End mills are widely used in industry for high-speed machining. End milling cutters are C H A P T E R 2

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1 C H A P T E R End Mill End mills are widely used in industry for high-speed mahining. End milling utters are multi-point milling utters with utting edges both on the fae end as well as on the periphery, and may be single or double end onstrution [33, 58, ]. The teeth of the utter may be straight (parallel to the axis of rotation or helial (at a helix angle. The utter may be righthand (to turn lokwise or left-hand (to turn ounter lokwise. The end mills have a straight or tapered shank for mounting and driving. End mills ombine the abilities of end utting, peripheral utting and fae milling into one tool. End mills an be used on vertial and horizontal milling mahines for a variety of faing, slotting, and profiling operations. End mills an be lassified based upon: (i Configuration of end profile - Flat, Chamfer, Radius, Ball, Taper, Bull Nose end mills and their ombinations.

2 Chapter. End Mill 5 (ii Shank type Srewed parallel shank, Plain shank, Weldon shank, Taper shank, Whistle noth parallel shank and Combination shank. (iii Mounting type - Cylindrial, Cylindrial threaded, Cylindrial power huk, Weldon and Weldon threaded. Geometry of utting flutes and surfaes of end mills is one of the ruial parameters affeting the quality of the mahining in the ase of end milling. In the present work, the results of the generi definitions of flat end mills proposed by Tandon et al. [8] have been further developed, validated and applied for finite element analysis. Here, a generi flat end mill is modeled as it an be generalized to other types of end mills with omplex shapes. A mathematial model of the geometry of a flat end mill is formulated in terms of its surfae pathes using the onept of surfae modeling. The model is generated keeping in mind that it is to be used for diret analysis, prototyping, manufaturing and grinding of the utters. The orientation of the surfae pathes is defined in a right hand oordinate frame of referene by three-dimensional angles, termed as rotational angles. The flutes of the flat end mill are modeled by sweeping the setional profile of the utter along the perpendiular diretion. Mapping relations are developed between the proposed three-dimensional nomenlature and traditional two-dimensional (D projeted geometry based nomenlatures. The output in the form of graphial model of the flat end mill is shown in OpenGL [50-5] for verifiation of the methodology. Besides, an interfae is also developed to diretly pull the detailed mathematial / parametri definition of the end mill in a ommerial CAD pakage for further validation (in the form of a solid model. This exat solid model is analyzed for stati and transient dynami load onditions. Also, the advantages offered by the detailed CAD model over an approximated CAD model are shown.. Surfae Modeling of Flat End Mill The geometry of a flat end mill projeted on two-dimensional orthographi planes with its tehnial features is shown in Figure.(a and the setional view of its flute is shown in Figure.(b. L and L are the lengths of the fluted shank and the end mill respetively. D and D s are the utter diameter and shank diameter of the end mill respetively. For an end mill,,,, and R are axial relief angle, end utting edge angle, radial relief angle, A e R R radial learane angle and radial rake angle respetively and are shown in Figure.. The geometry of an end mill an be divided into three groups, namely

3 Chapter. End Mill 6 Geometry of fluted shank. End surfae geometry. Shank geometry. The fluted shank geometry onsists of irumferential surfae pathes formed by helial sweeping a profile of a setion of the fluted shank. The sweep operation is a ombined rotational and parallel sweep, perpendiular to the axis of the utter. The end geometry is dependent on the onfiguration of the flat end profile. A single tooth of a flat end mill may be modeled with the help of nine surfae pathes, labeled to 9 as shown in Figure.. They are Rake fae (, Peripheral land, Rake fae extension ( ( 9. Shank geometry of the utter may be a straight shank or tapered shank with surfae pathes modeled as axisymmetri surfae of revolution along with a flat end surfae. (a (b Figure.: (a Two-Dimensional Projeted Geometry of a Flat End Mill (b Setional View of End Mill Flutes

