Mechanisms of 10 Links, 13-Joints, 1-F Kinematic Chains of Group B

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1 IJSTE Intrntonl Journl o Sn Tnoloy & Ennrn Volum 2 Issu 08 Frury 201 ISSN (onln): X Mnsms o Lnks, 13Jonts, 1F Knmt Cns o Group B Dr. Al Hsn Dprtmnt o Mnl Ennrn Jm Mll Islm, Nw Dl Astrt Autor s otv s to vlop nw, sy, rll, n nt mto to tt somorpsm n prpr tlou o x lnk n ts orrsponn quvlnt lnks n t stnt mnsms n knmt ns o Group B. It wll lp t nw rsrrs / snrs to slt t st mnsm knmt n n mnsm to prorm t sr tsk t t onptul st o sn. T propos mto s prsnt y omprn t struturl nvrnts sum o t solut vlus o t rtrst polynoml onts [SCPC] n mxmum solut vlu o t rtrst polynoml ont [MCPC] o [JJ] mtrs. Ts nvrnts my us to tt somorpsm n t mnsm knmt n vn smpl onts. T mto s xpln wt t lp o xmpls o plnr knmt n vn smpl onts. Kywors: Knmt Cn; Fx Lnk; Equvlnt Lnk; SCPC; MCPC NOTATIONS USED F.L.Fx Lnk, E.L.Equvlnt Lnk, n5 n4 n3 n2 Pntonl, Qutrnry, Trnry, Bnry Lnks I. INTRODUCTION Ovr t pst svrl yrs mu work s n rport n t ltrtur on t struturl syntss o knmt ns n mnsms. Untt somorpsm rsults n uplt solutons n unnssry ort. Tror, t n or rll n nt lr mto or ts purpos s nssry. Intyn somorpsm mon knmt ns usn rtrst polynomls o ny mtrs o orrsponn knmt ns r smpl mtos [Ur n Ru 1975, Mrutyuny n Rvn 1979, Yn n Hll, 1981]. But t rllty o ts mtos ws n qustons s svrl ountr xmpls wr oun y Mrutyuny [Mrutyuny, 1987]. T tst propos y Mrutyuny [Mrutyuny, 1987] s s on rtrst onts o t Dr mtrx o t rp o t knmt ns. T lmnts o t r mtrx wr sum o t r o vrts (r or typ o lnks) or unty n lnklnk ny mtrx. Ltr on ts tst ws lso oun unrll. Krsnmurty [Mrutyuny, 1987] propos t rprsntton polynoml or ttn somorpsm twn two knmt ns. T rprsntton polynoml s t trmnnt o t nrlz ny mtrx, ll rprsntton mtrx o t knmt n. But t rprsntton mtrx rqurs t us o lr numr o symols, t lulton n omprson o t rprsntton polynomls s not s sy s tt o t rtrst onts o t ny mtrx. Blsurmnn n Prtsrty [Blsurmnn n Prtsrty 1981] propos t prour s on t onpt o t prmnnt o mtrx or t purpos. Tn n Lu [Tn, Lu, Tyn, 1993] prsnt mto s on r o s mnsm ntr. Svrl otr mtos lk ommnts [Cullo n Wn, Jno, 2005], [Hsn A., 2007, 2009, 20, 2012] r lso n us. II. THE JOINTJOINT [JJ] MATRIX Ts mtrx s s upon t onntvty o t onts trou t lnks n n, s squr symmtr mtrx o sz n x n, wr n s t numr o onts n knmt n. [JJ] = L n x n (1) Wr L = Dr o lnk twn t n t onts tos r rtly onnt =0, ont s not rtly onnt to ont O ours ll t onl lmnts L = 0 All rts rsrv y 155

