feed angle ( \u03b1\u03c8 \u2212= 0180 ) In this and the following case the cutter has a plunging motion with respect to feed vector. When the feed angle \u03b1\u03c8 \u2212= 0180 is plugged into Eqs. (5.44-a, b), the following results are obtained 0L 2,1 180m=\u03b8 (5.47-a) \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ed \uf8eb \u2212 = \u2212 \u03b1 \u03b2\u03b8 tan tan cos 12,1 m U (5.47-b) In Eq.(5.47-a), the lower limit of \u03b8 has two real solutions 0180m . These two solutions and the lower limit of\u03c6 , ( )\u03b1\u03c6 \u2212= 090L generate only one point on the lower surface boundary of the Chapter5. Feasible Contact Surfaces 122 corner-torus. This point which also belongs to the envelope boundary is located on the back face. The ratio inside the parenthesis of Eq. (5.47-b) is greater than -1 and therefore there are two distinct real solutions for the upper limit of \u03b8. These solutions and the upper limit of\u03c6 , ( )\u03b2\u03c6 \u2212= 090U produce two distinct points on the upper surface boundary of the corner-torus. These points which are members of the envelope boundary are located on the back face. Therefore by changing \u03c6 between its limits other envelope boundary points are generated on the back face. In Figure (5.11-b), B2 represents the envelope boundary for this motion type. The limits of the point set CWEK are defined by the envelope boundary on the back face and the portion of the upper surface boundary on the front face. Thus this proves the proposition (5.2). B2 in Figure (5.12) illustrates the point set CWEK with shaded area. \u0001 Cutter has a feed angle ( \u03b2\u03c8 \u2212= 0180 ) With this feed angle Eqs. (5.44, a-b) take the following form \uf8f7\uf8f7 \uf8f8 \uf8f6 \uf8ec\uf8ec \uf8ed \uf8eb \u2212 = \u2212 \u03b2 \u03b1\u03b8 tan tan cos 1L2,1 m (5.48-a) 0U 2,1 180m=\u03b8 (5.48-b) Because the ratio in Eq.(5.48-a) is less than -1, there is no real solution for the lower limit of \u03b8. The upper limit has two solutions in Eq. (5.48-b). These solutions and the upper limit of\u03c6 , ( )\u03b2\u03c6 \u2212= 090U produce only one point on the upper surface boundary of the corner-torus. B4 in Figure (5.11-b) shows this point on the back face. For this motion it can be said that even there is only one envelope boundary point, the whole surface of the corner-torus contains points from the set CWEK. B4 in Figure (5.12) illustrates the point set CWEK with shaded area. For clearly seeing this situation the tangency function in Eq. (5.18) is utilized for the corner torus. Under the 3-axis machining the tangency function of the corner-torus takes the following form ( ) ( ) ( ) ( )L LL sinsin sincoscos,, y\u00b7f x\u00b7fz\u00b7f \u03c6\u03b8 \u03c6\u03b8\u03c6\u03c6\u03b8 + +\u2212=tf (5.49) Chapter5. Feasible Contact Surfaces 123 When the Eq. (5.49) is evaluated for the feed angle \u03b2\u03c8 \u2212= 0180 , it yields ( ) ( )\u03b8\u03b2\u03c6\u03b2\u03c6\u03c6\u03b8 cossinsincoscos,, += ftf (5.50) where ]2,0[ pi\u03b8 \u2208 and thus \u03b8cos \u2208[-1, 1]. Note that because the feed vector f and yL axis of TCS are orthogonal, their dot product is equal to zero. According to Eq. (5.43) it can be said that \u03b2\u03c6 sincos \u2265 and \u03c6\u03b2 sincos \u2265 . When these inequalities are evaluated in Eq. (5.50) the tangency function becomes ( ) 0,, \u2265tf \u03c6\u03b8 , where ]90,90[ 00 \u03b2\u03b1\u03c6 \u2212\u2212\u2208 . Therefore the whole surface of the corner-torus contains the point set CWEK. Thus this proves the proposition (5.2). B4 in Figure (5.12) illustrates the point set CWEK with shaded area. Also the upper and the lower cone surface patches of the generic cutter may have feasible contact surfaces (FCS). The distribution of FCS on these surfaces can be analyzed using the same steps taken for the corner-torus. Table (5.2) shows the distribution of FCS on each surface patch of the generic cutter with respect to the motion types defined in Table (5.1). CWEK,U, CWEK,T and CWEK,L represent the feasible engagement point sets of the upper-cone, the corner-torus and the lower-cone surface respectively. If for a given feed angle, the surface patch does not contain any point from the FCS, the symbol { } is used for representing this situation. For example, in Table (5.2) for the feed angle in F2 only the upper-cone and the corner-torus surfaces have FCS and thus during the material removal only these surfaces can contact the in-process workpiece. Table 5.2: CWEK point sets of the generic cutter under different tool motions. Chapter5. Feasible Contact Surfaces 124 5.5 Discussion In this chapter the engagement behaviors of NC cutter surfaces under varying tool motions have been presented by introducing the feasible contact surfaces (FCS) terminology. The description of the FCS has been made by utilizing the kinematically feasible engagement points contained in the set CWEK. In a typical NC cutter the constituent surface geometries show differences and because of this during the machining the engagement characteristics for these surfaces become different. Under this consideration the set CWEK has been partitioned among the constituent surfaces of a cutter. As mentioned in this chapter, the Cutter Workpiece Engagements (CWEs) are the subsets of the FCS and for calculating CWEs the boundaries of the FCS must be identified. In this work these boundaries which are limits of the set CWEK have been defined by the cutter surface boundaries and the envelope boundaries. Under the rigid tool motions the cutter surface boundaries stay fixed. But on the other hand the envelope boundaries may change. For modeling the envelope boundaries a tangency function defined by the surface normal and the tool velocity has been utilized. Later by changing the tool velocity direction the distributions of the FCS on the cutter have been analyzed. Also it has been shown with examples that only the certain parts of the cutter surfaces have contacts with the in-process workpiece. The results from this chapter will be used in the CWEs generation. 125 Chapter 6 Cutter Workpiece Engagements In this chapter the methodologies for obtaining Cutter Workpiece Engagements (CWEs) in milling are presented. Cutting forces are a key input to simulating the vibration of machine tools (chatter) prior to implementing the real machining process. These forces are determined by the feed rate, spindle speed, and CWEs (captures the depth of cut). Of these finding the CWE is most challenging due to the complex geometry of the in-process workpiece and varying tool motions. The CWE geometry defines the instantaneous intersection boundary between the cutting tool and the in-process workpiece at each location along a tool path. Figure (6.1) summarizes the steps involved in CWE extraction. Inputs from CAD\/CAM include the tool paths in the form of a CL Data (cutter location data) file, geometric description of the cutting tool and a geometric representation (B-rep, polyhedral, vector based model) of the initial workpiece. The key steps which are the swept volume generation and the in-process workpiece update have been presented in the previous chapters. CAD\/CAM Model of Swept Volume Model of In-process Workpiece CWE Extractor CWE Extractor End of Tool Path ? End of Tool Path ? Next Tool Path ? Next Tool Path ? Model of Cutting Tool DoneoneD 1 3 no yes yes no In-Process Geometry Generator In-Process Geometry Generator Swept Volume Generator Swept Volume Generator 2 1 CLData Initial Workpiece Cutter\/Workpiece Engagements Process Modeling & Optimization Cutting Force Prediction Stability Analysis Speed and Feed Optimization Static Deflection Analysis Static Deflection Analysis CWE Extraction Model of Removal Volume Figure 6.1: CWE Extraction Steps Chapter6. Cutter Workpiece Engagements 126 In this chapter the CWEs are calculated for supporting the force prediction model described in [5]. This model finds the Cartesian force components by analytically integrating the differential cutting forces along the in-cut portion of each cutter flute. In this model CWE area with a fixed axial depth of cut is defined by mapping the engagement region on the cutter surface onto X - Y plane which represents the engagement angle versus the depth of cut respectively (Figure 6.2). Figure 6.2 has been removed due to copyright restrictions. The information removed is CWE area for the force prediction, [5]. In this chapter the methodologies for finding the CWEs are developed based on the mathematical representation of the workpiece geometry. The workpiece geometries are defined by a solid, a polyhedral and a vector based modeler respectively. Although polyhedral and vector based approaches require a shorter computational time than does the solid modeler based approach, the accuracy of these approaches depends greatly on the resolution of the workpiece. There is always a tradeoff between computational efficiency and accuracy in these approaches. For example the solid modeler has an advantage in accuracy because it provides an accurate geometric representation for the workpiece. Therefore in the solid modeler based approach the most accurate representations of the CWEs are obtained. But on the other hand this modeler uses numerical techniques that are limited primarily by efficiency and robustness. The Cutter Workpiece Engagements in Solid Models are presented in section (6.1). This section contains two methodologies: Engagement Extraction Methodology in 3-Axis Milling (section 6.1.1) and Engagement Extraction Methodology in 5-Axis Milling (section 6.1.3) respectively. In the 3-axis milling methodology the cutter surfaces are decomposed with respect to the feed direction as explained in chapter (5), and then these decomposed surfaces are intersected with their removal volumes for obtaining the closed CWE area. In the 5-axis methodology the similar approach is utilized with one difference. In this case the in-process workpiece is used instead of cutter removal volumes. The developed solid modeler based methodologies are implemented using ACIS solid modeling kernel. The both methodologies are supported by examples. Chapter6. Cutter Workpiece Engagements 127 The Cutter Workpiece Engagements in Polyhedral Models are presented in section (6.2). Also this section contains two methodologies. In the 3-axis CWE methodology (section 6.2.1) the chordal error problem described in chapter (1) is addressed. For finding the CWEs a polyhedral representation of the removal volume is mapped from Euclidean space into a parametric space. Thus CWE calculations are reduced to line-plane intersections. In this methodology the formulas are developed for the linear, circular and helical toolpaths. For the 5-axis milling a direct triangle\/surface intersection approach (section 6.2.2) is developed. In this approach for generating CWE area, without doing mapping the cutter surface is intersected with the triangular facets obtained from the in-process workpiece. The Cutter Workpiece Engagements in Vector Based Models are presented in section (6.3). In this methodology the workpiece geometry is broken into a set of evenly distributed discrete vectors and also the cutter is discretized into slices perpendicular to the tool axis. For generating CWEs the intersections are performed between discrete vectors and cutter slices. Finally the chapter ends with the discussion in section (6.4). 6.1 Cutter Workpiece Engagements in Solid Models This section presents Solid modeling methodologies for finding Cutter Workpiece Engagements (CWEs) generated during 3 and 5-axis machining of free \u2013 form surfaces using a range of different types of cutting tools and tool paths. Figure (6.3) summarizes the steps involved in CWE extraction using B-rep based solid modeler. Inputs from CAD\/CAM include the tool paths in the form of a CL Data (cutter location data) file, geometric description of the cutting tool and a geometric representation (B-rep) of the initial workpiece. Key steps include swept and removal volume generation for each tool path. Although computational complexity and robustness, for limited applications, remain issues that need to be addressed, solid modelers have been recognized as one approach to finding CWE geometry. Solid modelers have an advantage in accuracy because they generate the exact mathematical representation for the intersections. The methodologies in this section have been implemented using a commercial geometric modeler (ACIS) which is selected to be the kernel around which the geometric simulator is built. In these approaches, in-process workpiece updating and cutter\/workpiece engagement extraction are performed using geometric and topologic algorithms within the solid modeler kernel. In these situations the Chapter6. Cutter Workpiece Engagements 128 solid modeling approach applies numerical surface intersection algorithms which are based on subdivision, and curve tracing (marching) methods. Figure 6.3: A B-rep Solid Modeler based CWE extraction 6.1.1 Engagement Extraction Methodology in 3-Axis Milling This section presents a B-rep Solid modeler based methodology for finding CWEs generated during 3 -axis machining. For this purpose cutter surfaces are decomposed with respect to the tool feed direction and then they intersected with their removal volumes for obtaining the boundary curves of the closed CWE area in 3D Euclidian space. Later these boundary curves are mapped from Euclidean space to a parametric space defined by the engagement angle and the depth-of-cut for a given tool geometry. As explained in chapter (5), a typical NC cutter has different surfaces with varying geometries and during the machining the engagement characteristics of each surface become different. Under this consideration in this section the kinematically feasible engagement points CWEK are divided into constituent sub sets. The primary task in finding the CWE geometry is finding the boundary of the engagement region. The boundary set of CWE is Chapter6. Cutter Workpiece Engagements 129 represented by bCWE. This defines the geometry required for input to process modeling (i.e. force prediction). Thus )()()( tCWEtCWEtbCWE K\u2282\u2282 (6.1) These three sets are illustrated in (Figure 6.4) for the Flat-End mill performing a helical tool motion. In this figure the sets bCWE, CWE and CWEK are shown separately for the side (cylindrical) face and the bottom (flat) face of the Flat-End mill. Figure 6.4: Point sets used in defining Engagements For the force model described in [5] in-cut segments of the cutting edges are needed. In this section for obtaining in-cut segments both bCWE and the cutting edges are mapped into 2D space defined by the engagement angle (u) and the depth of cut (v). Then the curve\/curve intersection is performed between bCWE and the cutting edges. The common milling cutters have different surface geometries i.e. a Ball-End mill is defined by two natural quadric surfaces \u2013 spherical and cylindrical. As a result of this the parameterization of the engagement area differs for each cutter surface. In the Flat-End mill, for example, the depth of cut v for an arbitrary point P \u2208 CWE is the distance between the location of P and the cutter tool tip along the tool rotation axis, and the engagement angle u is measured from the yL axis of TCS (Figure 6.5). Chapter6. Cutter Workpiece Engagements 130 Figure 6.5: CWE parameters of an arbitrary point P. The engagement parameters for different cutter surfaces are illustrated in Figure (6.6). Because the Flat-End, the Taper-End and the Fillet-End cutters have a flat surface at their bottom, also in this figure the engagement parameters of the cutter bottom surface are shown. As explained in chapter (5), when the feed angle is in the range of (B1 to B4) the cutter bottom surface can have contact with the workpiece during the machining. (a) (b) (c) (d) (e) Figure 6.6: Defining CWE parameters u and v on (a) torus, (b) sphere, (c) frustum of a cone, (d) cylinder, and (e) flat bottom surfaces of common milling cutters. It has been shown in chapter (5) for the 3-axis machining that according to the feed angle ranges the cutter can have three types of motion: Ascending type (F1 to F4), Horizontal type (F5) and Descending type (B1 to B4). These motion types effect the construction of the kinematically feasible engagement point set CWEK and consequently this changes the angular ranges of the engagement parameter u as follows \u0001 If the feed angle is in the range of (F1 to F5) only the front face of the cutter can have contact with the workpiece and the engagement angle u covers ],0[ pi range i.e. pi\u2264\u2208\u2264 )|(0 CWEu PP . Chapter6. Cutter Workpiece Engagements 131 \u0001 If the feed angle is in the range of (B1 to B4), both the front and the back faces of the cutter can have contact with the workpiece. This time the range of the engagement angle u becomes: pi2)|(0 \u2264\u2208\u2264 CWEu PP The two cases above show that based on the kinematics of the 3-axis machining the cutter surfaces differently contribute to the CWE extraction. For this reason cutter surfaces are broken down into their constituent sub surfaces. It has been shown in chapter (5) that, the front and the back surfaces are separated by an unbounded plane which passes through yL and zL axes of TCS. Also if the cutter has a flat bottom surface then this surface lies on another unbounded plane which passes through xL and yL axes of TCS. This decomposition is shown for the Taper-End and Flat-End mills in Figure (6.7), where GBack, GBottom and GFront represent the constituent sub surfaces of the cutter geometry G. Table (6.1) shows the decomposition of the surface geometries for the common milling tools. Figure 6.7: Geometric decomposition of the cutter surfaces Table 6.1: Constituent surfaces of cutter geometries after geometric decomposition in 3-axis milling. Each constituent surface of a cutter geometry given in the Table (6.1) generates its kinematically feasible engagement point set with respect to the cutter feed vector. For example when a Fillet-End mill has a descending motion described by (B1 to B4) all three Chapter6. Cutter Workpiece Engagements 132 constituent surfaces of the toroidal part: GBack , GBottom and GFront generate the sets CWEK,Back, CWEK,Bottom and CWEK,Front respectively (see Figure 6.8). Thus the following equation can be written for a toroidal part performing this motion FrontKBottomKBackKK CWECWECWECWE , * , * , \u222a\u222a= (6.2) where *\u222a and later *\u2229 represent regularized Boolean union and intersection set operations respectively. Figure 6.8: Decomposing the point set CWEK of the torus into three parts. Note that depending on the motion type some of these subsets can be empty. For example in the ascending motion (F1 to F5) only the front part of the cutter engages with the workpiece and therefore only the set CWEK,Front is considered as a full set. Table (6.2) shows these three sets for different cutter geometries with respect to the feed angle ranges. The symbol \u00d8 represents an empty set for a given feed angle range. Table 6.2: Feasible engagement points for cutter surfaces with respect to the tool motions Chapter6. Cutter Workpiece Engagements 133 The following properties (6.1) and (6.2) motivate finding the Cutter Workpiece Engagements in 3-axis milling. Property 6.1: Given a tool path Ti for the ith tool motion and the cutter surface geometry: a) Sweeping the point sets CWEK,Back, CWEK,Bottom and CWEK,Front along Ti generates the swept volumes of the corresponding cutter surfaces i.e. SVi (Back), SVi (Bottom) and SVi (Front) respectively. For a given tool motion and the cutter surface geometry the constituent parts of the set CWEK are chosen from the Table (6.2) for sweeping. For example Figure (6.9) illustrates this for the Fillet-End mill which follows a linear toolpath in descending motion (B1 to B4). (a) (b) (c) Figure 6.9: Swept volumes generated by the kinematically feasible engagement point sets. b) Intersecting the swept volumes from the property 1(a) with the in-process workpiece Wi-1 generates the removal volumes as follows )()(*1 BackRVBackSVW iii \u2192\u2229\u2212 )()(*1 BottomRVBottomSVW iii \u2192\u2229\u2212 (6.3) )()(*1 FrontRVFrontSVW iii \u2192\u2229\u2212 where RVi (.) represents the corresponding removal volume of each swept volume. Note that for the ith tool motion the total removed material from the in-process workpiece can be denoted as follows )()()( ** FrontRVBottomRVBackRVRV iiii \u222a\u222a= (6.4) Chapter6. Cutter Workpiece Engagements 134 Property 6.2: The intersection of constituent surfaces GBack, GBottom and GFront with their corresponding removal volumes (.)iRV generates the cutter workpiece engagement boundary bCWEi for the given cutter location as follows )}({ )}({ )}({ * ** ** FrontRVG BottomRVG BackRVGbCWE iFront iBottom iBacki \u2229 \u222a\u2229 \u222a\u2229= (6.5) The 3-axis CWE extraction methodology is summarized in Figure 6.10. The inputs are ith toolpath segment Ti, in-process workpiece Wi-1 and the constituent surfaces of the cutter geometry G. Figure 6.10: Procedure for obtaining the CWEs The reported method has been implemented using a commercial geometric modeler (ACIS) which is selected to be the kernel around which the geometric simulator is built. In the described methodology for generating the bCWEs, the face\u2013face intersections between cutter surfaces and the removal volume surfaces are performed. For this purpose a given removal volume is decomposed into its faces by using ACIS function api_get_faces. The faces obtained from this function are intersected with the cutter surface which is also Chapter6. Cutter Workpiece Engagements 135 represented as a FACE Entity. These intersections are performed by using the function api_fafa_int. For obtaining the cutter surface i.e. GFront, first the circular wires which are perpendicular to the tool axis are generated along the tool axis and then they are skinned by using the function api_skin_wires. It has been shown in chapter (1) that ACIS representational hierarchy contains BODY, LUMP and SHELL etc. Therefore a body can be represented by LUMPs, a LUMP can be represented by SHELLs and so on. The function api_fafa_int generates intersection curves as a BODY Entity. For obtaining the properties of these curves i.e the start and end coordinates, these BODYs are decomposed into LUMPs, SHELLs, WIREs and EDGEs respectively. The Algorithm (6.1) for this process is shown below in the C++ format. The inputs are RVi(Front), GFront and a cutter location point PCL. It generates the closed boundary set of the CWE area for the given cutter location point. Input: RVi (front), GFront, PCL ( xCL, yCL, zCL) Output: bCWEi at PCL ENTITY_LIST face_list; \/\/ Container of removal volume faces FACE *ff = api_get_faces(RVi (front), face_list); \/\/points the first face of the face_list FACE *cutter_face = GFront; BODY *int_curve = NULL; \/\/a member from bCWE LUMP *rem_vol_lump = NULL; SHELL *rem_vol_shell = NULL; WIRE *rem_vol_wire = NULL; EDGE *rem_vol_edge = NULL; while( (ff = (FACE*)face_list.next()) != NULL) { \/\/iterates until all faces of the \/\/removal volume is processes api_fafa_int(cutter_face, ff, int_curve); rem_vol_lump = int_curve\u2192lump(); while(rem_vol_lump != NULL){ rem_vol_shell = rem_vol_lump\u2192shell(); while(rem_vol_shell != NULL){ rem_vol_wire = rem_vol_shell\u2192wire(); while(rem_vol_wire != NULL){ rem_vol_edge = rem_vol_wire\u2192coedge()\u2192edge(); \/\/ process the edge here for obtaining bCWEs rem_vol_wire = rem_vol_wire\u2192next(); } rem_vol_shell = rem_vol_shell\u2192next(); } rem_vol_lump = rem_vol_lump\u2192next(); } ff = ff\u2192next(); } Algorithm 6.1: Obtaining the closed boundaries of the CWEs Chapter6. Cutter Workpiece Engagements 136 6.1.2 Implementation Figures (6.11) and (6.12) show Flat-End and Taper-End mills removing material along a linear ramping tool path i.e. the tool moves in all three axes simultaneously. The removal volumes associated with the different cutter surfaces are separated. In the case of the Flat- End mill this gives the material removed by the cylindrical at the side and the flat at the bottom surfaces. Plots of CWEs at different Cutter Locations (CLs) are illustrated. These are obtained by intersecting the constituent surfaces of the cutter with their corresponding removal volumes. Two formats are used for plotting the CWEs. The first is an XY plot of depth-of-cut v (as measured from the tool tip point) versus engagement angle u. The second plot shows the engagement area of the cutter bottom surface in a polar format: cutter bottom radius r versus engagement angle u. In Figure (6.11) the Flat-End mill is performing a descending motion in which the feed angle is in the range of (B1 to B4). Thus both the cylindrical and the bottom surface remove material. For this motion type of the Flat End mill only the CWEK,Front and the CWEK,Bottom point sets are active. The XY plots show the engagement between zero and pi for the front face, and polar plots show the engagement between zero and 2pi for the bottom flat face. Figure 6.11: CWEs for the Flat End mill performing a linear 3-axis descending motion. In Figure (6.12) because the Taper-End mill is ramping up in which the feed angle is in the range of (F1 to F5), only the point set CWEK,Front is active during the material removal. Thus Chapter6. Cutter Workpiece Engagements 137 for this motion type, only the front face of the cutter contributes to the CWE extraction procedure with the engagement angle u changes between zero andpi radian. Figure 6.12: CWEs for the Taper-End mill performing a linear 3-axis ascending motion. In the last example (Figure 6.13) a Flat-End mill is following a helical toolpath (plunging cutting) for enlarging a hole. In this example both the side and the bottom faces of the Flat- End mill are removing material. The cutter performs three half turns each corresponding to a 00 1800 \u2212 range i.e. third turn has the starting angle 0360 and ending angle 0540 . In this motion of the Flat-End mill the point sets CWEK,Front and CWEK,Bottom are active. The total removed material RV (Front) and RV (Bottom) are shown for each half turn with different colors. For the cutter location CL1 both types of plots are shown because the bottom of the cutter is removing material at this location also. For CL2 only the side (cylindrical) face of the cutter has engagement with the workpiece. Chapter6. Cutter Workpiece Engagements 138 Figure 6.13: Helical Tool Motions with a Flat-End Mill and CWEs 6.1.3 Engagement Extraction Methodology in 5-Axis Milling This section presents a B-rep Solid modeler based methodology for calculating CWEs in 5-axis milling operation. Many of the steps defined in the previous section for the 3-axis CWE methodology are applicable for the 5-axis CWE methodology in this section. There is only one main difference between these two methodologies. In 5-axis CWE methodology for obtaining bCWEs the feasible contact surface of a cutter will be intersected with the in- process workpiece instead of the removal volume which