EIGENVALUES AND EIGENMODES OF AN INCLINED HOMOGENEOUS TRUSS IN A ROTATIONAL FIELD

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1 IVAUS AD IMODS OF A ICID HOMOOUS RUSS I A ROAIOA FID S. VAS RASIVAIA Unrsty of Braşo RO-536 B-dul rolor 9 Romana -mal: slas@untb.ro Rcd January 4 4 h tchncal alcatons of th last dcads ar assocatd to mchancal systms hch or th orful forcs at hgh locts. h rgd lmnts hyothss hch s usd n most ngnrng alcatons dos not or n ths stuaton hr th lastcty of th bods lads to snsbly dffrnt bhaors hr bg dformatons ntns bratons and mchancal nstablty hnomna can ars. In ths ar am to us fnt lmnts n ordr to obtan th modl of a homognous slong truss n a rotatonal fld. Ky ords: gnalus fnt lmnt mthod lnar lastc lmnts agrang s quatons on-dmnsonal fnt lmnt rotatonal fld.. IRODUCIO Any mchancal systm s comosd of sold mor or lss lastc lmnts. h rgd lmnts hyothss hch s usually usd hn a mchancal systm s studd rrsnts an ntal aromaton n th dynamc analyss and t could or could not b satsfactory dndng of th lmnts lastcty th stng locts and forcs dtrmnd by th oratng mod of th studd mchancal systm. If ths locts and forcs ha sgnfcantly bg alus th lastcty bcoms sgnfcant also and t can qualtatly chang th dynamc rsons of th systm. h bratons hnomna th bg dformatons hch modfy th gomtry and th loss of stablty ll rrsnt n fact forms of manfstaton of th systms lastcty and can dtrmn th damag or th malfuncton of th systm. hortcally sang th study of such a systm can b dlod usng contnuous mathmatcal systms usng th fundamntal thorms of th contnuum mchancs. h man dsadantag of ths aroach s rrsntd by th rsultng artal drats dffrntal quatons hch ar dffcult to ntgrat [ ]. A much mor connnt ay to aroach ths roblm coms from usng numrcal mthods to dscrt th lastc doman. h fnt lmnt mthod rrsnts undr ths aroach th most connnt study mthod. h adantags of ths aroach rsult from [8 ]. Rom. Journ. Phys. Vol. 59 os. 7 8 P Bucharst 4

2 7 S. Vlas h ars hch dlod ths aroach usd on-dmnsonal fnt lmnts n a lan moton [ 5 7 3] to-dmnsonal fnt lmnt [4] or n a satal moton [ 3]. h ars [9 8] synths th rsults n ths doman. h am of ths ar s to aly th fnt lmnts mthod n ordr to study th bhaor of a nclnd truss n a rotatonal fld. h shar ffct on bndng s also consdrd.. MOIO S QUAIOS.. DSCRIPIO OF H MCHAICA MOD In th follong ll dtrmn th moton quatons hch dscrb th bhaor n tm of a truss fnt lmnt n a rotatng moton around an as along th th jost hch t dscrts Fg.. Fg. Inclnd truss lmnt n a rotatonal moton. In ordr to obtan th quatons of moton ll us th agrang quatons. hs mans that nd to dtrmn th ntc nrgy th ntrnal nrgy hch arss du to th lastc bhaor of th jost and th mchancal or of th dstrbutd and concntratd forcs n ordr to b abl to dtrmn th agrangan of th studd mchancal systm. h ndndnt coordnats of th nods hch ll rt th moton quatons for dnd both on th ty of th chosn fnt lmnt and on th sha

3 gnalus and gnmods of an nclnd homognous truss 7 functons. If chos a fnt truss lmnt th to nods th ndndnt coordnats ll b th ctors of th nodal dslacmnts hch ll nclud th dslacmnts undr th thr man drctons and thr drats hch ha th sgnfcaton of som rotatons or of som curaturs. h numbr of chosn ndndnt coordnats ll dnd on th chosn bar hyothss and on th tys of th chosn fnt lmnts. W ll obtan th gnral moton quatons for a fnt lmnt n a cntrfugal fld by customng th rsults obtand n []... FORM OF H ARAIA W ll us th hyothss that th lastc dformatons ar small nough not to nflunc th gnral moton of th systm. W ll suos that no th angular locty and th angular acclraton for th rotaton. ts consdr a truss fnt lmnt th and j bng th nd nods. h and j ctors rrsnt th ndndant dslacmnts of th trusss nds Fg.. h fnt lmnt ll b rfrncd to a local coordnats systm Oy hch s n a non rotatng moton. W can dtrmn th locty and th acclraton for th orgn of th mobl coordnats systm n raort th th fd coordnats systm OXYZ. Fg. On-dmnsonal fnt lmnt th a thr-dmnsonal moton.

