Prediction of Milling Forces by Integrating a Geometric and a Mechanistic Model

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1 Predicion of Milling Forces by Inegraing a Geomeric and a Mechanisic Model S. Abainia, M. Bey, N. Moussaoui and S. Gouasmia Absrac In milling processes, he predicion of cuing forces is of grea imporance for machining free form surfaces. Due o he coninuous curvaure variaions of hese surfaces, cuing forces are variable. The predicion of cuing forces helps o minimize ool deflecions and vibraions in order o increases ool life, o avoid ool breakage and o obain a good surface finish. The aim of his work is o predic he milling forces while finishing free form surfaces on 3-axis CNC milling machines using ball end milling ools by inegraing a mechanisic cuing force model and a geomeric model. Index Terms Dexel, STL model, Free Form Surface, Cuing Forces, Mechanisic model I. INTRODUCTION N manufacuring, he appropriae cuing condiions mus I be seleced in order o reduce ool wear, ool deflecions, vibraions and o have a sable machining and finally o obain a good surface finish. When machining free form surfaces, conac regions beween ball end mills/surface are variables which make cuing forces vary coninuously along he ool pah due o he coninuous curvaures variaion of hese surfaces. This fac is considered by many researchers. In [1], a model is developed o predic cuing forces applied o ball end mill by considering several parameers. In [], he influence of dynamic radii, radial and axial dephs on milling forces is sudied. In [3], geomeric and mechanisic milling models are inegraed o predic cuing forces and feedrae scheduling for five-axis machining. In [4], a comparison is done beween graphical mechanisic model wih simulaion mehod and experimenal resuls. In [5], finie elemen mehod is used o subsiue experimenal works o predic he cuing forces. In [6], feedrae for hree axis machining is seleced by combining geomeric and mechanisic models. In [7], an analyical milling force model is used o opimize cuing condiions. The aim of his paper is o predic cuing forces applied on ball end mill ools for each ool posiion and for each ooh along ool pah while finishing free form surfaces on hree-axis CNC milling machines by inegraing a geomeric model using dexels and a mechanisic force model. II. PROPOSED APPROACH The proposed approach conains five seps (Figure 1). The differen seps are deailed in he following subsecions. 1-STL File Srucuring Line i+1 5-Cuing Forces Predicion Line i Y Sep_y -Modele Geomeric Generaion of Dexel Fig. 1. Seps of he proposed approach. A. STL File Srucuring The STL model represens he ouer skin of objecs by a lis of riangles, where heir number and heir size depend on he objec geomery and he olerances of approximaion. Each riangle is defined by he componens of is uni normal vecor N oriened o he ouside of he objec and by he coordinaes X, Y and Z of is verices P 1, P and P 3 B. Generaion of he Dexel Model Dexels are used o approximae solid objecs by maerial columns parallel o he Z-axis where heir bases can be square or recangular. Dexels are creaed from hese seps: Cells creaion : o accelerae he inersecion calculus beween riangles and dexels, wo liss are creaed (Figure ) : Uniform grid of cells for grouping riangles enirely conained in one cell ; Supplemenary cell for grouping riangles belonging a leas o wo cells. Cell P min P max Sep_x Col j Col j+1 X max_cel Y max_cel X min_cel Y min_cel X 3-Machining Simulaion 4-Conac Region Localizaion Fig.. Cells grid creaion. S. ABAINIA. Cenre de Développemen des Technologies Avancées (CDTA), Cié Aoû 1956, BP N 17 Baba Hassen, Algiers, Algeria, ( s_abainia@yahoo.fr). M. BEY. Cenre de Développemen des Technologies Avancées (CDTA), Cié Aoû 1956, BP N 17 Baba Hassen, Algiers, Algeria, ( bey_mohamed@yahoo.com). Dexels creaion: he cener of each dexel (X, Y ) and is wo exremums poins along he Z-axis (Z min and Z max ) o define is heigh H are calculaed (Figure 3).

Heigh dexels calculus: he dexel heigh is calculaed from he deerminaion of he exremums poins inersecion beween he verical line passing by is cener wih he se of riangles. C.

c. No inersecion. Fig. 5. Inersecion cases of verical sraigh line and a sphere.

done. So, Z maerial = Z inersecion (Figure. 6.a). Case : if (Z maerial < Z inersecion ), hen no updae is required (Figure. 6.b).

