Key words: path synthesis, joint clearances, Lagrange s equation, Differential evaluation (DE), optimization.

Transcription

1 Rub Mshra, T.K.Naskar, Sanb Ahara / nternatonal Journal of Engneerng Researh and Applatons (JERA) SSN: Vol., ssue, Januar -Februar 0, pp.9-99 Snthess of oupler urve of a Four Bar Lnkage wth ont Clearanes Rub Mshra *, T.K.Naskar ** and Sanb Ahara ** * Shool of Mehanal Engneerng, KT Unverst, BBSR- 750, nda ** Mehanal Engneerng Departments, Jadavpur Unverst, Kolkata-7000, nda Abstrat n ths paper, snthess of lnkages wth ont learanes to generate a desred oupler urve s studed. These learanes are treated as a mass less lnk and the equaton of ts moton were found b usng Lagrange s equaton. Dnam parameters are used n Lagrange s equaton to defne the dreton of moton of learane. Deferental evaluaton (DE) method was used to analze the optmzaton of lnk parameters for mnmzng the error between desred and atual path due to learanes at ont of oupler and follower.. Ke words: path snthess, ont learanes, Lagrange s equaton, Dfferental evaluaton (DE), optmzaton.. NTODUCTON Jont learane s a mportant fator that affet the dnam stablt and the performane of mehanal sstem. t s generall exst n all knds of planar mehansm. Clearane an auses vbraton and nose and also affet the dnam propertes of the mahne. So learane ma pla a mportant role to defne the knemat and dnam behavor of mehansms. Wthout onsderng learane we assume the lnkage s alled rgd lnk and when learane s assumed n onts, the resultng lnkage s alled flexble lnkage. A no. of works s reported n ournals on flexble lnkages and modelng of ont learane. The ntal framework for the stud of planar flexble lnk mehansms has been provded b Boronka and Me [] analzed lnkages whose pvots were replaed b flexble onts. Dubowsk [] nvestgated the effets of ont learane n a slder rank mehansm onsderng mpat model. Burns and Crossle [] n ther nvestgaton of the strutural permutatons of flexble lnk -bar hans havng revolute pars. Ths led to the development of tehnques for the dmensonal snthess of deves that satsfed spef moton requrements b undergong large elast deformatons. T. Furuhash, N. Morta and M. Matsuura [-5] determned angular dretons of ont learanes b assumng ontnuous ontat n revolute onts wth learane and usng Lagrange funton. Tng et al [6] presented an approah to dentf the poston and dreton errors due to the ont learane of lnkages and manpulators. Jont learane was modeled lke a small lnk wth length equals to one half of the learane. A geometral model was used n ther method to assess the output poston or dreton varaton, to predt the lmt of poston unertant and to determne the maxmum learane. Mallk and Dhande [7] ntrodued a stohast model of the four-bar, path-generatng lnkage wth tolerane and learane to analze the mehanal error n the path of a oupler pont. Toleranes and learanes were assumed to be random varables. Shwab et al [8] studed dnam response of mehansms and mahnes affeted b learane n revolute ont and a omparson was made between several ontnuous ontat fore models and an mpat model. Tsa and La [9] performed poston analss of a planar -bar mehansm wth ont learane b usng looplosure equatons. Clearane was treated as a vrtual lnk n ths work. Flores and Ambroso et al [0] performed dnam analss of mehanal sstems onsderng realst ont haratersts lke onts wth learane and lubraton. The work analzed ontat mpat fores of the knemats and dnam sstem. The ont learane was modeled as a ontat par wth dr ontat as well as lubraton. For the lubrated ase, the theor of hdrodnam ournal bearngs was used to ompute the fores generated b lubraton aton. Tsa and La [] used equvalent knematal par n mult-loop lnkage to model the moton freedoms obtaned from ont learanes. The learanes were onsdered to be vrtual lnks. Erkaa and Uzma [] studed the effets of ont learane on qualt of path generaton and transmsson angle. n the paper GA approah was used to determne the dreton of ont learane relatve to nput lnk of a -bar lnkage. n ths paper, the nonlnear behavor n a four bar mehansm wth ont learane at the oupler and follower and ts effet on path generaton s analzed. Dfferental evaluaton method usng langrage s equaton s appled to desrbe the dreton of ont learanes. And also usng ths method an obetve funton s defned and parametr relatons between nput varable and dreton of ont learanes are onsdered as the onstrants for solvng nonlnear dfferental equaton.usng DE optmzaton of the mehansm lnk parameters s also performed to mnmze the 9 P a g e

