ON THE NONLINEAR RESONANCE WAVE INTERACTION

Transcription

1 U.P.B. Sci. Bull., Series, Vol. 7, Iss. 3, ISSN 3-77 ON THE NONLINER RESONNCE WVE INTERCTION Pere P.TEODORESCU, Veuria CHIROIU ceasă lucrare sudiază ieracţiuea diamică a uei o bare lieare dispersive aşezaă pe u mediu elasic coiuu, cu u dispoziiv eliear care ese slab coeca la capăul di dreapa. Su sudiae ieracţiuile rezoae ale dispoziivului cu uda icideă care se propagă pri bară folosid meoda coidală. Soluţiile su scrise ca o suma îre o superpoziţie lieară şi ua eliiară de vibraţii coidale. This paper is sudyig he dyamic ieracio of a dispersive liear rod resig o a coiuous elasic foudaio, wih a oliear ed aachme ha is wealy coeced o is righ ed. The resoa ieracios of he aachme wih icide ravelig wave propagaig i he rod are sudied by usig he coidal mehod. The soluios are wrie as a sum bewee a liear ad a oliear superposiio of coidal vibraios. Keywords: resoa wave ieracio, resoace capure, eergy pumpig.. Iroducio Resoa wave ieracio is a oliear process i which eergy is rasferred bewee differe aural modes of a sysem by resoace. For a oliear sysem, he moio is o simply a summaio of he liear modes, bu cosiss of he liear harmoics plus heir oliear couplig [], []. Uder resoace codiios, he oliear couplig bewee differe modes may lead o exciaio of eural modes. ieresig siuaio occurs i sysems of coupled a mai srucure wih a oliear aachme, where isolaed resoace capures are resulig as a cosequece of he eergy pumpig [3-5]. The eergy pumpig is a irreversible rasfer of vibraio eergy from he mai srucure o is oliear aachme. I is ieresig o oe ha his rasie resoa ieracio resuls i broadbad passive absorpio of eergy by he aachme, i coras o he liear vibraio absorber whose effec is arrowbad [6]. The ieracio of icide ravelig waves wih a local defec ca lead o pheomea, such as, speed up or slow dow of he ravelig wave, scaerig of he wave o muliple idepede wave paces, or rappig of he wave a he defec i he form of a localized wave [6], [7]. Prof., Uiversiy of Buchares, Faculy of Mahemaics ad Iformaics, Romaia, pere_eodorescu@yahoo.com Researcher, Isiue of Solid Mechaics of Romaia cademy, Buchares, Romaia

2 8 Pere P. Teodorescu, Veuria Chiroiu I his paper, sarig from he resuls obaied i [6], he eergy exchage bewee a rod ad a oliear ed aachme is aalyzed, for a exeral sie exciaio applied o he assembly. The aalyical soluios of he problem are obaied by usig he coidal mehod [].. The model Cosider a elasic rod of legh L coeced o a grouded local aachme of ui mass, viscous dampig ad siffess olieariy. The coecio bewee he rod ad he oliear ed aachme is made o he poi x = x by meas of a wea liear siffess. Le us assume ha v () is he displacemes of he aachme, he rod is iiially a res ad ha a exeral force f (, ) = siω is applied a he origi O of he coordiae sysem, a =. The displaceme yx (, ) of he rod a he poi of aachme, i he direcio of v (), ca be wrie as [6] yx (, ) = f(, τ) g ( τ)dτ O ( ) ε yx (, τ) v( τ) f(, τ) g ( τ)d τ,, (.) where he Gree s fucio g is he displaceme a poi of he rod i he direcio of v (), due o a ui impulse applied a he same poi ad he same direcio, ad he Gree s fucio g O is he displaceme a poi of he rod i he direcio of v (), due o a ui impulse applied a origi O i he direcio of he exeral force. Cosequely, he moio equaio of he aachme is give by p v () + λv () + α v() = ε[ yx (,) v ()], v() = v () =, (.) = where ε scales he wea couplig, λ deoes he viscous dampig coefficie, ad α, =,..., p, he coefficies of he siffess olieariy. Subsiuig (.) i (.), he followig equaio for he oscillaio of he aachme is obaied p N = = v () + λv () + α v() = ( ) ε Δ, (.3)

