THE VEINING STRUCTURE METHOD, THE FINITE ELEMENT METHOD IN THERMAL DEFORMATION DETERMINATION FOR THE MAIN SPINDLE AT NUMERICAL CONTROL LATHES

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1 THE VEINING STRUCTURE METHOD, THE FINITE ELEMENT METHOD IN THERMAL DEFORMATION DETERMINATION FOR THE MAIN SPINDLE AT NUMERICAL CONTROL LATHES by Adran BUT and Ioan LUMINOSU Orgnal scentfc paper UDC: : BIBLID: , 9 (2005), 1, In the de ter m na ton of the process accuracy on computer numercal control (CNC) lathe [1] t s very m por tant to know the ther mal de for ma tons. Ths ar t cle pres ents a new so lu ton to ob tan the ther mal de for ma tons at the prn c pal ar bor (the most m por tant part of a prn c pal ac tu a ton for a CNC lathe) by us ng the el e ment f nte method. The frst part of the the o ret - cal as pect pres ents the steps of ths method used es pe cally for the prn c pal ar bor. The ex per men tal part pres ents how ths method was used to mea sure and pro cess the sg nals. The ex per men tal tests were ap pled n two df fer ent st u a tons: (a) when ma te ral was catch only n prn c pal ar bor, and (b) when ma te ral was catch n prn c pal ar bor and chuck. The NASTRAN MSC pro gram was used, whch has the pos s bl ty to de ter mne the tem per a ture val ues n the whole mass of the prn c pal ar bor and later the de for ma ton tem per a ture of ev ery pont of the en sem ble was ob taned. In or der to an a lyze the ther mal stran n flu ence on the man spn dle, the f - nte el e ment method, the ven ng struc ture of a body was used. Ths s a method through whch the mother (com plex) struc ture was splt n com - pos ng parts con sd ered as n de pend ent (sub struc tures). The lnk be tween df fer ent sub struc tures n whch the com plex struc ture was splt, s made on the com mon con tour of sub struc tures. The sub struc ture be comes ths way an n de pend ent struc ture wth edge con d ton, con strant by as sem bly con t - nu ty of n tal struc ture. Key words: fnte element, thermal deformaton, man spndle Introducton The ven ng struc ture s prob lem pre sented n ths ar t cle, us ng the f nte el e - ment method, spec fes that f elm nate the move ments as a rgd body, the sub struc ture s be hav or s unque de ter mned by the out lne s move ments, rep re sent ng the bor der be - tween sub struc tures, where we can fnd the so-called com mon node [2]. Us ng the ds - place ment method, the sub struc tures be hav or s de scrbed by r gd ty ma trx re duced to the bor der s nodes and as re sult the sub struc ture can be con sd ered as a com plex element. 99

2 THERMAL SCIENCE: Vol. 9 (2005), No. 1, pp Methodology The r gd ty ma trx as sem bly s made two-stage: (1) the r gd ty ma tr ces of sub struc ture el e ments are as sem bled, and re duced to com mon nodes, and (2) af ter that the struc ture as sem bly n struc ture r gd ty ma trx s made [3]. Ex cept ng the sub struc ture s bor der nodes, all other nodes can be de ac t vated be cause for those can be wrt ten f nal equ lb rum equa tons, thus the nodes on the sub - struc ture edge are be com ng n for ma ton car r ers to wards the rest of the struc ture, and the re spec tve un knowns re mans ac tve. Those re man ng ac tve nodes are called mas ter. In con clu son re solv ng com plex struc tures by splt tng them n sub struc tures s made two-stages: (1) the de for ma ton state for ev ery sub struc ture, and (2) the con nec ton of sub struc tures so that n the n ter sec ton zones the equ lb rum and con t nu ty of ds - place ment con d tons to be ob served. In the ven ng struc ture the nodes on the com mon edges of negh bor sub struc - tures are de fned as con tact nodes c and the oth ers are called n te ror nodes. Af ter the struc ture splt tng n sub struc tures, the con tact nodes wll be blocked us ng lm t ng de - vces n or der to pre vent the ds place ments we want to an a lyze. In ths way the sub struc - ture can be re solved by draw ng out the equa ton sys tem for the or der sub struc ture [4]: where, [ M ]{ r } { V } (1) [ M, ] structure rgdty matrx (wth blocked nteror nodes), { r } column vector for the dsplacements of nteror nodes, and { V } column vector for external loads appled n nteror nodes. By re solv ng ths sys tem re sults that the re ac tons vec tor n blocked nodes gen er - ated by re ac ton vec tors from all sub struc tures n ter faces wth other sub struc tures. The equ lb rum con d ton of n ter faces > k wll be: where, V k,,, e k e k k { R } { R } { V } 0 (2) L { } column vector for loads on nterface > k. L Af ter the sub struc tures con nec ton and con tact nodes un block ng the ds place - ment con t nu ty con d ton are wrt ten: where { } [ e][ T ] (3) r c { r c } column vector of all contact nodes dsplacements for all substructure nterfaces, wrtten n the local reference system (X, Y, Z), 100

