Process model for the design of bent 3-dimensional free-form geometries for the three-roll-push-bending process

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Vae*, Raoul Pleke Manufacuing Technology, Egelandsaße 13, 9158 Elangen, Gemany * Coesponding auho. Tel.: +49-9131-85-2774; fax: +49-9131-85-27141. E-mail addess: pee.vae@fau.

In he scope of his pape he pocess behavio fo he manufacuing of 3-dimensional geomeies is invesigaed. A FE model which consides all of he elevan influences, suc has been developed.

1 Available online a Pocedia CIRP 7 (213 ) Foy Sixh CIRP Confeence on Manufacuing Sysems 213 Pocess model fo he design of ben 3-dimensional fee-fom geomeies fo he hee-oll-push-bending pocess Pee H. Vae*, Raoul Pleke Manufacuing Technology, Egelandsaße 13, 9158 Elangen, Gemany * Coesponding auho. Tel.: ; fax: addess: pee.vae@fau.de Absac Thee-Roll-Push-Bending is a highly flexible manufacuing pocess fo he poducion of 3-dimensional fee-fom ben ubes. In he scope of his pape he pocess behavio fo he manufacuing of 3-dimensional geomeies is invesigaed. A FE model which consides all of he elevan influences, suc has been developed. A vaian simulaion invesigaing he impac of he pocess paamees on he shape of he esuling helical sucues has been caied ou. This allows a mehodical pocess design fo he manufacuing of 3-dimensional fee-fom geomeies. 213 The Auhos. Published by by Elsevie Elsevie B.V. B.V. Open access unde CC BY-NC-ND license. Selecion and/o pee-eview unde unde esponsibiliy esponsibiliy of Pofesso Pofesso Pedo Filipe Pedo do Filipe Camo do Cunha Camo Cunha Keywods: Thee-oll-push-bending; fee-fom bending; 3D; pocess design; FE; 1. Inoducion Moden fee-fom bending pocesses ae poducion mehods which suie he inceasing demands o ceae complex bending geomeies. In conas o bending geomeies manufacued by common ool-bound bending pocess, such fee-fom geomeies can have cuvaue and osion disibuions which can vay coninuously ove he longiudinal axis of he pa. Complex shaped ubes ae equied in vaious fields of applicaion as fo example in he auomoive indusy o in moden achiecue and design. Those ubes and pofiles can be used fo he anspo of media o as sucual componens. Fee-fom geomeies allow a moe efficien sauaion of he consucion space and moe feedom in consucion while poducion cycles can be inceasingly shoened. Fuhemoe, ool coss can be educed significanly as he foming is done kinemaically and only one ool se is needed fo a given ube diamee. Thee-oll-push-bending is one of such pocesses, capable of manufacuing 3-dimensional fee-fom geomeies. In his pocess, a ube is clamped by a colle a he fon of a feeding am and is pushed hough a bending die consising of olls. In 2-dimensional bending he cuvaue is deemined by he posiion of he seing oll in elaion o he ohe olls, namely he bending oll and he holding olls (Fig. 1). Fo sabiliy easons, wo holding olls ae applied usually. An inceasing infeed of he seing oll esuls in a highe bending momen and a smalle coesponding bending adius. Afe a bending opeaion he bending die can be opened and he ube can be oaed by he feeding am esuling in a oaion of he bending plane fo he subsequen bending opeaion. A oaion of he feeding am duing he feeding movemen leads o a oque load on he ube by which 3-dimensional helical sucues can be ealized. Feeding am B axis Y axis Holding olls Bending oll Seing oll Fig. 1. Bending die of a 3-oll-push-bending machine U axis C axis The Auhos. Published by Elsevie B.V. Open access unde CC BY-NC-ND license. Selecion and pee-eview unde esponsibiliy of Pofesso Pedo Filipe do Camo Cunha doi:1.116/j.poci

