FSP Synthesis of an off-set five bar-slider mechanism with variable topology

Transcription

1 FSP Synthsis of an off-st fiv bar-slidr mchanism with variabl topology Umsh. M. Daivagna 1*, Shrinivas. S. Balli 2 1 Dpartmnt of Mchanical Enginring, S.T.J.Institut of Tchnology, Ranbnnur, India 2 Dpt. of Mchanical Enginring, Basavshwar Enginring Collg, Bagalkot India * Corrsponding author (mail:daivagna_um@rdiffmail.com) Abstract A mthod to synthsiz an off-st fiv-bar slidr mchanism with variabl topology is suggstd. Synthsis is carrid out in two phass for function gnration for thr finitly sparatd positions in phas-i and two finitly sparatd positions in phas-ii. A dyadic complx numbr mthod is usd to writ th quations of motions. Numrical xampls ar providd. Kywords: fiv-bar slidr, variabl topology, complx numbrs 1 Introduction Four bar-slidr crank linkags hav bn widly usd in industrial applications. Currnt work on fiv-bar linkags is mainly focusd on rvolut joint typ. Littl work has bn don on fiv-bar slidr crank linkags. Fiv bar-slidr crank paralll manipulators can b usd as ffctiv singularity-fr path gnrators if thir dimnsions and slidr input rangs ar proprly slctd. [1]. In this papr an offst fiv bar-slidr mchanism is synthsizd by variabl topology mthod using complx numbrs. A planar off-st fiv bar-slidr mchanism has two dgrs of frdom; it has on rotary and on linar indpndnt as input shown in fig.1. A planar off-st fiv barslidr mchanism with variabl topology is a mchanism of two dgrs of frdom that oprats in two phass. During Phas-I of opration a link adjacnt to th prmanntly fixd link i.. crank O a A [fig.2] is tmporarily fixd and that rsult in an off-st singl slidr crank mchanism of on dgr of frdom. Th Synthsis procdur is thn carrid out for thr finitly sparatd positions for th function gnration task. Th unknown paramtrs dtrmind in Phas-I synthsis ar considrd whil solving th rmaining dimnsions in Phas-II [2]. In Phas II th slidr is tmporarily fixd in position 3 and th crank is rlasd [fig3]. Th rsulting mchanism is a four bar mchanism of singl dgr of frdom. Now th synthsis is carrid out for th dtrmination of th othr unknown paramtrs [2, 3 and 4]. To start with an ovrviw of th variabl topology mchanism is givn to form th basis of th mthod dvlopd in th prsnt work. Ros [6] mad indirct rfrnc of fiv-bar variabl topology mchanism with th hlp of graphical mthods. Rawat [7] stablishd a synthsis tchniqu for fiv-bar topology mchanism oprating in two phass. Joshi t.al. [8] usd graphical synthsis of a fivbar slidr variabl topology mchanism. Balli and Chand [3] [4] [5] dal with aspcts lik transmission angl control, dfcts and solutions rctification of fiv-bar variabl topology mchanism. Chand and Balli [2] proposd a mthod of synthsis of a svn-link mchanism with variabl topology. Gadad, Daivagna and Balli prsntd combind triad and dyad synthsis of svn-link variabl topology mchanism [9].Daivagna and Balli prsntd synthsis of fiv-bar slidr with variabl topology [10]. W. Z Guo t. al [11] dal with a nw typ of controllabl mchanical prss for mtal forming applications. Abhijit Nagchaudhuri [12] dals with dynamic modling and analysis of a crank slidr mchanism. H.Zhou t.al [1] dal th path gnration with singularity avoidanc for fiv-bar slidr-crank paralll manipulators. Ers Soylmz [13] dalt with Classical transmission angl problm for slidr- crank mchanisms. In this papr an off-st fiv-bar slidr with variabl topology is synthsizd. A variabl topology synthsis mthod is suggstd as a tool to th graphical mthod of synthsis suggstd by Joshi and Amaranth [8]. Many mthods of synthsis lik algbraic mthod, loop closur tchniqu and graphical mthods ar availabl for th offst singl slidr crank mchanisms and four bar mchanisms. Th mthod of variabl topology suggstd in this papr rducs th cumbrsom calculations of th algbraic mthods. It has an incrasd accuracy ovr th graphical mthods. 345