No any updae. Fig. 6. Updae of dexels heigh. D. Conac region localizaion The deerminaion of he conac regions beween cuing ools and surfaces for a ool posiion passes by hese seps: Fig. 4.

2 Fig. 3. Dexel definiion. Dexel cener calculus: he cener of each dexel is calculaed from he dimensions and he coordinaes of he exremums poins of he raw par using he specified seps along X-axis (pas_x) and Y-axis (pas_y). Heigh dexels calculus: he dexel heigh is calculaed from he deerminaion of he exremums poins inersecion beween he verical line passing by is cener wih he se of riangles. C. Machining simulaion This sep permis o simulae he finishing machining using ball end mills for any machining sraegy (Parallel Planes, Isoparameric, Z-Consan, ec.). i passes by: a. One poin. b. Two poins. c. No inersecion. Fig. 5. Inersecion cases of verical sraigh line and a sphere. Dexels heigh updae: afer calculaing he inersecion poins, an updae of he heigh dexels is necessary by considering he wo following cases: Case 1: if (Z maerial > Z inersecion ), hen an updae mus be done. So, Z maerial = Z inersecion (Figure. 6.a). Case : if (Z maerial < Z inersecion ), hen no updae is required (Figure. 6.b). Addiion of he sock allowance o each dexel wih he condiion ha he op face of he dexel mus no exceeds he superior face of he raw par (Figure 4). The added sock allowance a. An updae is necessary. b. No any updae. Fig. 6. Updae of dexels heigh. D. Conac region localizaion The deerminaion of he conac regions beween cuing ools and surfaces for a ool posiion passes by hese seps: Fig. 4. Addiion of he sock allowance. Dexel-sphere inersecion: he conac regions for ball end mills are deermined from he calculaion of he inersecion poins beween ball end mills and dexels for each ool posiion. This necessiaes he following seps: For a ool posiion : Recuperae he coordinaes of he sphere cener C(xc,yc, zc) and is radius R; Deermine he limis of he sphere envelope; Deermine he dexels having inersecions wih he sphere envelope; Deermine he inersecion poins beween he sphere and he verical lines passing by dexels ceners (X, Y ) from he following equaions sysem : ( X xc) + ( Y yc) X = X Y = Y + ( Z zc) = R Three cases mus be considered: One inersecion poin: his poin is reained (Figure. 5.a) Two inersecion poins: he lower poin is reained (Figure. 5.b). No inersecion poins (Figure. 5.c). (1) Cuing ool segmenaion: consiss in subdividing he acive par (spherical par) of he ball end mill ino a se of discs wih he same hickness for each ool posiion. The limied acive par by his slicing is deermined from Z min and Z max of he inersecion poins lis (Figure. 7). Once disc hickness is specified, disc cener, disc radius and limis (z min_disc and z max_disc ) for each disc are deermined. Fig. 7. Segmenaion of he acive par. Affecaion of poins o discs: he inersecion poins associaed o each disc are deermined based on he Z coordinae of he inersecion poins and he limis of he disc (z min_disc and z max_disc ). Nex, he posiion angle for each poin is calculaed according o he X-axis (Figure. 8). This angle is beween and 36. Fig. 8. Posiion angle of poins.

Creaion of he conac regions for each disc: afer he affecaion of he inersecion poins o discs, he differen conac regions for each disc and he enrance angle and he exi angle associaed o each region are

9) : Soring he inersecion poins based on heir posiion angles; Affecaion of inersecion poins o conac regions : Run hrough he inersecion poins lis: Tes he difference of he posiion angle beween wo

affeced o his new region; For each conac region: The enrance angle φ en is equal o he posiion angle of he firs poin of his region.

