Time domain simulation of torsional-axial and lateral vibration in drilling operation

Transcription

1 Proceeding of Inernaional onference on pplicaions and Design in Mechanical Engineering (IDME) Ocober 2009, au Ferringhi, Penang,MLYSI Time domain simulaion of orsional-axial and laeral vibraion in drilling operaion.m.imani 1, S.G.Moosavi 2 1 Faculy of Engineering, Ferdowsi Universiy of Mashhad, Mashhad, Iran 2 Faculy of Engineering, Ferdowsi Universiy of Mashhad bsrac- Drilling process is one of he common radiional machining operaions. Due o ime inefficiency of he drilling process, i is considered as a criical operaion ha considerably affecs he ime of he enire machining process. Therefore, finding opimal condiions which resul in increase in maerial removal rae (MRR) is of grea imporance. In order o increase cuing velociy, widh of cu and feed, drilling sysem should be dynamically analyed. Sabiliy of he operaion should also be invesigaed in differen operaional condiions. In his research a new force model has been proposed which are validaed using experimenal ess. Simulaion of orsional-axial vibraion is carried ou using ayly s vibraion model. This model is based on he fac ha when a wis drill unwiss, i exend in lengh. In oher words, boh wising and axial deflecion are coupled. oundary condiions which are considered in simulaion of laeral vibraion are decoupled in o wo pars so ha hey can mimic boundary condiions of he drilling process more realisically. oundary condiions in sar of drilling operaion are considered o be clamped-free. They changed o clamped-pin condiions for he res of operaion when drill is compleely engaged in he hole. pplying hese boundary condiions, dynamic undeformed chip hickness is also deermined. Simulaion of laeral vibraion is carried ou using he proposed force model. Simulaion resuls are verified by experimens. Keywords: Time domain simulaion, dynamic undeformed chip hickness, laeral vibraion, Edge force. I. INTRODUTION One of he widely used approaches in sabiliy analysis is ime domain simulaion, which allows a beer insigh ino he dynamics of cuing. lso, he effec of basic nonlineariy can be aken ino accoun and a more accurae esimaion of cuing forces can be obained [1]. In order o simulae drilling vibraion, i is imporan o predic cuing force and vibraions of he sysem when boundary condiions for drilling vibraion are applied. Simulaion of drilling vibraion has recenly been sudied by Roukema and linas[2]. They obained cuing pressure for differen widhs of cu as well as differen feeds for specific drill geomery while spindle speed was consan. They found a second and hird order funcion describing cuing pressure in erm of unformed chip hickness and a drill radius, respecively. Force model proposed by Roukema and linas can use for differen spindle speeds and drill wih differen radius. Mahemaical model of he drill ha used was a clamped-free beam wih four degree of freedom (wo degree for laeral and one for axial and one for orsional direcion). Tooraj rvajeh and Fahy Ismail [1, 3] used a specific drill, feed and spindle speed o deermine cuing pressure for differen widhs of cu. They obained cuing pressure along he cuing edge as a second order funcion of drill radius. Their force model did no include drill wih differen radius, spindle speed and feed. The mahemaical model for laeral vibraion ha hey used was a beam wih clamped-pin boundary condiion. Empirical funcions describing he drilling cuing force have been esablished for l6061-t6 workpiece and HSS ool by R.F.Hamade,.Y.Seif and F.Ismail [4]. This cuing force model uses he empirical cuing pressures which are represened in he form: F b1 c1 d1 (1) a1h V (1 sin( )) F b2 c2 d 2 a 2 h V (1 sin( )) (2) Where and F are cuing pressures and cuing force in direcion of drill axis respecively and, F are cuing pressure and cuing force in angenial direcion of he cuing edge respecively. h is undeformed chip hickness, V is linear velociy of cuing elemen, is rake angle of cuing elemen and b is widh of cu. oefficiens (a, b, c, d) are deermined