Articl citation info: Świć A, Tarannko W Adaptiv control of machining accurac of axial-mmtrical low-rigidit part in latic-dformabl tat Ekploataca i Nizawodnoc Maintnanc and Rliabilit 4 (3): 5- Antoni Świć Wiktor Tarannko Adaptiv control of machining accurac of axialmmtrical low-rigidit part in latic-dformabl tat Strowani adaptacn dokładnością obróbki części oiowo-mtrcznch o mał ztwności w tani prężści-odkztałcalnm* Th author dvlopd a mthod of corrction coniting in th introduction, in th control tm, of an additional poitiv fdback rlativ to th forc of milling Adaptiv control wa applid for axial fd and for additional forc action cauing th latic-dformabl tat, which prmit th limination of tatic rror of control ffct and intrfrnc in th control of qualit paramtr word: adaptiv control, machining accurc, low-rigidit part,latic-dformabl tat Opracowano poób korkt utawinia układu tchnologiczngo obrabiarki polgaąc na wprowadzniu do układu trowania dodatkowgo dodatnigo przężnia zwrotngo względm ił krawania Zatoowan trowani adaptacn pouwm wzdłużnm oraz dodatkowmi oddziałwaniami iłowmi, wwołuącmi tan prężści-odkztałcaln, umożliwia wliminowani błędów tatcznch oddziałwania truącgo oraz zakłócaącgo prz trowaniu paramtrami dokładności obróbki części o mał ztwności w tani prężści-odkztałcalnm Słowa kluczow: trowani adaptacn, dokładność obróbki, części o mał ztwności, tan prężści-odkztałcaln Introduction In th machin-building indutr, aximmtrical part contitut approximatl 34% of th total numbr of part, and among tho up to % can b claifid a low-rigidit haft Such part ar charactrid b diproportional rlation of ovrall dimnion and b low rigidit in pcific ction and dirction Stringnt rquirmnt ar alo pod rlativ to th paramtr of gomtric form, mutual poitioning of urfac, linar dimnion, and urfac qualit Th pcific charactr of machining of imilar part cau that it i difficult to achiv th rquird paramtr of accurac of form, dimnion and urfac qualit Th low inhrnt tiffn and th rlativl low rigidit of th haft compard to th tiff ambli of th machin tool cau th apparanc of vibration, undr pcific condition In th cour of machining thr appar numrou factor that intrfr with and dtabili th proc of machining (larg fr dformation of haft, tool, fixtur, having, dut, tc) Thi mak it ncar to arch for nw mthod of controlling th machining of low-rigidit haft Poitiv olution of problm latd with improvmnt of qualit of machining of part of milling machin for mtal dpnd dirctl on furthr improvmnt of mthod for digning and tting of automatic control tm (AS), machin tool and paramtr of th dnamic tm Th dnamic tm (DS) of th proc of machining i an MT (machin tool-grip-fixtur-tool) tm it compri th ma-diipativ-latic (MDE) tm of th machin tool and th ralizd proc of machining (turning, grinding, drilling, milling) [, 7, 9, ] In principl, th mthod of anali and nthi of contmporar automatic control tm ar bad on olving a problm with notabl implification of th phical and mathmatical rlation that charactri th proc includd in th tm Thi i largl a rult of imprfction of th rarch apparatu mplod, and of complxit of acquird in prior information on th dnamic and tatic charactritic of th obct of control and of xtrnal intrfrnc factor Th xiting mthod of nthi of automatic control tm tak into account th poibl uncrtaint concrning th charactr and xtnt of intrfrnc onl to a limitd dgr Evn l dvlopd ar th iu of cattr of control obct paramtr, though at prnt conidrabl attntion i dvotd to thm Digning of AS capabl of oprating at uncontrolld variabilit of control