is used in the 3-axis CWE methodology. At any given instance of the 5-axis tool motion the bottom center and top center of the rigid cutter may move in directions that do not lie in the same plane. For example in Figure (6.14-a), the top velocity vector VTop and the bottom velocity vector VBottom point to the different directions. An arbitrary velocity V on the tool axis can be calculated by linearly interpolating VTop and VBottom as follows ( ) BottomTop 1 VVV uu \u2212+= (6.6) where u\u2208[0,1]. On the other hand in 3-axis milling the top and bottom centers of the cutter move in the same direction (Figure 6.14-b). As explained in chapter (5), a cutter contacts in-process workpiece through the set of CWEK. Most of the points in this set lie towards the front of the cutter and are machined away as the tool leaves its current position. Only those points for which the motion direction Chapter6. Cutter Workpiece Engagements 139 is perpendicular to the cutter surface normal are left behind on the machined surface as a curve. As explained in chapter (5), these points define the envelope boundary of the cutter which describes the geometric limits of the set CWEK. (a) (b) Figure 6.14: (a) Envelope boundary in 5-axis milling, and in (b) 3-axis milling respectively. As explained in chapter (5), for defining the boundaries of the feasible contact surfaces (FCS) envelope boundary set is used. As illustrated in Figure (6.14-b) for the Flat-End mill performing 3-axis machining, because the top and the bottom velocity vectors of the cutter point to the same direction the envelope boundary has a linear characteristic. But on the other hand in the 5-axis tool motions the velocity vector on the cutter axis continuously changes and as a result the envelope boundary looses its linearity (see Figure 6.14-a). In this case the envelope boundary curves are approximated by splines. For example in the case of a Flat- End mill, these boundaries are represented by the helix like curves [12]. Because of the approximation in the envelope boundary curves the CWE methodology described for the 3- axis milling does not properly work for the 5-axis milling. The intersections obtained numerically by the B-rep solid modeler become non robust. For solving this problem a methodology based on the intersections between the FCS and the in-process workpiece is presented in this section. This methodology is explained for an impeller machining using a Taper-Ball-End mill and it can also be applied to other milling cutters performing 5-axis tool moves. In Figure (6.15-a) a Taper-Ball-End mill is shown for a given Cutter Location (CL) point. At this CL point the feasible contact surface defined by the envelope and cutter surface boundaries is constructed (Figure 6.15-b). In the next stage (Figure 6.15-c) we offset this surface in the (+\/-) directions of the velocity vector V with an infinitesimal amount ( 2\/\u03b5 ). This process makes this surface a volume which is called as BODY. To minimize the error Chapter6. Cutter Workpiece Engagements 140 introduced by this offsetting the location of the vector V is taken in the middle of the tool axis. This causes a geometric error at both VTop and VBottom in equal magnitude and zero error halfway between these two points. (a) (b) (c) Figure 6.15: Offsetting the feasible contact surface As illustrated in Figure (6.16) there are three main steps for generating the CWEs. In the first step the BODY obtained by offsetting the feasible contact surface is intersected with the in-process workpiece at a given cutter location point. This intersection generates a removal volume. In the second step the removal volume is decomposed into its constituent faces and between these faces and the feasible contact surface face\/face intersections are performed. Each one of these intersections generates a curve. The full boundary of the CWE area, bCWE is the combination of each individual curve obtained from face\/face intersections. Finally in the last step, bCWE represented in 3D Euclidian space is mapped into 2D space defined by the immersion (engagement) angle and the depth of cut. Note that these 3 steps are performed for each cutter location point on a toolpath segment. Chapter6. Cutter Workpiece Engagements 141 Figure 6.16: CWE steps in 5-axis milling methodology The implementation of the 5-axis CWE methodology is illustrated in Figures (6.17) to (6.19) respectively. For this implementation the solid modeler kernel (ACIS) and C++ is used. The ACIS function api_get_faces decomposes the faces of the in-process workpiece. Then each one of these faces is intersected with the feasible contact surface utilizing the function api_fafa_int. For obtaining the closed boundaries of the CWEs the same procedure described in Algorithm (6.1) is followed. In Figures (6.17) and (6.18) for the given three cutter location points XY plots of depth-of-cut versus immersion angle are plotted. These examples are illustrated for the first and second passes of the cutter. Also in these figures the removal volumes for each passes are shown. From these two figures it can be seen that the amount of the removed material in the first pass is more than that of the second pass. In Figure (6.19) the in-process workpiece is shown after the third and fourth passes respectively. Chapter6. Cutter Workpiece Engagements 142 Figure 6.17: 5-axis CWEs during the first pass of the impeller machining Figure 6.18: 5-axis CWEs during the second pass of the impeller machining Chapter6. Cutter Workpiece Engagements 143 Figure 6.19: In-process workpieces after the third and fourth passes respectively. 6.2 Cutter Workpiece Engagements in Polyhedral Models Polyhedral models provide the advantage of simplifying the workpiece surface geometry to planes which consist of linear boundaries. Thus the intersection calculations reduce to line \/ surface intersections. These can be performed analytically for the geometry found on cutting tools. For obtaining CWE area, facets which contain linear boundaries are intersected with the surface of the cutter and then the intersection points are connected to each other. If the cutter surface has the second order equation, e.g. cylinder, cone or sphere natural quadric surfaces, each line \u2013 surface intersection gives two roots. If the cutter surface has the fourth order equation e.g. torus, each line surface intersection gives four roots. In most cases only one of these roots are needed for obtaining the CWEs and the rest is redundant. CWE extraction algorithms must be robust enough to handle the complete set of intersection cases between the cutting tool and a triangular facet (see Figure 6.20). Figure 6.20: Typical intersections between a facet and a cutting tool A major consideration in CWE extraction is the form of the in-process geometry. Beyond there being differences based on the type of model being used there is also a choice of using the in-process workpiece or the removal volume. Either can be generated by applying 4 Intersection points 1 Tangent point Cutting tool 2 Intersection points 1 tangent and 2 intersection points Chapter6. Cutter Workpiece Engagements 144 Boolean operations between the swept volume for a tool path and the initial workpiece or in- process workpiece generated by the previous tool path (subtraction or intersection). The use of the removal volume instead of the in-process workpiece has a number of advantages: \u0001 The size of the geometry model that must be manipulated by the CWE extraction algorithms is significantly smaller for the removal volume. \u0001 The use of the removal volume model better supports parallel computation strategies for CWE extraction. The CWEs for each removal volume can be extracted independently. An agent based methodology that does just this is described in [85]. For these reasons the removal volume is used in the research described in this chapter to capture the in-process geometry. Since a polyhedral model is an approximate representation of the exact analytical surface from which it was generated it would seem that similar accuracy issues to those found in the discrete modeling approaches exist. One example is illustrated in (Figure 6.21). Shown is the polyhedral model of a removal volume machined from the workpiece by a cylindrical end mill. The faceting algorithm that generates this model approximates surfaces to a specified chordal error. As can be seen from the 2D view this results in facets that lie outside the tool envelop at a given location even though the cutting tool is in contact with the actual removal volume surface. This facet should be considered in finding the CWE boundary but would be difficult to detect since it does not intersect with the tool geometry. Figure 6.21: The edges of facets deviate from the real surface In this section CWEs are calculated for the common milling cutters performing 3-axis and 5-axis tool movements. In calculations the cutting tool geometries are represented implicitly Chapter6. Cutter Workpiece Engagements 145 by natural quadrics and toroidal surfaces. Natural quadrics (Figure 6.22-a) consist of the sphere, circular cylinder and the cone. Together with the plane (a degenerate quadric) and torus these constitute the surface geometries found on the majority of cutters used in milling. For example a ball nose end mill (BNEM) is defined by two natural quadric surfaces \u2013 spherical and cylindrical. Other examples are shown in Figure (6.22-b). (a) (b) Figure 6.22: (a) Constituent surfaces of milling cutters and (b) some typical milling cutter surfaces. In developing the CWE methodologies for the polyhedral models the implicit equations of the cutter geometries will be used. These are Sphere: ( ) ( ) 02 =\u2212\u2212\u2022\u2212 rCPCP (6.7-a) Cylinder: ( ) ( ) ( )[ ] 022 =\u2212\u2022\u2212\u2212\u2212\u2022\u2212 rnBPBPBP (6.7-b) Cone: ( )[ ] ( ) ( ) 0)(cos 22 =\u2212\u2022\u2212\u2212\u2022\u2212 VPVPnVP \u03b1 (6.7-c) Torus: [ ] 0])[(4( 222222 =\u2212+\u2212\u2022 rzzRRr C--)C-P)C-P( (6.7-d) Plane: 0)( =\u2022\u2212 nBP (6.7-e) Chapter6. Cutter Workpiece Engagements 146 where P represents the position of an arbitrary point on the cutter\u2019s surface and the coordinates of C are (xC, yC, zC). 6.2.1 Engagement Extraction Methodology in 3-Axis Milling This section presents a methodology for calculating CWEs from polyhedral models that addresses the chordal error problem described in section (6.2) and which reduces the problem to line-plane intersections for the common milling cutter geometries and move types. In this methodology for finding the CWEs in 3-Axis milling, a polyhedral representation of the removal volume is mapped from Euclidean space into a parametric space. The nature of the swept geometry and the goal of engagement extraction points towards a preferred parameterization. As shown in Figure(6.23), the engagement(immersion) angle (\u03c6), depth of cut (d) for points on the cutter surface and the tool tip distance (L(t)) make up this parameterization, P(\u03c6, d, L(t)) where (0 \u2264 t \u2264 1). Figure 6.23: CWE parameters This mapping as will be seen has the effect of reducing the cutting tool geometry to an unbounded plane. Thus the boundary of the CWE is found by performing first order intersections between the planar representation of the cutting tool and the planar facets in the polyhedral representation of the removal volume. Chapter6. Cutter Workpiece Engagements 147 General Terminology Ti : Tool path for the ith tool motion RVi : Removal volume generated for ith tool motion in E3. M : A transformation that maps RVi from E3 to P(\u03c6, d, L) i.e. M: E3 \u2192 P(\u03c6, d, L) P iRV : The map of RVi in P(\u03c6, d, L). N : Surface normal of a point on cutter. CWEK(t): The set of all points on the surface of a cutter at location t along a tool path where f \u2022 N \u2265 0. These are kinematically feasible engagement points for a given instantaneous feed vector f. CWE(t): The set of all points on the surface of a cutter at location t along a tool path that are engaged with the workpiece. bCWE(t): Points on the boundary of CWE(t). In finding engagement geometry the following property of the representations for removal volumes in the parametric space P(\u03c6, d, L) motivate finding the mapping M: E3 \u2192 P(\u03c6, d, L). Property 6.3: Given PiRV a representation of the removal volume RVi for tool path Ti in P(\u03c6, d, L), its intersection with an unbounded plane Q generates a closed set of points CWE(t) that constitutes all points in engagement with the workpiece at location t along the tool path. Of interest is the boundary set of CWE(t), bCWE(t). This defines the geometry required for input to process modeling (i.e. force prediction). Thus bCWE(t) \u2282 CWE(t) \u2282 CWEK(t) (6.8) These three sets are illustrated in Figure (6.24). Chapter6. Cutter Workpiece Engagements 148 Figure 6.24: Point sets CWEK(t), CWE(t) and bCWE(t) used in defining engagements The property given above point to a novel approach for finding the engagement geometry assuming that the mapping M can be constructed: Given PiRV generated by applying M to RVi , CWE(t) and bCWE(t) can be found by intersecting PiRV with an unbounded plane for each cutter location defined by t (Figure 6.25). The use of an unbounded plane in finding engagements eliminates the problem highlighted in Figure (6.21) where the chordal error in the polyhedral representation of a removal volume introduces uncertainty in the intersection calculation. Further the reduction of the cutter surface geometry to a first order form simplifies