by drilling experimens. This force model can be used for predicion of cuing force wih differen feed, speed, rake angle and widh of cu. The developmen of orsional-axial vibraion model by ayly has been a major milesone in undersanding chaer in machining using wis drills. ayly s model is based on he fac ha when a wis drill unwiss, i exends in lengh. In oher words, boh wising and axial deflecion are coupled [4]. ayly considered he lumped mass model, depiced in Fig. 1, in which he orsional deflecion,, creaes he axial elongaion,. ccording o ayly, he equaion of moion of he coupled orsional-axial mode can be wrien as: m ( c ( k( F ( ( r) F ( (3) Where m, c and k are he mass, damping and siffness; F and F are he dynamic pars of hrus and angenial force, respecively; and (r) represens he orsional unwinding of he drill cuing edge due o one uni of axial compression [5]. 9G-1

2 Proceeding of Inernaional onference on pplicaions and Design in Mechanical Engineering (IDME) Ocober 2009, au Ferringhi, Penang,MLYSI harge amplifier KISTLER Fig. 1. Lumped model for orsional-axial vibraion [5] II. FORE MODEL Geomerical parameers (rake angle, inclined angle, helical angle and widh of cu vary grealy along he cuing edge of he wis drill herefore i is necessary o divide he cuing edge o smaller elemens wih consan value for hese parameers. In R.F.Hamade,.Y.Seif, F.Ismail [4] geomerical parameers were obained only heoreically for each elemen (i.e. hese parameers were no verified by experimen. In his paper we make a more realisic assumpion by considering variaion of rake angle inclined angle, helical angle and widh of cu. Geomerical parameers of drill are funcion of is radius. These parameers are he same for drills wih differen radius and equal radius raio (r/r) [6]. Therefore we propose he following equaions for cuing pressures. F b1 c1 d1 a1h V ( r / R) (4) F b2 c2 d2 a2h V ( r / R) (5) The proposed force model (4), (5) considers effecs of geomerical parameers which are modeled by he erm (r/r). r is radial disance of cuing elemen from drill axis and R is drill radius. Values of a 1 -d 1 and a 2 -d 2 in (4), (5) are obained by experimens ess. In hese equaions geomerical parameers can be obained by experimens alhough geomerical parameers used in (1) and (2) are obained heoreically. onsidering inheren errors of heoreical obaining of hese values, i can be saed ha he proposed model can be have a more realisic predicion of cuing pressure.. Drilling ess ll drilling ess are performed using a lahe and he workpiece maerial is l7075. In order o measure forces applied o he drill during drilling operaion, a dynamomeer (KISTLER ype 9225) along wih Mulichanel harge mplifier ype 5070 and P based daa acquisiion card wih MTL for sampling are used. The dynamomeer has capabiliy of measuring forces in hree orhogonal direcions (Fx, Fy, F). In order o measure orque, a fixure is designed and manufacured which is shown in Fig. 2. The fixure has been calibraed before saring experimens. huck Fig. 2. onsruced fixure for holding and measuring orque TLE I DRILL GEOMETRIL PRMETERS Drill diameer (D) 10.5mm hisel hickness(2) 2mm Poin angle (2k 118 o Helix angle () 29 o hisel angle () 135 o Drill lengh (l) 280mm. Edge force Measured axial forces and orques from experimenal ess are sum of hree componens: 1- uing force and cuing orque (F c, M c ). 2- The force and orque aribued o ploughing a he main cuing edge (F e,m e ). 3- Force and orque from rubbing acion a he margin (fricion beween helical edge of he drill and workpiece) (F m, M m ). F Fc Fe Fm (6) M M c M e M m (7) Where he subscrips c, e and m sand for cuing, edge, and margin respecively [4]. In order o obain main cuing edge and margin conribuion for axial force and orque we follow his procedure. Firs, we approximae axial forces and orques measured for differen feed and specific predrilled wih line. Nex, he by exrapolaing he obained line for feed=0 mm/rev he main cuing edge and margin conribuions o axial force and orque for specific predrilled are deermined. We applied R.F.Hamade mehod [3] by using measured cuing forces and orque o deermined coefficiens a, b and c in (4) and (5). In order o deermine d 1 and d 2, cuing forces and orque are measured during drilling operaion wih differen diameer of predrilled and consan feed and spindle speed. The value of axial force and orque for wo specific predrilled diameers are subraced and divided by widh of cu beween wo predrilled diameers. For example we measure axial cuing force for 3 and 5 mm predrilled (widh 9G-2