4 7 S. Vlas A random ont M has ts dslacmnt ctor f u hch can b rttn n th trms of nodal dslacmnts as: and: u f α α α α h nodal dslacmnts ctor : ; h matr contans sha functons. h lns of th matr corrsondng to th dslacmnts u and ar namd u and : W can thn rt: u 3 u f u 3. 4 h quatons for th rotatons of th nds of th jost β and ar rttn n corrlaton th th dslacmnts of th truss undr Oy and rsctly O drctons []: d d β and. 5 d d If consdr th sha functons th rotatons of th nds of th jost ll ha th form: d d β ; ; d In th nd can rt: 6 d

5 gnalus and gnmods of an nclnd homognous truss 73 ; β α α β α hr: β α α. W notd: β and. In hat follos nd s ald to a ctor or a matr th th alus rssd n th global rfrnc systm and nd s ald to a ctor or a matr th th alus rssd n th local rfrnc systm. h oston ctor of ont M bcoms n th dformd oston M []: u u o M M M r r r r 7 or th rsct to th global rfrnc systm: o o M M u u r R R r R R r R r 8 h matr R rsss th transton of th ctors from th mobl rfrnc systm Oy to th fd rfrnc systm OXYZ. h ntc nrgy rsson for th hol fnt lmnt fnt s: M M c d A I r r ρ 9 hr: y y yy I I I I I I. yy I and I rrsnt momnts of nrta of th bar cross scton about coordnat as Oy and O rsctly of a rfrnc systm th ts orgn n th mass cntr of th lmnt Ad dm ρ ρ - dnsty; y I s th cntrfugal momnt

6 74 S. Vlas of nrta and I s th nrta momnt about th co-ordnat as O. Snc ha chosn y and as rncal drctons of nrta I y ha: I I I y I For smlcty notd: I I Iyy Iy I I. In th rttn rlatons can notc th occurrnc of drats of th matr for ostonal rlaton to th global rfrnc systm R. hs drats rrsnt angular locts and acclratons. For a bttr undrstandng shall rss ths drats accordng to []: ε ε ; ε. ε h angular locty and acclraton ctors shall ha n th local rfrnc systm th comonnts: sn α ; cos α ; ε sn α ; ε cos α. 3 h angular locty of th lmnt dm s: α snα β cos α hr: s th comonnts of th angular locty ctor along th O as and. α β h matr orator angular locty ll b n ths cas: ~ c c s s 4 ; 5 6

7 gnalus and gnmods of an nclnd homognous truss 75 n th local coordnats systm and: ~ 7 ~ R R ; ~ R R. 8 W shall also ha th angular acclraton tnsor ε ~ dfnd as: ε ~ R R R R ; ε ~ R R R R. 9 In ordr to obtan th agrangan of th systm nd to dtrmn th dformaton nrgy. h nrgy du to th bndng of th lmnt can b rttn usng th rsson: d d y yβ d d I I d I I d If consdr 4 lctly rttn: thn: ;. or: I y I d hr: I y I h nrgy du to aal dformaton s: d. a du A d d u u Ad 3

8 76 S. Vlas or: hr: h dformaton nrgy du to torson s: a a 4 u u Ad a. 5 or: t dα I d d I d α α 6 t 7 t hr: t α α I d. 8 h dformaton nrgy du to th bndng tracton-comrsson and torson s: hr: t 9 a a t t. 3 h aal forc P hch aars n th transrsal scton of th truss ll g n hyothss that n a frst aromaton th aal dformaton can b nglctd and th dformaton nrgy s: tot d 3 d d a P d d hr P tot s th aal forc hch rsds n th bar cross scton at th dstanc from th nd of th bar. h forc comonnts actng at th rght bar nd consdrd n th local coordnat systm ar rrsntd by P P y P. Bsd ths comonnts th alu of P and th comonnts of th nrta forcs