2 Pee H. Vae and Raoul Pleke / Pocedia CIRP 7 ( 213 ) Bu, he pocess design is vey complex as hee ae many influences o be consideed. Due o he incemenal naue of he pocess, hee is no possibiliy o measue and adjus a manufacued pa wihin he same pocess sep if i does no comply wih he given oleances. So, an exac pocess design based on a pecise pedicion of he pa geomey is highly desiable. To cay ou a mehodological pocess design fo 2- dimensional bends a chaaceisic map had o be deemined, descibing he coesponding bending adius fo each seing oll posiion [1]. Using his, he suiable pocess paamees can be eieved fo evey ben secion pio o poducion by doing a backwad seach. Fo 2-dimensional bends seveal numeical and analyical pocess models have been developed in ecen yeas. Gelach [2] developed an FE-model in Abaqus fo he ecangula coss-secions wih igid ool design. Hagenah e al. [1] implemened an FE model fo cicula coss-secions which consideed he deflecion of he olls. The calculaions wee solved by explici compuaion. The model has been validaed and showed a good ageemen wih expeimenal esuls. The deviaion of he measued bending adius was below 3% wihin he ange of 8 mm o 2 mm [1]. Meklein e al. [3] showed ha an exac pedicion becomes moe difficul fo lage bending adii, as inhomogeneous maeial popeies caused by bach vaiaions play an impoan ole. Hee, he elasic pa of he deflecion dominaes, so even slighly diffeen flow behavio can cause a high scae of he cuvaue afe sping back. An FE model fo he elaed hee-oll-bending pocess was se up in Ansys v11 by Selvaggio e al. [4]. Analyical appoaches o calculae he deflecion as well as he sping back fo ecangula pofiles have been pesened by Gelach [2]. Thee, he maeial model by Ludwik was used. Engel e al. [5] esablished a model fo cicula ubes. The deeminaion of he momencuvaue elaionship using he sess-sain law by Swif was caied ou using numeical invesigaions. Fuhe analyical eflecions abou he bending behavio of ubes unde he load of a pue bending momen can be found in he woks of Al-Queshi [6]. Hee, an elasic ideal-plasic sess-sain elaionship accoding o Pandl-Reuß was applied. The influence of axial loads as hey appea a he hee-oll-push-bending pocess is aken ino accoun in he conibuion by Yu e al. [7]. Fo 3-dimensional bending he effec of he feeding am oaion has o be consideed addiionally. A sysemaic appoach fo his has been pesened based on expeimens by Pleke e al. [8]. In his wok he coelai he ube was shown. A mehodology was pesened o measue and descibe he geomey of helical shaped componens as well as o deemine a chaaceisic map holding he pocess behavio in he 3-dimensional case. This elaion was expessed by he osion adjusmen coefficien B / Y (1) whee B denoes he oaion angle of he feeding am in adian measue, Y he feeding disance and esuling osion. Theeby, i expesses he aio he ube mus be ovewised duing he feeding movemen o fom a given osion. A compehensive invesigaion of he pocess behavio houghou he enie paamee space has no been published ye. Fuhemoe, hee is a lack of deailed examinaions of he facos influencing he osion. 2. Developmen of a FE model fo 3D-bending In he scope of his wok, an advanced FE model has been newly developed in Abaqus 6.9EF1. The main chaaceisics ae based on he model pesened by Hagenah e al. [1]. Due o he moe complex pocess sae in 3-dimensional bending, a seies of influences had o be consideed which have no been coveed ye in exising models. Addiionally, he shapes of he die elemens as well as he kinemaic have been changed Tool componens & kinemaics The componens of he bending die wee modeled accoding o he ool layou and he shape of he olls of The kinemaic has been adoped accodingly. The ools have been meshed wih 5 discee igid elemens aound he cicumfeence. They have been placed on cylindes which can be displaced by he pocess foces wihin he bending plane (see secion 2.4) Chaaceisics of he semifinished poduc The ube has a lengh of 8 mm, an oue diamee of 2 mm and a wall hickness of 1. mm. The fon and back end of he ube secions do no undego elevan foming, bu ae needed o guaanee a smooh pocess behavio a he beginning and he end of he feeding. A secion of 6 mm undegoes significan foming and hus is finely meshed by shell elemens of he ype S4R and a size of 2. x 2. mm. Thee ae five inegaion poins ove he shee hickness. The modeled maeial was egula cabon seel S235JR, fomely known as S37-2. The modulus of elasiciy E = 21 GPa was aken fom lieaue. The flow behavio was ecoded in a uniaxial ension es accoding o he guidelines of

242 Pee H. Vae and Raoul Pleke / Pocedia CIRP 7 ( 213 ) 24 245 DIN EN ISO 6892:27 and DIN EN ISO 1127:1996 [9, 1]. This esuled in he yield sengh of = 31 MPa and he flow cuve picued in Fig. 2. 8 - individual oll.