4 Chapter. End Mill 7.. Modeling of Fluted Surfae Geometry of End Mill Surfaes to 6 are the surfae pathes on the fluted shank. These surfaes are formed when a omposite urve in XY plane is swept with a sweeping rule, omposed of ombined rotational and parallel sweep, known as helioidal surfaes. Figure.3 shows the omposite setional urve V..., onsisting of six segments. Out of these six, three segments V V, ( V7 V V 3 and V 6V7 orrespond to the three land widths, namely peripheral land, heel and fae, and are shown as straight lines on a two-dimensional projeted plane. The other three segments V 3V 4, V 4V5 and V 5V6 are irular ars of radii r 3,r and R respetively, orresponding to fillet, bak of tooth and blending surfae. The surfae pathes ould better be visualized from the unfolded view of an end mill tooth (Figure. (a. The tip geometry of the flat end mill is shown in Figure. (b. In modeling the ross-setional profile of fluted shank in a two-dimensional plane, the input parameters are (i widths of lands i.e. fae, peripheral land and heel or seondary land given by l,l and l 3 respetively, (ii 3D angles obtained to form fae (, land ( and heel ( 3 about Z axis, (iii radii of fillet (R, bak of tooth ( r and blending surfae ( r 3, (iv diameter of utting end of the end mill ( D and (v number of flutes (N. The position Figure. (a: Unfolded View of an End Mill Tooth Figure. (b: Tip Geometry of a Flat End Mill

5 Chapter. End Mill 8 vetors of the end verties of different setions of the omposite profile urve V... and ( V7 enter points of the three irular ars,, are evaluated [8] to parametrially model the profile urve. ( 3 The ross-setional profile of a fluted setion of an end mill onsists of three parametri linear edges and three parametri irular ars, namely, p ( to p (. Edges p s, p ( and p ( are straight, while p s, p ( and p ( are irular in two-dimensional ( s 6 s ( 3 4 s 5 s s 6 s spae. The generi definition of the setional profile p (s in XY plane in terms of parameter s may be represented by, p s f ( s f ( 0. i( i i s The fluted surfae is obtained by ombined rotational and parallel sweeping and is parametrially desribed by, p( s, p( s. T s i, where transformation matrix is os sin 0 0 sin os 0 0 L Ts for 0. P P 0 0 In the above equation, L is the length of fluted shank for flat end mills, be the parameter denoting the angular movement and P.is the pith of the end mill. Figure.3: Composite Setional Curve of an End Mill Flute

6 Chapter. End Mill 9 Sweeping rules The fluted setion of an end mill an have right helix or left helix. If the flute's spiral has a lokwise ontour when looked along the utter axis from either end, then it is a right helix else the helix is left [30, 33]. For a right helix utter, the ross-setion urve rotates by an angle + about the axis in right hand sense. Three different sweeping rules an be formulated for the fluted shank and the end profile of the utter. These rules are for (i Cylindrial Helial Path (ii Conial Helial Path (iii Hemispherial Helial Path Cylindrial Helial Path - The path when the omposite profile urve is swept helially along a ylinder is known as ylindrial helial path. For a helial utter let be the parameter denoting the angular movement, P the pith of the helix, D the ylindrial utter diameter and L the length of the fluted part of utter, then the mathematial definition of the helix is, x ( D y ( D / os / sin z ( P /(, where 0 (L / P (. Conial Helial Path - The helial path along a frustum of one of utting end diameter D C and shank side diameter D S x ( D / os y ( D / sin z ( P /(, is defined by, where D D ( D s D z / L and 0 (L / P. (. This is valid for both types of frustum of ones i.e. when D Ds and when Ds D. Hemispherial Helial Path - The helial path along the hemispherial objet of diameter is given as, x ( D / os os y ( D / os sin z ( D / ( sin, where 0 D / P and 0 /. D (.3