9 gnalus and gnmods of an nclnd homognous truss 77 actng uon th orton of th bar btn and ar bng dtrmnd. h nflunc of th nrta forcs n th transrsal dformaton s nglctd. In ordr to calculat thm dtrmn th acclraton of th currnt ont of th bar n th rgd moton fg.3: sn ϕ a ao ε snϕ 3 snϕ cosϕ Fg. 3 Dtrmnaton of th aal nrta forc. h nrtal forc n th scton hch strtchs th truss s gn by th formula: s sn ϕ F adm ao ρas d ε ssnϕρ As d ρ As d s sn cos ϕ ϕ X sn ϕ o ε Y o ρa sn ϕρ A ρa 33 snϕcosϕ µ X o sn ϕ ε { µ } µ y ρa Y o sn ϕρ A ρa µ snϕcosϕ λ X o λ λ yρay o ; 34 λ { }

10 S. Vlas 78 {} sn sn sn cos y A A ν ϕ ε ν ν ϕ ρ ρ ν ϕ ϕ. 35 W ll obtan thn th ntrnal nrgy du to th aal forc hch arss n th bar du to th rotaton: a d β β ν λ µ 36 or: a 37 hr d β β ν λ µ d d λ µ d ν 38 hr th notatons for ar obous. h total ntrnal nrgy s: t a. 39 h trnal or of dstrbutd loads s: [ ] d m m m d m m m u W y y y y β α 4

11 gnalus and gnmods of an nclnd homognous truss 79 and that of concntratd loads q n th nods s: c W q 4 Usng all of ths can no buld th agrangan for th studd fnt lmnt: W W c c a SI M S MOIO QUAIOS h moton quatons n trms of ndndnt coordnats ar obtand by alyng th agrang quatons [ ]: If not: d. 43 dt mt ρ A d nrta of translaton; m ρ ; j3; j j Ad mr ρ I d ρi α α ρi y β β ρi d ; nflunc of nrta ffcts at cross-scton rotaton c m m m ; ~ ρad ; R R ρad t r ε ~ ε ρad ; ~ ρad ; ~ mo ρa d; d m ; m ρad ; j3. [ m m m ] d q y y ; 44 th moton quatons tan from th local coordnats systm can b rttn n a sml form:

12 7 S. Vlas m q c q ε q ε q m I ε m o R r o OBA COORDIA SYSM h consdrd truss s comosd of fnt lmnts lnd btn thm usng nods. h forcs hch ars n ths nods ll b rmod durng th assmbly rocss. h unnon aramtrs hch aar n an lastc systm ar of to nds: nodal dslacmnts and lason forcs. h solng mthods for th dffrntal and algbrac systms of quatons DA focus on tryng to rmo th algbrac unnons [4 9 ] hch usually ar lason forcs; n ths stuaton th only unnons lft ar th nodal ndndnt dslacmnts. h mchancal lns btn dffrnt fnt lmnts ar usually dfnd by lnar rlatonshs btn th nodal dslacmnts as: A q 46 or for th hol mchancal systm hr hch rrsnts th global coordnats systm ll b dscardd n th follong quatons: n A A A q An In rl.47 rrsnts th ctor of nodal dslacmnts rrsnts th ctor of nodal dslacmnts of an lmnt th th comonnts rssd n th fd coordnats systm and q s th ctor of ndndnt coordnats: [ q q ] q. q s 47 If dffrntat 47 to tms obtan: Aq 48 h quaton hch dtrmns th ln btn th comonnts of th nodal dslacmnts ctor n th fd coordnat systm and th mobl coordnat systm s: R. 49