2 Rub Mshra, T.K.Naskar, Sanb Ahara / nternatonal Journal of Engneerng Researh and Applatons (JERA) SSN: Vol., ssue, Januar -Februar 0, pp.9-99 path errors between desred and atual mehansms s analzed.. Modelng of Jont learane t s assumed that lnks of mehansm are onneted to eah other b revolute onts wth learane. Jont learane, shown n Fg.Jont learane r s defned as the dfferene between the rad of the pn and hole, rb and r J respetvel. When the ournal gves mpat on the bearng wall, normal and tangental fore ours. Also, f the frton s neglgble, the dreton of ont learane vetor ondes wth the dreton of normal fore at the ontat pont. When the ontnuous ontat mode between ournal and bearng at ont s onsdered, the learane ma be modeled as vetor whh orresponds to mass less vrtual lnk wth the length equal to ont learane. The equvalent learane (learane vetor) an be defned n the form r r B r J lnk r J r () r B lnk Fg.. Equvalent learane lnk. Snthess of four bar path generator wth ont learane. Dmensonal snthess of -bar lnkage for path generaton nvolves A) The determnaton of the dmenson of the lnkage related to the desred path B) The mnmzaton of strutural error whh s subet to a set of sze and geometr onstant... -Bar mehansm wth double learane A four-bar mehansm wth double learane, as shown n Fg (), s onsdered as an example to determne the effet of ont learane between rank and oupler on path generaton. Knemat equatons are derved from the vetor onfguraton of the mehansm, shown n Fg () Fg..vetor representaton of mehansm From the lose loop vetor relaton e Le Le Le re re L B separatng Eq. (), nto ts real part and magnar part and usng trgonometr relatons, and wth ont learane an be expressed as a funton of and respetvel: tan B B AC A os L L os r os L L 0 ( ) () os r os...() Where the supersrpt denotes the value wth learane. A, B, C terms are gven, wth ont learane respetvel. A L L L os r os r os L r os L r os r r os L L B C L L L L r r...(5) L sn r sn r sn...(6) L L L r os L L os r os r os L r os r r os L L L L r r L L As shown n Fg., the poston of the oupler pont P x, relatve to the rank pvot A s gven wth ont learane, respetvel: Px L os r os AP os (7) (8) P r sn APsn. (9) L sn Fg.. Shemat representaton of a -bar mehansm wth learane x Where P, P denote the X andy oordnate values for the path of the oupler pont n onsderng the ont learane. n the knemat analss of the -bar mehansm wth double ont learane, t s neessar to determne the poston of mass entre for movng lnks and then ther orrespondng velotes and aeleratons. So n the ase of ont 9 P a g e