obct paramtr ha ld to th apparanc adaptiv control (A) in machin tool [, 3, 8] In pit of it flxibilit and poibilit of auranc of rquird qualit of tranition proc for a broad rang of obct, in man ca th application of A on machin tool i difficult du to th ncit of continuou maurmnt of charactritic of tchnological tm and intrfrnc Rrv for incraing th accurac and qualit of machining ma b dfind in th dign of optimum tructur of control tm, a th AS of latic dformation (b man of rlativ mov of th cutting tool and th machind part) of th DS i tatic in trm of both control ffct and intrfrnc, and thir chang cau rror of mutual poitioning of th part and th cutting dg [4, 8] Th thortical foundation for improvmnt of machining qualit with adaptiv control, conidrd in thi rport, ar bad on mathmatical dcription of th obct of control in th DS prntd in rfrnc [, 4, 5, 7, ] and ma b applid in DS anal dcribd mathmaticall diffrntl than th prntd hrin and taking into account a pcific charactr of th control obct (*) Tkt artkułu w polki wri ęzkow dotępn w lktronicznm wdaniu kwartalnika na troni wwwinorgpl Ek p l o a t a c a i Ni z a w o d n o c Ma i n t n a n c a n d Rliabilit Vo l4, No 3, 5
Som iu of invariabilit in th control of latic dformation of th dnamic tm In principl, tm of automatic control of tchnological proc compri th obct of control - th dnamic tm of th machin tool, mad up of th MDE tm of th machin tool and th working proc involvd in thir intraction, and th rgulator Anali of connction within th dnamic tm of th machin tool indicat that it i a multi-contour clod tm, and th intraction of th fundamntal componnt i dirctd b th intraction btwn th working proc, taking plac via th MDE tm onl MDS [, 6] Sparation of th zon whr th working proc tak plac, b man of th lmnt of th latic tm prmit rlativ impl tranition to quivalnt concpt of th dnamic tm of th machin tool a a ingl-contour tm Th tranfr function of an quivalnt MDE tm ha th form: whr: ( ), p T T u() + + p prntation, acc to Laplac, of latic dformation and forc of machining,, T n cofficint of T m proportionalit and tim contant of MDE tm of ma chin tool m, n, rducd ma, cofficint of attnuation and rigidit of MDE tm Th tranfr function of lmnt of adaptiv control tm (ig a) hav th following form: --for th MDE tm, rlativ to th control ffct p T + T+ --for th MDE tm, rlativ to intrfrnc ffct T + T + --for th proc of machining k --for th xcutiv mchanim w p p k, d Tk+ w w, Tw+ whr: T m, T n,, f, T w, tim contant and cofficint of proportionalit w of tranfr function of th xcutiv mchanim Th accurac of opration of th tm of adaptiv control of latic dformation of DS i charactrid b an rror [, 7]: whr: Error qual to o + f, () t, rror caud b control ffct f rror caud b intrfrnc ffct f ( t ) o and f ar dtrmind from th rlation: ig Block diagram of th tm of tabiliation of latic dformation of tchnological tm a chmatic of tm of adaptiv control of latic dformation of tchnological tm b functional chmatic of adaptiv control c nod of poitiv fdback rlativ to th forc of machining d i + i k b i 4, () k k w a + + i f i d i 4,(3) k k w a f f f + + whr: a k k w b a b T w + kw + T k + T a b T kt w + T T w + T T k + T a3 b3 TT w + T T kt w + TT k a4 b4 TT kt w + + k 4 6 Ek p l o a t a c a i Ni z a w o d n o c Ma i n t n a n c a n d Rliabilit Vo l4, No 3,