intersection calculations particularly when the removal volume is polyhedral. Figure 6.25: CWE calculations in the parametric domain P(\u03c6, d, L). Chapter6. Cutter Workpiece Engagements 149 6.2.1.1 Mapping M for Linear Toolpath In this section derivations for the linear tool path will be shown for the Ball Nose End Mill (BNEM) and formulas for the other cutter types can be found in Appendix A-2. A BNEM is made up of a hemi-spherical and a cylindrical surface. Based on the kinematics of 3-axis machining subsets of these surfaces denoted CWEK,S(t) and CWEK,C(t) respectively will contribute to the set CWEK(t) as defined in chapter (5) i.e. CWEK (t) = CWEK,S(t) \u222a CWEK,C(t) Transformation formulas will be derived for the mapping of the hemi-spherical surface, Msphere. The methodology and many of formulae developed apply equally to the mapping for the cylindrical surface of the cutter, Mcyl. This will be further explained later in this section. Derivation of Msphere for the BNEM In addition to the geometric definition of the surface the location of a point on the cutter geometry as it moves along a tool path is also required. To do this the following terms are introduced (see Figure 6.26). Figure 6.26: Description of a point on a cutter moving along a Tool Path WCS: Workpiece Coordinate System (i, j, k). This is where the geometry of the removal volume and tool paths are defined. TCS: Tool Coordinate System (u, v, w) positioned at the tool tip with w along the cutter axis. It is assumed that the x axis of the local tool coordinate system is aligned with the feed vector direction. Chapter6. Cutter Workpiece Engagements 150 T(t): The position of the tool tip point (xT, yT, zT, t) at t along tool path Ti in the WCS. I: The position of a point (xI, yI, zI) on the cutter surface that belongs to the set CWEK(t) at t along tool path Ti in the WCS. F(t): The position of a reference point (xR, yR, zR, t) on the cutting tool axis at t along tool path Ti . Note that in this section the components of the (TCS) are defined by u,v and w which are different than those defined in chapter (5). The reason for this is to maintain the clarity in the variable symbolization i.e. to use less subscripts. The possible 3-axis linear tool motions are shown in Figure (6.27): horizontal, ascending and descending (from left to right) respectively. Figure 6.27: 3-axis Linear Tool Motions with a BNEM For motions A and B the engagement angle of a point at )( , tCWE SK\u2208I must lie within [0,pi] i.e. pi\u03d5 \u2264\u2208\u2264 ))(|(0 , tCWE SKII . At C both the front and the back sides of the hemi-spherical surface of the cutter have engagements and the total engagement area covers the full [0,2pi] range i.e. pi\u03d5 2))(|(0 , \u2264\u2208\u2264 tCWE SKII .The mapping will be developed for the most general case C where the hemi-spherical surface engages the workpiece in two regions \u2013 the front contact face CWEK,s1(t) and the back contact face CWEK,s2(t) where 2,1,, )( sKsKSK CWECWEtCWE \u222a= Figure (6.28) shows the engagement regions and parameters (\u03c6, d, L) for a point at I on the hemi-spherical surface. Chapter6. Cutter Workpiece Engagements 151 (a) (b) Figure 6.28: Engagement regions of the (a) front and (b) back contact faces The mapping methodology is summarized in Figure (6.29) and consists of 4 steps. The inputs are a point on the removal volume I(xI,yI,zI)\u2208E3, the implicit representation of the cutter surface geometry G and the parametric form of the tool path Ti. Figure 6.29: Procedure for performing Mapping MG:E3 \u2192 P(\u03c6, d, L). Step 1: In this step the parameter value t along Ti is found. For a linear tool path, the tool tip coordinates with respect to tool path start VS(xS, yS, zS) and end VE(xE, yE, zE) points are given by, Chapter6. Cutter Workpiece Engagements 152 ( ) tt )( SES VVVT \u2212+= , 10 \u2264\u2264 t (6.9) For a BNEM the reference point at F is chosen to be the center of the sphere. Its location can be expressed using the cutting tool tip coordinates as, ( ) nTF rtt += )( (6.10) where r and n are the radius of the hemi-sphere and unit normal vector of the tool axis respectively. When the hemi-spherical surface moves along a tool path, a family GS(t) of surfaces is generated. An expression for this family of surfaces is obtained by substituting in the coordinates of the reference point at F from Eq.(6.10) into the implicit form of a sphere given by Eq.( 6.7-a). ( ) ( ) 0:)( 2 =\u2212\u2212\u2022\u2212 rtGS FPFP (6.11) When the point at P belongs to the set CWEK,S(t), Eq.(6.11) can be rewritten as, ( ) ( ) 0:)( 2 , =\u2212\u2212\u2022\u2212 rtCWE SK FIFI (6.12) Given a point at I that is known to be an engagement point, Eq.(6.12) can be expanded to take on the following quadratic form, A2 t2 + A1 t + A0 = 0 (6.13) where the coefficients A2 , A1 and A0 are given by, 2 2 SE VV \u2212=A ( ) ( )[ ]SES VVnVI \u2212\u2022\u2212\u2212\u2212= rA 21 (6.14) 22 0 rrA \u2212\u2212\u2212= nVI S Chapter6. Cutter Workpiece Engagements 153 Solving for t in Eq.(6.13) gives the position of the cutter tool tip when the point at I is an engagement point between the cutter and the workpiece. In finding the correct tool position for an intersection point I the following property is used. Property 6.4: Given a linear tool path Ti and the cutter surface geometry G, for 3-axis machining, a) The maximum number of tool positions where the cutter touches a point in space is equal to the degree of the cutter surface geometry G, i.e. sphere is a degree of two surface. b) If a point belongs to the removal volume then there is at least one cutter location where the cutter surface touches this point. c) If the point is on the boundary of the swept volume SVi of the cutter there is a single tool position where the cutter surface touches this point. The solution for Eq.(6.13) gives two real roots (Property 6.4-a,b) t1 and t2 where t1 \u2264 t2. These roots represent two possible cutter locations where the intersection point I lies on the spherical surface as shown in Figure (6.30-a). To differentiate between the two choices and make the correct selection for t, the sign of the dot products f \u2022 N1 and f \u2022 N2 between the feed vector f and the surface normals N1 and N2 for the two positions of the cutter is used. A negative value indicates that I is in the shadow of the cutter and so cannot be a member of CWEK(t). For example, as shown in the figure at location t2 the dot product is negative while at t1, it is positive. It is easy to see that for linear tool motions the smaller root of Eq.(6.13) should always be used. For the special case where I lies on the boundary of the swept volume of the cutter such that the scalar product f \u2022 N1, N2 = 0, the cutting tool has only one cutter location (Property 6.4-c) as shown in Figure (6.30-b). The solution to Eq. (6.13) yields a repeated root and also I satisfies the following system of equations as shown at [82] 0),,,( III =tzyxGS , 0),,,( III =\u2202 \u2202 t tzyxGS (6.15) Chapter6. Cutter Workpiece Engagements 154 (a) (b) Figure 6.30: Different cutter locations for I \u2208 CWEK(t). Step 2: Given the value of the tool path parameter t = tI when the cutter surface passes through I, a transformation is created to map the global coordinate system (WCS) to a local tool coordinate system (TCS) at this location on the tool path. With the w axis of cutter being one of the axes of the TCS the v axis (see Figure 6.28) is obtained from the cross product of w and the instantaneous feed direction vector f(t=tI). The third axis u is perpendicular to the first two. ||\/)( fwfwv \u00d7\u00d7= (6.16) ||\/)( wvwvu \u00d7\u00d7= where kjif )()()( SESESE zzyyxx \u2212+\u2212+\u2212= (6.17) Using Eqs.(6.16) the transformation of a point in the TCS to one in the WCS is given by, [ ] Itt = = JI'I (6.18) where [ ] II I TTTT w v u J tt www vvv uuu tt tt tztytx zyx zyx zyx t == = \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb \uf8f9 \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 \uf8ee \u2261 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb \uf8f9 \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 \uf8ee = 1)()()( 0 0 0 1 0 0 0 )( (6.19) Chapter6. Cutter Workpiece Engagements 155 and I' are the coordinates of I in the TCS. Step 3: To find \u03d5(t) and d(t) the coordinates of an engagement point in the TCS is required. This is obtained from Eq.(6.18) as, [ ] 1 I ' \u2212 = = ttJII (6.20) Given that [ ] Itt= J is orthogonal, its inverse is defined by, [ ] I I 1)()()( 0 0 0 1 tt wvu wvu wvu tt ttt zzz yyy xxx = \u2212 = \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb \uf8f9 \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 \uf8ee \u2022\u2212\u2022\u2212\u2022\u2212 = TwTvTu J (6.21) Expanding equation (6.20) using (6.21) gives the coordinates of I'( III ,, zyx \u2032\u2032\u2032 ). These are, )( II ttx =\u2022\u2212\u2022=\u2032 TuuI )( II tty =\u2022\u2212\u2022=\u2032 TvvI (6.22) )( II ttz =\u2022\u2212\u2022=\u2032 TwwI Step 4: Given the coordinates of I' on the hemi-spherical surface of the BNEM, its tool engagement angle \u03d5(t) is obtained using spherical coordinates. The Engagement angle for the front contact face CWEK,s1(t) is given by \uf8f7 \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ec \uf8ed \uf8eb \u2032+\u2032 \u2032 == \u2212 2 I 2 I I1 I cos)( yx y tt\u03d5 (6.23-a) and for the back contact face CWEK,s2(t) is given by Chapter6. Cutter Workpiece Engagements 156 \uf8f7 \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ec \uf8ed \uf8eb \u2032+\u2032 \u2032 \u2212== \u2212 2 I 2 I I1 I cos2)( yx y tt pi\u03d5 (6.23-b) The depth of the engagement point for the spherical part is the angle from T(t) to I' such that: \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ed \uf8eb \u2032 \u2212== \u2212 r z ttd I1I 1cos)( (6.24) and finally the parameter L(t) of I is obtained using, SE VV \u2212== tttL )( I (6.25) Derivation of Mcyl for the BNEM The mapping for the cylindrical surface of the Ball end cutter is obtained following the same steps just developed for the hemispherical cutter surface. It can be seen that except for the first, all steps and equations to accomplish the mapping remain the same. For the first step, the geometry of the cylinder changes Eqs.(6.11) and (6.12) to, [ ] 0)()()(:)( 22 =\u2212\u2022\u2212\u2212\u2212\u2022\u2212 rtGC nFPFPFP (6.26) and, [ ] 0)()()(:)( 22 , =\u2212\u2022\u2212\u2212\u2212\u2022\u2212 rtCWE CK nFIFIFI (6.27) respectively (see Figure 6.31). Where r and n are the radius of the cylindrical surface and the unit normal vector of the tool rotation axis respectively. As with Eq.(6.13), Eq.(6.27) results in a quadratic equation in t when I and F are substituted. The solution to this and the remaining steps in the mapping procedure lead to the representation of a point I on the cylindrical surface of the BNEM in (\u03d5, d, L) coordinates. Detailed transformation formulas for the cylindrical end mill can be found in the Appendix A.2. Chapter6. Cutter Workpiece Engagements 157 Figure 6.31: Cylindrical contact face CWEK,C(t) of BNEM 6.2.1.2 Mapping M for Circular Toolpath In this section we highlight the mapping procedure for a circular tool path. The steps summarized in (Figure 6.29) can be followed with some important differences. One in particular is that the feed direction changes as a function of the tool path parametric variable t. This impacts the transformation from the WCS to the TCS. The parametric representation for the tool tip point for a circular tool path of radius R centered at ),,( CCC zyx=C is given by ( ) edCT )sin()cos( tRtRt ++= (6.28) where [ ]pi2,0\u2208t (see Figure 6.32). d(xd, yd, zd) and e(xe, ye, ze) are two orthogonal unit vectors defining the plane of the circular tool path. Thus, this representation for T is general even though circular interpolation on most milling machines is restricted to the XY, XZ and YZ planes. Chapter6. Cutter Workpiece Engagements 158 Figure 6.32: Moving coordinate frame for circular tool path As with linear tool paths the local tool coordinate system TCS defined with u, v and w has its origin at the tool tip point. The reference point F is expressed in terms of the cutting tool tip coordinates using Eq.(6.10). Following the same steps outlined in section (6.2.1.1) Eq.(6.28) is substituted into Eq.(6.10) then into the implicit equation of the hemispherical surface of the cutter (6.7-a) with the coordinates of a point I belonging to the CWEK(t) obtained from the removal volume. The resulting equation can be expressed in terms of the parameter t in the following form 0)sin()cos( 012 =++ AtAtA (6.29) where A2, A1 and A0 are constants for the current intersection point I given by the following expressions ( ) ( ) ( )[ ]ZCIYCIXCI nrzznryynrxx \u2212\u2212\u2212\u2212\u2212\u2212=a , (6.30) ( )da \u2022\u2212= RA 22 , ( )ea \u2022\u2212= RA 21 , 2220 rRA \u2212+= a Eq.(6.29) in t can be written in the following form 0)sin( 02221 =+++ AtAA \u03b8 (6.31) Chapter6. Cutter Workpiece Engagements 159 where ( )1212 ,tan AA\u2212=\u03b8 . This leads to the following general equation for t ( ) pi\u03b1\u03b8 nt n +\u2212+\u2212= 1 , \uf8f7 \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ec \uf8ed \uf8eb + \u2212 = \u2212 2 2 2 1 01sin AA A \u03b1 (6.32) where n is integer. Define n1 to be the smallest n such that t > t0. Then n1 can be calculated as \uf8f7\uf8f7 \uf8f8 \uf8f6 \uf8ec\uf8ec \uf8ed \uf8eb \uf8fa \uf8fa \uf8f9 \uf8ef \uf8ef \uf8ee \u2212 ++ \uf8fa \uf8fa \uf8f9 \uf8ef \uf8ef \uf8ee +\u2212 = 2 1 2 2, 2 2min 001 pi \u03b1\u03b8 pi \u03b1\u03b8 tt n (6.33) After n1 is found it can be incremented by unit steps to calculate the next tool tip point T for the given I. There can be up to two cutter location points in the interval for a given engagement point I (Figure 6.33). As explained in section (6.2.1.1) the minimum root is selected. If I is on the envelope surface of the cutter (Property 6.4-c), Eq.