3 Creaion of he conac regions for each disc: afer he affecaion of he inersecion poins o discs, he differen conac regions for each disc and he enrance angle and he exi angle associaed o each region are calculaed. The deerminaion of hese parameers passes by hese seps (Figure. 9) : Soring he inersecion poins based on heir posiion angles; Affecaion of inersecion poins o conac regions : Run hrough he inersecion poins lis: Tes he difference of he posiion angle beween wo consecuive poins o a predefined angle θ: If his difference is less han θ, hen he wo poins belong o he same region; If his difference is greaer han θ, hen a new region is creaed and he second poin is affeced o his new region; For each conac region: The enrance angle φ en is equal o he posiion angle of he firs poin of his region. The exi angle φ exi is equal o he posiion angle of he las poin of his region. The predefined angle θ is a consan ha depends on he dexel seps pas_x and pas_y, disc radius R and a real coefficien Coef. This angle is given by: Max( pas _ x, pas _ y ) θ =.Coef () R The cuing forces componens df x, df y and df z in caresian sysem are given by: dfx sinφ sinκ cosφ = dfy cosφ sinκ sinφ dfz cosκ sinφ cosκ df cosφ sinκ dfr sinκ dfa Fig. 1. Cuing forces for a ball end mill. Wih φ is he angular posiion of a ool poin. An elemenary disc is defined by is radius R(z) and is posiioning angle κ given by (Figure 11): (4) κ = arcsin(( R( z )) / R ) (5) Where R is he ool radius. Fig. 11. Elemenary disc parameers. In he caresian sysem (X, Y, Z), he cuing forces are prediced using he following parameers: Average chip hickness for a disc given by: h = f.n a (cosφ cosφ )cosθ cos β (6) ni sor en i= 1 φen φsor Fig. 9. Deerminaion of he conac regions for a disc. E. Cuing forces predicion The proposed approach uses a mechanisic force model o predic he cuing forces for each ool posiion along he ool pah in hree axis machining. For a given poin on he ball end mill, he differenial componens cuing forces (angenial df, radial df r and axial df a ) in a local caresian sysem linked o he cuing ool (Figure. 1) corresponding o an infiniesimal elemen edge lengh of he cuing ool is given by [6]: df = Kc( ha ) dfr = Krc( ha ) df = K ( h ) a ac a p1 p p3 h( φ,z )ds h( φ,z )ds h( φ,z )ds Wih: h a : average hickness of he chip. h(φ, z) : insananeous hickness of he chip. ds : edge cuing lengh. K c, K rc, K ac, p 1, p and p 3 : consans deermined by experimenal ess. They depend on edge geomery, ool and workpiece maerial. (3) Wih : θ : angular posiion of an elemenary disc (Figure. 1). n i : number of regions of a disc. β : angle beween he displacemen vecor D for wo consecuives posiions and he X-axis (Figure. 1). f : ooh feedrae (mm/ooh). n : eeh number. Tooh feedrae is given by he following formula: f f = (7) n Ω Wih: f : feedrae of he cuing ool (mm/min). Ω : spindle speed (r/min). Fig. 1. Geomeric parameers of a disc.

Insananeous chip hickness given by [6] : f h φ, z) = [ cosθ sinφ cos β sinθ sin β ] ( (8) n Ω Cuing edge lengh given by (Figure. 13): ds = R.dθ = R.

The predicion of he cuing forces for a ool posiion passes by he following seps: 1. Calculus of he cuing forces for a conac region.. Calculus of he cuing forces for a disc. 3.

For each poin corresponding o a posiion angle φ: Calculae he angles θ and β. Calculae he average chip hickness h a. Calculae he insananeous chip hickness h(φ,z). Calculae he cuing edge lengh ds.

