Analysis of kinematic motion deviations of machining centers based on geometric tolerances

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1 Bulltin of th JSME Journl of vn Mhnil Dsign, Sstms, n Mnufturing Vol.8, No.4, 4 nlsis of kinmti motion vitions of mhining ntrs bs on gomtri tolrns tsushi TKSI*, rt YOSID*, Wiroj TSN* Nobuhiro SUGIMUR*, Koji IWMUR* n Yoshitk TNIMIZU* * Grut Shool of Enginring, Osk Prftur Univrsit - Gkun-ho, Nk-ku, Ski, Osk , Jpn E-mil: sugimur@m.oskfu-u..jp Riv 4 Mrh 4 bstrt Th objtivs of th prsnt rsrh r to invstigt rltionships btwn kinmti motion vitions of mhin tools n gomtri tolrns of thir omponnts, to propos mthmtil mols rprsnting kinmti motion vitions of fiv-is mhining ntrs, n to ppl th mols to nlsis of th kinmti motion vitions of fiv-is mhining ntrs. st of mols is propos to rprsnt kinmti motion vitions of both th linr tbls n th rotr tbls, bs on th gomtri tolrns of th gui-ws onnting th omponnts. B ombing th mols, thr mols of th fiv-is mhining ntrs r vlop n ppli to th nlsis of th stnr vitions of th shp gnrtion motions of th tools n th workpis. Th propos mols provis us with sstmti mtho to nl n to stimt th kinmti motion vitions of th fiv-is mhining ntrs, bs on th gomtri tolrns of th gui-ws onnting th omponnts. K wors : Fiv-is mhining ntrs, Shp gnrtion motions, Gomtri tolrns, Kinmti motion vitions. Introution Mhining ur is on of th most importnt hrtristis of th CNC mhin tools for gnrting th prouts with th high ur n th omplit gomtris. Som rsrhs hv bn rri out to nl th mhining ur of th mhin tools bs on th shp gnrtion motions btwn th tools n th workpis Sugimur, t l., 98; Rshtov n Portmn, 988; Skmoto n Inski, 994; Sugimur, t. l., 998. owvr, th kinmti motion vitions of th mhin tools r pl influn b th gomtri tolrns of th omponnts, suh s gui-ws n brings. Thrfor, it is rquir to lrif th rltionships btwn th kinmti motion vitions of th mhin tools n th gomtri tolrns of th omponnts, from th viwpoints of th sign n th mnufturing of th mhin tools n thir omponnts. Mn rsrhs hv lso bn rri out to l with th imnsionl tolrns n th gomtri tolrns, im t rliing sstmti nlsis n sign mthoologis for th thr imnsionl mhin prouts Ro, t l., 99; Volkr, 99; Nigm n Turnr, 995; Ngoi, t l.,. Th rltionships btwn th motion vitions n th gomtri tolrns of th omponnts hv not t bn lrifi. Th objtivs of th prsnt rsrh r to stblish mthmtil mols rprsnting th kinmti motion vitions of th mhin tools, on th bsis of th gomtri tolrns of th omponnts, n to ppl th mols to thortil nlsis of th kinmti motion vitions of th mhin tools. In th prvious pprs Stonk, t l., 7; Stonk, t l., 8, mol hs bn propos to rprsnt th kinmti motion vitions of th linr tbls, n th propos mol hs bn ppli to th nlsis of th stnr vitions of th shp gnrtion motions of th thr-is mhining ntrs. Th mols of th tow-is rotr tbls hv lso bn propos, n Ppr No.4-4 [DOI:.99/jmsm.4jmsm6 ] 4 Th Jpn Soit of Mhnil Enginrs