7 Chapter. End Mill 30 Here, is the angle about Z axis and is the angle with XY plane and the relation between them is, sin P.. D.. Modeling of End Surfae Geometry The end surfae geometry of a flat end mill onsists of three planes and one blending surfae (Figure.. (a Fae Land ( 7 (b Minor Flank ( 8 and ( Rake Fae Extension ( 9 are the three planes whereas the blending surfae, 9 blends surfae path 8 of the first tooth with surfae path 9 Fae land ( 7 of the seond tooth of the end mill. is formed when an XY plane given by 0 u is transformed 7 v 7 through rotation by a 3D rotational angle 7 about X axis R x, ], followed by rotation by an angle about Z axis R ] and is defined as, [ z, [ 7 p u, v ( u os v os sin ( u sin v os os v sin (.4 7( [ 8 Minor Flank ( 8 is formed when the XY plane is rotated by an angle 8 about X axis R x, ], followed by an angle about Z axis R z, ], and then translated by a distane d8 l os along Y axis and 83 l os sin 7 [ d along Z diretion T ] and is given as, [ yz p u, v ( u os vos sin ( u sin v os os d ( v sin (.5 8( d83 an angle 9 Rake Fae Extension ( 9 is formed when a ZX plane (u 9 0 w9 is rotated by about X axis R x, ] and by an angle about Z axis R z, ]. Here, helix angle [ 9 [ * * tan ( P/ DC, 90 and (5 5 but 90. The surfae 9 satisfies the relation, 9 p u, w ( u os w sin sin ( u sin w sin os ( w os (.6 9(

8 Chapter. End Mill 3..3 Modeling of Shank Surfae Geometry The shank surfae of a flat end mill may be modeled as a ombination of two surfae pathes (a ylindrial surfae of revolution 50 (for straight and tapered shank and (b planar end surfae 5(as shown in Figure.4. The ylindrial surfae of shank in terms of parameter w, may be parametrially defined by, D' D' p 50( w, os sin { L w( L L } (.7 for 0, 0 w. The term L stands for overall length of end mill and diameter D' may be given by, modeled as, Ds ( D /, / w( D s D /, for straight shank for tapered shank The planar end surfae forming the end opposite to utting end is parametrially p u, v u v (.8 5( L for u, v. A unit width, 45 hamfer between shank surfae pathes 50 and 5 is the only blending surfae on the shank body of the utter. The hamfer 50, 5 is modeled by revolving the edge having end point oordinates as (( D s / 0.707, 0, L and ( D /, 0, ( L s about Z axis and is given by, ( Ds Ds 50,5 u, { 0.707( u}os { 0.707( u}sin ( L 0.707u (.9 for 0 u and 0.

9 Chapter. End Mill 3. Mapping Relations In this work, relations to map the proposed 3D rotational angles, used to define the geometry of the flat end mills, into onventional D projeted angles and vie versa are developed. The former is alled forward mapping while the latter is known as inverse mapping [8]. The onventional angles are formed by projeting the surfae pathes of the end mill on planes of projetion and the traditional geometry of end mill an be referred in the literature [30, 33, 49, 65, 69]. Table. shows the forward mapping relations for a flat end mill. The solution of these forward mapping relations establishes inverse mapping that helps to evaluate the 3D rotational angles if tool angles speified by onventional nomenlatures are known. Table. presents the inverse mapping for an end mill. Figure.4: Modeling of Shank Table.: Forward Mapping Relations for Flat End Mill Conventional D Angles Radial Rake Angle, R Radial Relief Angle, R Radial Clearane Angle, R 3 Axial Relief Angle, A End Cutting Edge Angle, e Rotational 3D Angles os os os 7 os sin 7 os 7 os 7 sin sin 7 os 7

10 Chapter. End Mill 33 Table.: Inverse Mapping Relations for Flat End Mill Rotational Angles 3 7 Conventional Angles R R R os os A.os e os A os e os A.os e.3 Modeling Algorithm Based on the above proposed generi definition, the parameters needed to ompletely desribe a four-flute helial flat end mill and the relevant data for these parameters to exemplify the methodology are shown in Table.3(a and.3(b. The algorithm that geometrially models a fluted helial end mill based on the proposed three-dimensional nomenlature (3D rotational angles and other dimensional parameters is as follows: Step : The omposite setional urve of the flute in two-dimensional Cartesian oordinate plane, with enter of the flat end mill's ross setion oiniding with global origin is generated as desribed in Setion... Step : The solid flute is generated by ombined rotational and parallel sweeping of omposite setional urve as illustrated in Setion... Step 3: The end geometry of flat end mill is formed through Boolean operations on the surfae pathes and the blending surfaes as mentioned in Setion... Step 4: The Shank is generated by applying Boolean operation on surfae pathes 50 and, and blending surfae as mentioned in Setion ( 50, 5 Step 5: The detailed and omplete CAD model of the end mill is generated. The details of the implementation depend on the speifi CAD environment, where they are implemented, and are disussed in Setion.4.