3 But, A., Lumnosu, I.: The Venng Structure Method, the Fnte Element Method n... [ e ] [ T ] s obtaned from the same nodes dsplacements wrtten n the global reference system (X, Y, Z), and the transformaton matrx for rotary axs dsplacements. Ths way, n the con t nu ty equa ton s un known only the con tact nodes ds - place ment, and makes nec es sary the re wrt ng of sub struc tures r gd ty ma trx by elm - nat ng n te ror nodes ds place ments. The method con ssts n splt tng the ma trx by the de gree of free dom of n te ror con tact nodes: and [ M, ][ k, ] c [ M ] [ kc, ][ kc, c ] { r } [ r ] { rc } Af ter un block ng, n the con tact nodes are ap pear ng f nal re ac tons and ex ter nal loads ap pled d rectly n these nodes { F }; the equa ton sys tem can be wrt ten as: L (4) (5) [ M, ]{ } [ M, c ]{ c r r } { F } [ M c, ]{ r } [ M c, c ]{ r c } { Fc } { R } (6) From here re sults the ds place ment ma trx of n te ror nodes: 1,, { r } [ M ] { F } [ H ]{ r } c (7) where { } the column vector of nteror nodes dsplacements when the contact nodes are blocked: r 1,,, c [ H ] [ M ] [ M ] (8) It re sults:, c [ r ] { r } [ H ]{ r } (9) c Ob tan ng: ([ M c, c ] [ M ][ H ]){ r } { F } { R } [ M ][ M c,, c c c ] { F } c,, 1 (10) 101

4 THERMAL SCIENCE: Vol. 9 (2005), No. 1, pp For the de vel op ment of ths pro cess we have to use the co or d nates trans for ma - ton of ro tary axs. The al go rthm of ths method s pre sented be low. (1) dvson of the mother structure n substructures, (2) defnng the structure topology, defnng structure and substructure nodes, (3) defnng nterfaces and markng out the contact nodes, and (4) calculatng transfer matrx from the local axs system to global axs system [ r ] and [ k c ], where [ r ] selects from the dsplacements vectors the one who matches wth Boolean type substructures nodes whch reflects the topologcal propretes of the structure that s the rotaton mode of nodes n the structures, and substructures. Notng wth [ g ] the drectors cosne matrx of axes accordng to the global axs X, Y, Z: [ g ] g g g g g g g g g XX XY XZ YX YY YZ ZX ZY ZZ Or, for nodes wth tree degrees of freedom wth planar axs rotaton: where q s the rotaton angle n the XY plane. The transformng matrx for a node s: (11) cos q sn q 0 [ g ] sn q cos q 0 (12) [ ][ ] [ S ] g 0 [ ][ ] 0 g (13) From whch results the structure transfer matrx referrng to a number of n contact nodes: T c T [ S ] T 0 [ S ] T 0 0 [ S ] [ S ] T type (6n 6n) (14) 102