3 242 Pee H. Vae and Raoul Pleke / Pocedia CIRP 7 ( 213 ) DIN EN ISO 6892:27 and DIN EN ISO 1127:1996 [9, 1]. This esuled in he yield sengh of = 31 MPa and he flow cuve picued in Fig individual oll. The conneco elemens wee configued accoding o he deemined chaaceisic line. The enhanced pocess velociy in he simulaion implicaed jieing of he olls so a damping faco minimizing he vibaion was chosen Evaluaion pocedue 4 2 S235R The simulaions wee solved by explici calculaion on Linux woksaions wih eigh quadcoe Xeon pocessos unning wih 2.93 GHz. An example of he esuling geomey can be seen in Fig Fig. 2. Flow cuve of he used maeial S235JR 2.3. Ficion behavio In 2-dimensional bending only olling ficion occus whose influence on he foming can be negleced. When oaing he ube duing he bending opeaion sliding ficion appeas in ansvesal diecion beween he ube In combinaion wih he high conac pessue his leads o high eacion foces which have an impac on he degee of foming [11]. In he axial diecion he ficion is sill low as he olls ae fee o oae. This leads o he need o conside diecion dependen ficion condiions. To allow a ficionless movemen in axial diecion he olls ae placed on cylindes wih ficionless conac condiions. Duing he feeding he olls ae able o oae feely. To descibe he ficion in angenial diecion he Coulomb model was applied. The ficion coefficien was aken accoding o he simulaive opimizaion of Pleke e al. [11] and was se o μ = Machine siffness Duing he foming pocess eacion foces of up o 6 kn occue. Alhough he displacemen of he olls is vey small, ignoing i can cause in a significan change of he bending adius [1]. Fo a moe pecise simulaion he siffness of he olls has been consideed using conneco elemens which fom a sping-dampe sysem. To model he laeal displacemen of he holding olls a conneco elemen was used each. Hee, foces and displacemens wee maginal in feeding diecion and wee negleced. The bending oll and he seing oll wee placed on wo conneco elemens each o descibe deflecion in boh laeal and feeding diecion. The focemachine was measued expeimenally a evey Fig. 3. Simulaion esul of bending pocess (C = 5, U = 16 mm, = 8.75 m -1 ) To evaluae he geomey of he simulaions he coodinaes of he ube elemens wee expoed. The middle line was calculaed fom he coodinaes of he nodes on he exados, he inados, he inne and he oue edge. The daa has been smoohed by calculaing he aihmeic mean of he coodinaes of five consecuive nodes o level numeical coaseness. As five elemens denoe a secion lengh of 1 mm which is half of he ube diamee he loss of infomaion was expeced o be insignifican. The values of he local cuvaue and osion () wee calculaed disceely accoding o he pocedue ecommended by Pleke e al. [8] by he Fene-See fomulas: 3 whee is he ac lengh and () is he veco of he coesponding poin on he cuve calculaed fom he oigin. As a esul he cuvaue and he osion disibuions of he bending geomey can be calculaed (Fig. 4). The beginning and he end of he ube show an unseady local foming. Fo fuhe evaluaion he secion beween 2 (2) (3)

4 Pee H. Vae and Raoul Pleke / Pocedia CIRP 7 ( 213 ) mm and 47 mm fom he ip of he ube has been evaluaed. To eieve compaable values fo evey se of pocess paamees he mean values of he cuvaue and osion disibuions wee calculaed fo he copped secions. Cuvaue / Tosion 2 m Fig. 4. Cuvaue and osion disibuion 2.6. Pecision consideaions Cuvaue Tosion 2 mm 6 Tube lengh To esimae he numeical accuacy of he model he numeical influences had o be checked. Fis, he impac of he binay pecision of he calculaion accoding o he IEEE 754 sandad was examined. The