d d ( T k T w) + d T ktw Th dtrmind valu of th rror () and (3) can b prntd a [8]: 3 4 d ( t) ( t) ( t) ( t) 3 4 ( t) ( t) d d d + + + +, dt! 3! 3 4! 4 dt dt dt 3 4 df ( t) f ( t) 3 f ( t) 4 f ( t) f ( t) f( t) d d d + + + +,(4) dt! 3! 3 4! 4 dt dt dt whr i, i (i,,,, 4) cofficint of rror, charactri ing th accurac of opration of th control tm and d pndnt on it tructur In th tructur of th control tm (ig ), th valu of th cofficint of rror ar dfind b th following rlation: b b a b a a, a,, a a d, a d a, a d a a a It hould b notd that th main rror i contitutd b th rror cofficint, and,, rfrrd to a th tatic and pd rror or a 6 machin tool, with tm paramtr of: 6, T, 6 4 6,, 6 mm N, (5) T, 3 3 4 T k, T w, f mm N, k bad on data (5), th valu of th rror cofficint wr dtrmind analticall, a follow:,3mm, 3,, mm 3 6,5 mm, mm, 6,6 mm, 9,73 mm z Total or partial limination of tho rror would prmit improvmnt of th accurac of opration of th control tm and, corrpondingl, of th machining of th part Th problm pod can b olvd through th introduction of an additional poitiv fdback rlativ to th forc of machining, with tranmittanc (), into th tm of control of latic dformation of th DS (ig b) [] In thi ca, rror () i dtrmind from th rlation: k z w k + (6) + k z w k w A follow from xprion (6), th rror introducd in th tm O of control of latic dformation of th TS b th ffct () t can b totall liminatd if th tructur and paramtr of th tranfr function ar prntd a follow: z + k k w (7) Exprion (7) can b calld th condition of total invariabilit of O th control tm rlativ to th control ffct () t If thi condition i mt, all th rror cofficint i (i,, 3, 4) ar qual zro Taking into account that mting th condition (7) baicall lad to tm that ar phicall unraliabl, and that ufficint accurac can b achivd in practic whn onl th tatic rror cofficint or and i ar qual to zro, it i nough to mt th condition of b and b Howvr, with b alo cofficint a qual zro, which cau that th tm lo tabilit Thrfor, in th ca undr conidration, it i onl poibl to liminat th tatic rror Error qual zro can b achivd with man that ar tchnologicall impl nough, through th introduction of an additional poitiv fdback rlativ to th forc of machining, with tranfr function of () z z Th valu of th cofficint of proportionalit of th fdback i dtrmind from rlation (7) that i ( ) b + k k z, (8) z + k k, howvr to maintain tabilit it mut b lowr than If condition (8) i mt, in th control tm with paramtr (5) conidrd arlir, th cofficint of rror ar qual to,33 mm Error introducd b an intrfrnc ffct can b prntd a: f k w z f (9) k w z + k w + k Rlation (9) can b alo ud to obtain th condition of total invariabilit of th tm undr th ffct of an intrfrnc factor z () k w B analog, condition () will b mt if th tructur of th tranfr function of th fdback i prntd a: z( ) z T z +, and it paramtr will b lctd from th quation: hnc d z k w, d T T T k + w z z w, z, k w T z k w T T kw In thi ca, th cofficint of rror in th xampl undr conidration will b qual to:,, 9 6, 5 mm ig c prnt a block diagram of a tm of adaptiv control ralizing th mthod of adutmnt of tting of th TS [8] Th control tm incorporat, in ri connction, th latic dformation Ek p l o a t a c a i Ni z a w o d n o c Ma i n t n a n c a n d Rliabilit Vo l4, No 3, 7