4 Insananeous chip hickness given by [6] : f h φ, z) = [ cosθ sinφ cos β sinθ sin β ] ( (8) n Ω Cuing edge lengh given by (Figure. 13): ds = R.dθ = R.( θ -θ1) θ1 = arcsin(( Zc Z θ = arcsin(( Zc Z max_ disq min_ disq ) / R )) ) / R )) Fig. 13. Geomeric parameers of a disc. The predicion of he cuing forces for a ool posiion passes by he following seps: 1. Calculus of he cuing forces for a conac region.. Calculus of he cuing forces for a disc. 3. Calculus of he resulan cuing forces. For each disc: For each conac region: (9) Subdivide each conac region ino a se of poins wih a consan angular incremen (Figure 14). For each poin corresponding o a posiion angle φ: Calculae he angles θ and β. Calculae he average chip hickness h a. Calculae he insananeous chip hickness h(φ,z). Calculae he cuing edge lengh ds. Calculae he differenials cuing forces df, df r and df a. Calculae he angle κ. Calculae he cuing forces df x, df y and df z. Sum all he cuing forces o obain he cuing force for he considered region. Sum all he cuing forces o obain he cuing force on each disc. Sum all he cuing forces for all discs o obain he resulan cuing force. Deermine he conac regions limis. Segmen he acive par of he ool ino a se of discs wih equal heighs (Figure.15); For he elemenary op disc; o Creae a se of poins wih a given angular sep. o For each poin: Calculae he corresponding poin for he ooh number k by: π φp = φref + ( k 1). (1) N Where φ ref is aken equal o. Deermine he corresponding conac regions; Calculae he insananeous hickness; Calculae he average hickness; Calculae he differenial forces df, dfa and dfr (Figure.16). Calculae he forces Fx, Fy and Fz applied on each ooh. For he oher discs: o Calculae he corresponding poins by: π anβ φk( z ) = φref + ( k 1) z (11) n R Where β is he helix angle. Deermine he corresponding conac regions; Calculae he insananeous hickness; Calculae he average hickness; Calculae he differenial forces df, dfa and dfr (Figure.16). Calculae he forces Fx, Fy and Fz applied on each ooh. Calculae all cuing forces applied on he ball end mill. Acive par Helix angle Fig. 15. Segmenaion of he acive par. Fig. 16. Cross-secions of 4 flue end mill. III. RESULTS Fig. 14. Posiion angle of poin for a disc. To predic he cuing forces in hree direcions X, Y and Z applied on each ooh, he helix angle is considered. For his, he necessary seps are: For a given complee roaion of he cuing ool: The proposed approach is implemened in objec-oriened sofware under Windows using C++ Builder and he graphical library OpenGL [8]. The validaion of his approach is performed on an STL model of a par generaed from a CAD model (Figure. 17). The dimensions of he raw par are 14mm 16.7mm 51.3mm. The ool pah is generaed using a ball end mill of a radius equal o 8 mm (Figure. 17.c).

a. CAD Model. Fig. 19. Sock allowance. b. STL Model. Fig.. Machining simulaion.

posiion are calculaed. Figure 1 shows he inersecion poins for he ool posiion number 64619. c. Machining ool pah.

The dexel model of he CAD par is generaed from a sep equal o.

a. Tool posiion number 64619. b. Inersecion poins. Fig. 1. Resuls for a specific ool posiion.

To predic he cuing forces, he fixed parameers are: Number of eeh=4; Angular incremen=1 ; Spindle

Table 1 gives he values of he consans used in he mechanisic cuing force model.

Figures from o 7 shows he differen resuls relaed o he cuing force componens for each ool posiion

5 a. CAD Model. Fig. 19. Sock allowance. b. STL Model. Fig.. Machining simulaion. During simulaion, he inersecion poins beween dexels and he sphere of he ball end mill for each ool posiion are calculaed. Figure 1 shows he inersecion poins for he ool posiion number c. Machining ool pah. d. Top view. Fig. 17. CAD and STL models and ool pah. The dexel model of he CAD par is generaed from a sep equal o. mm along X-axis and Y-axis (Figure 18). a. Inersecion poins beween verical lines and riangles in YZ plane. a. Tool posiion number b. Inersecion poins. Fig. 1. Resuls for a specific ool posiion. Nex, he conac regions of each disc are deermined. To predic he cuing forces, he fixed parameers are: Number of eeh=4; Angular incremen=1 ; Spindle speed= 15 r/min; Feedrae=5 mm/min. Table 1 gives he values of he consans used in he mechanisic cuing force model. Kc Krc Kac P1 P P Table 1. Values of he experimenal consans [3]. Figures from o 7 shows he differen resuls relaed o he cuing force componens for each ool posiion number. 15 b. Dexel Model. Fig. 18. Creaion of he dexel model. To simulae he machining, he sock allowance is fixed equal o 1 mm (Figure. 19). Figure shows he ool while machining simulaion. Cuing forces componens (N) Number of Cuing ool posiion Fig.. Cuing force componens variaions. Fx Fy Fz Fresul