5 But, A., Lumnosu, I.: The Venng Structure Method, the Fnte Element Method n... (5) calculus for every substructure, referrng to local reference system of rgdty matrx: [ M ]; [ M ]; [ M ],, c c, c, (6) calculus for every substructure, referrng to global reference system of rgdty matrx, (7) defnng every substructure of reactons n blocked contact nodes, at every nterface, k R,,[ k ] and addtonal loads {F A }, (8) calculus of load matrx accordng to global reference system: A { F } [ T ]{ F } (15) XYZ (9) calculus of condensed matrx, (10) assembly of structure rgdty matrx and equvalent load vector [5]: and c A XYZ n cc c cc c T 1 [ M ] [ W ][ M ][ W ] n c c c 1 F [ W ][ M ] { F } (11) solvng the equaton system and determne nodes dsplacement n every msubstructure. c, A (16) (17) Thermal stran test on the man spndle Re gard ng the clas s cal pro ce dure whch les n ds place ment method (pre sented above) we wll pres ent a meth od ol ogy based on the fol low ng con sd er atons: and [ ] [ e ][ e] (18) r 0 [ M ]{ e} [ F ] (19) cc c Sup pos ng t can be made a dgtzaton of the whole struc ture lm ted by the max mum mem ory of a com puter, from whch we can ob tan a ds place ment feld n the sub struc tures splt tng zone, we ob tan a pro to type on whch we can make test mea sure - ments. Mea sure ments wll be fo cused on the tem per a ture mea sure ments n the de sred zones; re sults wth whch we wll cor rect the val ues ob taned us ng the f nte el e ment method. The man spn dle can be con sd ered as a spe cal case of space struc ture formed by plates so that ev ery plate can be con sd ered as a f nte el e ment. In or der to ob tan ds - place ments re sults were used a cal cu lus pro gram as ssted by f nte elements NASTRAN. 103

6 THERMAL SCIENCE: Vol. 9 (2005), No. 1, pp Such a pro gram con tans the fol low ng se quences: data preprocessng: b-d men sonal or three-d men sonal de sgn draw ng, smulaton (dgtzaton, materal choosng, border condtons, calculus analyze), draw ng m port/ex port [6] model f nte el e ment, and spe cal cal cu lus as: sg nals pro cess ng, fa tgue cal cu lus, dy namc and con tact anal y ss, tem per a tures, etc.; data processng: ef fec tve n ter ac tve cal cu lus or d rectly by post pro cess ng data, the eval u a ton of anal y ss re sults, the re sults pro cess ng (graph cal, tab u lar), and ds play ng re sults or prnt ng re sults. Next wll be pre sented the steps fol lowed n us ng NASTRAN pro gram n or der to ob tan the cal cu lus model wth the log cal scheme for solv ng the prob lem as spec fed n fg. 1 [2]. 104 Fgure 1. The logcal scheme for solvng the analyss wth element fnte method

7 But, A., Lumnosu, I.: The Venng Structure Method, the Fnte Element Method n... Data preprocessng (1) Start ng from the man spn dle shop draw ngs for the SP 630 NC lathe de - sgned n AUTOCAD, af ter t was made data m port from AUTOCAD nto the NASTRAN de sgn pro gram. It was cre ated the source pro ect wth the sut able folder. Ths pro ect and ths folder wll be found dur ng the ex per ment. (2) The dates are m ported n 2D n for ma ton that through cut tng and re com - bn ng led to 3D struc ture. Ev ery part of the struc ture has been sep a rately de sgned and n tro duced nto the pro gram part cat a log. When all the parts are de sgned the f nal as sem - bly s made, ob tan ng the 3D struc ture [7]. (3) In the de fned struc ture were n tro duced the real ma te ral s char ac ter s tcs (OLC 45). (4) The f nte el e ment model was de sgned, us ng as sup port the struc ture re al - zed n 3D. Us ng the pro gram s fa cl tes n dg tz ng, by choos ng some sec tons through the spn dle whch were frst mul t pled, and af ter that oned, so that at the end the f nte el e ment model was ob taned. The struc ture pres ents 4813 nodes. (5) The dg tz ng qual ty ver f ca ton was made, for large struc tures. It s ab so - lutely nec es sary nodes over wrt ng ver f ca ton, and free edges ver f ca ton, re sult ng af - ter the dgtzaton whch means un de sred net work ds con t nu tes. Af ter the ver f ca tons s made a re num ber ng of nodes and an op t m za ton of the ma trx elements. Data processng Af ter cre at ng the f nte el e ment model, de fn ng the de for ma ton and the anal y - ss we wll pass to data pro cess ng n n ter ac tve d rect or n d rect sys tem. Data postprocessng A so lu ton anal y ss s made. Those re sults can be rep re sented wth ther value n nodes, on el e ments or on the whole struc ture. The model de for ma ton evo lu ton, or the model de for ma ton evo lu ton n suc ces sve, steps can be se lected. The pur pose of the re search s the ther mal de for ma tons test of the prod uct fxed be tween the chuck and the talstock spn dle sleeve ver t cally and hor zon tally. The mea - sure ments have been made at the spn dle head stock, spn dle sleeve and talstock. There were n stalled four dal gauges, two n a ver t cal plane (1 and 2) and other two n hor zon - tal plane (3 and 4), whch n d cates the de for ma tons along the two planes (fg.2). The spn dle drve speed n = 2000 rot./mn., dur ng 180 mn utes (fg. 3), pe rod n whch were mea sured the tem per a tures and de for ma tons ev ery 30 mn utes (fg. 4). 105