rgulator, comparing lmnt to on of th input, connctd to which i th latic dformation nor 8, corrcting lmnt 3, amplifir 4, powr tranducr 5, longitudinal fd motor 6, tchnological tm 7, machining forc nor 9, nod of poitiv fdback Nod, of th poitiv fdback rlativ to th machining forc p (ig d) i mad a a diffrntial chain of condnr, ritor a R3 and a voltag dividr - ritor R and R In th proc of work, prliminar dtrmination i mad of th initial valu of latic dformation, in th form of ignal U z, b man of rgulator Th tru valu of latic dformation i maurd b nor 8, and th rult of th maurmnt, in th form of voltag U z, i ummd up algbraicall on th comparing lmnt with th rgulatd voltag U z At th am tim, nor 9 maur th chang in th machining forc p, caud b chang in machining condition (matrial hardn of mi-finihd product, allowanc amount, width of machind lar, initial rror) Th chang in th valu of machining forc p in th tchnological tm cau, in turn, dviation of th latic dformation from th adoptd valu Signal from nor 9, of machining forc, i upplid to th input of nod, of th poitiv fdback, from which ignal U f( z ) ar takn off from th output of th voltag plittr, on ritor R and R, and U f( T ) from th output of th diffrntial chain z and R3 Th tting of paramtr z and T z and th corrponding ignal U and U for a pcific tchnological tm i ralizd b man of th adutabl ritor R and R3 Signal U f( z ) and U f( Tz ) ar upplid to th input of th comparing lmnt, whr th ar algbraicall ummd up with U z and U z Th rror ignal U Uz + U+ U Uz, via th corrction lmnt 3, i upplid to th input of amplifir 4, and thn, through th powr tranducr, to th longitudinal fd motor 5 hang in th rotation pd of motor 6 cau a chang in th valu of th longitudinal fd that i a control ffct for th tchnological tm 7, and thu a corrction i mad to th rlativ poitioning of th machind part and th cutting dg, taking into account th chang in latic dformation and th machining forc p 3 ontrol of paramtr of latic-dformabl tat of low-rigidit haft in turning Th abov conidration concrning th cration of a tm of adaptiv control of accurac paramtr in machining with control ffct in th form of longitudinal fd can alo b gnralid onto th dnamic tm of profiling of latic-dformabl low-rigidit haft Rfrnc [8, 9] prnt a mathmatical dcription of tm in turning and grinding of latic-dformabl low-rigidit haft, th dnamic proprti of linarizd modl bing approximatd b man of tranfr function of tpical dnamic lmnt Th rultant modl of DS and of th paramtr of th obct of control prmit ubtantiatd ralization of th arch for optimum algorithm of control, lction of control tm tructur, and nthi of corrctiv dvic ig prnt a gnralid tructural chmatic of th obct of control for th ca of controlling th latic dformabl tat of low-rigidit part through th application of tnil forc Th tranfr function of th dnamic tm, taking into account th aumption and rult of thortical and xprimntal tudi, can b prntd a: 4 k k m t ( ) t + k + bz znz+ xnx+ n+ ( + x + bz z + ) ( k x )( bz z + ) k m nx nz n nx nz m ig 3 prnt a tructural chmatic of a tm of adaptiv control of latic dformation of an latic-dformabl part in DS with an additional fdback () rlativ to th machining forc p, that z raliz th condition (7) of invariabilit in ordr to liminat th tatic rror rlatd to th control ffct () with poitiv fdback and to incra tabilit, pd of opration and innitivit to a chang in matrial allowanc with ngativ fdback In ig 3 th xprion for 5 () can b prntd a: 5 t ( ) ( x k) k t + k m( ) m nx mn n () Taking into account rlation () and (), th tranfr function of th corrctd control tm can b writtn a: (3) Φk t bz znz xnx n n w k m t bz znz xnx n ( ) w z n ( + xnx + bz znz + n + n ) w z + ( x z )( ) w z x k ( nz + m m )] w z, k ( ) ] + + + + + + + + + + + k m + xn + bz zn + n + + n + xn + ( xnx ) bz z ( xnx ) bz znz m m w z + + + and th tatic rror of th tm rlativ to th control ffct i dtrmind from th rlation: + bz znz + xnx + n n w z (4) + bz znx + n + n + w + n z A follow from rlation (4), th tatic rror ( ) N, introducd into th tm of