6 Fx Cuing force componen Fy (N) Cuing force componen Fx (N) Number of Cuing ool posiion Fig. 3. Cuing force componen Fx variaions. Fx Fy Figure 8 shows he variaions of he cuing force per ooh versus he ool posiion. Cuing force per ooh (N) Cuing ool posiion Fig. 8 Variaions of cuing forces per ooh. The resuls show ha he cuing forces vary coninuously from a ooh o anoher. This variaion is more influenced by he par curvaures. F_ooh1 F_ooh F_ooh3 F_ooh4 Cuing force resul (N) Cuing force componen (N) Number of Cuing ool posiion Fig. 4. Cuing force componen Fy variaions Fz Number of Cuing ool posiion Fig. 5. Cuing force componen Fz variaions Fresul Number of cuing ool posiion 3 Fig. 6. Cuing force resul variaions. IV. CONCLUSION In his paper, an approach is proposed and implemened for predicing he cuing forces applied on he end ball mill during he finishing of free form surfaces defined by heir STL model on 3-axis CNC milling machines. In his approach, he dexels elemens are used o approximae he solid model of he par, machining simulaion is used for deermining he effecive conac regions beween ball end mills and surfaces and a mechanisic force model is used o predic cuing forces for each ool posiion along any machining ool pah. The resuls show he influence of he surfaces curvaures variaions on he conac region beween ball end mills and surfaces and consequenly on he cuing forces. This can help o anicipae he force peak a any posiion along he ool pah and herefore o avoid he breakage of he cuing ool and he machine elemens. In perspecive, he deerminaion of he opimal cuing condiions permiing o have a sable machining and a good surface finish will be considered. 5 Cuing force per ooh (N) Number of cuing ool posiion Fig. 7. Variaions of cuing force average per ooh. Many remarks can be exricaed from he differen graphs: For each cuing ool posiion, Fy is more greaer han Fx and Fz also Fz is greaer han Fx. The cuing force Fx vary in he inerval [-71 N, 79 N]. The cuing force Fy vary in he inerval [ N-836 N]. The cuing force Fz vary in he inerval [-115 N, 13 N] The cuing force resulan vary in he inerval [ N, 159 N] The abrup flucuaions of he cuing forces along he ool pah is due o he coninuous suddenly variaions of he surface curvaures. F_ooh REFERENCES [1] Ismail Lazoglu, Sculpure surface machining: a generalized model of ball-end milling force sysem, Inernaional Journal of Machine Tools & Manufacure, , Manufacuring, Auomaion & Research Cenre, Koc, Universiy Isanbul, Turkey, 3. [] Wen-Hsiang Lai, Modeling of Cuing Forces in End Milling Operaions, Universiy of Kansas, Tamkang Journal of Science and Engineering, Vol. 3, No. 1, pp. 15-,. [3] Liqiang Zhang, Jingchun Feng, Yuhan Wang & Ming Chen, Chen, Feedrae scheduling sraegy for free-form surface machining hrough an inegraed geomeric and mechanisic model, In Journ Adv Manuf Technol, 4: , DOI 1.17/s , 9. [4] D. Roh, F. Ismail, S. Bedi, Mechanisic modelling of he milling process using complex ool geomery, In Journ Adv Manuf Technol 5: , DOI 1.17/s , 5. [5] O. Gonzalo, H. Jauregi, L. G. Uriare & L. N. López de Lacalle, Predicion of specific force coefficiens from a FEM cuing model, In Journ Adv Manuf Technol, DOI 1.17/s [6] B. K. Fussell R. B. Jerard J. G, Hemme, Robus Feedrae Selecion for 3-Axis NC Machining Using Discree Models, Universiy of New Hampshire, Journal of Manufacuring Science and Engineering MAY 1, Vol. 13 / 1, DOI: / [7] E. Budak, Analyical models for high performance milling. Par I: Cuing forces, srucural deformaions and olerance inegriy, Inernaional Journal of Machine Tools & Manufacure , Sabanci Universiy, Isanbul, Turkey, 6. [8] M. Dixon e M. Lima, OpenGL programming guide, Addisson- Wesley Publishing Company, 1997.

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