3 Proceeding of Inernaional onference on pplicaions and Design in Mechanical Engineering (IDME) Ocober 2009, au Ferringhi, Penang,MLYSI xure of cu is 2mm and posiion of elemen is 4mm from he cener of drill). Therefore, radius raio of elemen (r/r) is calculaed (4/10.5). The differenial axial force is calculaed by subracing he value measured a 5 mm predrilled from ha of 3 mm predrilled. Nex, his value is divided by 2 mm (elemen widh of cu. This value is ploed versus raio of (r/r) for his elemen (4/10.5). The same procedure is followed for orque. These values can be obained for differen raios of (r/r) from differen predrilled workpieces. Fig. 4 shows he effec of geomerical parameers on axial cuing force and orque wih spindle speed 1000 rpm. For spindle speeds of 250, 500, 710 and 1400 rpm, experimens are performed and coefficiens a, b, c and d are obained and lised in Table 2. Fig. 5. omparison of measured forces in experimenal ess wih prediced forces III. DYNMI UNDEFORMED HIP THIKNESS Fig. 4. Effec of geomerical parameers on () axial cuing force () cuing orque TLE 2 OEFFIIENTS FOR UTTING PRESSURE DISTRIUTION PER (4) ND (5) a 1 b 1 c 1 d a 2 b 2 c 2 d The difference beween measured forces in experimenal es and prediced forces by using new force model is shown in Fig. 5. Mainly wo ypes of boundary condiions have been used by differen researchers for mahemaical modeling of drilling vibraion [1, 2]. oundary condiions in real drilling operaion have wo sages: oundary condiions a he sar of drilling operaion which considered o be clamped-free. ferwards, boundary condiion is changed o clamped-pin for he res of operaion, when drill is compleely engaged in he hole. Definiion of undeformed chip hickness depends on boundary condiions of drilling operaion. In he following, dynamic undeformed chip hickness for clamped-free and clamped-pin boundary condiion is defined.. Dynamic undeformed chip hickness wih clamped-free boundary condiion Laeral moion of drill ip in clamped-free boundary condiion is shown in Fig. 6. hip hickness on each edge of drill wih clamped-free boundary condiion depends on: 1- x(: drill radial run ou defined as he deviaion from he spindle axis in he direcion of he cuing edge. 2- h p : difference in edge heigh called axial run ou. 3- e : edge angle error. 4- (: axial vibraion. 5- (h=f sin(k )/2): feed. Therefore, he un-deformed chip hickness for each edge of drill wih clamped-free boundary condiion can be express as: h h ( ( ( )) sin( k ) ( x( x( )) cos( k ) d1 h ( r / sin( k )) an( e) p h h ( ( ( )) sin( k ) ( x( x( )) cos( k ) d 2 ( r / sin( k )) an( e) Where 2k is he ip angle of drill, h is chip hickness equal o (f/2) sin(k ) and ( is axial ool ip deflecion. h d1 and h d2 are dynamic chip hickness for edge1 and edge 2, respecively. Unequal chip hicknesses on wo edges of drill (h d1 isn equal o h d2 ) cause laeral forces in drilling operaion. In oher (8) 9G-3

4 Proceeding of Inernaional onference on pplicaions and Design in Mechanical Engineering (IDME) Ocober 2009, au Ferringhi, Penang,MLYSI words, balanced chip hicknesses on wo edges generae only axial force and orque. Unlike unequal chip hickness, balanced chip hickness doesn creae laeral forces. F x ( Fy ( F ( F ( ) dfx dfy df dm r r o (, ) / (14). Dynamic undeformed chip hickness wih clamped-pin.. Fig8 shows he drill roaed abou is ip in clamped-pin.. Fig. 6. () Variaion of chip hickness for each cuing edge of drill wih clamped- free.. 1 () oomed area in Fig. 6() Fig. 8. ()Variaion of chip hickness in clamped-pin.. () oomed area in Fig. 8 () Fig. 7. Elemenal