control of paramtr of th latic-dformabl tat of part in DS b th controlling ffct, can b liminatd if th tructur and th paramtr of tranfr function of poitiv fdback z () i lctd a follow: z bz zn xn n n w + z+ + (5) If w lav out th ffct of componnt f of machining forc on th incra of latic dformation along coordinat, th tructural chmatic of th obct of control dnamic tm of latic-dformabl low-rigidit haft can b tranformd to th form prntd in ig 4a, and th tranfr function of th obct i thn dfind b th rlation: t + m k( ) 6 t (6) + bz znz+ n+ m k( + bz znz+ n) + ( bz zmz+ m)( k nx x) Th tructural chmatic of A with introducd fdback rlativ to th forc of machining i prntd in ig 4b, whr th tranfr function 5 () i dfind b rlation () () 8 Ek p l o a t a c a i Ni z a w o d n o c Ma i n t n a n c a n d Rliabilit Vo l4, No 3,
ig Structural chmatic of dnamic tm in turning of latic-dformabl low-rigidit haft ig4 Structural chmatic of control obct without th incluion of th ffct of componnt p of machining forc on incra of latic dformation along coordinat a tructural chmatic of adaptiv control tm with fdback rlativ to machining forc p - b ig3 Structural chmatic of adaptiv control tm with additional poitiv fdback rlativ to th machining forc Th tatic rror of th control tm, introducd b a controlling ffct, i dfind b th rlation: + bz znz + n n w z bz znz n w n z + + + + To liminat th tatic rror (7) introducd into th tm b th controlling ffct, th tructur and paramtr of tranfr function of poitiv fdback z z hould b dfind a follow: nz n + bz z + n w (8) In th ca whn w lav out th ffct of latic dformation along coordinat z and x on th chang in machining dpth (along coordinat ), th tructural chmatic for a pcific modl of th tchnological tm of turning of latic-dformabl haft can b prntd a in ig 5a, whr th tranfr function 5 i dfind b th rlation () ig 5b prnt th tructural chmatic of cor- ig 5 Structural chmatic of pcific modl of tchnological tm of turning of latic-dformabl low-rigidit haft a tructural chmatic of adaptiv control tm b Ek p l o a t a c a i Ni z a w o d n o c Ma i n t n a n c a n d Rliabilit Vo l4, No 3, 9
rctd tm of adaptiv control of th paramtr of th latic-dformabl tat of a low-rigidit haft, that prmit th limination of th tatic rror with th introduction of an additional poitiv fdback Th tranfr function for th corrctd tm for th pcific modl i dfind b th rlation: t + n+ n w z + m Φk t + n+ x w z + n + m (9) k + ( + ) w z n k xnx + k k ( + ) ( ) w w z + + k n xnx + k Th rror of th control tm introducd b a controlling ffct i dfind from th rlation: + n n w z n w n z + + + () In th ca of dtrmining th tructur and paramtr of fdback rlativ to th forc of machining in accordanc with th rlation: + n z, () n w( ) th tatic rror valu of zro rlativ to th controlling ffct aum th ig 6 Structural chmatic of tm of adaptiv control of paramtr of latic-dformabl tat in longitudinal grinding 4 ontrol of paramtr of latic-dformabl tat of low-rigidit haft in grinding Th gnralid modl of dnamic tm of turning of laticdformabl part, taking into account th pcific charactr of profiling th har ction in grinding of latic-dformabl low-rigidit haft with longitudinal fd, wa ud to driv th tructur of th obct of control and th mathmatical dcription of th tm [9] Th tranfr function of DS in longitudinal grinding i dfind b th rlation: 7 t + mx x( ) ( t ) + xnx+ n+ mx x x mn x mn x () ig 6 prnt th tructural chmatic of a tm of adaptiv control of th paramtr of latic-dformabl tat of low rigidit haft in TS in longitudinal grinding, whr: 8 