FINITE ELEMENT ANALYSIS OF A TWO-DIMENSIONAL LINEAR ELASTIC SYSTEMS WITH A PLANE RIGID MOTION

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1 FINIE ELEMEN ANALYSIS OF A WO-DIMENSIONAL LINEAR ELASIC SYSEMS WIH A PLANE RIGID MOION S. LASE, C. DĂNĂŞEL, M.L. SCUARU, M. MIHĂLCICĂ RANSILANIA Univrsity of Braşov, RO , B-dul Eroilor, 9, Romania, svlas@unitbv.ro Rcivd January 4, 014 Discrtizing a mchanical structur as part of finit lmnt analysis maks it ncssary to us a varity of finit lmnts, according to th shap and gomtry of th analyzd systm. wo-dimnsional finit lmnts ar widly usd in finit lmnt analysis and using thm is gnrally asy and fast. For mchanical systms that also xhibit a gnral rigid plan motion, dynamic analysis nforcs th us of numrous additional trms that can no longr b nglctd. In this papr w dtrmin motion quations for a two-dimnsional finit lmnt and w distinguish th additional trms that can vntually chang th qualitativ bhavior of th systm. h motion quations ar obtaind using Lagrang s quations. Ky words: finit lmnt, nonlinar systm, lastic lmnt, shll finit lmnt. 1. INRODUCION Any multibody mchanical systm consists, mor or lss, of lastic lmnts. A first approximation in th study of tchnical systms is givn by th rigid lmnts hypothsis. h hypothsis works vry wll in th dynamic analysis of systms which ar sufficintly slow or mildly loadd. If, howvr, th vlocitis and th loads ar high, th lastic bhavior of th componnts will gnrally ngativly influnc th opration of th systm. W also nd to study th phnomna of rsonanc and loss of stability as manifstations of lasticity. h analytic approach of such problms, using th classical thorms of th mchanics of continuous systms, lads to practically unsolvabl systms of diffrntial quations. his lavs us with th numrical mthods, rprsntd mainly by th finit lmnt mthod. h advantags of this approach rsult from Grstmayr and Schöbrl, 006, Khulif, 199, las, h motion quations yildd by th finit lmnt mthod hav a complx form, ar strongly nonlinar and contain a numbr of additional trms rsulting from th rigid motion insid th systm. Prvious paprs on this topic analyzd systms with a singl dformabl lmnt and a plan motion (s for xampl Bagci, 1983, Bahgat and Rom. Journ. Phys., ol. 59, Nos. 5 6, P , Bucharst, 014

2 Finit lmnt analysis of a two-dimnsional systm 477 Willmrt, 1976, Clghorn t al., 1981, Fanghlla t al., 003, Nath and Ghosh, 1980, Sung, 1986) or mor complx systms. h rsults obtaind wr synthtizd by Ibrahimbgović t al., 000, hompson and Sung, For a gnral thrdimnsional motion of systms modld with on or thr-dimnsional finit lmnts w mntion las, 01, 013a. h rsarchs in th domain also targtd som mor complx aspcts of th problm, about th calculus, xprimntal vrification and control for th mchanical systms, usually having som form of simplicity (imposd by th difficulty of th numrical approach - Dü t al., 008, Mayo and Domínguz, 1996, Piras and Mills, 005). h influnc of damping, th stability and th us of composit matrials in such systms wr also aspcts which wr studid (s Nto t al., 006, Shi t al., 001, Zhang and Erdman, 001, Zhang t al., 007) along with th thrmal ffcts (s Hou and Zhang, 009). h main difficulty is constitutd by th symbolic rprsntation of th quations of motion and by th adquat slction of intgration mthods. Modls for th bi and tri-dimnsional motion wr dvlopd by hompson and Sung, his papr aims to prform th dynamic analysis and to dtrmin th quations of motion for a two-dimnsional finit lmnt, having a rigid planparalll motion. It is prviously considrd that th fild of vlocitis and acclrations was alrady dtrmind for all th lmnts of th systm (considrd rigid). A finit lmnt in a "rigid" motion along with th body which it discrtizs will b analyzd. In ordr to dtrmin th quations of motion, th Lagrang quations will b usd. For this, it is ncssary to dtrmin th kintic nrgy and th intrnal nrgy for th lmnts of th mchanical systm (considrd linar lastic). If th obtaind quations will b compard with th stady stat rspons quations, nw trms will b found. hs appar du to th rlativ motion of indpndnt coordinats rlativ to th mobil rfrnc systms which ar attachd to th lmnts of th mchanical systm (Coriolis ffcts) and thy will dtrmin changs in stiffnss and in th inrtial trms along with th apparanc of som consrvativ damping.. MOION S EQUAIONS Lt's considr a two-dimnsional finit lmnt with a plan paralll motion. h typ of finit lmnt which will b usd will dtrmin th shap functions and th final form for th matrix cofficints. In what follows it is considrd that th dformations ar small nough not to influnc th gnral rigid motion of th systm. Both th problm of th rigid motion of th systm and th fild of vlocitis and acclrations for ach two-dimnsional lmnt of th multicorp systms ar considrd to b solvd (s Pnnstri, 009). h finit lmnt is rlatd to th