2 2 Variabl Topology of Fiv-Bar Slidr Mchanism A planar off-st fiv-bar slidr mchanism has two dgrs of frdom. It is to b synthsizd to prform thrposition function gnration in Phas-I and two-position function gnration in Phas-II. 2.2 Phas-II In Phas-II, th slidr is fixd tmporarily at P 3 at an offst distanc d 2 from th prmanntly fixd pivot O a and th crank O a A is rlasd, rsulting mchanism is a four bar mchanism of singl dgr of frdom. Th input link i.. th crank O a A movs through an anglθ 34, link PB movs through an angl β 34. Th trac point B movs from B 3 to B 4. Figur 1: Off-st Fiv-bar slidr Mchanism 2.1 Phas-I Thr positions of th slidr is considrd for th synthsis of th unknown paramtrs Z 3, Z4 and Z 5.. Link O a A is fixd tmporarily and th lin of action of th slidr is offst at a distanc of d 1 from th tmporarily fixd pivot A. Hr off-st fiv-bar slidr VTM of two-dgr frdom bcoms off-st slidr crank mchanism of on-dgr frdom. Considr th trac point as P, whn th slidr movs from P 1 to P 2 and from P 1 to P 3, th link AB movs through th angls φ 12 and φ 13 rspctivly. Whn th slidr movs to position P 3, furthr motion is casd. Th slidr dos not mov furthr. In this position th initial position of th Phas-II is considrd. Figur 2: Off-st Fiv-bar slidr mchanism with variabl topology in phas-i Figur 3: Off-st Fiv-bar slidr mchanism with variabl topology in phas-ii 3 Synthsis It is rquird to synthsiz a planar fiv-bar slidr variabl topology mchanism; on can hav two options as follows: (i) On nd i.. th crank is fixd tmporarily. (ii) Th slidr is fixd tmporarily. Th convntions to b followd to dnot angls in Phas-I and in Phas-II ar dscribd in Tabl No.1. and Tabl No.2 Tabl 1: Paramtrs usd in Phas I. Links in trms of Vctors O a A = Z 2 AB= Z 3 BP = Z 4 AP = Z 5 O a P = Z 1 ρ 12 = Z 5H + X 12 / Z 5H Z 5H + X 13 / Z 5H Phas-I from 1 st Position to 2 rd Position & from Position 1 st to 3 rd Position Tmporarily fixd link Angls Φ 12, Φ 13. (Givn paramtrs) Angls β 12, β 13 (Assumd paramtrs) Vctor from A to P Vctor from fixd point O a to Slidr position. ρ 12 & ρ 13 ar Strtch ratios X 12 & X 13 ρ 13 = (prscribd paramtrs) Numbr of solutions 2 346