2 Mnsms o Lnks, 13Jonts, 1F Knmt Cns o Group B (IJSTE/ Volum 2 / Issu 08 / 027) III. CHARACTERISTIC POLYNOMIAL OF [JJ] MATRIX D (λ) vs t rtrst polynoml o [JJ] mtrx. T mon polynoml o r n s vn y quton (2). (JJ λ I) = λ n + 1λ n 1 + 2λ n 2 + n 1λ + n (2) Wr; n = numr o smpl onts n knmt n n 1, 1, 2, n1, n r t rtrst polynoml onts. T two mportnt proprts o t rtrst polynomls r T sum o t solut vlus o t rtrst polynoml onts (SCPC) s n nvrnt or [JJ] mtrx n1 + n = nvrnt T mxmum solut vlu o t rtrst polynoml ont (MCPC) s notr nvrnt or [JJ] mtrx. IV. STRUCTURAL INVARIANTS [SCPC] AND [MCPC] T vlus o rtrst polynoml onts r nvrnts or [JJ] mtrx. To mk ts [JJ] mtrx rtrst polynoml onts s powrul snl numr rtrst nx, nw ompost nvrnts v n propos. Ts nvrnts r SCPC n MCPC. Ts nvrnts r unqu or [JJ] mtrx n my us s ntton numrs to tt t somorpsm mon smpl ont knmt ns. T rtrst polynoml onts vlus r t rtrst nvrnts or t knmt ns. Mny nvsttors v rport osptrl rp (nonsomorp rp vn sm En sptrum). But ts En sptr (En vlus or rtrst polynoml) v n trmn rom (0, 1) ny mtrs. T propos [JJ] mtrx provs stnt st o rtrst polynoml onts o t knmt ns vn osptrl rps. Tror, t s vr tt t struturl nvrnts SCPC n MCPC r pl o rtrzn ll knmt ns n mnsms unquly. Hn, t s possl to tt somorpsm mon ll t vn knmt ns. V. ISOMORPHISM OF KINEMATIC CHAINS 1) Torm: Two smlr squr symmtr mtrs v t sm rtrst polynomls. 2) Proo: Lt t two knmt ns r rprsnt y t two smlr mtrs A n B su tt B = P 1 AP, tkn nto ount tt t mtrx λi ommuts wt t mtrx P n P 1 = P 1. Sn t trmnnt o t prout o two squr mtrs quls t prout o tr trmnnts, w v B λ I = P 1 A P λ I = P 1 (A λ I) P = P 1 (A λ I) P = A λ I Hn, D (λ) o A mtrx = D (λ) o B mtrx. D (λ) =rtrst polynoml o t mtrx. It mns tt D (λ) o two [JJ] mtrs rprsntn two knmt ns s sm, tr struturl nvrnts SCPC n MCPC wll lso sm n t two knmt ns r somorp otrws nonsomorp ns. VI. ILLUSTRATIVE EXAMPLE 1 (IDENTIFICATION OF COSPECTRAL GRAPHS) T nonsomorp knmt ns v t sm rtrst polynomls usn (0, 1) ny mtrs n tr knmt rps r ll s Cosptrl rps. But t rtrst polynomls o su ns rv rom [JJ] mtrs r stnt. Tror, t struturl nvrnts [SCPC] n [MCPC] r lso stnt. Ts s prov wt t lp o xmpls o two knmt ns wt rs, 12 onts, tr r o rom s sown n F 1.T tsk s to xmn wtr ts two ns r somorp. t struturl nvrnts o ts two ns r s ollows: For n 2(): [SCPC] = , [MCPC] = For n 2(): [SCPC] = , [MCPC] = Our mto rports tt n 2() n 2() r nonsomorp s t st o vlus o [SCPC] n [MCPC] r rnt or ot t knmt ns. Not tt y usn otr mto ommnts [Cullo n Wn, Jno, 2005], t sm onluson s otn. All rts rsrv y 15

3 Mnsms o Lnks, 13Jonts, 1F Knmt Cns o Group B (IJSTE/ Volum 2 / Issu 08 / 027) F. 1: Tn r KC wt 3 o VII. RESULTS T propos nvrnts [SCPC] n [MCPC] r us s t ntton numr o t knmt ns vn smpl onts. T ntton numrs o ll 1o knmt ns up to Lnks r wt t utor. Ts nvrnts r lso l to tt somorpsm mon t knmt ns wt multpl onts lso. Fx Lnks n Equvlnt Lnks n Dstnt Mnsms o Lnks, 13 Jonts, Snl r o rom Knmt Cns Group B r lst n Tl 1(B). VIII. CONCLUSIONS In ts ppr, smpl, nt, n rll mto to nty somorpsm s propos. By ts mto, t somorpsm o mnsms knmt ns n sly nt. It norports ll turs o t knmt ns n s su, volton o t somorpsm tst s rtr ult. T mto s n oun to sussul n stnusn ll known 1 knmt n o 8lnks, 230 knmt n o lnks vn 1F. T vnt s tt ty r vry sy to omput usn MATLAB sotwr. It s not ssntl to trmn ot t struturl nvrnts to ompr two ns, only n s t [SCPC] s sm tn t s n to trmn [MCPC]or ot knmt ns. T [JJ] mtrs n wrttn wt vry lttl ort, vn y mr nspton o t n. T propos tst s qut nrl n ntur n n us to tt somorpsm o not only plnr knmt ns o on r o rom, ut lso knmt ns o mult r o rom. Tl 1(B) Dstnt mnsms o lnks, 13 Jonts, 1F Knmt Cns GROUP B Knmt Cn n5 n4 n3 n2 EZDV F. L. E. l. D. M. SCPC(B1) = MCPC(B1) = SCPC(B2) = MCPC(B2) = All rts rsrv y 157