force acing on he cuing edges of a wo- flued drill shown in () x plane () xy plane Differenial cuing force componens for drill wih clamped-free.. are shown in Fig. 7. The dynamomeer and fixure seup can measure orque (angenial force F (, can be calculaed by dividing orque o elemen disance from he drill cene and axial force, F (. F ( is verical componen of normal force, F n (. Horional componen of normal force, F r ( is shown in (Fig. 7). Tangenial componen of F ( along he cuing edge, F ( and is normal componen, F n ( is shown in Fig. 7(). Therefore orque and force componens can be wrien as. dfx ( dfr 2 dfr 1 ) ( df 2 df1(, ) sin( k ) (9) dfy dfn1(, dfn 2 (10) df df1(, df 2 ( df1(, df 2 ) cos( k ) (11) dm o r df 1( r df 2( (12) Where, dfr df co( k ) df df sin( i) (13) dfn df cos( i) Subsiuing (4) and (5) in o (13) and subsiuing (13) and (8) in o (9), (10), (11) and (12) elemenal force componens can be deermined. The oal cuing force componens are evaluaed by summaion of elemenal forces disribued along he cuing edges. Therefore componens of cuing force can be deermined as: 1... : boundary condiion hip hickness on each edge of drill wih clamped-pin boundary condiion consiss of: 1- (:Drill roaion angle abou drill ip. 2- h p :The difference in edge heigh is called axial run ou. 3- e :Edge angle error. 4- (:xial vibraion. 5- (h=f sin (k )/2): Feed. Undeformed chip hickness for each edge of drill wih clamped-pin boundary condiion can be wrien as: 1(, ) ( ( ) ( )) sin( ) ( / sin( )) an( ( ) ( )) hd r h k r k hp ( r / sin( k) an( e) hd 2 h ( ( ( )) sin( k ( r / sin( k) an( ( ( )) ( r / sin( k) an( e) (15) In order o simulae drilling vibraion wih clamped-pin boundary condiion, drill is assumed as a clamped-pinned beam whose mass is lumped a middle a =l/2 as shown in Fig. 9 [1]. s can be seen in he Fig. 9, when he drill deflecs in he X- direcion or he Y-direcion, he drill ip roaes abou he Y- axis or he X-axis respecively. The roaion of he cuing edges causes a variaion in cuing forces beween he cuing edges which consequenly creaes a variaion in momens of he cuing forces abou he ip of drill in all direcions; around X, Mx, around Y, My and round Z, M. uing forces componens for drill wih clamped-pin boundary condiion are shown in Fig. 10. Momens of cuing forces abou X, Y and Z axis can be wrien as: dmx dfn1(, r co( k ( ) dfn 2( r co( k ( ) (16) dmy(, ( df 2 df1(, df1(, cos( k ( ) df 2 cos( k ( )) r ( dfr 2 df 2( sin( k ( )) r co( k ( ) (1 ( dfr 1 df1(, sin( k ( )) r co( k ( ) 7) dm r df 1( r df 2( (18) o 9G-4

5 Proceeding of Inernaional onference on pplicaions and Design in Mechanical Engineering (IDME) Ocober 2009, au Ferringhi, Penang,MLYSI Where df n, df and df r are obained by using (13) and axial force, F is obained from: df, df 1 df 2 ( df 1 df 2 ) cos( k ) (19) ( Fig. 9. Lumped mass model of drill for laeral vibraion [1] V. TIME DOMIN SIMULTION Time domain simulaions are carried ou by using ayly s model, Fig. 1 for orsional-axial vibraion and Ismail model, Fig. 9 for laeral vibraion. Equaions of moion used in simulaion can be expressed as follows: x( x ( x( F x( M y( y ( K y( Fy ( (22) ( ( ( F ( ( r ) F ( Where M,, K conain he mass, damping and siffness, respecively. x( and y( are he deflecions of he middle of he drill and ( is he axial elongaion of he drill ip. The righ hand side of equaion shows dynamic forces acing on he drill obained by (14). ased on boundary condiion of he drill (9), (10), (11) and (12) for clamped-free. or (18), (19), (20) and (21) for clamped-pin., dynamic forces will be obained. Resuls of measured axial and laeral forces in experimenal drilling es and FFT of axial and laeral forces daa are shown in Fig. 11. In his experimen, spindle speed is 1000rpm, predrilled is 5mm and feed is 0.11 mm/rev. Fig. 10. uing force componens () axial force () angenial force Since he drill is assumed as a lumped mass a he middle, =l/2, he momens abou poin (Fig. 9) around X-axis, Mx and around Y-axis, My, is replaced by forces F y and F x, according o [1]: My F x (20) l / 2 Mx F y (21) l / 2 Subsiuing (4) and (5) in o (13) and subsiuing (13) and (15) in o (16), (17), (18) momen of cuing forces abou X, Y and Z axis are deermined. Then by using (20), (21) and (19) componens of cuing force in X, Y, Z direcions can be found. The oal cuing forces are evaluaed by summaion of elemenal forces disribued along he cuing edges (14). TE III DYNMI PRMETERS oundary condiion lamped-pin lamped-free naural Frequency in axial direcion 2600[H] 2600[H] naural Frequency in 475[H] laeral direcion 132 [H] Damping raio Laeral frequency xial frequency D IV. DYNMI PRMETERS OF DRILL uing and dynamic sysem parameers are needed for ime domain simulaion of cuing process. The cuing parameers are obained using cuing force model, and dynamic parameers are obained using finie elemen model in conjuncion wih he impac modal esing. The naural frequencies in differen mode shapes wih differen boundary condiions are obained from finie elemen mehod. These values are nex compared wih he values obained from modal esing. Damping raios are measured experimenally using modal esing. The obained dynamic parameers for he drill wih geomerical shown in able1, is lised in able3. xial frequency Fig. 11. () ime hisory experimenal axial forces () FFT of force daa in axial direcion () Time hisory Experimenal laeral forces (D) FFT of force daa in laeral direcion Fig. 11() shows he ime race of axial force. Since axial vibraions ampliude grows (a ime 15 o 22 sec), unsabiliy occurred a axial mode. Fig. 11 () shows FFT of force daa in axial direcion, as can be seen wo srong peaks a he frequency around 450 H (laeral frequency) and a he 9G-5

6 Proceeding of Inernaional onference on pplicaions and Design in Mechanical Engineering (IDME) Ocober 2009, au Ferringhi, Penang,MLYSI frequency around 2600 H (axial frequency). Fig. 11() shows he ime race of laeral force. Since ampliude of vibraions in laeral direcion grows herefore a laeral mode is unsable. Fig. 11(D) shows FFT of force daa in laeral direcion, as can be seen a large peak a he frequency around 2600 H (axial frequency). I should be noed ha laeral frequency appear in axial vibraion and axial frequency appear in laeral vibraion, Fig. 11() and Fig. 11(D). Simulaions of axial and laeral vibraion are carried ou by using he proposed force model. To solve he equaions of moion, (22) fourh order Rung-Kua mehod is used. Simulaion of axial force ime hisory is shown in Fig. 12. Time domain simulaion, Fig. 12() and Fig. 12() shows unsabiliy in axial direcion for spindle speed 1000 rpm, predrilled 5mm and feed 0.11 mm/rev. FFT of simulaed axial force daa, Fig. 12() shows wo peaks a he frequency around 450 H and frequency around 2700H. The simulaion resuls of axial force wih new force model and dynamic undeformed chip hickness in Fig. 12 are in close agreemen wih experimenal resuls Fig. 11. engaged wih he workpiece and laeral vibraion is unsable, boundary condiion in his region is clamped-free (region 3 in Fig. 13). Fourh region drill compleely engaged wih workpiece and laeral vibraion is sable.oundary condiion in fourh region is clamped-pin (region 4 in Fig. 13). Therefore differen boundary condiions for simulaions of hird and fourh regions are imporance. Simulaion resuls for laeral vibraion wih clamped-pin and clamped- free boundary condiion are shown in Fig. 14. s can be seen in Fig. 14 laeral vibraion is unsable in clamped-free boundary condiion and is sabile in clamped-pin boundary condiion wih he same spindle speed, predrilled and feed. These simulaion resuls agree wih he hird and fourh region in experimenal es, Fig. 13. Simulaion xial frequency Fig. 14. Time domain simulaion wih () clamped-free () clamped-pin boundary condiion for spindle speed 500 rpm, predrilled 5mm and feed 0.05 mm/rev Laeral frequency Fig. 12. () Resul of ime domain simulaion of axial force and comparison wih experimenal resul for spindle speed 1000 rpm, predrilled 5mm and feed 0.11 mm/rev () Frequency excied of simulaed axial force Fig. 15. Frequency conen of laeral force simulaion Fig. 13. Differen regions