3 478 S. las t al. 3 local coordinats systm Oxy, which is mobil and which participats to th whol motion (fig. 1), so it has a known rigid motion. Lt's considr v o( X o, Y o,0) bing th vlocity and a o( X o, Y o,0) bing th acclration of th origin of th mobil rfrnc systm rlatd to th fixd rfrnc systm OXY, to which th whol motion of th mchanical systm rlats. W will considr ω (0, 0, ω z ) as bing th angular vlocity of th solid containing th finit lmnt and ε (0, 0, ε ) as bing th angular acclration of th sam solid. h chang in th valus of th position vctor r of a random point M of th finit lmnt whn going from th local Oxy rfrnc systm into th global fixd OXY rfrnc systm is obtaind using a rotation transformation matrix R. If w considr r M, G as bing th position vctor of th M point, w hav: r M, G M, G = r O, G + r G = r O, G + R r L, (1) whr w considr G dfining th vctorial ntitis which hav thir componnts rlating to th global rfrnc systm and L dfining th vctorial ntitis which hav thir componnts rlating to th local rfrnc systm. Fig.1 A bidimnsional finit lmnt with plan motion. h rotation transformation matrix R has, in this situation, a vry simpl form: R cos θ sin θ = sin θ cos θ. () If th M point is subjctd to a displacmnt f L, transforming into th M' point, w will hav:

4 4 Finit lmnt analysis of a two-dimnsional systm 479 r M ', G = r O, G + R ( r L + f L ) (3) whr r M ', G is th position vctor of th point M' with its componnts rlating to th global rfrnc systm. h continuous displacmnt fild f (x,y) L is approximatd in th finit lmnt mthod, dpnding on th nodal displacmnts, using th rlationship: ( x, y) ( t) f = N δ (4) L L whr th lmnts of th N matrix (which contains th intrpolation functions) dpnd on th typ of th chosn finit lmnt. h vlocity of th M' point, rlatd to th fixd coordinat systm, will b: vm,g = r0 + RrL + RfL + Rf L = r0 + RrL + RN δ,l + RNδ, L (5) h vlocity componnts ar dfind rlatd to th global coordinats systm. h quations of motion will b obtaind in th local coordinats systm. h kintic nrgy of th considrd lmnt will b dtrmind using th rlationship: 1 1 = ρ = ρ v M, L Ec v d ' v M, L ' d. (6) Whn w dtrmind th kintic nrgy, w could hav bn rprsnting th vlocity in th global coordinats systm, th rsult for th kintic nrgy rmaining th sam. his happns bcaus th scalar product of two vctors rmains th sam, no mattr which rfrnc systm w us. h dformation nrgy is: E p = 1 d σ ε. (7) For th as of undrstanding, w rmmbr that th gnralizd Hook law is: σ = D ε, (8) whr, for a two-dimnsional finit lmnt (plan stat of dformation) w hav: D 1 µ 0 0 E = 0 1 µ 0 (1 µ )(1 µ ) 1 µ 0 0 (9)