2 Tkhshi, Yoshi, Thsn, Sugimur, Iwmur n Tnimiu, Journl of vn Mhnil Dsign, Sstms, n Mnuft, Vol.8, No.4 4 ppli to th stimtion of th stnr vitions of th rotr motions Sugimur, t l.,. Th issus isuss in th ppr r summri in th followings. Molling of kinmti motion vitions for rotr tbls n linr tbls, Molling of kinmti motion vitions for thr tps of fiv-is mhining ntrs, n pplitions of propos mols to nlsis of kinmti motion vitions of fiv-is mhining ntrs. Th objtivs of th prsnt ppr is to stblish mthmtil mols rprsnting th kinmti motion vitions of th 5-is mhining ntrs, on th bsis of th gomtri tolrns of th omponnts, n to ppl th mols to th stnr vition nlsis of th kinmti motions of th 5-is mhining ntrs. Th kinmti motion vitions r stimt bs on th mols in th vrious onitions of th gomtri tolrns of th omponnts n th vrious tbl positions, to invstigt th influn of th gomtri tolrns of th omponnts on th kinmti motion vitions of th 5-is mhining ntrs. Min issus in th ppr r summri s follows. Moling of gomtri vitions of th 5-is mhining ntrs. Invstigtion of influns of gomtri tolrns of gui-ws to tbl motions. Invstigtion of influns of tbl motions to kinmti motions of mhin tools.. Gomtri Tolrns n Dvitions of Fturs Stonk, t l., 7 Th gomtri tolrns of th fturs spif th llowbl rs nm tolrn ons, whih onstrin th position n orinttion vitions of th ssoit fturs ginst th nominl fturs, s shown in Figur. Th ssoit fturs n th nominl fturs mn th fturs of th mnuftur prouts n th il fturs fin in th sign phs, rsptivl. Th gomtri vitions of th ssoit fturs from th nominl fturs r rprsnt b sts of prmtrs nm vition prmtrs. For mpl, on position prmtr w n two rottionl prmtrs α n β r rquir to rprsnt th gomtri vitions of th ssoit pln fturs ginst th nominl pln fturs, for th s whr th tolrn on is givn b th r btwn pir of prlll plns. In th rsrh, th followings r ssum for th s of th moling n th nlsis of th gomtri vitions. Th vition prmtrs δ i rprsnting th position n orinttion vitions of th ssoit fturs follow th norml istribution Nμ i,σ i, n μ i =. Whr, μ i n σ i r th mn vlus n th stnr vitions, rsptivl. Th mnufturing prosss of th omponnts r wll ontroll, n th proportion of th non-onforming omponnts is s smll s vlu P ll prnt ftiv. r, th non-onforming omponnts mn th omponnts, in whih th tolrn fturs th tolrn ons. Eqution rprsnts th rltionships btwn th stnr vitions σ i of th vition prmtrs of th tolrn fturs n th sis of th tolrn ons. whr, i : Inis of vition prmtrs of th tolrn ftur. : Tolrn limits of th vition prmtrs i, if th othr vition prmtrs j =, i j. Tolrn on t w Nominl Ftur L L ssoit Ftur Fig. Dfinition of gomtri tolrn of pln fs [DOI:.99/jmsm.4jmsm6 ] 4 Th Jpn Soit of Mhnil Enginrs