3 x G Rub Mshra, T.K.Naskar, Sanb Ahara / nternatonal Journal of Engneerng Researh and Applatons (JERA) SSN: Vol., ssue, Januar -Februar 0, pp.9-99 learane these postons are derved from the vetor representaton of the mehansm n Fg.. Due to moton transmsson from rank lnk to follower, ont learane between rank and oupler has mportant role on the path generaton b oupler pont P. f the rank pvot A s taken as the referene pont, the mass entre postons for movng lnks are gven as follows. A 0 G os (0) G A 0 G sn x G L os r os AG os G () () L sn r sn AG sn () These dfferental equatons determne the moton of the sstem. Smlarl we an determne the dreton of. The are used to determne the dreton of the ont learane wth respet to nput and are solved b DE on MATLAB software. 5. Case stud A ase stud s onduted four bar lnkage wth learanes n the revolute ont onnetng the nput lnk and the oupler. Values of dnam parameters used n Lagrange s equaton are gven n Table. Table.. Assumed values of dnam parameters G x G L os r os L os r os BG os () L sn r sn L sn r sn BG sn. Lagrange s equaton The Lagrange s equaton s used to determne, the dreton of the mass less learane lnk. d dt K. E. For ths spef purpose the equaton an be expressed as (5) K. E K. E P. E D. F 0 (6) Where, the knet energ ( KE),.. potental energ ( PE).. and dsspaton funton ( DF..) are: P. E. m g D. F C m x G C The Eqs. (7) together wth Eq. (6) gve, G G xg G G 0 m xg G gm C C (7) (8) 6. DE to determne dreton of ont learane The equaton of moton of the sstem has a non lnear harater as t s obvous from Eq. (8). The dreton of the learane lnk and s determned as a funton of the dreton of the nput lnk b optmzaton approah usng DE. The obetve funton s expressed as [], MnmzeF x f m xg h xg G G x 0, x x x, x X, (9) l u x 0, x x x, x X, (0) G gm C C, xg G G MnmzeF x f m xg G gm C C, h Desrptons parameters values MO of the lnk 5. 0 Kg m MO of the lnk Kg m MO of the lnk.58 0 Kg m Mass of the lnk m Mass of the lnk m Mass of the lnk m Gravtatonal aeleraton nput velot Dampng oeffent Dampng oeffent l u angular 0.Kg.08Kg 0.07Kg g 9.8m / s 6.8rad / s 6 C 0.0 Kgm s/ rad 6 C C 0.0 Kgm s/ rad, Where h ( x) are the equalt onstrants that depend on parametr relatons between dreton of learane lnk and nput varable; X s a vetor that omprses the desgn varables orrespondng to dreton of learane lnk, that s, 95 P a g e

4 Rub Mshra, T.K.Naskar, Sanb Ahara / nternatonal Journal of Engneerng Researh and Applatons (JERA) SSN: Vol., ssue, Januar -Februar 0, pp.9-99, and xl and xuare the lower and upper 7. Result onsderng dfferent values of ont learane. bounds of the desgn varables, respetvel. The obetve funton s the sum of two terms. The frst part omputes the poston error (also alled the strutural error) as the sum of squares of the Euldan dstanes between eah desred ponts along the oupler urve ( Px, Pd ) that d has to be traed b the mehansm and the orrespondng generated ponts ( Pxg, Pg ) b the desgned mehansm. The desred ponts are a set of target ponts along the oupler urve ndated b the desgner and should be met b the oupler pont of the mehansm and the generated urve s the urve that s atuall obtaned b the oupler pont of the desgned mehansm. For mnmzng the error between desred and generated urves, the obetve funton s gven b MnmzeF( x) subet to h x N 0, x x x, wheren num berof presonponts l N Pxd Pxg Pd P u g () The nequalt onstrants h x onsst of ondtons of Grashof s rule that spefes rank roker stuaton. x l and xu are respetvel the lower and upper bounds of the desgn varables. The onsst of lnk lengths L, ( Px, Pd ) and the d strutural angle, the angle that the lne AP makes Fg. Coupler urve for r = mm and r = mm AB wth the lne of entre of the oupler as seen n to obtan the o-ordnates of the generated Fg. postons of the oupler pont dsplaement analss s essental. The suessful applaton of the developed methodolog for snthess of mehansms s shown through some ase studes. All the data presented here are n a onsstent set of unts,.e., all lnear dmensons are n the unt of length and the angular dmensons are n degrees unless otherwse stated. For eah ase stud an ntermedate result and the fnal result are gven. Fnall, a summar of the results of all the ases ndatng the values of the desgn varables obtaned through DE s presented. The optmzaton algorthm s appled usng MATLAB verson 7.. All the lnkages shown n the followng setons are obtaned b usng the desgn varables. Here the tehnque of geometr entrod of preson ponts (GCPP) [] has been used to defne the ntal bounds of the desgn varables. The GCPP s obtaned b evaluatng the mean oordnates X and Y g g (mass enter poston) of the desred preson ponts n X and Y dretons respetvel. Fg.5. Coupler urve for r = mm and r = mm Fg.6. Coupler urve for r = mm and r = mm 96 P a g e

5 Rub Mshra, T.K.Naskar, Sanb Ahara / nternatonal Journal of Engneerng Researh and Applatons (JERA) SSN: Vol., ssue, Januar -Februar 0, pp.9-99 Fg.0 (θ ) for r = mm and r = mm Fg.7. Coupler urve for r = mm and r = mm Fgs. (8-) Dreton of ont learane wth respet to poston of nput lnk (θ ) Fg.8. for r = mm and r = mm Fg.. for r = mm and r = mm Fgs. (-5) Path error between desred and atual mehansm and path error between desred and optmsed mehansm. Fg. for r = mm and r = mm Fg.9 for r = mm and r = mm Fg. for r = mm and r = mm 97 P a g e