4 Mnsms o Lnks, 13Jonts, 1F Knmt Cns o Group B (IJSTE/ Volum 2 / Issu 08 / 027) SCPC(B3) = MCPC(B3) = SCPC(B4) = MCPC(B4) = Tl 1(B) ontnu. Knmt Cn n5 n4 n3 n2 EZDV F. L. E. l. D. M. SCPC(B5) = MCPC(B5) = SCPC(B) = MCPC(B) = All rts rsrv y 158

5 Mnsms o Lnks, 13Jonts, 1F Knmt Cns o Group B (IJSTE/ Volum 2 / Issu 08 / 027) SCPC(B7) = MCPC(B7) = SCPC(B8) = MCPC(B8) = Tl 1(B) ontnu. Knmt Cn n5 n4 n3 n2 EZDV F. L. E. l. D. M. SCPC(B9) = MCPC(B9) = SCPC(B) = MCPC(B) = ,, SCPC(B11) = MCPC(B11) = All rts rsrv y 159

6 Mnsms o Lnks, 13Jonts, 1F Knmt Cns o Group B (IJSTE/ Volum 2 / Issu 08 / 027) SCPC(B12) = MCPC(B12) = Tl 1(B) ontnu. Knmt Cn n5 n4 n3 n2 EZDV F. L. E. l. D. M. 9 SCPC(B13) = MCPC(B13) = SCPC(B14) = MCPC(B14) = SCPC(B15) = MCPC(B15) = TOTAL NUMBER OF DISTINCT MECHANISMS OF GROUP B = 12 8 REFERENCES [1] [Arwl n Ro,1987] Arwl V.P. n Ro J.S., T Molty Proprts o Knmt Cns, M. M. Tory, Vol. 22, pp , [2] [Arwl n Ro,1987] Arwl V.P.n Ro J.S., Struturl Clsston o Knmt Cns n Mnsms, M. M. Tory, Vol. 22, pp , [3] [Amkr, 1987()] Amkr A. G. n Arwl V. P., Cnonl Numrn o Knmt Cns n Isomorpsm Prolm: Mn. Co, M. M tory, Vol. 22, No 4, pp 45341, [4] [Amkr, 1987()] Amkr A. G. n Arwl V. P., Intton o Knmt Cns, Mnsms, Pt Gnrtors n Funton Gnrtors Usn Mn. Cos, M. M tory, Vol. 22, No 4, pp 43471, [5] [Ermn n Snor, 1988] Ermn A. G. n Snor G. N., Avn Mnsm Dsn, Vol. I n II, Prnt Hll o In, [] [Gos n Mllk, 1988] Amt Gos n Asok Kumr Mllk, Tory o Mnsms n Mns, Est Wst Prss Pvt. Lt., Nw Dl, pp. II Eton 1988 [7] [Gson n Mrs,1989] Gson C.G. n Mrs D., On t Lnk Vrts o Wtt Br MnsmsI. Bs Gomtry M. M. Tory, Vol. 24, pp. 5113, [8] [Hn, 197] Kurt Hn, Appl Knmts MGrwHll, Nw York,197.[Hrtnr n Dnvt, 194] Hrtnr S. n Dnvt J., Knmt Syntss o Lnks MGrwHll, Nw York, 194. [9] [Hsn A.,2009] Intton o Isomorpsm mon Knmt Cns n Invrsons Usn Lnk Any Vlus,Intrntonl J. o M. n Mtrls Ennrn (IJMME), pp , No.3, Vol. 4(2009) All rts rsrv y 10