5 480 S. las t al. 5 and th rlation btwn th spcific dformations and th finit dformations can b xprssd using th rlationship: whr a is a diffrntiation oprator: a ε = a f, (10) 0 0 x = 0 0 y 0 y x Considring all this, for th dformation nrgy w hav: with k bing th stiffnss matrix: (11) E = 1 p, L δ k δ d, (1) L, k = N a D a N d. (13) If w considr p = p (x,y) as bing th vctor for th distributd forcs, thn th xtrnal machin work of thos forcs is: L ( L d ), W = p f d = p N δ. (14) L L h nodal forcs q giv an xtrnal machin work: W h Lagrangan for th considrd lmnt is: L= E E + W + W = c p c c = q, L δ, L. (15) ( L ) 1 1 = 0 0 d,, d ρr r γ L L + d,l+,l,l. k δ p N δ q δ h quations of motion ar obtaind by applying th Lagrang quations. Aftr som ordring, w can writ th quations of motion for th finit lmnt lik this: (16)

6 6 Finit lmnt analysis of a two-dimnsional systm 481 ( N N ρd) δ, L + ( N R R N ρd) δ, L +( k + N R R N ρd) δ, L = (17) = q + N p d ( L N ρd) R r O N R R r ρd W can writ th quations of motion in concntratd form: ( ) ( ) ( ε ) ( ω i ) 0 m δ +,L cδ,l + k + k ε + k ω δ =,L = q * i i + q,l q,l q,l m R r0. whr w considrd: L d ; m i O = N ρd ; N q *,L = p m = N N ρd = m 11 + m ; c (ω)= N R R N ρd ; (19) k (ε) + k (ω ) = N R R N ρd ; q i, L (ε) + q i, L (ω) = N R R r ρd. h matrix products whr th rotation transformation matrix appars can asily b calculatd, as this matrix is dtrmind by only on lmnt, th angl θ. h anti-symmtric matrix: ω = R R ω = =ω ω (0) rprsnts th angular vlocity oprator corrsponding to th angular vlocity ω. W also hav: R R =ε 1 0 ω 0 1. (1) In both th global and local rfrnc systms, th angular vlocity and angular acclration vctors hav th sam componnts. If w considr N (1) and N () bing th rows of th matrix N, with th notations: m 1x = N (1) x ρd ; m 1 y = N ( 1) y ρd ; m 11 = N 1) m x = N ( ) x ρd ; m y = N ( ) y ρd ; m = N ) ( N (1) ( N () ρd ; (18) ρd ; ()

7 48 S. las t al. 7 w can obtain th quations of motion for th finit lmnt, with xplicit dpndncis to th angular vlocity ω and angular acclration ε. Prviously, w solv th intgrals with th givn notations and w can obtain a dtaild form for th quations of motion. hrfor, w hav: m = ( m 11 + m ) ; c (ω) =ω ( m k (ε) + k (ω ) q i, L (ε) + q i L = ε ( m ) 1 m 1 m1 1 ) ω ( m11 m ), (ω) = ε( ) - ( ) ω x + m y m x m1 y h quations of motion will b, in this cas: = q + liaison xt ( m 11 + m ) δ m1, L + ω ( m 1 m 1 ) δ, L + ω m + m ] +[k + ε( m ) - ( ) 1 m 1 q ε( ) + ( ) ω + m x m1 y 11 m1 x m y, L (3) δ = (4) m i R O r O. h unknowns in th finit lmnt analysis of such a systm ar of two kinds: nodal displacmnts and contact forcs. Using a corrct assmbly of th quations of motion, writtn by ach finit lmnt, th algbraic unknowns (th contact forcs) can b rmovd (s Blajr and Kołodzijczyk, 011, Khang, 011, las, 1987a,1987b). h quations of motion for th whol multibody systm will b possibl to b writtn as a systm of nd ordr non-linar diffrntial quations. h matrix cofficints of this systm of quations hav th following proprtis: h inrtial matrix m is symmtric; h damping matrix c is a skw symmtric matrix. h trms dfind by this matrix rprsnt th Coriolis acclrations, du to rlativ motions of nodal displacmnts with rspct to th mobil rfrnc coordinat systm, linkd to th moving parts of th systm studid; h stiffnss matrix k contains both symmtric and skw symmtric trms. Morovr, this matrix can hav singularitis du to th rigid motion of th systm that hav to b rmovd bfor conducting th study of th systm. 3. MODAL ANALYSIS OF A ROAING DISK Lt s considr a rotating disk. If w considr a stationary motion, th disk will rotat with an angular vlocity ω. In a vry small intrval t w may considr that th angular vlocity is constant and w intnd to achiv th modal analysis of rotating disk. W will us th mthod prviously prsntd for carrying out th calculation and compar th rsult with that obtaind by applying th classic vrsion of th mthod of finit lmnts. W considr two cass:

8 8 Finit lmnt analysis of a two-dimnsional systm 483 A. h rotating disk has a plan motion and th finit lmnt is in a plan displacmnt fild. h ignvalus ar prsntd in abl 1 and th ignvctor in Fig. Fig. 6. MODE NUMBER (WIHOU RIGID BODY MODE) abl 1 h ignvalus of th doubl cardan joint CLASSIC MODEL EIGENALUES (HZ) PROPOSED MODEL EIGENALUES (HZ) Fig. Eignmods 4,5 and 6. Fig. 3 Eignmods 7,8 and 9.

9 484 S. las t al. 9 Fig. 4 Eignmods 10,11 and 1. Fig. 5 Eignmods 13, 14 and 15. Fig. 6 Eignmods 16,17 and 18. B. h rotating disk has a plan motion and th finit lmnt is in a gnral displacmnt fild. h ignmods ar prsntd in Fig.7-Fig.11.

10 10 Finit lmnt analysis of a two-dimnsional systm 485 Fig. 7 Eignmods 7,8 and 9. Fig. 8 Eignmods 10,11 and 1. Fig. 9 Eignmods 13,14 and 15. Fig.10 Eignmods 16,17 and 18.

11 486 S. las t al. 11 Fig. 11 Eignmods 19,0 and CONCLUSIONS h additional trms in th quations of motion will influnc th dynamic rspons of th systm. It may happn that a rsonant stat or a loss of stability stat to b rachd. h modifications of ignvalus considring a rotating disc (along its own axis) is prsntd in th papr. An upwards displacmnt for all th ignvalus can b obsrvd - this happns bcaus an incras in stiffnss taks plac in this situation (rotation), du to th inrtial forcs. h vibration mods rmain practically th sam, whil a small chang in th amplituds can b obsrvd. h prsnc of inrtial and Coriolis ffcts can significantly modify, in som situations, th dynamic rspons of th systm. REFERENCES 1. C. Bagci, Elastodynamic Rspons of Mchanical Systms using Matrix Exponntial Mod Uncoupling and Incrmntal Forcing chniqus with Finit Elmnt Mthod. Procding of th Sixth Word Congrss on hory of Machins and Mchanisms, India, p. 47 (1983).. Bahgat, B.M., Willmrt, K.D., Finit Elmnt ibrational Analysis of Planar Mchanisms. Mchanism and Machin hory, vol.11, p. 47 (1976). 3. Blajr W., Kołodzijczyk K., 011, Improvd DAE formulation for invrs dynamics simulation of crans, Multibody Syst Dyn 5: Clghorn, W.L., Fnton, E.G., abarrok, K.B., Finit Elmnt Analysis of High-Spd Flxibl Mchanism. Mch.Mach.hory, 16, p. 407 (1981). 5. Dü J.-F., Galucio A.C., Ohayon R., 008, Dynamic rsponss of flxibl-link mchanisms with passiv/activ damping tratmnt. Computrs & Structurs, olum 86, Issus 3 5, pp Fanghlla P., Galltti C., orr G., 003, An xplicit indpndnt-coordinat formulation for th quations of motion of flxibl multibody systms. Mch. Mach. hory, 38, p Grstmayr, J., Schöbrl, J., A 3D Finit Elmnt Mthod for Flxibl Multibody Systms, Multibody Systm Dynamics, olum 15, Numbr 4, (006). 8. Hou W., Zhang X., 009, Dynamic analysis of flxibl linkag mchanisms undr uniform tmpratur chang. Journal of Sound and ibration, olum 319, Issus 1, Pags