sampling poin wih he seing oll posiion C = 4, U = 16 mm oay velociy was = m -1. The simulaion esuls wee compaed beween a simulaion un wih single pecision floaing-poin calculaion (32 bi) and a double pecision floaing-poin calculaion (64 bi). In he fis case, he bending adius was mm. In he second case he adius deceased by.2 % o 136. mm. The compuing ime was 55 minues and did no change noiceable which can be explained by he fac ha 64-bi CPUs wee used. The second faco examined was he elemen size. To esimae he influence he ube was meshed wih diffeen elemen sizes. In he fis un, he edge lengh was 2. mm x 2. mm. In he second un i was 1. mm x 1. mm. As sampling poin he paamee combinaion was configued as menioned above. In he fis case, he adius amouned o mm. In he second case he bending adius deceased by 1.4 % o mm. Due o he compuaion ime which inceased o 4.5 hous an elemen size of 2. x 2. mm was consideed as sufficien. Regading he selecion of he elemen ype Engel e al. showed ha fully inegaed volume elemens of he ype S4 led o a deviaion of he bending adius of abou,46 % in conas o shell elemens of he ype S4R [5]. Theefoe, shell elemens wee consideed as saisfacoy. 3. Vaian calculaions Fo a deepe invesigaion of he foming behavio a numeical vaian calculaion using he developed model was caied ou. A full-facoial design maix was se up coveing he majoiy of he usable paamee space [12]. Fo he seing oll nine posiions have been seleced which wee known o esul in epesenaive bending adii. These seing oll posiions wee eached by hee diffeen inclinaions of he bending am o C = 3, C = 4 and C = 5. A each inclinaion angle he seing oll was emoved U = 1 mm, U = 16 mm and U = 22 mm fom he iniial posiion whee i had conac wih he ube. These seing oll posiions wee combined wih five oay speeds of he feeding am which wee B = m -1, 2.19 m -1, 4.38 m -1, 8.75 m -1 and 17.5 m -1. The uni m -1 esuls fom he oaion of he feeding am in adian measue pe feeding disance measued in mee. This led o a design maix wih an oveall numbe of 45 sampling poins. The calculaions wee compued as descibed befoe (see secion 2.5). 4. Resuls 4.1. Impac of feeding am oaion on cuvaue The conduced invesigaions show he following esuls. As picued in Fig. 5, he cuvaue was 2.47 m -1 o m -1 ( = 85 mm o 417 mm). The cuvaue values emained oughly consan so he impac of he oaion on he cuvaue was ahe low. A clea dependency on he oay velociy could no be shown. Fo sligh cuvaues up o 6 m -1, an incease of he cuvaue is obsevable fo lage feeding am oaions. This effec does no occu fo highe cuvaues. Thee a mino decease is seen up o modeae oaions followed by an incease fo highe oay velociies. The minima ae no locaed a he same posiion and can be beween = 2.19 m -1 and 8.75 m -1. No egulaiy can be noiced in he shif of he minima. The maximum deviaion of he cuvaue beween he sampling poin wihou oaion and he one wih he highes oaion occus a he seing oll posiion U = 22 mm, C = 3. Thee, he cuvaue ises by 37.5 % fom 2.4 m -1 o 3.3 m -1. The lowes deviaion is obseved a he seing oll posiion a U = 16 mm, C = 5 whee he highes cuvaue value exceeds he lowes one a = 8.75 m -1 by 2 % Impac of feeding am oaion on osion The oay velociies of m -1 o 17.5 m -1 have been invesigaed (Fig. 6). Fo bending opeaions wihou any feeding am oaion, plane bends wihou any osion ae fomed, expecedly. A combinaion of he feeding