11 Chapter. End Mill 34 Table.3(a: Geometri Parameters (Dimensional and Input Data for an End Mill Dimensional Parameters Number of flutes (N 4 Values Cutter diameter ( D C 5.0 mm Length of utter ( L 3.0 mm Length of flutes ( L 7.0 mm Root diameter ( D R.0 mm Shank side diameter ( D S 5.0 mm Pith (P mm Width of primary land ( l.0 mm Width of heel ( l mm Width of rake fae ( l 7.0 mm Fillet Radius (R.0 mm Radius of bak of tooth ( r 5.0 mm Radius of blending surfae ( r 3.0 mm Table.3(b: Geometri Parameters (Angular and Input Data for an End Mill Rotational Angles Value (degrees Helix angle ( Implementation of Algorithm The algorithm has been implemented in OpenGL environment [5], a general purpose graphial modeling kernel, as well as in a ommerial CAD environment (here, CATIA V5 [3, 6]. The rendering of the end mill utter (input data as in Table.3(a and.3(b by using the 3D parameters proposed in the work, diretly in OpenGL and CATIA V5, validates the modeling methodology.

12 Chapter. End Mill Implementation in OpenGL Environment OpenGL is an Appliation Programming Interfae (API. It is graphial software that uses boundary representations to desribe models. To diretly render the helial flat end mill using 3D geometri parameters as proposed in this work, a tool design program is developed in C programming language. All geometri primitives are defined in terms of verties. The 3D geometri parameters were supplied as input in a text file. The program developed in C easily all funtions in the API for rendering. The libraries like GLU, GLX, GLUT, Open Inventor, et, simplify our programming tasks for rendering the utter in OpenGL [63]. The output is the rendered image of the flat end mill in OpenGL graphial environment as shown in Figure.5(a. OpenGL file formats are used for rendering purpose only. It is diffiult to aurately export the rendered image of OpenGL file format to a ommerial CAD/CAE/CAM environment for any meaningful tehnologial appliation..4. Implementation in Commerial CAD Environment To fully utilize the aurate and omprehensive parametri definition of the utter, there is a need to pull this model in a ommerial modeling pakage so as to take advantages of the tools of the modeler for further proessing. An interfae is developed here to diretly generate the model in a solid modeling environment (CATIA V5 using the omprehensive definition of the utter. This leads to a wide range of downstream appliations. Figure.5: Rendering of an End Mill in (a OpenGL environment and (b CATIA V5

13 Chapter. End Mill 36 CATIA V5 is a ommerial solid modeler that allows users to reate 3D models in a Graphial User Interfae (GUI. Solid models an also be reated using Visual Basi funtion alls through an appliation programming interfae (API [8, 6]. A Visual Basi, C++ program is developed here that uses this API to generate the model of the helial flat end mill based on the user defined 3D geometri parameters diretly in CATIA. The model is not reated using the GUI of the software but uses the parametri definitions of the surfaes developed, as desribed in Setion.. The users an hange the geometri parameters of the end mill in a ustom developed user interfae and based on the hosen parameters, the end mill model is diretly generated in CATIA modeling environment (as shown in Figure.5(b. Using the GUI, if one has to reate another model for a different set of parameters, one has to rebuild the model, may be, from srath. With the parametri definitions and the interfae developed here, the user an reate a set of models by simply hanging the input values. It is possible to save the geometri model of the end mill in a variety of file formats, inluding CAT PART, IGES, and STL. The CATIA implementation gives more ontrol to the user beause the end mill is modeled diretly in a modeling environment where further modifiations and embellishments an be easily done..5 Appliation of Helial Flat End Mill Models Numerous tehnologial down stream appliations may be performed on the geometri models of the helial flat end mill utters developed by the above defined tehnique. One suh tehnologial appliation is finite element based engineering analysis (FEA of the end mill. This setion presents an exerise on FEA on the 3D model of the helial end mill. This exerise highlights the advantages and utilities available, one a omprehensive 3D definition of the utter is available. The purpose here is not to present any detailed analysis of the end mill during mahining. The end mills are generated in CATIA as desribed in Setion.4. and exported to Generative Strutural Analysis module in CATIA Analysis and Simulation pakage for Strutural Stati and Transient Dynami Analysis. For the auray of simulation, the utting fore data taken in the work are based on results of the utting tests onduted by Li et al. [7]. For our simulation, we have hosen the utter material and the utting onditions as follows:

14 Chapter. End Mill 37 Cutter: A four-fluted M4 High Speed Steel with a helix angle = 30 and other geometri parameters as mentioned in Table.3. Material properties of the High Speed Steel utter [68]: It is Molybdenum based high speed steel with Weight % of C =.05-.5, Mn = , Si = , Cr = , Ni = 0.3, Mo = , W =.5-.85, V = , Co = , Cu = 0.5, P = 0.03, S = It is an isotropi material with Young s modulus =. e+n/m, Poisson s ratio = 0.3, density = 7980 kg/m 3, thermal expansion =.33 e- 005 K and tensile strength = 970N/mm. Work piee: Plain arbon steel whih has a hemial omposition of approximately 0.5% C, 0.5% Mn, 0.5% Si. Cutting Parameters: Axial depth of ut mm, spindle speed 600 rpm, feedrate 40 mm/min i.e. the feed per tooth per revolution = 0. mm. The end mill is meshed using paraboli tetrahedral elements. The tetrahedral element is a ten node iso-parametri solid element having 3 (translational degrees of freedom. It has a quadrati displaement behavior and is well suited to model irregular meshes. Quadrati tetrahedral (QT element is preferred for the development of smooth automati mesh FE model for the geometry of omplex solid models like femur, fluted utters, et. [04, 47]. For automati mesh generation, meshes omprising entirely of tetrahedral elements (or alltetrahedral mesh are frequently generated as opposed to a mesh omprising of purely hexahedral elements (or all-hexahedral mesh. This is beause a physial domain, as in the ase of geometrially omplex strutures, annot always be deomposed into an assembly of hexahedral elements. It an however be easily represented as a olletion of tetrahedral elements, as these elements are geometrially more versatile [4]. Infat, performing FEA using QT mesh models is effiient in terms of both auray and CPU time [8, 47]. Also, QT finite element models are more stable and less influened to the degree of refinement of the mesh, when modeling omplex solids [04]. The element also has plastiity, reep, swelling, stress stiffening, large defletion, and large strain apabilities. The meshing information of the end mill is presented in Table.4. Figure.6 shows the finite element mesh of the end mill utter. The next step is to restrain the end mill model by lamping the shank of the tool followed by load appliation.

15 Chapter. End Mill 38 For our simulation, load has been applied on a single flute in radial, tangential and axial diretions for an axial depth of mm. Figure.7 shows the oordinate system and the utting fore diretion on the flute of the end mill utter. The simulated and measured utting fores in the time domain of two revolutions of utter rotation as worked out by Li et al. [7] is shown in Figure.8. Using the utting fores mentioned in Figure.8, the fores ating in radial, tangential and axial diretions on the end mill flute an be alulated from the angle domain onvolution integral theory mentioned by Zheng et al. [6]. Table.4: Meshing Information of an End Mill Cutter Parameters Values Global mesh size 7 mm Loal mesh size mm Jaobian 0.3 Nodes 489 Elements 764 Warp 60 Absolute sag 0.3 mm Proportional sag 0. mm Figure.6: Meshing of an End Mill Cutter Figure.7: Coordinate Systems of the End Mill Cutter

16 Chapter. End Mill Strutural Stati Analysis For stati analysis at any partiular instantaneous time or utter angle rotation (, fores are onsidered on a single flute in radial, tangential and axial diretions for an axial depth of mm. For 3, the applied fores in radial, tangential and axial diretions are F 57.5N, F 38.77N and F 49.77N. Figures.9 -. shows the result for stati r t a analysis with Von mises stress, deformed mesh and translational displaement respetively for the applied load. Here, maximum equivalent stress reahed is 47. MPa and maximum total displaement is 0.0mm..5. Strutural Transient Dynami Analysis For transient dynami analysis, only time fator has been onsidered. Vibrations and other dynami fators are not inluded in this work. The transient dynami analysis is arried out for a total time domain of 0.05 se (for utter angle rotation of 90, with a time interval of se. Table.5 shows the fores ating on the flute for a time profile of se. Figures.,.3 and.4 shows the result for transient dynami analysis with Von mises stress, deformed mesh and translational displaement respetively for the applied load. For the dynami ase, the result shown is maximum equivalent stress of 978 MPa and maximum total displaement of mm at time 0.05 se. Table.5: Fores Ating on a Partiular Flute of a Helial End Mill θ Time (se F t (N F r (N F a (N