(6.29) gives two repeated roots. Having solved for the parameter t value when I is an engagement point, it is substituted into equation (6.28) to find the tool tip coordinates. The direction of the feed vector at this location is given by k jiTf ))cos()sin(( ))cos()sin(())cos()sin(()( ed ededI ztztR ytytRxtxtR dt d tt +\u2212+ +\u2212++\u2212=== (6.34) Figure 6.33: Parameter values of the cutter tool tip Chapter6. Cutter Workpiece Engagements 160 Given f and t, Eqs.(6.23-a,b) and (6.24) are used to find ( )d,\u03d5 parameters for I and L(t) of I is obtained using tRttL I == )( (6.35) the mapping for the cylindrical portion of the cutter follows the same steps using the implicit surface for a cylinder. 6.2.1.3 Mapping M for Helical Toolpath A special case of 3-axis machining is helical milling shown in Figure (6.34). In this operation a cutting tool is feed along its tool axis (Z-axis) as a circle is interpolated by the other two (X and Y axes). The cutter follows a helical trajectory. This operation is useful for (1) contour milling of cylindrical protrusions or for enlarging of pre-machined or pre-formed holes, (2) for hole machining into solid stock. Figure 6.34: Sweeps for Helical Milling In this section derivations of the mapping M for the helical toolpath will be shown for the Flat-End mill. The methodology and many of formulae developed apply equally to the mapping for the other types of the cutter geometries. A Flat end mill is made up of a cylindrical surface for its side and a flat surface for its bottom. Based on the kinematics of 3- axis machining these surfaces denoted CWEK,c1(t) and CWEK,c2(t) respectively will contribute to the set CWEK(t) as defined before i.e. CWEK (t) = CWEK,c1(t) \u222a CWEK,c2(t) Chapter6. Cutter Workpiece Engagements 161 As explained in section (5), during machining each cutter surface contributes to CWEs differently. Because of this the removal volumes generated by the side and flat surfaces of the Flat-End mill are separated in this research. The Side face (Figure 6.35-a) has an engagement in the range of zero topi, and the bottom face (Figure 6.35-b) in the range of zero to 2pi . (a) (b) Figure 6.35: Removal volumes of (a) side face and (b) bottom face. In this section the mapping methodology to be described is applied to the removal volume generated by the cutter side face only. This is because directly intersecting a plane which is perpendicular to the tool rotation axis with the removal volume of the bottom face gives the CWEs of the bottom face. For the cylindrical surface tool reference point F equals to the tool tip point T (see Figure 6.36) and the parametric representation for the tool tip point for a helical tool path of radius R centered at C( xc, yc , zc ) is ( ) \uf8f7 \uf8f7 \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ec \uf8ec \uf8ed \uf8eb + + + == tcz tRy tRx tzyxt C C C TTT )sin( )cos( ,,,)( TF (6.36) where t \u2208[0, 2pi ] and 2pic is a constant giving the vertical separations of the helix\u2019s loops where c < 0. Chapter6. Cutter Workpiece Engagements 162 Figure 6.36: Parameters describing a helical tool motion for the Flat-End mill Following the same steps outlined in section (6.2.1.1), Eq.(6.36) is substituted into the implicit equation of the cylindrical surface of the cutter (6.7-b) with the coordinates of a point I belonging to the CWEK(t) obtained from the removal volume. The resulting equation can be expressed in terms of the parameter t in the following form 0)sin()cos( 012 =++ AtAtA (6.37) where ( ) RxxA CI \u2212\u2212= 22 ( ) RyyA CI \u2212\u2212= 21 ( ) ( ) 22220 rRyyxxA CICI \u2212+\u2212+\u2212= Eq.(6.37) in t can be solved by applying the Eqs.(6.32) and (6.33). There can be up to two cutter location points in the interval for a given engagement point I (Figure 6.37). As explained in Property (6.4) the minimum root is selected. Chapter6. Cutter Workpiece Engagements 163 Figure 6.37: Different cutter locations for an engagement point I \u2208 CWEK(t). As with linear tool paths the local tool coordinate system TCS defined with u, v and w has its origin at the tool tip point. The direction of the feed vector for the parameter t value from Eq.(6.36) is given by kjiTf ctRtR dt d tt I ++\u2212=== )(cos)(sin)( (6.38) Using t, the parameter L(t) of I is obtained by 22)( cRtttL II +== (6.39) Given f and t, the Eq.(6.23-a) is used to find \u03d5 parameter for I and finally, the depth of the engagement point as defined by the distance from T(t) to I' measured along the tool axis vector is simply the z-coordinate of I'. )()( III ttzttd =\u2022\u2212\u2022=\u2032== TwwI (6.40) 6.2.1.4 Implementation In this research the mappings developed in the previous sections are applied directly to a polyhedral representation of the removal volume. The specific representation utilized is the STL (\u201cStereo Lithography\u201d) format though other representations (VRML, .jt, .hsf etc.) can as easily be adopted. As explained in chapter (1), in the STL representation the geometry of surfaces is represented by small triangles called facets. These facets are described by three Chapter6. Cutter Workpiece Engagements 164 vertices and the normal direction of the triangle. To test the methodology for CWE extraction, a prototype system has been assembled using existing commercial software applications and C++ implementations of the mapping procedure described above. This system is shown in (Figure 6.38). Simulation of the CLData is performed using VERICUT a commercial NC verification application. This application uses a voxel-based model to capture changes to the in-process workpiece. STL representations of in-process workpiece states can be generated from the voxel representation prior to any tool path motion in the CLData file. Not currently available through the programmable APIs provided for customization is a function for outputting STL representations of removal volumes. However, this is probably an easy extension to implement. Thus, to obtain removal volumes for a given tool motion, (ACIS) is utilized to model a B-rep solid of the associated swept volume. Figure 6.38: Implementation of CWE extraction methodology These swept volumes are exported to STL format. A Boolean intersection between this swept volume and the current in-process workpiece output from Vericut is performed using the polyhedral modeling Boolean operators implemented in a third commercial application, Magics X [51].Though currently a manual process this prototype system creates STL models Chapter6. Cutter Workpiece Engagements 165 of removal volumes generated during the machining of complex components. It utilizes the same CAD\/CAM data generated for machining the part. In this section examples of the application of the mapping M to removal volumes generated by different types of cutting tools and tool motions are presented. The first set of examples is designed to demonstrate the generality of the approach with respect to tool geometry and linear, circular and helical tool motions. The second set comes directly from applying the prototype system described in the previous section to CAD\/CAM data created in machining a gearbox cover. This demonstrates the practicality of the methodology. Example Set 1 In this example set different cutting tools and linear, circular and helical tool motions are presented to demonstrate the generality of the approach. Figures (6.39) to (6.41) show Ball, Flat and Tapered-Flat End mills removing material along a linear ramping tool path i.e. the tool moves in all three axes simultaneously. In each case the original and transformed removal volumes are given. The removal volumes associated with the different cutter surfaces are separated. In the case of the flat end mill this gives the material removed by the cylindrical (Side Face) and flat (Bottom Face) cutter surfaces. Plots of CWEs at different Cutter Locations (CL) along the tool path are illustrated. These are obtained by intersecting the transformed removal volume with a plane representing the cutting tool. Two formats for plotting the CWEs are used. The first is an XY plot of depth-of-cut (as measured from the tool tip point) versus engagement angle. The second plot shows the engagement area of the cutter bottom surface (Figure 6.40) in a polar format. In the linear ramping example of the Ball-End mill the plots show that engagement of the cutter occurs over the full range i.e. [0, 2pi]. Between zero and pi the engagement is due to the front contact face (the CWE in this zone is divided to illustrate this) and between pi and 2pi due to the back contact face. For the Flat-End mill XY plots show the engagement between zero and pi for the side face and polar plots show the engagement between zero and 2pi for the bottom face. Because the Tapered-Flat-End mill is ramping up only its front face has engagement with the workpiece and it changes between zero andpi. Chapter6. Cutter Workpiece Engagements 166 Figure 6.39: CWEs for Ball-End mill performing a linear 3-axis move Figure 6.40: CWEs for Flat End mill performing a linear 3-axis move Chapter6. Cutter Workpiece Engagements 167 Figure 6.41: CWEs for Tapered-Flat-End mill performing a linear 3-axis move The next example shows a Flat-End mill removing material along a helical tool path (Figures 6.42 \u2013 a,b,c). In each case the original and transformed removal volumes are shown. In Figure (6.42-d) the transformed removal volumes for the side face are shown and for the given CLs (cutter location points) CWEs are obtained by intersecting a plane representing the cutting tool with the transformed removal volume. Two formats for plotting the CWEs are used. The first is an XY plot of the depth of cut (DOC) and immersion angle which is generated by the side face of the flat end mill. The engagement of this surface of the cutter occurs over the [ ]pi,0 range as can be seen. The second is a polar plot of the cutter radius versus the immersion angle which is generated by the bottom surface of the flat end mill. The engagement for the bottom face of the cutter occurs over the [ ]pi2,0 range. For CL1and CL2 both types of plots are shown because the bottom of the cutter is removing material at these locations. For the other cutter locations only the side face of the end mill is engaging the workpiece. Chapter6. Cutter Workpiece Engagements 168 Figure 6.42: Helical Tool Motions with a Flat-End mill and CWEs The final example (Figure 6.43) demonstrates the transformation as applied to a circular 2\u00bdD tool path. These results together show that the mapping methodology reduces various Chapter6. Cutter Workpiece Engagements 169 combinations of cutter and tool path geometry to a generic form to which a single intersection operator can be applied. Figure 6.43: CWEs for Flat-End mill performing a circular move Example Set 2 The examples in this section are created using the steps outlined in Figure (6.38). The Figure (6.44) shows model of a gearbox cover to be machined from rectangular stock. The CAD model and tool paths for machining were both created using Unigraphics NX3. In this figure, it is shown in-process workpiece and the removal volume for the next tool path generated by Vericut, ACIS and Magics. The removal volume generated by the ball nose end mill clearly shows the complicated removal volume shapes and resulting CWEs that can be generated when machining complex parts. Chapter6. Cutter Workpiece Engagements 170 Figure 6.44: CWEs for Ball-End mill performing a linear 3-axis move Figure (6.45) shows a helical milling operation for enlarging a hole. In this example both the side and the bottom faces of the Flat-End mill are removing material. In this figure side face removal volumes are shown. The cutter performs four half turns each corresponding to a 00 \u2013 1800 range i.e. third turn has the starting angle 3600and ending angle5400along the tool path. Figure 6.45: Helical tool path application with removal volumes for each half turn. It can be seen from Figure (6.46) that starting from the first turn the material removal rate is constantly increasing. Then this rate becomes constant for the third and the fourth turns. Chapter6. Cutter Workpiece Engagements 171 Figure 6.46: CWEs of the Helical Tool Motions 6.2.2 Engagement Extraction Methodology in 5-Axis Milling In 3-axis milling cutter translates along a tool path with a fixed tool axis vector. For the 3-axis CWE methodology described in the previous section the mapping is performed with respect to the Tool Coordinate System (TCS) in which the z \u2013axis was fixed. In this methodology it is assumed that the size of the facets of the removal volume must be small enough. Thus during the mapping of the removal volume the triangle deformation stays small enough. Later this will be discussed in the discussion section. On the other hand in 5-axis tool motions the direction of the tool axis vector continuously changes. When the mapping described in the previous section is applied to the removal volume obtained from the 5-axis tool motions, the deformation on the triangles becomes big. One of the solutions for reducing the deformation is to use much smaller triangular facets but this brings