3 Tkhshi, Yoshi, Thsn, Sugimur, Iwmur n Tnimiu, Journl of vn Mhnil Dsign, Sstms, n Mnuft, Vol.8, No.4 4 : onstnt rprsnting th rtio of th mimum vlus im n th stnr vitions σ i. It is ssum tht hs sm vlu for ll th vition prmtrs of on tolrn ftur. Lt us onsir s shown in Figur, s n mpl. Th tolrn limit im r givn s follows, unr th onitons tht t is vr smllr thn L n L. δ = w, δ =α, δ =β δ m = t/, δ m = t/l, δ m = t/l whr, L, L : Lngth n with of th pln ftur. t : Tolrn vlus,.g. th istn btwn two plns rprsnting th tolrn ons. Th following inqulit givs th onitions tht th pln fturs r inlu within th tolrn on btwn pir of plns. -t/ < + L /+ L / < t/ Th probbilit tht th tolrn fturs r inlu within th tolrn ons is givn b th following qution. p 4 whr,,, If th prnt ftiv P is lss thn.7%, th onstnt n b stimt s = 5.8 for th ss of th pln fturs. s rgrs th linr fturs, =5.6. Th vlus for th pln n th linr fturs r ppli in th following nlsis, in orr to stimt th stnr vitions of th vition prmtrs of th gui-ws, bs on th tolrn vlus.. Kinmti Motions n Dvitions of Rotr Tbls n Linr Tbls Figur shows tpil mpl of rotr tbls with vrtil is utili for th mhin tools. bs supports n guis rotr tbl b pln f n linril f ll gui-ws, whih r init b n b. In th s, th rltiv position n orinttion of th tbl ginst th bs r givn b th following qution, if th gui-ws oini with th nominl fturs. i = - i+ = b b - i+ 5 whr, i : Inis rprsnting th onnting squns mong th moving omponnts of th mhin tools. i : Position vtor of point P in th bs oorint sstm. i+ : Position vtor of P in th tbl oorint sstm. j j =, b : 4 4 homognous trnsformtion mtris rprsnting th positions n orinttions of th gui-ws j in th bs oorint sstm. j j =, b : 4 4 homognous trnsformtion mtris rprsnting th positions n orinttions of th gui-ws j in th tbl oorint sstm. owvr, th rltiv positions of th tbl ginst th bs r influn b th position n orinttion vitions of th gui-ws, if th gui-ws o not oini with th nominl fturs. Th positions of th tbl ginst th bs r rprsnt b th following qution orrsponing to th iniviul gui-ws. i = j D j D j - j - i+ j =, b 6 [DOI:.99/jmsm.4jmsm6 ] 4 Th Jpn Soit of Mhnil Enginrs

4 Tkhshi, Yoshi, Thsn, Sugimur, Iwmur n Tnimiu, Journl of vn Mhnil Dsign, Sstms, n Mnuft, Vol.8, No.4 4 R Tbl Bs b b l Fig. Vrtil rotr tbls C-is tbl n thir gui-ws whr, D j, D j : 4 4 homognous trnsformtion mtris rprsnting th position n orinttion vitions of th pir of fturs onstituting th gui-ws j. Eqution 6 mns tht vrious vlus of th rltiv position of th tbl ginst th bs r obtin for th iniviul gui-ws. sstmti mtho ws propos b onsiring th priorit mong th gui-ws to trmin th rltiv positions of th tbls ginst th bs. Th followings summri th mols of th rotr tbls n th linr tbls Stonk, t l., 8; Sugimur, t l.,. Vrtil Rotr Tbls Th kinmti motion vitions of th vrtil rotr tbls shown in Figur r formult in Eq. 7, unr th ssumption tht th priorit mong th gui-ws is > b. Whr, > b mns tht th highr priorit is givn to th gui-w ginst b to trmin th positions n orinttions of th tbls to th bss onsiring th gomtri vitions of th gui-ws. 6 os V os V V V V V V 7 V b b os b V b b os b V b os b b V V l b os V l b V b os b l b l b os b l b l b b os b b os b whr, j j=,,, 6 : Motions of linr tbls in -,, n -irtions n ons of rotr tbls roun -,, n -s. Th tils will b shown in Eq.. θ: Rottion ngl of tbl. ij, ij, ij, : Orinttion vitions th j-th gomtri ftur of gui-w i roun, n is of th rotr tbls. ij, ij, ij : Position vitions th j-th gomtri ftur of gui-w i long, n is of th rotr tbls. oriontl Rotr tbls For th ss of th horiontl rotr tbls shown in Figur, th kinmti motion vitions r formult in Eq. 8, unr th ssumption tht th priorit mong th gui-ws is = > = f. Whr, = mns tht thr is not n priorit btwn th gui-ws n to trmin th positions n orinttions of th tbls to b b [DOI:.99/jmsm.4jmsm6 ] 4 Th Jpn Soit of Mhnil Enginrs 4