6 Rub Mshra, T.K.Naskar, Sanb Ahara / nternatonal Journal of Engneerng Researh and Applatons (JERA) SSN: Vol., ssue, Januar -Februar 0, pp.9-99 Fg.. for r = mm and r = mm Fg.5 for r = mm and r = mm 7.. Optmzed desgn varables for dfferent values of ont learane The mnmzed error between desred urve and generated urve are shown n table for dfferent learane value. Table.Optmzed desgn varables for dfferent values of ont learane Frst mm mm mm mm learane Seond mm mm mm mm learane L (mm) L (mm) L (mm) L (mm) A'P (mm) β (degree) Error (normalzed) (mm) Conluson n ths stud, for determnng the effet of ont learane on oupler urves the onts wth learane at the ontat pont of rank and oupler and ontat pont of oupler and follower have onsdered. Here the optmum lnk lengths are found out wth an obetve to mnmze the error n the path generaton onsderng the gven ont learanes. t s observed from the result that after dong optmzaton we an redue the error between the desred and optmzed urve. These results an be seen as approprate optmzaton tehnque for mnmzng the atual error n the presene of learanes. The values of the learanes ma var due to preson to be mantaned. Therefore optmum results are extrated for dfferent set of values of the learanes to stud the effet of ont learanes on the mnmzed error. From the result t an be adapted to smlar sstems to mprove the mehanal workng ondtons. Referenes [] R.H. Burns and F.R.E. Crossle, Strutural Permutatons of Flexble Lnk Mehansms, ASME Paper, 66-Meh-5. [] T. G. Boronka and C. Me, Analss and Desgn of Multple nput Flexble Lnk Mehansms, Journal of Mehansms 5 (970), pp.9-0. [] S. Dubowsk, On predtng the dnam effet of learanes n planar mehansms, ASME Journal of Engneerng for ndustr 9B (97) () pp. 7-. [] T. Furuhash, N. Morta and M. Matsuura, Researh on dnams of four-bar lnkage wth learanes at turnng pars (st Report, General theor of ontnuous ontat model), Bulletn of the JSME (978), pp [5] N. Morta, T. Furuhash and M. Matsuura, Researh on dnams of four-bar lnkage wth learanes at turnng pars (nd Report, Analss of rank-level mehansm wth learane at ont of rank and oupler usng ontnuous ontat model), Bulletn of the JSME (978), pp [6] A.K. Mallk and S.G. Dhande, Analss and snthess of mehanal error n pathgeneratng lnkages usng a stohast approah, Mehansm and Mahne Theor (987), pp. 5-. [7] K.W. Tng, J. Zhu and D. Watkns, The effets of ont learane on poston and orentaton devaton of lnkages and manpulators, Mehansm and Mahne Theor 5 (000), pp [8] A.L. Shwab, J.P. Meaard and P. Meers, A omparson of revolute ont learane models n the dnam analss of rgd and 98 P a g e

7 Rub Mshra, T.K.Naskar, Sanb Ahara / nternatonal Journal of Engneerng Researh and Applatons (JERA) SSN: Vol., ssue, Januar -Februar 0, pp.9-99 elast mehanal sstems, Mehansm and Mahne Theor 7 (00) [9] M.J. Tsa and T.H. La, Knemat senstvt analss of lnkage wth ont learane based on transmsson qualt, Mehansm and Mahne Theor 9 (00), pp [0] P. Flores, J. Ambroso, J.C.P. Claro, H.M. Lankaran and C.S. Kosh, A stud on dnams of mehanal sstems nludng onts wth learane and lubraton, Mehansm and Mahne Theor (006), pp [] M.J. Tsa and T.H. La, Aura analss of a mult-loop lnkage wth ont learanes, Mehansm and Mahne Theor (008) 57. [] S. Erkaa and. Uzma, Determnng lnk parameters usng genet algorthm n mehansms wth ont learane, Mehansm and Mahne Theor (009). 99 P a g e