17 Chapter. End Mill 40 Figure.9: Von Mises Stress for Stati Load Condition Figure.0: Deformed Mesh for Stati Load Condition Figure.8: Simulated and Measured Cutting Fores for Up Milling, with axial depth of Cut = mm. (a Fore in X Diretion, (b Fore in Y Diretion, ( Fore in Z Diretion [7] Figure.: Translational Displaement for Stati Load Condition

18 Chapter. End Mill 4 Figure.: Von Mises Stress for Transient Dynami Load Condition at 54 Figure.3: Deformed Mesh for Transient Dynami Load Condition at 54 Figure.4: Translational Displaement for Transient Dynami Load Condition at 54 Figure.5: Deformed Mesh of approximated CAD Model at 36.6 Results and Disussions This Setion highlights the advantages offered by the detailed CAD model over an approximated CAD model. Figure.5 shows an approximated CAD model with one of its flute, analyzed for finite element simulation under the load onditions at utter angle rotation ( of 36 (same as for exat CAD model. Using the analytial utting fore model and stress relationships for end milling operations, desribed by Chiou et al. [5] and Armarego et

19 Chapter. End Mill 4 al. [5], the shear stresses are alulated at one of the utting flutes under the load onditions. Table.6 presents the values of shear stresses alulated analytially as well as that obtained by performing finite element simulation on the same flute using approximated CAD model and detailed CAD model. The analytial values of the shear stress are ompared with the finite element simulation results of the two types of CAD models at 3 and 36. The differene in the values of the shear stresses between the analytial and two finite element omputations is tabulated in the form of error analysis in Table.6. The error analysis proves that the results of the FEA simulation on the detailed CAD model are loser to the analytial data than the results of approximated CAD model. It has also been observed while performing the finite element analysis using the approximate CAD model that the stress values sharply overshoots at some points on the utter flute as shown at the tip of utting edge (Table.6. This may be due to the fat that approximate CAD model approximates the geometry at ertain loations. It is evident that the detailed CAD model is onsistently in better agreement with the analytially alulated data rather than the approximated model and an lead to aurate design and fabriation of the utters. Table.6(a: Comparative Results of Finite Element Simulation at 3 3 At tip of the utting edge, Global Maxima At flank of the side utting edge, Loal Maxima & Analytial Shear Stress (MPa Approximated CAD Model Error (% Von Mises Stress (MPa Translational Displaement (mm Von Mises Stress (MPa Detailed CAD Model Error (% Translational Displaemen t (mm

20 Chapter. End Mill 43 Table.6(b: Comparative Results of Finite Element Simulation at At tip of the utting edge, Global Maxima At flank of the side utting edge, Loal Maxima & Analytial Shear Stress (MPa Approximated CAD Model Von Error Translational Mises (% Displaement Stress (mm (MPa Von Mises Stress (MPa Detailed CAD Model Error (% Translational Displaement (mm Chapter Summary Traditionally, the geometry of utting tools has been defined using the priniples of projeted geometry (D nomenlature approah. This hapter desribes in detail the methodology to model the geometry of a flat end mill in terms of three-dimensional parameters. The proposed model defines the end mill in terms of three-dimensional rotational angles rather than the onventional two dimensional angles. The model is validated while rendering the utter diretly in OpenGL environment in terms of three dimensional parameters. Further, an interfae is developed that diretly pulls the proposed three-dimensional model defined with the help of parametri equations into a ommerial CAD modeling environment (here CATIA V5. Besides, the modeled tool is used for finite element simulations to study the utting flutes under stati and transient dynami load onditions. The results show that the methodology offers a simple and intuitive way of generating aurate surfae models for use in mahining proess simulations.