Efficient experimental detection of milling stability boundary and the optimal axial immersion for helical mills

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Efficien expeimenal deecion of milling sabiliy bounday and he opimal axial immesion fo helical mills Daniel BACHRATHY Depamen of Applied Mechanics, Budapes Univesiy of Technology and Economics

, Budapes, H-1111, Hungay ABSTRACT In his pape, an opimal axial immesion is defined and i is calculaed fo he milling pocess. The analyical calculaion is based on one degee of feedom model.

We explain his wih he help of he Fouie seies of he ime-peiodic cuing foce. To suppo ou analyical esuls measuemens wee caied ou.

In his way, he axial immesion was inceased coninuously duing he cuing pocess. The sabiliy bounday and he opimal axial immesions could be found wih a single es a each spindle speed.

1 Efficien expeimenal deecion of milling sabiliy bounday and he opimal axial immesion fo helical mills Daniel BACHRATHY Depamen of Applied Mechanics, Budapes Univesiy of Technology and Economics Muegyeem kp 3., Budapes, H-1111, Hungay and Gabo STEPAN Depamen of Applied Mechanics, Budapes Univesiy of Technology and Economics Muegyeem kp 3., Budapes, H-1111, Hungay ABSTRACT In his pape, an opimal axial immesion is defined and i is calculaed fo he milling pocess. The analyical calculaion is based on one degee of feedom model. A helical ool is consideed duing he calculaion. Axial immesion values ae found whee lage maeial emoval ae can be achieved while negligible and oleable vibaion occus. We explain his wih he help of he Fouie seies of he ime-peiodic cuing foce. To suppo ou analyical esuls measuemens wee caied ou. Insead of making numeous measuemens wih disinc spindle speeds, we used wedge-shaped wok-pieces o speed-up he lenghy expeimens. In his way, he axial immesion was inceased coninuously duing he cuing pocess. The sabiliy bounday and he opimal axial immesions could be found wih a single es a each spindle speed. We found ha he vibaion ampliudes wee significanly educed in case of he heoeically pediced opimal axial immesions. Keywods: Chae, Sabiliy, Helical mills, Time delay, High-pefomance milling cuen posiion of he cuing edge and he pas posiion of he pevious/neighbouing cuing edge of he mill. Consequenly, he govening equaion of he model is a ime delayed diffeenial equaion. In ode o avoid his ype of vibaion, he so-called sabiliy cha mus be ceaed as a funcion of he echnological paamees. This calculaion can be caied ou in seveal ways like using he semi-disceizaion mehod [3,4,5], empoal finie elemens [6], mulifequency analysis appoach [7,8] o ime domain simulaion [9]. Duing ou analysis, he semi-disceizaion mehod was used o deec he sabiliy bounday of he milling pocess. The second ype of vibaions is caused by he peiodic non-smooh cuing foce. This foced vibaion can have lage ampliudes a esonan spindle speeds. Thee ae some egions of he echnological paamees whee lage maeial emoval aes can be achieved, while egeneaive chae is also avoided. Howeve, in mos cases, foced vibaions can sill occu a hese paamees even close o esonance. These vibaions lead o poo suface qualiy. In [10,11,1] he qualiy of he milled suface was pediced by means of analyical calculaion. The suface was geneaed in [13,14] accoding o numeical ime domain simulaion. In case of helical ools, we found ha his kind of vibaion can also be eliminaed by using he opimal axial immesion ha depends on he helix pich of he ool and on he spindle speed [15,16]. In he pesen sudy measuemen esuls ae given o suppo ou analyical appoach and pedicions. The mehod of he efficien expeimenal deecion of milling sabiliy bounday and he opimal axial immesion is also pesened in deails.. MODEL Fig.1: Poo suface qualiy caused by self-excied vibaion 1. INTRODUCTION Vibaions in milling pocesses may cause poo suface popeies and shoens he lifeime of he machine ool (see Fig.1.). Thee ae wo ypes of hese vibaions which ae essenial. The fis is he so-called chae ha is a kind of a self-excied vibaion caused by he egeneaive effec of he machined suface [1,]. The chip hickness depends on he In ou analysis, a 1 DoF flexible ool and igid wok-piece model was used (Fig..). In finishing opeaions, only he sable pocess is accepable, which can be deemined by a linea sabiliy calculaion. Thus, he cuing foce F of he h ooh is appoximaed as a linea funcion of he chip hickness h a he saionay cuing. The angenial and he adial cuing foce componens can be given as: F ( = w K h (, F ( = w K h (, (1)

analysis. On he igh-hand-side of Eq. (4), he oal cuing foce compises wo pas, he saic foce w N K Fsa ( = τ v g ) T1 ) dz, (5) 0 = 1 and emaining pa is he dynamic foce, whee Fig.