13 gnalus and gnmods of an nclnd homognous truss 7 If consdr 4547 aftr sml mathmatcal oratons ll obtan th moton quatons for th hol systm: Q t Q t K K ε K M C Q lason Q lason Q If consdr and 5 can rt th quatons n rlaton to th ndndnt coordnats q : Q t Q t Q K K ε K M A q C A q lason Q nrta nrta If r-multly th rlaton 5 by A [4 9 ] thn obtan: A MA q A A A Q Q t t A A CA q A Q Q t t A A K K ε K Q Q lason nrta A Q lason A q A A q In th rous quaton 5 consdrd that th lason forcs ll dsaar as on of thr orthogonalty ffct th th dslacmnts [4] []. Fnally ll gt: m q c q q Q 53 h systm of dffrntal quatons hch ll obtan s non-lnar th matr coffcnts dndng at ry momnt on th gomtrcal confguraton of th studd mchancal systm. h matr coffcnts for th dffrntal quatons hch obtand ha th follong rorts: th nrtal matr m s symmtrc dfnd as ost th c matr s s-symmtrc and th matr rrsnts th sum of to symmtrc matr dfnd as non-ngat th a s-symmtrc matr. h rsultng systm s strongly non-lnar th matr coffcnts dndng on th tm aramtr. h asst ay to sol th systm s by lnaraton hr consdr som trmly small tm ntrals so that th gnral moton can b consdrd as bng th sum of som motons hch ta lac n a ry small tm ntral. On ths ry small tm ntrals obtand can aly modal analyss so can dtrmn th gnctors and th gnalus. Q nrta IVAUS AD IVCORS h roblm of dtrmnng th gnctors and th gnalus for a mchancal systm s on of a artcular ractcal mortanc. If n 53 nglct th damng Corols matr and th trnal forcs th gnalus roblm rducs to:

14 7 S. Vlas dt m 54 If consdr th Corols damng ffct thn th quatons systm dfnd n 53 can b brought to a frst ordr dffrntal quatons systm. Usng th substtuton: q X ; q X 55 X and X ; 56 X obtan th systm of dffrntal quatons: X X m m hch has th corrsondng gnalus roblm: dt m m X c X ; 57 λ. 58 c h systm of dffrntal quatons hch obtand 53 has th follong mortant rorty: In th Raylgh Quotnt th gnalus Do ot Dnd Drctly on th Damng Matr ts consdr a ry short tm ntral hn th matr coffcnts of th systm can b consdrd constant. ts consdr a soluton for th systm 53 n form of: q A cos t ϕ 59 By succss dffrntatons ll obtan: q A sn t ϕ ; q A cos t ϕ 6 and f rt ths n 53 ll obtan: m Acos t ϕ ca sn t ϕ Acos t ϕ 6 hn dtrmn th gnalus us Q. If r-multly th rlatonshs th A obtan: A m Acos t ϕ A c A sn t ϕ A Acos t ϕ 6

15 gnalus and gnmods of an nclnd homognous truss 73 and bcaus c s s-symmtrc obtan: A c A 63 from hch obtan th ulsaton th Raylgh quotnt [6]: A A 64 A m A hrfor th c matr hang th sgnfcanc of a damng matr ll not brng any ral damnng n th systm. h s symmtry sgnfy that ths trm dos not lad to any nd of tror nrgy dssaton. h rol of ths matr s to modfy th consrat nrgy nsd th systm. 5. COCUSIOS In cas of th moton of an nclnd lastc truss n a cntrfugal fld th artcular natur of th angular locty and angular acclraton ctors dtrmns a smlfcaton of th quatons of moton and consquntly th ossblty to asly dtrmn th gnalus and th gnctors for such a roblm. h locts and acclratons fld bcoms a artcular fld dntcal to that corrsondng to th rotaton of th fd-as rgd. hs s a ry common nd of roblm n th tchncal fld for aml at th hlcotr rollrs ll. For ths rason th ffct dfnton of th moton quatons for ths artcular stuatons has grat ractcal alu. Furthrmor th scfc stuatons hch ars n ractcal ngnrng alcatons lad to ths quatons to ha rmarabl rorts and artcularts hch hl as th calculus ffort n ordr to dtrmn th gnalus. RFRCS. Bagc C. lastodynamc Rsons of Mchancal Systms usng Matr onntal Mod Uncoulng and Incrmntal Forcng chnqus th Fnt lmnt Mthod Procdng of th Sth Word Congrss on hory of Machns and Mchansms Inda Bahgat B.M. Wllmrt K.D. Fnt lmnt Vbratonal Analyss of Planar Mchansms. Mchansm and Machn hory ol Bancoln M.. Brutt C. Pnnstr. Valntn P.P. Dynamc mchancal ffcncy and fatgu analyss of th doubl cardan homontc jont Journal of Vhcl Dsgn ol. 3 n. 3/ Blajr W. Kołodjcy K. Imrod DA formulaton for nrs dynamcs smulaton of crans Multbody Syst Dyn Clghorn W.. Fnton.. abarro K.B. Fnt lmnt Analyss of Hgh-Sd Flbl Mchansm Mch.Mach.hory