7 Mnsms o Lnks, 13Jonts, 1F Knmt Cns o Group B (IJSTE/ Volum 2 / Issu 08 / 027) [] [Hsn A.,20] Isomorpsm Intton o Compoun Knmt Cn n Tr Mnsm,mnr s Journl on Mnl Ennrn, Vol. 2, No. 1,pp 715, Nov Jn [11] [Hsn A., 2007] Systmt Dvlopmnt o Knmt Cns n Mnsms rom vn Assortmnt o Lnks,Journl o Insttuton o Ennrs (In), Vol. 88,pp.1519, 2007 [12] [Hsn A., 2007] Isomorpsm n Knmt Cns Usn Pt Mtrx,Journl o Insttuton o Prossonl Ennrs (IPENZ), Nw Zln,2007 [13] [Hsn A.,2009] Isomorpsm n Invrsons o Knmt Cns up to Lnks,Journl o Insttuton o Ennrs (In), Vol. 90, pp.14, 2009 [14] [Hunt, 1978] Hunt K.H., Knmt Gomtry o Mnsms Oxor Ennrn Sn Srs, [15] [Hwn n Hwn, 1992] Wn Mn Hwn n Y Wn Hwn, Computr A Struturl Syntss o Plnr Knmt Cns wt Smpl Jonts, M. M. Tory, Vol. 27, pp , 1992 [1] [Mo,1998] Mo A. Knmt ns struturl molln n orrlton wt mn prormn, M.T. Dssrtton, IIT Dl,1998 [17] [Mrutyuny, 1984()] Mrutyuny T. S., A Computrz Mtooloy or Struturl Syntss o Knmt Cns, Prt 1, Formulton, M. M. Tory, Vol. 19, No., pp , 1984 [18] [Mrutyuny, 1984()] Mrutyuny T. S., A Computrz Mtooloy or Struturl Syntss o Knmt Cns, Prt 2, Applton to svrl ully or Prtl Known Css, M. M. Tory, Vol. 19, No., pp , 1984 [19] [Mrutyuny n Blsurmnum, 1987] Mrutyuny T.S. n Blsurmnum H.R., In Qust o Rll n Ent Computtonl Tst or Dtton o Isomorpsm n Knmt Cns, M. M tory, Vol. 22, No 4, pp , [20] [Mrutyuny n Rvn, 1979] Mrutyuny T. S. n Rvn M. R., Struturl Anlyss o Knmt Cns n Mnsms s on Mtrx rprsntton, Trnston o t ASME, Journl o Mnl Dsn, Vol. 1, pp , [21] [Prn W. Jnsn, 1991] Prn W. Jnsn, Clssl n Morn Mnsm or Ennrs n Invntors, Mrl Dkr, In, Nw York (1991) [22] [Ru, 1974] Ru A., Mtrs Assot wt Knmt ns rom 3 to 5 Mmrs, M. M. Tory, Vol. 9, pp , [23] [Ro, 1989] Ro A.C, Slton o Groun, Input n Output Lnk n Mnsms, pp. A Quntttv Appro, Trns, CSME, Vol. 13, No. 1/2, pp 23 30, [24] [Ro, 1997] Ro A. C., Hmmn Numr Tnqu 2, Gnrton o n plnr knmt Cns, M. M. Tory, Vol. 32, No. 4, pp , [25] [Ro, 2000] Ro A. C., Applton o Fuzzy Lo or t Stuy o Isomorpsm, Invrsons, Symmtry, Prlllsm n Molty n Knmt Cns, M. M. Tory, Vol. 35, pp , [2] [Ro,t.l.,200] Ro..s., Srnt A. n Ro A.C., Syntss o Plnr Knmt Cns, IE (I), vol.8, pp , 200. [27] [Ro n Ro, 1993()] Ro A. C. n Ro C. N, Loop Bs Psuo Hmmn VluI, Tstn Isomorpsm An Rtn Knmt Cn., M. M. Tory, Vol. 28, No. 1, pp , [28] [Ro n Ro, 1993()] Ro A. C. n Ro C. N, Loop Bs Psuo Hmmn VluII, Invrsons, Prrr Frms n Atutors, M. M. Tory, Vol. 28, No. 1, pp , [29] [Ro t l, 2001] Ro A. C., Prtp n B. Dsmuk, Computr A Struturl Syntss o Plnr Knmt Cns Ovtn t Tst o Isomorpsm, M. M. Tory, Vol. 3, pp , [30] [Tuttl,1975] Tuttl E.R, Gnrton o Plnr Knmt Cns, M. M. Tory, Vol. 31, pp , 199. [31] [Ukr n Ru, 1975] Ukr J. J. n Ru A., A mto or t Intton n Ronton o Equvln o Knmt Cns, M. M. Tory, Vol., pp , [32] [Yv, 199()] Yv J. N., Prtp C. R.n Arwl V. P., Mnsms o knmt n n r o struturl smlrty s on t onpt o lnk pt o, M. M. Tory, Vol. 31, pp , 199. [33] [Yn n Hll, 1982] Yn H. S. n Hll A. S., Lnk Crtrst Polynomls: Assmly Torms, Unqunss ASME, Journl o Mnl Dsn, Vol. 4, pp. 1120, [34] [Yn n Hwn, 1991] Yn n Hwn, Splzton o Mnsms, M. M. Tory, Vol. 2, No., pp , [35] [Zn n L, 1999] Zn W. J. n L Q., On A Nw Appro to Mnsm Topoloy Intton ASME, Journl o Mnl Dsn, Vol. 121, pp. 574, All rts rsrv y 11

Isomorphism In Kinematic Chains

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