2 can be obained. Afe a saighfowad calculaion, he govening equaion is given by ( y( y( )) m& y ( + cy& ( + ky( = Fsa ( + A( τ, (4) whee he mass m, damping faco c and he siffness k ae idenified fom expeimenal modal analysis. On he igh-hand-side of Eq. (4), he oal cuing foce compises wo pas, he saic foce w N K Fsa ( = τ v g ) T1 ) dz, (5) 0 = 1 and emaining pa is he dynamic foce, whee Fig. : 1DoF model of he flexible machine ool whee w is he axial immesion, K and K ae he specific cuing foce coefficiens, while k denoes hei aion, k =K / K. The chip hickness compises wo pas, he saic and he dynamic chip hicknesses. The saic pa is geneaed by he consan feed moion v, while he vibaion of he ool descibed by he y coodinae geneaes he dynamic pa, so he chip hickness can be esimaed as ( φ ) + ( y( y( τ )) cos( φ ) h = τ v sin. () Hee φ is he angula posiion of he h ooh, he egeneaive ime delay is equal o he ooh-pass fequency (τ =/ ΩN), N is he numbe of he eeh and Ω is he angula velociy of he ool. The em τv is equal o he feed pe ooh. If helical ools ae consideed in he model, like Fig. 3., han he cuen angula posiion of he edge also depends on he axial coodinae of he ool z. whee p is he helix pich. z φ ( z, = Ω +, (3) N p w N K A( = g ) T ) ) dz. (6) 0 = 1 The sep funcion g shows whehe he h ooh is in conac wih he wok-piece o no. The poecion coefficiens T 1 and T ae given by T1 T ) = 1 cos( φ ( z, ) k sin ( φ ( z, ) ( φ ( z, ) = sin ( φ ( z, ) k k cos( φ ( z, ). I can be seen easily ha he saic cuing foce is consan in ime if he axial immesion (7) p w = k, k (7) N is used. In his case, hee is no foced vibaion. The govening equaion Eq. (5) can be sepaaed ino wo pas by using he new vaiables φ and η. such ha foced moion φ ( saisfies y( = φ ( + η( (8) m & φ ( + c & φ( + kφ( = Fsa (. (9) Subsiuing Eq. (8) and Eq. (9) ino Eq. (4) we ge he delayed paameically foced pa of he govening equaion. ( k A( ) η( = A( η( ) m & η ( + c & η( + τ (10) Fig. 3: Geomey of a helical edged ool and he helix pich. In each coss-secion of he ool, he local cuing foce can be calculaed by using Eq. (1) (3). By he inegaion of hese local cuing foces along he ool axes, he oal cuing foce which descibes he sabiliy of he ool moion. If i is unsable, he ool vibaion ampliude ends o infiniy and hee is no need o calculae he suface popeies, his is aleady unaccepable. When he ivial soluion η( 0 of Eq. (10) is sable, he paicula moion of he ool cene is given by he soluion of he foced moion Eq (9). This always has a peiodic soluion because of he small damping. By using his soluion, we can compue he moion of he eeh, which geneaes he suface afe he seled ansien moion. The paamee domains of sable cuing wee calculaed fom Eq. (10) using he semi-disceizaion mehod, which is descibed in deail in [4,15].

3 3. OPTIMAL AXIAL IMMERSION Efficien echnical paamees fo sable cuing can be chosen fom he sabiliy chas, bu in mos cases, foced vibaions sill occu a hese paamees. In hese cases, he vibaion is close o esonance because he fequencies of highe hamonics of he cuing foce ae close o he naual fequencies of he sysem ω n. In case of helical ool we can define he opimal axial immesions whee hese hamonics disappea. This phenomenon is explained below by he Fouie ansfomaion of he cuing foce signal, because he specum of he cuing foce does no conain he naual fequency of he sysem. In ohe wods, he esonan Fouie componen of he cuing foce is zeo along hese lines. The local cuing foce on a single elemenay secion of he helical ool is N K f ( φ ( z, ) = τ v g ) T1 )dz (11) = 1 cu ( Since ( φ) ω = f ) = f e d ( θ Ω) ( Ω) e d gˆ ( ) ω w i i e d 1 e (16) ω f is a peiodic funcion, is Fouie ansfom is a discee funcion. I can be seen fom Eq. (16) and also in Fig. 4, ha he esonan Fouie componen of he cuing foce cu ( ) is zeo if eihe o ω n k = kωn, τ Ω k N w = k w = k, k k (17) (18) whee, he angula posiion φ ( z, of he edge is given by Eq. (3). Funcion f ( φ) is a peiodic funcion in ime and is peiod equals o he ooh-pass fequency1 / τ. By using Eqs. (3) and (5), he Fouie ansfomaion of he cuing foce is given by w z cu ( ω) = f Ω dz e d (1) p 0 The axial coodinae z is subsiued by he new vaiable z = ( θ ), (13) π and he inegaion by subsiuion leads o he convoluion inegal. cu ( ω ) = f ( θ Ω) dθ e d w ( ) ˆ ( ) iω = f θ Ω g θ dθ e d, (14) wih he sep funcion gˆ () 1 = 0 w if 0 < < p Ω ohewise. (15) The Convoluion Theoem saes ha he Fouie ansfom ansfoms a convoluion ino a muliplicaion (see [17]), hus Eq. (14) can be wien as Fig.4: Repesenaion of he ( Ω) f and ĝ () as a funcion in ime and is Fouie ansfom If his componen is zeo hen hee is no esonance. The lines ( w (Ω) ) descibed by Eq. (18) ae he same lines as he doed slaning ones in Fig 9, while he ohe condiion (3) is idenical o (11) deemined ealie and shown a p/z in Fig 9.