12 1 Finit lmnt analysis of a two-dimnsional systm Ibrahimbgović, A., Mamouri, S., aylor, R.L., Chn, A.J., Finit Elmnt Mthod in Dynamics of Flxibl Multibody Systms: Modling of Holonomic Constraints and Enrgy Consrving Intgration Schms, Multibody Systm Dynamics, olum 4, Numbrs 3, (000). 10. Khang N.., Kronckr product and a nw matrix form of Lagrangian quations with multiplirs for constraind multibody systms, Mchanics Rsarch Communications, olum 38, Issu 4, pp , Khulif, Y.A., On th finit lmnt dynamic analysis of flxibl mchanisms. Computr Mthods in Applid Mchanics and Enginring, olum 97, Issu 1, Pags 3 3 (199). 1. Mayo J, Domínguz J., 1996, Gomtrically non-linar formulation of flxibl multibody systms in trms of bam lmnts: Gomtric stiffnss. Computrs & Structurs, olum 59, Issu 6, Pags P.K. Nath, A. Ghosh, Kinto-Elastodynamic Analysis of Mchanisms by Finit Elmnt Mthod, Mch.Mach.hory, 15, pp. 179 (1980). 14. Nto M.A., Ambrósio J.A.C, Lal R.P., 006, Composit matrials in flxibl multibody systms. Computr Mthods in Applid Mchanics and Enginring, olum 195, Issus 50 51, p Pnnstri', E., d Falco, D., ita, L., An Invstigation of th Influnc of Psudoinvrs Matrix Calculations on Multibody Dynamics by Mans of th Udwadia-Kalaba Formulation, Journal of Arospac Enginring, olum, Issu 4, pp (009). 16. Piras G., Clghorn W.L., Mills J.K., 005, Dynamic finit-lmnt analysis of a planar high spd, high-prcision paralll manipulator with flxibl links. Mch. Mach. hory, olum 40, Issu 7, p Shi Y.M., Li Z.F., Hua H.X., Fu Z.F., Liu.X., 001, h Modlling and ibration Control of Bams with Activ Constraind Layr Damping. Journal of Sound and ibration, olum 45, Issu 5, p Sung, C.K., 1986, An Exprimntal Study on th Nonlinar Elastic Dynamic Rspons of Linkag Mchanism. Mch. Mach. hory, 1, p hompson, B.S., Sung, C.K., A Survy of Finit Elmnt chniqus for Mchanism Dsign. Mch.Mach.hory, 1, nr. 4, p (1986). 0. las, S., Contributions to th lastodynamic analysis of th mchanisms with th finit lmnt mthod. Ph.D., RANSILANIA Univrsity (1989). 1. las, S., A Mthod of Eliminating Lagrangan Multiplirs from th Equations of Motion of Intrconnctd Mchanical Systms. Journal of applid Mchanics, ASME ransactions, vol. 54, nr. 1 (1987a).. las, S., Elimination of Lagrangan Multiplirs. Mchanics Rsarch Communications, vol. 14, pp. 17 (1987b). 3. las S., Dynamical Rspons of a Multibody Systm with Flxibl Elmnts with a Gnral hr Dimnsional Motion, Rom. J. Phys., 57 (3 4), , las, S., odorscu, Elasto-dynamics of a solid with a gnral rigid motion using FEM modl. Part I, Rom. J. Phys., 58 (7 8), , Zhang X., Erdman A.G., 001, Dynamic rsponss of flxibl linkag mchanisms with viscolastic constraind layr damping tratmnt. Computrs & Structurs, olum 79, Issu 13, p Zhang X., Lu J., Shn Y., 007, Simultanous optimal structur and control dsign of flxibl linkag mchanism for nois attnuation. Journal of Sound and ibration, olum 99, Issus 4 5, p