3 Tabl 2: Paramtrs usd in Phas II Links in Phas-II from 3rd Vctors Position to 4 th Position O a A = Z 2 θ 34 (prscribd paramtr) AB= Z 3 φ 34 (Assumd paramtrs) BP = Z 4 β 34 (prscribd paramtr) AP = Z Vctor from fixd point O a to 6 Slidr position P 3 Numbr of solutions Total Numbr of solutions 1 3 Phas-I and Phas-II dscribd in Tabl-1 and Tabl 2 ar usd to writ th dyad quations of th displacmnts of th slidr for function gnration. A function gnration mchanism is a linkag in which rlativ motion or a forc btwn links connctd to ground is takn into account. It is rquird to coordinat th sliding motion of th input i.. th slidr for thr spcifid dsign positions to th angular position of th out put link AB [15, 16]. 4.1 Phas-I Synthsis In function gnration problm, th input and output motions ( X, φ 12 12, X 13, φ 13 ) ar prscribd. β12 and β 13 ar fr choics. Thrfor thr will b 2 numbr of solutions. Thn th unknowns Z 3 and Z 4 and Z5 ar dtrmind in Phas-I as follows: Using loop closur vctor and dyad quations and P as th tracing point w writ: Z Z = 0 (1) 5 5V Z 5H 3 + Z 4 Z 5V Z 5H = 0 j iβ + j 3 Z 4 Z 5V Z 5H ρ j Z (2) ( ) ( ) = 0 Z (3) Subtracting (3) from (2),w gt ( j iβ Z 1 ) + ( j 3 Z 4 1) Z 5V Z 5H ρ j = 0 (4) Writing quation (4) for 3 positions of th slidr, w hav, i φ ( ) ( 12 1) 12 iβ Z + Z ρ = 0 (5) Z 5H 12 φ 13 iβ Z 4 1 Z 5H ρ13 = ( ) ( ) 0 i Z (6) R writing in th following form, i φ 12 iβ Z 1 + Z 12 1 = ρ (7) ( ) 4 ( ) 5H 12 φ ( ) ( ) 13 iβ 1 + Z H ρ13 3 Z i Z 3 = Z (8) Writing in th matrix form, ( 12 1) ( 1β12 1) Z 3 X 12 = (9) ( 13 1) ( iβ13 1) Z 4 X 13 ( 12 1β 1) ( 12 1) ( ) ( ) 13 iβ Ltδ = (10) Thn, Z 3 and Z 4 ar dtrmind as followd X Z 3 = X β ( 12 1) ( ) iβ 13 1 Z 4 = ( ( ) ) 12 1 X X / δ (11) / δ (12) 13 From loop closur quation w hav, Z 3 + Z 4 = Z 5 (13) Z is dtrmind. 5 Th constant vctor Z 5V for which magnitud is qual to th off-st distanc d 1 and th variabl vctor Z5H ar dtrmind as follows. Z 5H = Z 5 iα Cosα (14) Z 5V = Z 5 iα Sinα (15) d 1 = magnitud of Z 5V Whr α = angl mad by th vctor Z5 with X-axis as shown in fig 2 Thus in phas-i, following vctors ar dtrmind. Z 3, Z 4, Z5V and Z 5H 4.2 Phas-II Synthsis Whn th mchanism movs from position 1 to position 3, it stops and switchs to Phas-II. Hr OaA is rlasd and th slidr is tmporarily fixd. Now th mchanism bcoms a four bar mchanism of singl dgr of frdom. Hr th input angl is θ 34, th out put link PB dscribs β 34. It is rquird to coordinat th angular motion btwn th input link 2 and th output link 4 for th function gnration. Hnc th prscribd paramtrs ar β 34 and θ 34. Assuming th couplr angl φ 34 for th link 3, th only unknown Z 2 i.. th link O a A and Z 1 ar dtrmind for two position function gnration by taking th tracr point as B. Th numbr of solutions is 1. Hnc total 3 numbr of solutions will b solutions. Writing dyadic quations, w hav Z 2 ( iθ 34 1) + Z 3 ( 1 φ 34 1) = Z 4 ( iβ 1 ) 34 (16) Hr th only unknown is Z 2 and it is dtrmind. 4 β φ 34 1 Z 2 = (17) iθ 34 1 Z ( i 1 ) 34 ( ) Z ( ) By loop closur quation, w hav, Z 2 + Z 3 ( 1φ ) 13 + Z 4 ( i β13) - Z 6 =0 (18) Hnc Z 6 = Z 2 + Z 3 ( 1φ ) 13 + Z 4 ( i β 13) (19) Th off-st distanc d 2 is dtrmind as following. d 2 = Z 6 ( i ( 90 γ ) ) (20) Whr, γ is th angl mad by th vctor Z 6 with x-axis. Also, Z 1 Vctor is dtrmind by th closd loop vctor quation as givn in (21). Z 1 = Z 2 + Z 3 + Z 4 (21) 347

4 Hnc all th dsign paramtrs of th off-st fivbar slidr mchanism ar dtrmind from Phas-I and Phas-II. 5. Advantags and Limitations 5.1 Advantags Simplicity and gnrality ar th attractions of th mthod as compard to othr mthod of synthsis of fiv bar-slidr mchanisms. Unlik graphical mthod, it is not limitd by drawing accuracy. Sinc th dimnsions dtrmind in phas-i ar carrid to phas-ii lading to lss calculations. Th mthod can b applid to synthsis of mchanisms for thr; four or fiv infinitsimally sparatd positions. 5.2 Limitations Th proposd mthod is applicabl only to complx numbr approach. Th synthsizd mchanism may suffr from branch, ordr, Grashof or circuit dfcts which can b rctifid sparatly. 6. Conclusions and Discussions Th prsnt work suggsts variabl topology mthod using dyad tchniqus for synthsizing a fiv-bar slidr. An analytical mthod for synthsizing a fiv-bar slidr crank with variabl topology for thr positions in Phas-I and two positions in Phas-II is suggstd for function gnration. Complx numbrs, which radily lnd thmslvs as an idal tool for modling linkag mmbrs as parts of planar mchanisms, ar usd for writing displacmnt quations for dyads. Th couplr point B is displacd from position on to position four that may b usd for th application of transfr of objct from plac on to plac four. Th mchanism may suffr from branch, ordr, Grashof or circuit dfcts which can b rctifid sparatly. As this is synthsizd work, th authors will considr th study of th prformanc of th mchanism in thir futur work. Z 4 = i = mm Similarly from quation (13), Z 5 = i = mm Th offst distanc is calculatd as 31.00mm using quation (14). Phas-II Assuming φ 34 = 7 0, from quation (17) w gt, Z 2 = i = mm Again from th quation of loop closur (21), w gt Z 1 = i = mm Th synthsizd mchanism is shown in figur Exampl 2: It is rquird to synthsiz a Fiv Bar slidr with variabl topology shown in Fig.1. Givn that φ 12 = CCW, φ 13 = CCW, X 12 =30.0mm X 13 =60.0mm for phas-i and θ 34 = CCW, β 34 = CCW for th phas-ii Solution: Phas-I Assuming β 12 = , β 13 = , From quation (11) w gt, Z 3 = ( i) = mm From quation (12) Z 4 = i = mm Similarly from quation (13) Z 5 = i = mm Th offst distanc is calculatd as 33.24mm using quation (14). Exampls 6.1 Exampl 1: It is rquird to synthsiz a Fiv Bar slidr with variabl topology shown in Fig.1. Givn that φ 12 = 22 0 CCW, φ 13 = CCW, X 12 =25.0mm, X 13 =50.0mm for phas-i and θ 34 =40 0 CCW, β 34 =29 0 CCW for th phas-ii Solution: Phas-I Assuming β 12 = , β 13 = , From quation (11) w gt, Z 3 = ( i) = mm From quation (12), Figur 4: Synthsizd off-st Fiv Slidr Mchanism with variabl topology for th xampl 1 348