4. EXPERIMENTAL VALIDATION An expeimenal seup was esablished o confim ou analyical invesigaions. To ealize his model, we manufacued a es ig wih high flexibiliy in one diecion (see Fig.

0 sofwae o deec he signal of he acceleomees (B&K4397) placed on he uppe pa. Fom he Impac es pocedue, he measued naual fequency is ω n = 44.1 ad/s, and he damping aio is ξ =1.14 %. In Fig.

Duing he measuemen he opimal immesions ae deeced as he local minimum poin of he vibaion ampliude. These poins ae denoed by whie cicles in Fig.6. We also deeced he sabiliy losing duing he measuemens.

4 4. EXPERIMENTAL VALIDATION An expeimenal seup was esablished o confim ou analyical invesigaions. To ealize his model, we manufacued a es ig wih high flexibiliy in one diecion (see Fig. 5) based on [8]. The mass of he uppe pa was 8.76 kg. We caied ou is modal analysis o check he naual fequency and he damping aio of he es ig. We use Pulse Fon-end and Pulse Labshop v11.0 sofwae o deec he signal of he acceleomees (B&K4397) placed on he uppe pa. Fom he Impac es pocedue, he measued naual fequency is ω n = 44.1 ad/s, and he damping aio is ξ =1.14 %. In Fig.6 he heoeical opimal axial immesions ae denoed by he doed hoizonal lines, whee he oal cuing foce is consan in ime, and coninuous slaning lines, whee he esonan Fouie componen on he cuing foce is zeo. Duing he measuemen he opimal immesions ae deeced as he local minimum poin of he vibaion ampliude. These poins ae denoed by whie cicles in Fig.6. We also deeced he sabiliy losing duing he measuemens. Those axial immesion whee he sabiliy losing occus, ae denoed by dak ed cicles. We can see ha he measued opimal immesion poins fi he heoeical ones vey well, which suppo ou analyical esuls. In he es measuemens, we used epoxy esin maeial. We used wedge-shaped wok-piece o speed-up he lenghy expeimens. In his way, he axial immesion was inceasing coninuously duing he cuing pocess. So he sabiliy bounday and he opimal axial immesions could be found wih one single es along an Ω = cons. line. Duing he measuemens a 5-flued ool was used wih helix pich p=11.5 mm and diamee D=Ø0 mm. Small adial immesion (0.5 mm) was used duing he up-milling pocess, which is a usual value in case of suface finishing pocesses. The feed ae was se o τv=0.15 mm/ooh, whee v is he feed velociy. The axial immesion is limied by he ool, so w max =35 mm. Fig.6: Measued opimal axial immesion (whie cicles), sabiliy bounday (dak ed cicles). The esonan fequencies ae denoed by veical dashed lines. The opimal axial immesions ae denoed by blue doed and coninuous lines. The appoximaed insable aeas ae shaded. 5. CONCLUSION Fig.5: The flexible holde and he wedge-shaped polyme wok-piece. In his epo, we invesigaed he effec of he paamees of a helical ool model o pedic he opimal axial immesions in high-speed milling. The mos impoan obsevaion is ha small vibaions ake place even fo esonan angula velociies ( Ω = / N, / N, / 3N, / 4N,... ) if appopiae axial immesions ae applied. The appopiae axial immesions ae no us he ivial ones whee he cuing foce is ime-independen, ha is, whee he axial immesions ae equal o an inege muliple of he helix pich p ove he numbe of cuing eeh N. Non-ivial appopiae axial immesions ae also found which depend linealy on he cuing speed. The idenificaion of hese new appopiae axial immesions is useful since us he esonan cuing speeds ae pefeed o achieve high maeial emoval aes wih sable machining. If non-appopiae axial immesions ae used a hese cuing speeds, he machined suface qualiy paamees migh be exemely poo. We caied ou measuemens o suppo ou heoeical esuls. Wedge-shaped wok-piece was used o speed-up he lenghy expeimens. We found ha he vibaion ampliude was significanly smalle in case of he opimal axial immesions, which confimed ou heoeical pedicions.