in laeral vibraion ecause of presence of erm F in (11) and (19) laeral frequency is appeared in axial vibraion, Fig. 12(). For simulaion of laeral force, differen regions vibraion menioned in secion (III) should be considered. Differen regions in laeral vibraion for spindle speed 500rpm, predrilled 5mm and feed 0.05 mm/rev are shown in Fig. 13. s can be seen in Fig. 13 four differen regions: in firs region drill has no ye engaged wih he workpiece (region 1 in Fig. 13). Second region drill is beginning o engage wih he workpiece (region 2 in Fig. 13). In Third region drill is Simulaion resul of laeral force wih clamped- pin boundary condiion for spindle speed 1000 rpm, predrilled 5 mm and feed 0.11 mm/rev is shown in Fig. 15. Fig. 15 shows unsabiliy in laeral vibraion ha is verified wih experimenal es, Fig 11(D). ecause of presence of erm F in (9) and (17) axial frequency is appeared in laeral vibraion Fig. 15. V. ONLUSIONS In his paper we proposed a new force model base on experimenal drilling ess. Effec of linear cuing velociy, feed and geomerical parameers are aken in o accoun in proposed force model. oefficiens of cuing pressures in new force model are compleely deermined from experimenal drilling ess wihou using heoreical equaions. In he proposed force model effec of geomerical parameers 9G-6

7 are modeled by he erm (r/r). lhough in (1) and (2) geomerical parameers are modeled by he erm rake angle () which obained heoreically. Inheren errors consider heoreical obaining of his value. Resuls of prediced forces by using new force model were in close agreemen wih forces measured from experimenal drilling ess wih specific drill geomery. The proposed force model can be verified for drill wih differen diameers and compared wih experimenal ess. This model can also be esed for high speed drilling. In his paper an equaion for dynamic undeformed chip hickness is presened based on a more precise model for boundary condiions of he drill. oundary condiions a he beginning of drilling operaion are modeled as clamped-free. Nex, boundary condiions changed o clamped-pin when drill is compleely engaged in workpiece. For each boundary condiion, componens of cuing force were deermined. Simulaion of orsional and axial vibraion carried ou by using ayly s model. Resul of simulaion is verified by experimenal es. The reason of appearance of laeral frequency in axial vibraion is explained by simulaion using he proposed force model. (Presence of componen of angenial force along he cuing edge in axial direcion.) The exisence of differen sages in drilling vibraion is jusified by considering differen boundary condiions for each sage. Resuls of laeral force simulaion based on differen boundary condiions in drilling operaion are verified by experimenal resuls. REFERENES [1] Tooraj rvajeh and Fahy Ismail, Machining sabiliy in high-speed drilling-par 1:Modeling vibraion sabiliy in bending, Inernaional Journal of Machine Tools & Manufacure 46(2006) [2] Jochem.Roukema,Yusuf linas, Generalied modeling of drilling vibraions, Par I: Time domain model of drilling kinemaics, dynamics and hole formaion Inernaional Journal of Machine Tools & Manufacure,2006 [3] Tooraj rvajeh, Fahy Ismail, Machining sabiliy in high speed drillingpar 2: Time domain simulaion of a bending-orsional model and experimenal validaions, Inernaional Journal of Machine Tool & Manufacure, 46 (2006) [4] R. Hamade,.Y. Seif, F. Ismail, Using drilling experimens o exrac specific cuing force coefficiens, Inernaional Journal of Machine Tools and Manufacure, 46 (2006) [5] Toraj rvajeh and Fahy Ismail, Sabiliy lobes of Twis drills including profiles of geomerical and cuing parameers along he cuing edge Deparmen of Mechanical Engineering, Universiy of Waerloo, IRP 2005 [6] V.handrasekharan, S.G. kapoor, R. E. DeVor Mechanisic pproach o predicing he uing Forces in Drilling: Wih pplicaion o Fiber- Reinforced omposie Maerials. Proceeding of Inernaional onference on pplicaions and Design in Mechanical Engineering (IDME) Ocober 2009, au Ferringhi, Penang,MLYSI 9G-7