16 74 S. Vlas 6. nscu M. Alandru C. Modlng and Smulaton of a 6 DOF Robot. Procdngs of Intrnatonal Confrnc on Otmaton of th Robots and Manulators OPIROB Snaa Romana 6 8 May. 7. rdman A.. Sandor.. Oabrg A. A nral Mthod for Knto-lastodynamc Analyss and Synthss of Mchansms. Journal of ngnrng for Industry. ASM rans rstmayr J. Schöbrl J. A 3D Fnt lmnt Mthod for Flbl Multbody Systms Multbody Systm Dynamcs Volum 5 umbr Ibrahmbgoć A. Mamour S. aylor R.. Chn A.J. Fnt lmnt Mthod n Dynamcs of Flbl Multbody Systms: Modlng of Holonomc Constrants and nrgy Consrng Intgraton Schms Multbody Systm Dynamcs Volum 4 umbrs Khulf Y.A. On th fnt lmnt dynamc analyss of flbl mchansms Comutr Mthods n Ald Mchancs and ngnrng Volum 97 Issu Pags Marn M. Harmonc bratons n hrmolastcty of mcrostrtch matrals. ASM Journal of Vbraton and Acoustcs Vol. 3.. Marn M. A doman of nflunc thorm for mcrostrtch lastc matrals onlnar Analyss: R. W. A. Vol ath P.K. hosh A. Knto-lastodynamc Analyss of Mchansms by Fnt lmnt Mthod Mch.Mach.hory Pnnstr. d Falco D. Vta. An Instgaton of th Inunc of Psudonrs Matr Calculatons on Multbody Dynamcs by Mans of th Udada-Kalaba Formulaton Journal of Arosac ngnrng Volum Issu Scutaru M.. Dlu h. uca Motoc D. Inrta drn rducd DO mchansm for aolan ums SM-7. Procdng of ICAM h 3th Intrnatonal Congrss n hortcal and Ald Mchancs Bjng. 6. Scutaru M.. Vlas S. Som Prorts of Moton quatons Dscrbng th onlnar Dynamcal Rsons of a Multbody Systm th Flbl lmnts Journal of Ald Mathmatcs Volum Artcl ID Smon B. On agrang multlrs n flbl multbody dynamc Comutr Mthods n Ald Mchancs and ngnrng Volum 95 Issus 5 5 Pags homson B.S. Sung C.K. A sury of Fnt lmnt chnqus for Mchansm Dsgn Mch.Mach.hory nr Vlas S. A Mthod of lmnatng agrangan Multlrs from th quatons of Moton of Intrconnctd Mchancal Systms Journal of ald Mchancs ASM ransactons ol. 54 nr Vlas S. lmnaton of agrangan Multlrs Mchancs Rsarch Communcatons ol Vlas S. Dynamcal Rsons of a Multbody Systm th Flbl lmnts th a nral hr-dmnsonal Moton Rom. Journ. Phys Vlas S. odorscu P.P. lastodynamcs of a Sold th a nral Rgd Moton usng FM Modl Part I. hortcal Aroach Rom. Journ. Phys Vlas S. odorscu P.P. Itu C. Scutaru M.. lastodynamcs of a Sold th a nral Rgd Moton usng FM Modl Part II. Analyss of a Doubl Cardan JOI Rom. Journ. Phys Vlas S. Danasl C. Scutaru M..MhalccaM. Fnt lmnt Analyss of a o- Dmnsonal nar lastc Systms th a Plan Rgd Moton Rom. Journ. Phys