5 5 Tkhshi, Yoshi, Thsn, Sugimur, Iwmur n Tnimiu, Journl of vn Mhnil Dsign, Sstms, n Mnuft, Vol.8, No Th Jpn Soit of Mhnil Enginrs [DOI:.99/jmsm.4jmsm6 ] th bss onsiring th gomtri vitions of th gui-ws. os os 4 8 os os os os f f } { } { os os os os l l whr, : Rottionl ngl of th tbl. Linr Tbls Th kinmti motion vitions of -is linr tbls shown in Figur 4 r formult in th following qutions, unr th ssumption tht th priorit mong th gui-ws is = > b =. l l i i w w b i i b u ub b 9 whr, : -irtion trvl. pq : Diffrns of th position n orinttion vitions of th fturs in bs si n tbl si onsisting of th q-th gui-ws of in th p-th irtions. p = u, v, w: Componnts of th position n orinttion vitions. Fig. oriontl Rotr tbls -is tbl Fig. 4 Linr motion tbls l f b Bs Tbl l l l b l l l b

6 6 Tkhshi, Yoshi, Thsn, Sugimur, Iwmur n Tnimiu, Journl of vn Mhnil Dsign, Sstms, n Mnuft, Vol.8, No Th Jpn Soit of Mhnil Enginrs [DOI:.99/jmsm.4jmsm6 ] q =, b,, : Gui-ws. i : Position n orinttion vitions of th fturs in bs si of th q-th gui-ws of in th p-th irtions. 4. nlsis of Kinmti Motion Dvitions of Fiv-is Mhining Cntrs 4. Moling of Kinmti Motions Th fiv-is mhining ntrs r ompos of two rotr tbls n thr linr tbls shown in Figurs, n 4. Thr tps of fiv is mhining ntrs shown in Figur 5 r onsir hr for th nlsis. Th shp gnrtion motions of th mhining ntrs r summri in Eq. Tp : t w φ r r Tp : t w φ r r Tp : t w φ r r whr, : -is linr motion : -is linr motion : -is linr motion 4 φ : -is rotr motion 6 : C-is rotr motion i : Trnsltion btwn th tbls i : Distns in -irtion btwn pirs of onstituting tbls. Th r summri in Tbl. 4. nlsis of Kinmti Motion Dvitions of Fiv-is Mhining Cntrs Th imnsions n th stroks shown in Tbl r givn to -, - n -tbls for thr tps of mhining ntrs. Th imnsions r sm for ll tps of mhining ntrs. Tbl summris th imnsions of th rotr tbls, whih r sign for th iniviul tps of mhining ntrs. Th kinmti motion vitions of th tools ginst th workpis r nl b ppling Eq., oring to th following prour. Tp :Workpi rottion tp b Tp :Tool & workpi rottion tp Tp :Tool rottion tp Fig. 5 Fiv-is mhining ntrs

7 Tkhshi, Yoshi, Thsn, Sugimur, Iwmur n Tnimiu, Journl of vn Mhnil Dsign, Sstms, n Mnuft, Vol.8, No.4 4 STEP : St th tbl positions Th tbl positions r st s shown in Tbl 4 to nl th kinmti motion vitions for th iniviul positions. STEP : St th initil tolrn vlus for ll th gui-ws ll th tolrn vlus r st tht t = 5 m for ll th gui-ws shown in Figurs, n 4. Ths vlus r ssum s th initil vlus. Th stnr vitions of th position n orinttion vitions of th tools ginst th workpis r stimt b ppling Eq., bs on th formuls givn b Equtions 7, 8 n 9 rprsnting th kinmti motions n thir stnr vitions of th rotr n th linr tbls. Th stnr vitions of ths tbl motions r lso stimt, bs on th stnr vitions of th position n orinttion vitions of th iniviul gui-ws givn in Eq.. STEP :St th smll tolrn vlu for h gui-w tolrn vlus of on of th gui-ws r st tht t = 5 m, whih is / of th initil vlus, in orr to lrif th ffts of th improvmnt of th tolrn vlus of iniviul gui-ws. Th stnr vitions of th position n orinttion vitions of th tools ginst th workpis r stimt in sm mnnr shown in, n th obtin stnr vitions r ompr b th following qution to lrif th ffts of th iniviul gui-ws. St5 St5 ir St 5 whr, St 5 : Stnr vitions obtin bs on initil tolrn vlus. St 5 : Stnr vitions obtin bs on improv tolrn vlus. Tbl Distns btwn pir of onnting tbls Sis mm Tp Tp Tp Tbl Sis n stroks of linr tbls for tps, n. Sis mm -tbl -tbl -tbl l 95 4 l l b l 95 4 l l b lngth mm 5 5 strok mm [DOI:.99/jmsm.4jmsm6 ] 4 Th Jpn Soit of Mhnil Enginrs 7