5 244 Pee H. Vae and Raoul Pleke / Pocedia CIRP 7 ( 213 ) movemen wih a oaion of he feeding am leads o osion values up o 17.1 m -1. A pedominanly linea elaion can be seen as he osion ises sicly monoonic in accodance wih he oaion of he feeding am. The behavio deviaes slighly a he diffeen seing oll posiions. Fo a deepe invesigaion of his elaionship he aio of he oaion o he osion was analyzed. As chaaceisic value he osion adjusmen coefficien was used (see secion 1.) Impac of feeding am oaion on osion adjusmen coefficien The osion adjusmen coefficien seves as a value fo he effeciveness of he feeding am oaion. I expesses he degee he ube mus be ovewised o ealize a given osion of he ube. A high osion adjusmen coefficien symbolizes an ineffecive osional shaping. If = 1 he esuling osion is equivalen o he oaion duing he foming pocess. Fig. 7 shows he disibuion of he osion adjusmen coefficien fo he invesigaed seing oll posiions and feeding am oaions. The osion adjusmen coefficien usually eaches values beween = 1. and 1.8 bu can decease o = can be even songe han he oaion induced by he feeding am. Moeove, i can be seen ha he osion adjusmen coefficien highly depends on he seing oll posiion fo low feeding am oaions. A a oay velociy of = 2.19 m -1 he highes value, 1.82, is moe han wice han he lowes value,.85. As he feeding am oaion inceases he osion adjusmen coefficiens assimilae in he egion of = 1.17 ± Coelaion of cuvaue and osion adjusmen coefficien Fuhemoe, Fig. 8 shows ha he osion adjusmen coefficien no only changes fo diffeen seing oll posiions, bu coelaes wih he aising cuvaue. An inceasing cuvaue demands fo highe feeding am oaions o fom an equivalen osion. This effec does Cuvaue 12 m U=1, C=5 U=16, C=5 U=1, C=4 U=22, C=5 U=16, C=4 U=1, C=3 U=22, C=4 U=16, C=3 U=22, C= m -1 Feeding am oaion Fig. 5. Cuvaue of diffeen oay velociies of he feeding am Tosion 16 m -1 8 C=3 C=4 C=5 U=1 U=1 U=1 4 U=16 U=16 U=16 U=22 U=22 U= m Feeding am oaion Fig. 6. Tosion of diffeen oay velociies of he feeding am Tosion adjusmen coefficien m -1 Feeding am oaion C=3 C=4 C=5 Fig. 7. Tosion adjusmen coefficien depending on oay velociies of he feeding am U=1 U=16 U=22 U=1 U=16 U=22 U=1 U=16 U=22 Tosion adjusmen coefficien U=22, C=3 U=16, C=3 U=22, C=4 U=1, C=3 U=16, C=4 U=22, C=5 U=1, C=4 U=16, C=5 U=1, C= m Cuvaue Fig. 8. Tosion adjusmen coefficien depending on cuvaue

6 Pee H. Vae and Raoul Pleke / Pocedia CIRP 7 ( 213 ) no occu as long as he seing oll is no moved. In his case, hee is no unifom end obsevable. Bu fo diffeen seing oll posiion, esuling in ising cuvaues, he osion adjusmen coefficien shows a endency o incease likewise. The seing oll posiion U = 22 mm, C = 3, leading o he lowes cuvaues abou = 2.4 m -1 o 3.3 m -1, equies a osion adjusmen coefficien beween =.85 and The seing oll posiion U = 1 mm, C = 5 whee he cuvaue is beween 1.9 m -1 and 11.7 m -1 demands fo osion adjusmen coefficiens fom 1.33 o The span beween he lowes and he highes value, excluded he sampling poins wihou feeding am oaion, is ~25 % of he lowes osion adjusmen coefficien a all of he invesigaed seing oll posiions. 5. Inepeaion Afe a caeful consideaion some basic insighs can be deduced. Fis, he oaion of he feeding am deemines he esuling osion of he ube decisively. An incease of he fe sicly monoonic incease of he osion. Fuhe on, seconday influences on he osion ae he posiion of he seing oll as well as he ficion condiions [11]. The seing oll posiion deemines he cuvaue of he bending geomey as well as he lengh of he foming zone and hus has a high effec on he shaping of he osion as well. The cuvaue is slighly influenced by he feeding am oaion bu he impac depends on he degee of oaion as well as on he seing oll posiion. A clea end canno be obseved. In geneal, he osion adjusmen coefficien inceases wih lage cuvaues. Unfounaely i seems no o be consan fo he individual seing oll posiions. Addiionally, he diffeence beween he lowes and he highes value obseved ises when cuvaue inceases. This makes i necessay o deemine a chaaceisic map wihin a 5-dimensional paamee space having hee inpu paamees and wo oupu paamees, ) o descibe he pocess behavio eniely. Fo a ough guess he osion adjusmen coefficien can be assumed o be consan on evey seing oll posiion o allow a quick pocess design in indusial applicaion. 