a heavy computational load to CWE extractions. Therefore, in this research, the mapping methodology is not applied for obtaining CWEs in the 5-axis milling. We developed a new methodology for the 5-axis CWE extractions. This methodology is explained in Figure (6.47) for the impeller machining by using a Taper-Flat-End mill and it has 3 main steps: Step 1: In this stage there are two components: the in-process workpiece and the BODY. Both of them are represented by triangular facets having vertices and normal vectors. For a given Cutter Location (CL) point the BODY is generated as a solid using the methodology described in section (6.1.3) and then it is exported to STL format for obtaining the tessellated representation. A Boolean intersection between this tessellated BODY and the current in- Chapter6. Cutter Workpiece Engagements 172 process workpiece is performed using the polyhedral modeling Boolean operator. This intersection process creates an STL model of the removal volume for a given CL point. Step 2: In this stage there are two components also: The removal volume in the STL format and the feasible contact surface in the solid format. The analytical surface\/line intersections are performed between the feasible contact surface and the triangles of the removal volume. For example in Figure (6.47)-steps 2, triangles of the removal volume are intersected with the cone and sphere parts of the feasible contact surface. Step 3: In this last step the connection of the intersection points obtained in the previous step generates the boundary of the CWE, bCWE. Then this boundary described in 3D Euclidian space is mapped into 2D space defined by the engagement angle and the depth of cut. Figure 6.47: CWE extraction steps for 5-axis milling. Although this 5-axis CWE methodology in polyhedral models is explained by using the Taper-Flat-End mill, the methodology applies equally for the other cutter surfaces also. If the Chapter6. Cutter Workpiece Engagements 173 cutter surface has the second order equation, e.g. cylinder, cone or sphere natural quadric surfaces, each line \u2013 surface intersection gives two roots. If the cutter surface has the fourth order equation e.g. torus, each line surface intersection gives four roots. In most cases only one of these roots are needed for obtaining the bCWEs and the rest is redundant. During the implementation the ACIS geometric kernel with C++ is used for creating and tessellating the BODY and the Magics is used for the polyhedral Boolean intersections and updating the in- process workpiece. These three steps are performed for each tool path segment. After processing a toolpath segment for CWE extractions, the material is removed from the in- process workpiece in the vicinity of the toolpath segment. 6.3 The Cutter Workpiece Engagements in Vector Based Model In this section for the CWE calculations the cutting tool geometries are represented implicitly by natural quadrics and the plane (a degenerate quadric). Natural quadrics consist of the sphere, circular cylinder and the cone. 6.3.1 Intersecting Segment Against Plane Given a point P with normal n on a planepi , the following equation can be written for all points X on this plane ( ) 0=\u2212 PX\u00b7n (6.41) In the above equation the vector (X-P) is perpendicular to the vector n. When the dot product in Eq.(6.41) is distributed across the subtraction the implicit representation of the plane takes the form of dX =\u00b7n where Pd \u00b7n= . Let a segment is given by ( ) vIIIvI aba \u2212+=)( for 10 \u2264\u2264 v . For finding the intersection point I (see Figure 6.48) between the segment and the plane the equivalent of I(v) is substituted for X in the plane equation and then this equation is solved for v ( )( ) dvIII aba =\u2212+\u00b7n (6.42) Chapter6. Cutter Workpiece Engagements 174 Isolating v in the above equation yields ( ) ( )( )aba IIIdv \u2212\u2212= \u00b7n\u00b7n . The expression for v can now be inserted the parametric equation of the segment I(v) for finding the intersection points I ( ) ( )( )[ ]( )ababaa IIIIIdII \u2212\u2212\u2212+= \u00b7n\u00b7n (6.43) Figure 6.48: Intersecting a segment against a plane 6.3.2 Intersecting Segment Against Sphere Let a segment is given by ( ) vIIIvI aba \u2212+=)( for 10 \u2264\u2264 v . Let the sphere surface defined by ( ) ( ) 2rCXCX =\u2212\u2212 \u00b7 , where C is the center of the sphere, and r is radius. To find the v value at which the segment intersect the sphere surface I(v) is substituted for X, giving ( )( ) ( )( ) 2rCvIIICvIII abaaba =\u2212\u2212+\u2212\u2212+ \u00b7 (6.44) Let CI a \u2212=k and ( ) abab IIII \u2212\u2212=d then the Eq.(6.44) takes the form of ( ) ( ) 2rvv =++ dk\u00b7dk . Expanding the dot product yields the following quadratic equation in v ( ) ( ) 0=\u2212++ 22 k\u00b7kd\u00b7k2 rvv (6.45) The solutions of this quadratic equation is given by Chapter6. Cutter Workpiece Engagements 175 ( ) ( ) ( ) ][22,1 2k\u00b7kd\u00b7kd\u00b7k rv \u2212\u2212\u2212= m (6.46) Solution of the above equation gives three outcomes with respect to the discriminant ( ) ( ) ][2 2k\u00b7kd\u00b7k r\u2212\u2212=\u2206 . If 0>\u2206 there are two real roots for which segment intersects the sphere twice (Figure 6.49-a), If 0=\u2206 there are two equal real roots for which the segment hits the sphere tangentially (Figure 6.49-b), and if 0<\u2206 there are no real roots which corresponds to segment missing the sphere completely (Figure 6.49-c). Although for the case given by 0>\u2206 there are two real roots, one of them can be false intersection (Figure 6.49-d,e). In this case the segment can start inside or outside sphere. One of the v values from Eq.(6.45), its value is in the range of ]1,0[\u2208v , gives the intersection point. (a) (b) (c) (d) (e) Figure 6.49: Different cases in segment\/sphere intersections: (a) Two intersection points, (b) intersecting tangentially, (c) no intersection, (d) segment starts inside sphere, and (e) segment starts outside sphere. 6.3.3 Intersecting Segment Against Cylinder A cylinder with an arbitrary orientation can be described by an axis which passes through points B and Q, and by a radius r (see Figure 6.50). Let X denotes a point on the cylinder surface. The projection of the vector (X-B) onto the cylinder axis defined the vector (Q-B) yields ( ) n\u00b7BXBP \u2212=\u2212 (6.47) where the unit vector ( ) BQBQ \u2212\u2212=n . Applying Pythagorean Theorem to triangle defined by the points B, X and P yields Chapter6. Cutter Workpiece Engagements 176 ( ) ( ) 222 rBPBXBXBX +\u2212=\u2212\u2212=\u2212 \u00b7 (6.48) Plugging Eq.(6.47) into Eq.(6.48) yields the following implicit representation for a cylinder surface ( ) ( ) ( ){ } 022 =\u2212\u2212\u2212\u2212\u2212 rBXBXBX n\u00b7\u00b7 (6.49) Figure 6.50: A segment is intersected against the cylinder given by points B and Q and the radius r. The intersection of a segment ( ) vIIIvI aba \u2212+=)( with the cylinder can be found by substituting I(v) for X into Eq.(6.49) and solving for v. ( )( ) ( )( ) ( )( ){ } 022 =\u2212\u2212\u2212+\u2212\u2212\u2212+\u2212\u2212+ rBvIIIBvIIIBvIII abaabaaba n\u00b7\u00b7 (6.50) Setting BI a \u2212=d and ab II \u2212=k in the above equation the above equation can be written in the following form A2 v2 + 2A1 v + A0 = 0 (6.51) where ( )22 n\u00b7k-k\u00b7k=A ( ) ( )n\u00b7dn\u00b7k-d\u00b7k=1A Chapter6. Cutter Workpiece Engagements 177 ( ) 220 rA \u2212= n\u00b7d-d\u00b7d Solving Eq.(X) for v gives 2 02 2 11 2,1 A AAAA v \u2212\u2212 = m (6.52) The sign of ( )0221 AAA \u2212 determines the number of real roots in the above equation. If the sign is positive, there are two real roots, which correspond to the line intersecting the cylinder in two points. If the sign is negative, there are no real roots, which corresponds to the line not intersecting the cylinder. When ( )0221 AAA \u2212 equals to zero there are two equal real roots signifying that the line tangentially touching the cylinder. 6.3.4 Intersecting Segment Against a Cone The cone (see Figure 6.51) is defined by a vertex V, unit axis direction vector n and half- angle\u03b1 , where ( )2\/,0 pi\u03b1 \u2208 . Let P define the points on the cone surface. The half-angle \u03b1 is between P-V and n, therefore the following expression can be written \u03b1cos=\uf8f7 \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ec \uf8ed \uf8eb V-P V-P \u00b7n (6.53) The above equation is squared for eliminating the square root calculations and this yields the following quadratic equation ( )( ) ( )22 cos V-PV-P\u00b7n \u03b1= (6.54) Chapter6. Cutter Workpiece Engagements 178 Figure 6.51: A cone with defining parameters The Eq.(6.54) represents a double cone which has the original cone and its reflection in the opposite direction of n. For the intersection calculations the original cone is needed. For eliminating the reflected cone the constraint ( ) 0>V-P\u00b7n is taken into account also. The quadratic cone equation given in Eq.(6.54) can be written in a quadratic form as ( ) ( ) 0=V-PV-P T M , where IM \u03b12cos\u2212= Tnn . Therefore the original cone which has the unit axis vector n can be defined by ( ) ( ) 0=V-PV-P T M , and ( ) 0>V-P\u00b7n (6.55) Let the line segment is defined by ( ) vIIIvI aba \u2212+=)( , where v\u2208R. For obtaining the intersection points between the line segment and the cone )(vI is substituted into Eq.(6.55) and this yields the following quadratic equation in one variable A2 v2 + 2A1 v + A0 = 0 (6.56) where ( ) ( )TT abab IIMIIA \u2212\u2212=2 ( ) ( )V-PT MIIA ab \u2212=1 ( ) ( )V-PV-P T MA =0 Chapter6. Cutter Workpiece Engagements 179 As explained in section (6.3.3) the sign of ( )0221 AAA \u2212 determines the number of real roots in the above equation. 6.3.5 Obtaining the Cutter Workpiece Engagements In the discrete vector approach calculating the CWEs are straight forward. By intersecting the line segments with the cutter geometries the intersection points are obtained. For mapping the intersection points into 2D domain represented by the engagement angle vs. depth of cut, cutter surfaces are decomposed into the grids (Figure 6.52). During simulation if any one of these grids contains an intersection point, then it is mapped into 2D space. The resolution of the grids on the cutter surfaces is adjusted with respect to the resolution of the discrete vectors of the workpiece. Figure 6.52: Decomposing cutter surfaces into grids. 6.4 Discussion The methodologies presented in this chapter target the important problem of finding Cutter Workpiece Engagements (CWEs) in the milling operations. The CWEs have been calculated for supporting the force prediction model which requires the CWE area in the format described by the engagement angle versus the depth of cut respectively. The developed methodologies can be classified into three categories based on the mathematical representation of the workpiece geometry: Solid modeler based, polyhedral modeler based and vector based methodologies. There is always a tradeoff between computational efficiency and accuracy in these approaches. For example the polyhedral and vector based approaches generate approximated CWE results and because of this they require a shorter Chapter6. Cutter Workpiece Engagements 180 computational time than does the solid modeler based approach. But on the other hand in the solid modeler based approach the most accurate CWEs are obtained. In section (6.1) the B-rep solid modeler based CWE methodologies for the 3 and 5-axis milling have been developed using a range of different types of cutters and tool paths. In the 3-axis methodology the decomposed cutter surfaces have been intersected with their removal volumes for obtaining the boundary curves of the closed CWE area. For this purpose a removal volume has been decomposed into its constituent faces and then the face\/face intersection have been performed between the feasible contact surface and these constituent faces at a given cutter location point. Examples have been presented to show that the approach generates proper engagements. Also it can be seen from the examples that in this methodology the computational load in terms of the required Boolean operations is higher for the descending motions described in chapter (5). For example in the ascending motion of a Flat-End mill only the cylindrical part removes the material from in-process workpiece and because of this one Boolean intersection is needed for obtaining the required removal volume in this methodology. But on the other hand the descending motion of the same cutter requires two Boolean intersections one for the cylindrical part and another for the bottom flat part. For the 5-axis milling we had to develop another methodology. As explained in section (6.1.3), because the envelope boundary for 5-axis tool motions are approximated by spline curves applying the same methodology described for the 3-axis milling generates non robust results. Because of this in 5-axis methodology a BODY obtained from the feasible contact surface has been intersected with the in-process workpiece at a given cutter location point. Then the face\/face intersections have been performed for obtaining the boundaries of the CWEs. In this methodology for minimizing the error introduced by offsetting the feasible contact surface infinitesimally, the offset direction vector has been chosen at the middle of the tool rotation axis. In section (6.2) the polyhedral modeler based CWE methodologies for the 3 and 5-axis milling have been developed. For the 3-axis milling a mapping methodology has been