5 Phas-II Assuming φ 34 = , from quation (17) w gt Z 2 = i = mm Again from th quation of loop closur (21), w gt Z = mm Th synthsizd mchanism is shown in figur 5. Th function curvs for phas-i and phas-ii for th xampls ar shown in figur 6 and figur 7 rspctivly. Figur 7: Function curvs for phas-ii Rfrncs [1] H. Zou, Kwun-Lon Ting, Path Gnration with singularity avoidanc for fiv-bar slidr crank paralll manipulators, Mchanisms and Machin Thory, Vol.40, pp [2] Shrinivas S Balli and Satish Chand, Synthsis of a Planar Svn Link Mchanism with Variabl Topology for Motion Btwn Two Dad Cntr Positions, Mchanism and Machin Thory, Vol. 38,2003,pp Figur 5: Synthsizd off-st Fiv Slidr Mchanism with variabl topology for th xampl 2 [3] Shrinivas S Balli and Satish Chand, Transmission angl in Mchanisms, Mchanism & Machin Thory, Vol.37, 2002, pp [4] Shrinivas S Balli and Satish Chand, Synthsis of Fiv Bar Mchanism with Variabl Topology for Motion btwn Extrm positions, Mchanisms and Machin Thory, Vol.36, 2001, pp [5] Shrinivas S Balli and Satish Chand, Dfcts in Mchanisms and Solution Rctification, Mchanisms and Machin Thory, Vol. 37, 2002, pp [6] Stanly E Ros, Fiv-bar loop synthsis, Machin Dsign,Octobr 12, 1961, pp [7] Yogsh R Rawat. Synthsis of Variabl Topology Mchanisms Graphical Mthod, MTch dissrtation, M.E.D.IIT, 1997, Bombay pp Figur 6: Function curvs for phas-i [8] S A Joshi, Synthsis of Variabl Topology Mchanisms for Circuit Brakr Applications, MTch Thsis, I.I.T Bombay, 1998, pp

6 [9] Gadad G M, U M Daivagna and S S Balli, Triad and Dyad Synthsis of Planar Svn-Link Mchanisms with Variabl Topology, Procdings of NaCoMM2005, 2005, pp [10] Daivagna U M and S S Balli, Synthsis of Fiv Bar- Slidr Mchanism with Variabl Topology, National Confrnc on Rcnt Advancs in Dsign Enginring (RADE 2007), Manipal Institut of Tchnology, Manipal (India), 2007; Papr No.MDS-04,pp [11] W. J.Guo, K.H and R.Du, A nw Typ of Controllabl Mchanical Prss: Motion Control and Exprimnt validation, Journal of Manufacturing Scinc and Enginring, 2005, Vol.127, pp [12] Abhijit Nagchaudhuri, Dynamic Modling and Analysis of a Crank Slidr Mchanism, Univrsity of Maryland, Eastrn Shor, Princss Ann, MD 21853, Sssion 1566, [13] Ers Soylmz, Classical transmission angl problm for slidr- crank mchanisms. Mchanisms and Machin Thory, Vol. 37, 2002, pp [14 A. G. Erdman, G.N.Sandor, Mchanism Dsign: Analysis and Synthsis, Vol-lI, 1999, Prntic Hall, Nw Dlhi, 1997, [15] A. G Erdman, Thr and Four prcision point Kinmatics synthsis of planar linkags, Mchanism and Machin Thory, Vol. 16, 1981, pp