6. Summay and oulook In he scope of his pape a FE model of he heeoll-push-bending pocess has been pesened. I is capable o pedic he esuling bending geomey in ems of he cuvaue and he osion. In paicula, 3- dimensional bends have been invesigaed by numeical simulaions houghou he enie elevan paamee space. The key pocess vaiable is he oaion angle of he feeding am in elaion o he feeding disance poceeded simulaneously. To gain a deepe undesanding he pocess behavio was examined using a chaaceisic value called he osion adjusmen coefficien which he feeding movemen and he esuling osion of he ube. The values of he osion adjusmen coefficien have been beween.85 and 1.8 bu have no shown a unifom end. The occuence of he osion highly depends on he posiion of he ool elemens. In fuue wok, fuhe deemining facos on he esuling osion, as hee ae he oll geomey and he maeial popeies, should be invesigaed moe deeply. A mehod fo an efficien deeminaion of chaaceisic maps should be esablished o enable a pecise pocess design fo complex 3-dimensional geomeies. Fuhemoe, he pocess boundaies should be examined, as well as mehods and possibiliies fo hei expansion. Refeences [1] Hagenah, H.; Vipavc, D.; Pleke, R.; Meklein, M.: Numeical Model of Tube Feefom Bending by Thee-Roll-Push-Bending. In. Conf. on Engineeing Opimizaion, 21, S [2] Gelach, C.: Ein Beiag zu Hesellung definiee Feifombiegegeomeien bei Rohen und Pofilen. Aachen: Shake, 21 [3] Meklein, M.; Vae, P. H.; Pleke, R.; Güne, M.: Einfluss de Fließkuve auf den Rohbiegeadius. Weksasechnik online, Düssledof: Spinge VDI, 1(212) S [4] Selvaggio, A.; Diksen, U.; Tekkaya, A. E.; Schikoa, M.; Kleine, M.: Inceasing he poducion accuacy of pofile bending wih mehods of compuaional inelligence. Evoluionay compuaion, MIT Pess, 17(29)4, S [5] Engel, B.; Kesen, S.: Analyical Models o Impove he Thee- Roll-Pushbending Pocess of Tube-Pofiles. Seel Reseach Inenaional, 211, S [6] Al-Queshi, H. A.: Elasic-plasic analysis of ube bending. In. J. of Machine Tools and Manufacue, 39(1999) 1, [7] Yu, T. X.; Johnson, W.: Influence of axialfoce on he elasicplasicbending and spingback of abeam. Jounal of Mechanical Wiking Technology, 6(1982), S [8] Pleke, R.; Vae, P. H.; Vipavc, D.: Basics of Pocess Design fo 3D feefom bending. Seel eseach inenaional: 14h In. Conf. on Meal Foming 212, S [9] N.N.: Nom DIN EN ISO 6892:27: Meallische Weksoffe - Zugvesuch - Püfvefahen bei Raumempeau. Beuh Velag, Belin, 27 [1] N.N.: Nom DIN EN ISO 1127:1996: Nichosende Sahlohe - Maße, Genzabmaße und längenbezogene Masse. Belin: Beuh Velag, 1996 [11] Pleke, R.; Vipavc, D.; Vae, P. H.: Influence facos of heeoll-push-bending. Tekkaya, A. E. (Hsg.): Poceedings of he 1s In. Tube and Pofile Bending Confeence, 211, S [12] Kleppmann, W.: Taschenbuch Vesuchsplanung. München: Cal Hanse Velag, 211

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGID ODIES In igid body kinemaics, we use he elaionships govening he displacemen, velociy and acceleaion, bu mus also accoun fo he oaional moion of he body. Descipion of he moion of igid