5 6. ACKNOWLEDGEMENT This eseach was suppoed by he Hungaian Naional Science Foundaion unde gan no. OTKA K REFERENCES [1] S.A. Tobias, W. Fiswick, Theoy of Regeneaive Machine Tool Chae, Engineeing, London, 58. [] J. Tlusy, M. Polacek, Beispiele de behandlung de selbseegen Schwingung de Wekzeugmaschinen FoKoMa, 1957 Hanse Velag, Munchen. [3] T. Inspege, G. Sépán, Sabiliy of High Speed Milling, Poceedings of he 000 ASME Inenaional Engineeing Congess and Exposiion, Olando, Floida, 000, AMD 41: [4] T. Inspege, G. Sépán, Updaed Semi-Disceizaion Mehod fo Peiodic Delay-Diffeenial Equaions wih Discee Delay, Inenaional Jounal of Numeical Mehods in Engineeing, 61/1: [5] J. Gadišek, E. Goveka, I. Gabec, M. Kalveam, K. Weine, T. Inspege, G. Sépán, On sabiliy pedicion fo low adial immesion milling, Machine Science and Technology, Vol.9, 005, pp [6] P.V. Bayly, J.E. Halley, B.P. Mann, M.A. Davies, Sabiliy of Ineuped Cuing by Tempoal Finie Elemen Analysis, Jounal of Manufacuing Science and Engineeing, 003, 15:0-5. [7] S.D. Medol, Y. Alinas, Muli Fequency Soluion of Chae Sabiliy fo Low Immesion Milling, Jounal of Manufacuing Science and Engineeing, 004, 16: [8] M. Zaaain, J. Munoa, G. Peigné, T. Inspege, Analysis of he influence of mill helix angle on chae sabiliy, Annals of he CIRP, Vol. 5, 006, pp [9] B. Balachandan, M. X. Zhao, A Mechanics Based Model fo Sudy of Dynamics of Milling Opeaions, Meccanica, 000, 35/: [10] B.P. Mann, K.A. Young, T.L. Schmiz, D.N. Diley, Simulaneous Sabiliy and Suface Locaion Eo Pedicions in Milling, Jounal of Manufacuing Science and Engineeing, Vol. 17, 005, pp [11] T. Inspege, J. Gadišek, M. Kalveam, G. Sépán, K. Weine, E. Goveka, Machine ool chae and suface qualiy in milling pocesses, Jounal of Manufacuing Science and Engineeing, Vol. 18, 006, pp [1] K. Shiase, Y. Alinas, Cuing foce and dimensional suface eo geneaion in peipheal milling wih vaiable pich helical end mills, Inenaional Jounal of Machine Tools and Manufacue, Vol. 36, No. 5, 1996, pp [13] G. Peigné, H. Pais, D. Bissaud, Suface Shape pedicion in high speed milling, Inenaional Jounal of Machine Tool & Manufacue, Vol.44, No. 4, 004, pp [14] K. Weine, T. Sumann, Geomeic model of he suface sucue esuling fom he dynamic milling pocess, 9h CIRP Inenaional Wokshop on Modelling of Machining Opeaions. Bled, Slovenia: 006. pp [15] D. Bachahy, A sudy of he milled suface in he dynamics of milling, Msc. Thesis, Univesiy of Bisol and Budapes Univesiy of Technology and Economics, 006 [16] D. Bachahy, M. Home, T. Inspege, G. Sépán, Suface popeies of he machined wok-piece fo helical mills, Machine Science and Technology, Submied [17] R.B. Randal, B. Tech, B.A., Fequency analysis ; 3 d ediion 1 s pin, Büel & Kae, 1987, 49-55

Lecture-V Stochastic Processes and the Basic Term-Structure Equation 1 Stochastic Processes Any variable whose value changes over time in an uncertain

Lecture-V Stochastic Processes and the Basic Term-Structure Equation 1 Stochastic Processes Any variable whose value changes over time in an uncertain Lecue-V Sochasic Pocesses and he Basic Tem-Sucue Equaion 1 Sochasic Pocesses Any vaiable whose value changes ove ime in an unceain way is called a Sochasic Pocess. Sochasic Pocesses can be classied as