8 Tkhshi, Yoshi, Thsn, Sugimur, Iwmur n Tnimiu, Journl of vn Mhnil Dsign, Sstms, n Mnuft, Vol.8, No.4 4 Tbl Sis of rotr tbls for tps, n. Tbls Sis mm Tps Tp Tp Tp C-is tbl is tbl 5 4 Tbl 4 Tbl positions to b st in nlsis Positions -is mm -is mm -is mm -is φ gr C-is θ gr Positoin Position Position- Th ir vlus obtin hr show th improvmnt rtio of th tolrn vlus of th iniviul gui-ws. Whn th ir vlu of on gui-w is high, th improvmnt of th tolrn vlus of th gui-w is vr fftiv to improv th whol kinmti vitions of th fiv-is mhining ntrs. Figurs 6, 7, n 8 summri th nlsis rsults of th ir vlus for ll th tbl positions n ll th tps of th mhining ntrs. It is foun tht th improvmnt rtio of th -is n C-is rotr tbls r highr thn ons of th linr tbls for ll ss of th nlsis onitions. In prtiulr, th gui-ws n of -is n th gui-w of C-is r lrgr ffts thn th othr gui-ws, ths gui-ws hv highr priorit to trmin th kinmti motion vitions for th rotr tbls, s shown in Eqs. 7 n 8. s rgrs th tps of mhining ntrs, th tp hs iffrnt hrtristis from th othrs. Th improvmnt rtio of th -is n -is linr tbls r rltivl high ginst th othr tps of mhining ntrs. Th tp mhining ntrs hv no rottionl s in th workpi si, n th ffts of th rottionl motion vitions in tool si m b ru, th hv smllr rm lngths btwn th rottionl ntr of th tool si n th utting points of th tools. It is unrstoo tht th propos mol is fftiv to nl th ffts of th tolrn vlus of th gui-ws to th kinmti motion vitions of th tools ginst th workpis in th fiv-is mhining ntrs. 5. Conlusions mthmtil mol hs bn propos to rprsnt th kinmti motion vitions of th fiv-is mhining ntrs. Following rmrks r onlu. st to mols r propos to stimt th kinmti motion vitions of th rotr tbls n th linr tbls, bs on th tolrn vlus of th gui-ws of th tbls. Th mols giv th mthmtil rltionships btwn th kinmti vitions of tbls n th gomtri tolrns of th gui-ws. [DOI:.99/jmsm.4jmsm6 ] 4 Th Jpn Soit of Mhnil Enginrs 8

9 Tkhshi, Yoshi, Thsn, Sugimur, Iwmur n Tnimiu, Journl of vn Mhnil Dsign, Sstms, n Mnuft, Vol.8, No.4 4 Position- b Position- Position- Fig. 6 nlsis rsults of tp mhining ntr [DOI:.99/jmsm.4jmsm6 ] 4 Th Jpn Soit of Mhnil Enginrs 9