developed that transforms a polyhedral model of the removal volume from Euclidean space to a parametric space which is defined by the cutter engagement angle, depth of cut and cutter location. To reduce the size of the data structure that needs to be manipulated the removal volume has been used instead of the in-process workpiece. This approach also has Chapter6. Cutter Workpiece Engagements 181 the potential of being implemented using a parallel processing strategy [85]. This mapping methodology brings some important advantages over other approaches [88,90]. First, the complexity in the CWE calculations is reduced to first order analytical plane-plane intersections. When compared to other polyhedral modeling approaches it has greater robustness because it addresses the chordal error problem found in intersecting polyhedral models. In this methodology it has been assumed that the size of the facets in the removal volume is sufficiently small so as not to introduce significant errors in the CWE boundaries. To study this, a comparison has been made between engagements obtained from intersecting the removal volume using a B-rep solid modeler (the most accurate approach) and those from the 3-axis polyhedral approach described in this chapter using different faceting resolutions. Examples of the original and transformed removal volumes are shown in Figure 6.53(a) and (b) respectively. CWEs are obtained for cutter locations CL1 to CL29. For CL5 the CWEs from the solid modeler (Figure 6.53(c)) and polyhedral modeler at facet resolutions of 2 mm (Figure 6.53(d)), and 6mm (Figure 6.53(e)) are shown. To compare the effect of the size of the facets the CWE area is decomposed using a QuadTree [22] spatial data structure (Figure 6.53(f)). Figure 6.53: Removal Volumes and CWEs for different resolutions Chapter6. Cutter Workpiece Engagements 182 The area of the CWE is obtained by accumulating the square areas that lie within the QuadTree representation of the CWE boundary. For each CL point the areas obtained from the different resolutions are compared with that of the B-rep solid modeler (Figure 6.54(a)). The graph shows a 4% error at 6mm and less than 1% at 2mm. Figure 6.54(b) shows the facet size vs. intersection time i.e. the time for obtaining the CWE area for a given cutter location point. While the absolute value would vary depending on the implementation, the trend should remain the same. This shows that there is a small increase in the intersection time as the resolution is decreased from 6mm to 2mm after which it increases significantly. Both the error and intersection time results point to 2mm being a practical limit for the facet size in this example. We point out that this limit will vary depending on the cutter size. An expression needs to be developed to calculate the resolution that considers this parameter. (a) (b) Figure 6.54: The effect of the facet resolutions Because in 5-axis tool motions the direction of the tool axis vector continuously changes applying the mapping described for the 3-axis milling has increased the distortions of the facets. Therefore an approach similar to that of the 5-axis solid modeler has been developed. In this approach the only difference was the application of the face-face intersections. These intersections have been performed between triangular facets of a BODY and a solid representation of the feasible contact surface. And finally in section (6.3) a vector based CWE methodology has been developed. In this methodology the cutter has been discretized into slices perpendicular to the tool axis. For obtaining the CWEs the intersections have been performed between discrete vectors and cutter slices. The intersection calculations are straight forward. 183 Chapter 7 Conclusions 7.1 Contributions and Limitations In this thesis new methodologies have been proposed to facilitate Cutter Workpiece Engagement (CWE) extractions in milling process modeling. This includes an analytical methodology for determining the shapes of the cutter swept envelopes in multi-axes milling, methodologies for updating the in-process workpiece surfaces, analysis of the feasible cutting faces and finally algorithms for extracting CWEs. More specifically the contributions are summarized as follows: \u0001 An analytical approach for determining the envelope of a swept volume generated by a general surface of revolution performing multi-axes machining has been developed. In this approach the cutter geometries are represented using canal surfaces and for describing the cutter envelope surfaces the two-parameter-family of spheres has been introduced. Analytically it has been proven that for cutter surfaces performing 5-axis tool motions any point on the envelope surface is also a member of the point set generated from the two-parameter-family of spheres formulation. Later the methodology is generalized for cutters with general surfaces of revolution which performs 5-axis tool motions. In this methodology, by describing the radius function and the trajectory of the moving sphere, different cutter surfaces can be obtained. In this sense the methodology is independent of any particular cutter geometry. The implementation of the methodology is simple. Especially when the cutter geometries are pipe surfaces, fewer calculations are needed for describing the cutter envelope surfaces. Although examples from the application of this methodology have been shown for common milling cutter geometries described by the 7-parameter APT model, this methodology can be also applied to rare cutter geometries. In some cases of 5-Axis milling the cutter swept volume maybe self-intersecting, which requires special processing to handle the topological and geometric problems due to the complex tool motions. Self intersections in this research have not been considered. Modifications to the methodology will need to be developed for this special case. Chapter7. Conclusions 184 \u0001 A discrete vector model based in-process workpiece update methodology has been developed. During machining simulation, for each tool movement modification of the in-process workpiece geometry is required to keep track of the material removal process. In this thesis in-process workpiece modeling (or updating) methodologies have been developed using a discrete vector representation. These vectors having orientations in the directions of the x,y,z-axes of R3 are intersected with tool envelopes. With this representation more vectors in different directions are used when compared to other discrete vector approaches. Therefore especially when the workpiece has features like vertical walls and sharp edges, the quality in the visualization of the final product has been increased. Also the localization advantage of the Discrete Vertical Vector approach has been preserved. For simplifying the intersection calculations the properties of the canal surfaces have been utilized. For cutter geometries defined by a circular cylinder, frustum of a cone, sphere and plane the vector intersection calculations for updating have been made analytically. Because of the complexity of the torus geometry the calculations in this case have been made by using a numerical root finding method. For this purpose a root finding analysis has been developed for guaranteeing the root(s) in the given interval. A typical milling tool path contains thousands of tool movements and during the machining simulation for calculating the intersections only a small percentage of all the discrete vectors is needed. For this purpose a localization methodology, based on the Axis Aligned Bounding Box (AABB) of each tool movement, has been developed. The best feature of the AABB is its fast overlap check, which simply involves direct comparison of individual coordinate values. As explained in chapter (4), for some workpiece geometries 3-axis machining is not suitable for updating the surfaces. Also exact 5-axis milling tool motions are not preferable in workpiece update simulations because the calculations require using the nonlinear root finding algorithms and therefore the computational time becomes high. Therefore in the developed methodologies tool motions using (3+2)-axis milling are considered instead. Using (3+2)-axis tool motions in which a cutter can have an arbitrary fixed orientation in space, 5-axis tool motions can be approximated. An example has been given for illustrating this situation. The Discrete Normal Vector (DNV) approach has not been considered in this research. The DNV approach can represent the workpiece surface features well with respect to a given tolerance. But because Chapter7. Conclusions 185 in this approach the directions of discrete vectors are not identical, localizing the cutter envelope surface during machining simulation becomes difficult. If the workpiece surfaces are represented by DNVs then an efficient localization methodology will need to be developed. \u0001 The engagement behaviors of NC cutter surfaces under varying tool motions have been analyzed. A typical NC cutter has different surfaces with varying geometries and during the material removal process restricted regions of these surfaces are eligible to contact the in- process workpiece with respect to the tool motions. In this thesis for representing these regions the terminology feasible contact surfaces (FCS) has been introduced. The word feasible has been used because although these surfaces are eligible to contact the in-process workpiece, they may or may not remove material depending on the cutter position relative to the workpiece. When the FCS contact the in-process workpiece the Cutter Workpiece Engagements (CWEs) are generated. Since CWEs are subsets of the FCS, formalizing the FCS helps us to better understand the CWE generation process. The FCS have been formulized by using the cutter surface and the cutter envelope boundaries. The cutter surface boundaries are fixed, but on the other hand the cutter envelope boundaries may change depending on the tool motion. For modeling the cutter envelope boundaries a tangency function defined by using the surface normal and the tool velocity has been utilized. Later by changing the tool velocity direction the distributions of the FCS on the cutter have been analyzed. The results from these analyses are later used in the development of the CWE extraction methodologies. \u0001 Methodologies for obtaining Cutter Workpiece Engagements (CWEs) in milling have been developed. A major step in simulating machining operations is the accurate extraction of the CWE geometries at changing tool locations. These geometries define the instantaneous intersection boundaries between the cutting tool and the in-process workpiece at each location along a tool path. The methodologies presented in this thesis target the important problem of finding CWEs in milling operations. The CWEs are calculated for supporting the force prediction Chapter7. Conclusions 186 model requires the CWE area in the format described by the engagement angle versus the depth of cut. In these methodologies a wide range of cutter geometries, toolpaths including 5- axis tool motions and workpiece surfaces have been used. The workpiece surfaces cover different surface geometries including sculptured surfaces. The developed CWE extraction methodologies can be classified into three categories based on the mathematical representation of the workpiece geometry: Solid modeler based, polyhedral modeler based and vector based methodologies. There is always a tradeoff between computational efficiency and accuracy in these approaches. In 3-axis solid and polyhedral model based approaches developed in this thesis to reduce the size of the data structure that needs to be manipulated the removal volume has been used instead of the in-process workpiece. In the 3-axis solid modeler methodology the cutter surfaces have been decomposed into different regions with respect to the feed vector direction. Then these surface regions have been intersected with their removal volumes for obtaining the boundary curves of the closed CWE area. Decomposing the cutter surfaces in this way allows CWEs to be obtained for different parts of a given cutter geometry, e.g. bottom flat or back side of a cutter. Using a solid modeler based representation the envelope boundaries generated by 5-axis tool motions are approximated by spline curve. Applying the solid model based methodology described for 3-axis tool motions to the removal volume obtained from 5-axis tool motions can generate un-expected results because of the non-robust surface\/surface intersections. Therefore in this thesis for 5-axis tool motions in-process workpiece has been used instead of the removal volume. In this approach the feasible contact surface generated at a given cutter location point has been offsetted linearly with an infinitesimal amount. As a result of this linear offsetting a surface volume has been generated. Then this volume has been intersected with the in-process workpiece. Later face\/face intersections have been performed for obtaining the boundaries of the CWEs. In this 5-axis solid model based methodology by linearly offsetting the feasible contact surfaces, the CWE extractions have became more robust. For addressing the chordal error problem in polyhedral models a 3-axis mapping technique has been developed that transforms a polyhedral model of the removal volume from Euclidean space to a parametric space defined by location along the tool path, engagement angle and depth-of-cut. As a result, intersection operations have been reduced to Chapter7. Conclusions 187 first order plane-plane intersections. This approach reduces the complexity of the cutter\/workpiece intersections and also eliminates robustness problems found in standard polyhedral modeling. Because in 5-axis tool motions the direction of the tool axis vector continuously changes applying the mapping described for the 3-axis milling increases the distortions of the facets. Therefore a CWE extraction approach similar to the 5-axis solid modeler based approach has been developed. In the polyhedral model based CWE extractions 2mm was the practical limit for the facet size. Still needed is an expression for finding the optimal facet size. 7.2 Future Work The future research work based on this thesis is summarized in the following: \u0001 In this thesis for the polyhedral model based CWE extraction methodologies a prototype system has been assembled using existing commercial software applications and C++ implementations. For future work a standalone polyhedral based CWE extraction approach that replaces the various commercial components will be developed. \u0001 The CWE areas must be decomposed for integrating with the force prediction model. More efficient decomposition techniques (e.g. Quadtrees) will be investigated as a way for representing the engagement geometry. 