8 THERMAL SCIENCE: Vol. 9 (2005), No. 1, pp Fgure 2. Measurement modes for the product fxed between the chuck and the talstock spndle sleeve Fgure 3. Gear box for a CNC lathe 106

9 But, A., Lumnosu, I.: The Venng Structure Method, the Fnte Element Method n... Fgure 4. Measure mode Conclusons Af ter the ther mal de for ma ton test for the prod uct fxed be tween the chuck and the talstock spn dle sleeve t was found that af ter the de for ma tons sta b l za ton the prod - uct moved ver t cally for mm at the end to wards the spn dle head stock and for mm at the end to wards spn dle sleeve. Hor zon tally the prod uct was de formed for mm at the end to wards the spn dle head stock and for 0.03 mm at the end to wards spn dle sleeve (fg. 5 and 6). Fgure 5. Temperature values n all nodes of the prncpal arbor at 2406 seconds from measurng start 107

10 THERMAL SCIENCE: Vol. 9 (2005), No. 1, pp Fgure 6. Temperature values n all nodes of the prncpal arbor at 7206 seconds from measurng start The tem per a tures were sta b lzed af ter 180 mn utes of func ton ng: at the man spn dle at 48 C, at the fx top at 44 C, at the spn dle head stock at 44 C, and at the spn dle sleeve at 50 C (fg. 7 and 8). 108 Fgure 7. Deformatons values n horzontal plane of the prncpal arbor after 180 mnutes

11 But, A., Lumnosu, I.: The Venng Structure Method, the Fnte Element Method n... Fgure 8. Temperature deformatons values n all the mass of the prncpal arbor after 180 mnutes The max mum tem per a ture emt ted n man spn dle gear s 70 C much more lke the tem per a ture n d cates n tech n cal lt er a ture [7]. The re sults that was ob taned ex per men tal and the meth od ol ogy who used the el e ment f nte, to ob tan the ther mal de for ma tons, helped to de ter mned the to tal de for - ma tons of the axs of prn c pal ar bor at the CNC lathe [2]. References [1] Basturea, G., Nu mer cal Con trol, Tech n cal Pub lsh ng, Bu cha rest, 1976 [2] Cuteanu, E., Mancov, R., El e ments F nte Method n Lathe Pro ect ng, Facla Pub lsh ng, Tmsoara, Ro ma na, 1980 [3] Cclov, D., Re ss tance and Fablty at Varables Strangs, Facla Pub lsh ng,tmsoara, Ro - ma na, 1975 [4] But, A., The Influence of CNC Lathe Con struc ton on Man u fac tur ng Accurancy, Poltehnca Pub lsh ng, Tmsoara, Ro ma na, 2003 [5] Acerkan, M., Ma chne-tools Man u fac tur ng and Cal cu la ton, Tech n cal Pub lsh ng, Bu - cha rest, 1978 [6] Rusu, E., Stuparu, A., Chlman, A., Com puter Nu mer cal Con trol for Ma chne-tools, n: AMC, Vol. 20A., Tech n cal Pub lsh ng, Bu cha rest, 1975, pp [7] Mltaru, C., Fablty and Accurancy n Buldng Technology of Ma chne-tools, Tech n - cal Pub lsh ng, Bu cha rest,

12 THERMAL SCIENCE: Vol. 9 (2005), No. 1, pp Authors' addresses: A. But Department of Mechancs, Poltehnca Unversty, M. Vteazul Bld., No Tmsoara, Romana I. Lumnosu Department of Physcs, Poltehnca Unversty of Tmsoara, Queen Mara Square No Tmsoara, Romana Correspondng author (A. But): E-mal: ad.but@qmal.com Paper submtted: September 1, 2004 Paper revsed: March, 15, 2005 Paper accepted: Aprl 6,