t ( )( x x) t + mx x( ) n mn mn + x x (3) Th tranfr function of th corrctd control tm, taking into account rlation () and (3), aum th form of: Φ k t + xnx + n + n 3 x w z + t + xnx + n+ 3 x w z + n + x x ( w z3 m )( n ) xnm x ( ) ( 3 ) + + + m n nm x x w w z x x (4) If th tructur and paramtr of th tranfr function of poitiv fdback z3 ar dtrmind from th rlation: z3 n n + x x+, (5) n w thn, a follow from rlation (), th tatic rror introducd in th adaptiv control tm in longitudinal grinding b th controlling ffct, c, can b liminatd 5 oncluion Anali of functioning of tm of automatic control of latic dformation in th tchnological tm how that undr dnamic condition th ar tatic rlativ to th ffct of both controlling and intrfrnc ffct Thi lad to rror of th rlativ poitioning of th machind part and th cutting dg Optimum tructur of control tm, including th dvlopd tm of adaptiv control, prmit improvmnt of th qualit (accurac) of machining Th author hav dvlopd a mthod for th corrction of th tting of th tchnological tm through th application of adaptiv control of longitudinal fd and additional forc ffct, inducing th latic-dformabl tat through th introduction into th control tm of an additional poitiv fdback rlativ to th machining forc, prmitting th limination of tatic rror of controlling and intrfrnc ffct in th control of paramtr of accurac and qualit Th condition wr dvlopd for th dtrmination of th tructur of paramtr of th additional poitiv fdback rlativ to th machining forc, that guarant th limination of tatic rror and impart to th tructur adaptiv proprti Rfrnc Hala W, Tarannko V, Swic A, Tarannko Invtigation of influnc of grinding rgim on urfac tnion tat Brlin, Hidlbrg: Springr Vrlag Lctur Not In Artificial Intllignc, 8 57: 749 756 Marchlk : Dnamika obrabiark, WNT: Warzawa, 99 3 uan Badania i analiza powtarzalności rozkładu odchłk gomtrcznch w proci obróbki krawanim Ekploataca i Nizawodnoc Maintnanc and Rliabilit 8 3(39): 45 5 4 Ratchv S, Liu S, Huang W, Bckr A A A flxibl forc modl for nd milling of low-rigidit part Journal of Matrial Procing Tchnolog Procding of th Intrnational onfrnc in Advanc in Matrial and Procing Tchnologi 4 53-54: 34-38 5 Ratchv S, Liu S, Huang W, Bckr A A Milling rror prdiction and compnation in machining of low-rigidit part Intrnational Journal of Machin Tool and Manufactur 4 5(44): 69-64 Ek p l o a t a c a i Ni z a w o d n o c Ma i n t n a n c a n d Rliabilit Vo l4, No 3,
6 Ratchv S, ovndr E, Nikov S Toward dflction prdiction and compnation in machining of low-rigidit part Procding of th Intitution of Mchanical Enginr - Part B (6): 9-34 7 Świć A, Tarannko W, Szablki J Modlling dnamic tm of low-rigid haft grinding Ekploataca i Nizawodnoc Maintnanc and Rliabilit, (5): 3 4 8 Tarannko W, Świć A Urządznia truąc dokładnością obróbki mazn o mał ztwności Lublin: Politchnika Lublka, 6 9 Tarannko W, Tarannko, Szablki J, Świć A Idntfikaca układu dnamiczngo zlifowania wałów o mał ztwności Modlowani Inżnirki 8, 4(35): 5 3 Tarannko, Tarannko W, Świć A, Szablki J Modlling of dnamic tm of low-rigidit haft machining Ekploataca i Nizawodnoc Maintnanc and Rliabilit, 4 (48): 4 5 Dr hab inż Antoni Świć, prof nadzw Intitut of Tchnological Information Stm Lubliln Univrit of Tchnolog ul Nadbtrzcka 36, -68 Lublin, Poland -mail: awic@pollubpl Prof dr hab inż Wiktor Tarannko Intitut of Tchnological Information Stm Politchnika Lublka ul Nadbtrzcka 36, -68 Lublin, Poland -mail: wtaran_ark@mailru Ek p l o a t a c a i Ni z a w o d n o c Ma i n t n a n c a n d Rliabilit Vo l4, No 3,