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[90] Yip-Hoi D., Peng X., \u201cR*-Tree Localization for Polyhedral Model Based Cutter\/Workpiece Engagements Calculations in Milling\u201d, ASME CIE, sept 4-7, 2007 [91] Yun Won-Soo, Ko Jeong Hoon, Lee Han UI, Cho Dong-Woo, Ehmann Kornel F., \u201cDevelopment of a virtual machining system, Part 3: Cutting process simulation in transient cuts\u201d, International Journal of Machine Tools and Manufacture, v 42, n15, p 1617-1626, 2002. 193 Appendix A Obtaining the grazing points for cutter geometries represented by pipe surfaces. As mentioned in chapter (3), the cylinder and torus surfaces are called as pipe surfaces. By definition a pipe surface is an envelope of the family of spheres with a constant radius. In this surface type the centers of the characteristic circle and the moving sphere equal to each other. In this section because of its complexity the property (3.2) is proven for the torus surface and for the cylindrical surface similar steps can be used. When the components of the Frenet frame given in Eq.(3.53) are plugged into Eq. (3.52), the torus surface is obtained as a set of the characteristic circles in the following form )),(sin),()(cos,(),(),,( utututRututK BMC \u03b8\u03b8\u03b8 ++= (A.1) where [ ]pi\u03b8 2,0\u2208 . The radius of the moving sphere for the pipe surfaces is constant. Therefore the partial derivative of the radius with respect to the toolpath parameter t equals to zero, i.e. ( ) 0=trt . Thus under this condition the radius and the center of the characteristic circle from Eqs. (3.50) and (3.51) become as rutR =),( , ),(),( utut mC = (A.2) Plugging the equations given in (A.2) into (A.1) yields )),(sin),((cos),(),,( ututrututK BMm \u03b8\u03b8\u03b8 ++= (A.3) For obtaining a point (so called grazing point) on the cutter envelope surface Eq.(A.3) is plugged into Eq. (3.55) and then the resultant equation representing the surface normal is plugged into Eq. (3.56). These yields ( ) ( ) 0sincos =+ B\u00b7mM\u00b7m uu \u03b8\u03b8 (A.4) AppendixA. Obtaining the grazing points for cutter geometries represented by pipe surfaces. 194 The trigonometric terms from Eq.(A.4) are extracted as follows ( ) ( )22cos B\u00b7mM\u00b7m B\u00b7m uu u + =\u03b8 , ( ) ( )22sin B\u00b7mM\u00b7m M\u00b7m- uu u + =\u03b8 or (A.5) ( ) ( )22cos B\u00b7mM\u00b7m B\u00b7m- uu u + =\u03b8 , ( ) ( )22sin B\u00b7mM\u00b7m M\u00b7m uu u + =\u03b8 Plugging \u03b8cos and \u03b8sin from Eq.(A.5) into Eq.(A.3) yields ( ) ( ) ( ) ( ) \uf8f7 \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ec \uf8ed \uf8eb + = 22 ),(),,( B\u00b7mM\u00b7m BM\u00b7m-MB\u00b7m m uu uurututK m\u03b8 (A.6) For simplifying the Eq.(A.6) the following cross product representation is used cb\u00b7a-bc\u00b7acba )()()( =\u00d7\u00d7 (A.7) Under the rule given in Eq.(A.7) the nominator and inside the parenthesis of Eq.(A.6) can be written as follows ( ) ( ) )( BMmBM\u00b7m-MB\u00b7m \u00d7\u00d7= uuu (A.8) Also from Eq.(A.7) the length of the nominator inside the parenthesis can be written as follows ( ) ( ) ( ) ( )22)(|| B\u00b7mM\u00b7mBMm||BM\u00b7m-MB\u00b7m uuuuu +=\u00d7\u00d7= (A.9) It can be seen that Eq.(A.9) equals to the denominator given inside the parenthesis of Eq.(A.6). Plugging equivalents of nominator and denominator given inside the parenthesis of Eq.(A.6). from Eqs.(A.8) and (A.9) yields AppendixA. Obtaining the grazing points for cutter geometries represented by pipe surfaces. 195 \uf8f7\uf8f7 \uf8f8 \uf8f6 \uf8ec\uf8ec \uf8ed \uf8eb \u00d7\u00d7 \u00d7\u00d7 = ||)(|| )(),(),,( BMm BMm m u urututK m\u03b8 (A.10) In the above equation the cross product of the normal and bi-normal unit vectors are equal to the tangent vector of the spine curve i.e. BMT \u00d7= . Thus Plugging the equivalent of of T from Eq.(3.53) into Eq.(A.10) yields two grazing points P1,2 \uf8f7\uf8f7 \uf8f8 \uf8f6 \uf8ec\uf8ec \uf8ed \uf8eb \u00d7 \u00d7 == ||||),(),,( 2,1 tu turutPutK mm mm m m\u03b8 (A.11) Thus the property (3.2) is proven for the torus which is also a pipe surface. 196 Appendix B Mapping parameters for milling cutter geometries B.1 Derivation of Mcyl A Flat end mill is made up of a cylindrical surface CWEK,c1(t) at the side and a flat surface CWEK,c2(t) at the bottom (Figure B.1-a,b). Based on the kinematics of the 3-axis machining these surfaces will contribute to the set CWEK(t) as defined in section (6.2.1) i.e. 2,1,)( cKcKK CWECWEtCWE \u222a= . For the cylinder surface the engagement angle of a point at )(1, tCWE cK\u2208I must lie within [0,pi] i.e. ( pi\u03d5 \u2264\u2208\u2264 ))(|(0 1, tCWE cKII ) and for the flat surface it must lie within [0,2pi] i.e. ( pi\u03d5 2))(|(0 2, \u2264\u2208\u2264 tCWE cKII ). (a) (b) Figure B.1: (a) Cylindrical CWEK,c1(t), and (b) bottom CWEK,c2(t), faces of the Flat End Mill B.1.1 Obtaining CWE Parameters for the Cylindrical Surface For the cylindrical surface tool reference point F equals to the tool tip point T. ( ) ( ) ttt )( SES VVVTF \u2212+== (B.1) For the first step, the geometry of the cylinder changes Eq.(6.11) and (6.12) to, { } 0)()()(:)( 22 =\u2212\u2022\u2212\u2212\u2212\u2022\u2212 rtGC nFPFPFP (B.2) AppendixB. Mapping parameters for milling cutter geometries. 197 and, { } 0)()()(:)( 22 , =\u2212\u2022\u2212\u2212\u2212\u2022\u2212 rtCWE CK nFIFIFI (B.3) respectively. As with Eq.(6.13), Eq.(B.3) results in a quadratic equation in t when I and F are substituted. In this case the coefficients 12 , AA and 0A are given by, ( ) ( ) ( )[ ]SISISI zzyyxx \u2212\u2212\u2212=a ( )[ ] 222 SESE VVnVV \u2212\u2022\u2212\u2212=A ( ) ( )[ ] ( ) ( )SESE VVaVVnna \u2212\u2022\u2212\u2212\u2022\u2022= 221A (B.4) ( ) 2220 rA \u2212\u2022\u2212= ana Two real roots of the Eq.(B.3) represent two cutter locations according to Property (6.4) the minimum of them is taken to be the correct tool position as explained in section (6.2.1.1). Using Eq.(6.23-a) and (6.25) the parameters \u03d5 and L are obtained. Finally, the depth of the engagement point as defined by the distance from T(t) to I' measured along the tool axis vector is simply the z-coordinate of I'. )()( tztd I TwwI \u2022\u2212\u2022=\u2032= (B.5) B.1.2 Obtaining CWE Parameters for the Flat Surface CWE parameters for the Flat surface are obtained without doing mapping. An unbounded plane which is perpendicular to the tool rotation axis is intersected with the original removal volume of the bottom surface for a given cutter location point. Engagement angle \u03d5 is found by Eqs.(6.23-a) and (6.23-b). The depth of the cut for the bottom surface is the distance between the intersection point I and the center of the bottom surface such that 22 II yxd \u2032+\u2032= (B.6) AppendixB. Mapping parameters for milling cutter geometries. 198 B.2 Derivation of Mcone Conical part of a Tapered Flat End Mill is made up of a front CWEK,co1(t) and back CWEK,co2(t) conical contact faces at the side and a flat surface CWEK,co3(t) at the bottom respectively (see FigureB.2). Based on the kinematics of the 3-axis machining these surfaces will contribute to the set CWEK,Co(t) as defined in section (6.2.1) i.e. )()()( 3,2,1,, tCWEtCWECWEtCWE coKcoKcoKCoK \u222a\u222a= (B.7) For the given tool motion type C in Figure (6.27), the cutter has engagements with all its surfaces and the total engagement area covers the full [0,2pi] range i.e. pi\u03d5 2))(|(0 , \u2264\u2208\u2264 tCWE CoKII . (a) (b) Figure B.2: (a) Front CWEK,co1(t), and (b) back CWEK,co2(t) conical faces of Tapered Flat End Mill CWE parameters for the flat surface are used from B.1.2. For the Tapered Flat End Mill tool reference point F equals to the tool tip point T, Eq.(B.1). For the first step, the geometry of the cone changes Eqs.(6.11) and (6.12) to, ( ){ } 0)(tan1)()(:)( 22 =\u2022\u2212+\u2212\u2212\u2022\u2212 nVPVPVP \u03b1tGCo (B.8) and AppendixB. Mapping parameters for milling cutter geometries. 199 ( ){ } 0)(tan1)()(: 22 , =\u2022\u2212+\u2212\u2212\u2022\u2212 nVIVIVI \u03b1CoKCWE (B.9) where\u03b1 , n and V are the cone half angle, unit normal vector of the tool rotation axis and cone vertex coordinates (see Figure 6.22-a) respectively. The relationship between V and F is given by n-FV \u03b1 r tan = (B.10) where r is the radius of the cone bottom surface. As with Eq.(6.13), Eq.(B.9) results in a quadratic equation in t when I and F are substituted. In this case the coefficients 12 , AA and 0A are given by, \uf8fa \uf8fb \uf8f9 \uf8ef \uf8f0 \uf8ee \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ed \uf8eb +\u2212\uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ed \uf8eb +\u2212\uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ed \uf8eb +\u2212= ZSIYSIXSI n r zzn ryynrxx \u03b1\u03b1\u03b1 tantantan a ( ) ( )[ ] 2222 tan1 SESE VVnVV \u2212\u2022+\u2212\u2212= \u03b1A (B.11) ( ) ( ) ( )( ) ( )[ ]SESE VVnnaVVa \u2212\u2022\u2022++\u2212\u2022\u2212= \u03b121 tan122A ( ) ( )2220 tan1 ana \u2022+\u2212= \u03b1A Two real roots of the Eq.(B.9) represent two cutter locations according to property (6.4) and the minimum of them is taken to be the correct tool position as explained in the section (6.2.1.1). Using Eqs.(6.23-a,b),(6.25) and (B.5) the parameters (\u03d5 , d, L) are obtained. B.3 Derivation of Mtorus A Toroidal portion of the end mill is made up of front CWEK,t1(t) and back CWEK,t2(t) contact faces at the side and a flat surface CWEK,t3(t) at the bottom respectively (see Figure (B.3)). Based on the kinematics of the 3-axis machining these surfaces will contribute to the set CWEK,T(t) as defined in section (6.2.1) i.e. AppendixB. Mapping parameters for milling cutter geometries. 200 )()()( 3,2,1,, tCWEtCWECWEtCWE tKtKtKTK \u222a\u222a= . (B.12) (a) (b) Figure B.3: (a) Front CWEK,t1(t),and (b) back CWEK,t2(t) contact faces of Toroidal End Mill CWE parameters for the flat surface are used from B.1.2. The reference point at F is chosen to be the center of the torus. Its location can be expressed by the cutting tool tip coordinates as, ( ) nTF rtt += )( (B.13) where r and n are the radius of tube (Figure (B.3-a)) and unit normal vector of the tool axis respectively. For the first step, the geometry of the torus changes Eqs.(6.11) and (6.12) to, ( ) ( ){ } ( ){ } 04:)( 222222 =\u2212\u2212+\u2212\u2212\u2212\u2022\u2212 rzzRRrtG CT FPFP (B.14) and ( ) ( ){ } ( ){ } 04: 222222 , =\u2212\u2212+\u2212\u2212\u2212\u2022\u2212 rzzRRrCWE CTK FIFI (B.15) respectively. Eq.(B.15) results in a quartic 4th order polynomial equation in t when I and F are substituted. 001 2 2 3 3 4 4 =++++ AtAtAtAtA (B.16) AppendixB. Mapping parameters for milling cutter geometries. 201 In this case the coefficients 01234 ,,, AAAAA are given by, ( )SE VVa \u2212= , ( )SV-Ib = 4 4 a=A , ( )baa \u2022\u2212= 23 4A ( ) ( ) ( ) 22222222 424 SE zzRRrA \u2212+\u2212\u2212+\u2022= baba (B.17) ( ) ( ) ( )SESI zzzzRRrA \u2212\u2212\u2212\u2212\u2212\u2022\u2212= )(84 22221 bba ( ) [ ]22222220 )(4 rzzRRrA SI \u2212\u2212+\u2212\u2212= b The Eq.(B.16) gives four roots and according to the property (6.4) toroidal surface has maximum four locations for I. For example, two CWE points 1I and 2I are shown in Figure(B.4). Along the tool path, toroidal surface touches the point 1I at four interference locations i.e. 4321 T,T,T,T thus for 1I the Eq.(B.16) gives four distinct real roots such that 4321 tttt <<< and as explained in section (6.2.1.1) minimum of them is taken to be the tool location when cutter touches this point. Figure B.4: Cutter interferences with a point in space Toroidal surface touches 2I at the interference location 5T and because 2I is on the envelope surface of the cutter, Eq.(B.16) gives repeated real roots for this point. Using Eqs.(6.23-a,b) and (6.25), (\u03d5 , L) parameters are obtained. The depth of the engagement point is obtained by AppendixB. Mapping parameters for milling cutter geometries. 202 \uf8f7 \uf8f8 \uf8f6 \uf8ec \uf8ed \uf8eb \u2032 \u2212= \u2212 r z td I1cos)( 1 (B.18)","attrs":{"lang":"en","ns":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","classmap":"oc:AnnotationContainer"},"iri":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","explain":"Simple Knowledge Organisation System; Notes are used to provide information relating to SKOS concepts. 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