10 Tkhshi, Yoshi, Thsn, Sugimur, Iwmur n Tnimiu, Journl of vn Mhnil Dsign, Sstms, n Mnuft, Vol.8, No.4 4 Position- b Position- Position- Fig. 7 nlsis rsults of tp mhining ntr [DOI:.99/jmsm.4jmsm6 ] 4 Th Jpn Soit of Mhnil Enginrs

11 Tkhshi, Yoshi, Thsn, Sugimur, Iwmur n Tnimiu, Journl of vn Mhnil Dsign, Sstms, n Mnuft, Vol.8, No.4 4 Position- b Position- Position- Fig. 8 nlsis rsults of tp mhining ntr [DOI:.99/jmsm.4jmsm6 ] 4 Th Jpn Soit of Mhnil Enginrs

12 Tkhshi, Yoshi, Thsn, Sugimur, Iwmur n Tnimiu, Journl of vn Mhnil Dsign, Sstms, n Mnuft, Vol.8, No.4 4 st of mols r propos to stimt th kinmti motion vitions for thr tps of th fiv-is mhining ntrs b ombining th mols of th rotr tbls n th linr tbls. Th mols propos hr provi us with sstmti mthos to stimt th stnr vitions of th position n orinttion vitions of th tools ginst th workpis, bs on tolrn vlus of th gui-ws Th kinmti motion vitions of th fiv-is mhining ntrs r stimt b ppling th propos mols. It is unrstoo tht th kinmti motion vitions r fft b th gomtri tolrns of th gui-ws n tht th gomtri tolrns of th gui-ws of rotr tbls hv highr ffts on th kinmti motion vitions thn th ons of linr tbls. Rfrns Ngoi, B.K.., Lim, B.. n ng, P.S., Nus mtho for stk nlsis of gomtri imnsioning n tolrning GDT problms, Intrntionl Journl of Proution Rsrh, Vol. 8, No., pp.-7. Nigm, S.D. n Turnr, J.U., Rviw of sttistil pprohs to tolrn nlsis, Computr-i Dsign, Vol. 7, No. 995, pp.6-5. Rshtov, D.N. n Portmn, V.T., 988, ur of mhin tools, PME Prss. Ro, R., Liu, C.R. n Woo, T.C., Rviw of imnsioning n tolrning: rprsnttion n prosg, Computr-i Dsign, Vol., No. 99, pp Skmoto, S. n Inski, I., Error nlsis of prision mhin tools, Proing of th r Intrntionl Confrn of Ultrprision in Mnufturing Enginring994, pp Stonk, N., Sugimur, N., Tnimiu, Y. n Iwmur, K., nlsis of kinmti motion vitions of mhining ntrs bs on gomtri tolrns, Proings of th 4th LEM-7, pp Stonk, N., Sugimur, N., Tnimiu, Y. n Iwmur, K., stu on moling n nlsis of kinmti motion vitions of mhin tools n Rport, Moling n nlsis of linr tbls n mhin tools, Trnstions of th Jpn Soit of Mhnil Enginrs, Sris C, Vol. 74, No. 778, pp.98-5in Jpns. Sugimur, N., Iwt, K., n Ob, F., Formtion of shp gnrtion prosss of mhin tools, Proings of IFC 8, Vol. XIV98, pp Sugimur, N. n Mitni, S., nltil pproh for sign of lignmnt rrors of mhin tools, Pro. of 998 Pifi Confrn on Mnufturing998, pp Sugimur, N., Wtbiki,., Thsn, W., Iwmur, K. n Tnimiu, Y., nlsis of kinmti motion vitions of rotr tbls bs on gomtri tolrns. Journl of vn Mhnil Dsign, Sstms n Mnufturing, Vol. 6, No. 7, pp.-4. Volkr,.B., urrnt prsptiv on tolrning n mtrolog, CRTD-Vol.7, Intrntionl Forum on Dimnsionl Tolrning n Mtrolog99, pp [DOI:.99/jmsm.4jmsm6 ] 4 Th Jpn Soit of Mhnil Enginrs