3 O he oliear resoace wave ieracio 8 Δ = (, )* ( )* ( ) + ( ) ( )* ( ). f go g v g The Gree fucios are expaded by a se of orhogoal polyomials ϕ () g() = c () w ϕ *, c = w() ϕ()d. = The polyomials ϕ ( ) saisfy he orhogoaliy ad compleeess codiios * w () ϕ() ϕm()d = wδm, ad also a recurrece formula. + w () δ( ) = ϕ ( ) ϕ ( ), (.4) * m = w ϕ () = b ( a ) ϕ () c ϕ (). (.5) The Hermie polyomials H() are used i his paper for expadig he Gree s fucios. The moio equaio of he rod is y ( x, ) + ω y( x, ) yxx( x, ) = f(, ), y ( x x, ) [ ( ) (, )] + ε v y x =, yx ( L, ) =, y( x,) = y ( x,) = v() = v () =, (.6) where x = L+ e( x = e ad x = L+ e are he eds of he rod) ad ω is he ormalized siffess of he elasic foudaio. 3. Soluios The coidal mehod was proposed i [] for solvig he oliear equaios, as a furher exesio of he Osbore mehod [8]. The se of equaios (.)-(.6) ca be reduced o equaios similar o Weiersrass equaio wih polyomials of higher order p θ = () θ, (3.) = wih θ a geeric fucio of ime, ad i > cosas. The do meas differeiaio wih respec o he variable x c, c a cosa. We ow ha (3.) admis a paricular soluio expressed as a ellipic Weiersrass fucio ha is reduced, i his case, o he coidal fucio c [9]

4 8 Pere P. Teodorescu, Veuria Chiroiu ( ) θ f () = e ( e e )c ( e e ); m, (3.) where e, e, e3 are he real roos of he equaio 4y g y g = wih e > e > e3, ad g, g R expressed i erms of he cosas i, i =,... 5, 3 ad saisfyig he codiio g 7 g >. The modulus m of he Jacobea e e3 ellipic fucio is m =. For arbirary iiial codiios e e 3 () = θ θ, () = p θ θ, (3.3) he soluio of (3.) ca be wrie as a sum bewee a liear ad a oliear superposiio of coidal vibraios where c ; = θ = θ + θ, (3.4) li θli = α ( ω m), θ oli oli = βc ( ωm ; ) = + γc ( ωm ; ) =, (3.5) where he moduli m, he frequecies ω ad ampliudes α depedig o θ, θ p ad. Therefore, he soluios of (.)-(.6) are wrie uder he form (3.5) v () = a c (; m) + = = y( ξ ) = a c ( ξ ; m ) + βc ( m ; ) = + γc ( m ; ) = βc ( ξ; m) = + γc ( ξ; m) =,, ξ = x ω, (3.6)

5 O he oliear resoace wave ieracio Resuls ad coclusios The calculaios are carried ou for L =, ω =., f =, λ =.4, e =. The respose v () of he aachme is displayed i fig.4.. Isaaeous frequecy of he oliear aachme is depiced i fig.4.. These figures pu io evidece he presece of four regimes of rasie resposes. The firs regime (- 5-9s) describe he ieracio of he oliear aachme wih icomig ravelig waves wih frequecy ω > ω. fer a shor rasiio, he aachme passes o periodic oscillaio of he secod regime (4-6s) wih frequecy early below ω, ad afer aoher shor rasiio o a wealy oscillaio of he hird regime (34-5s) wih frequecy early above ω. The periodic moio of he secod ad hird regimes are he cosequece of eergy pumpig where he aachme egages i resoace capure wih a liear srucural mode [6]. The las regime (55-8s) cosiss i a wealy modulaed periodic moios i he eighborhood of ω. The rasiio bewee he hird ad fourh regimes describes he case whe he aachme ca o loger susai resoace capure, ad escape from resoace capure occurs. The eergy is radiaed bac o he rod ad he isaaeous frequecy decreases uil i reaches a frequecy. By comparig our resuls wih hose obaied i [6] for impulse exciaio ad sep iiial displaceme disribuio, we observe ha i he case of a sie exeral force, four regimes are depiced, ad o hree as i [6]. This ca be explaied by a oscillaory irreversible rasfer of vibraio eergy from he rod o is oliear aachme. Two seps of eergy pumpig for - resoace capure wih he liear srucural mode are depiced, for wo wealy modulaed periodic moios wih frequecy early equal (below ad aboveω ). We ca erm his pheomeo as a oscillaory eergy pumpig. Fig. 4.. The respose of he aachme.

6 84 Pere P. Teodorescu, Veuria Chiroiu cowledgeme Fig. 4.. Isaaeous frequecy of he oliear aachme. The auhors graefully acowledge he fiacial suppor of he Naioal uhoriy for Scieific Research (NCS, UEFISCSU), hrough PN-II research proec code ID_39/8. R E F E R E N C E S [] C.R.Gilso ad M.C Raer, Three-dimesioal hree-wave ieracios: a biliear approach, i J. Phys. : Mah. Ge., vol.3, 998, pp [] L.Mueau ad S.Doescu, Iroducio o Solio Theory: pplicaios o Mechaics, Boo Series Fudameal Theories of Physics, vol.43, Kluwer cademic Publishers, 4. [3] L. I Maevich, O.Gedelma,.Musieo,.F Vaais ad L.. Bergma, Dyamic ieracio of a semi-ifiie liear chai of coupled oscillaors wih a srogly oliear ed aachme, Physica D., vol.78, o. -, 3, pp. 8. [4]. F. Vaais, L. I. Maevich, O. Gedelma, L.Bergma, Dyamics of liear discree sysems coeced o local esseially oliear aachmes, J. Soud Vibraio, vol. 64, 3, pp [5].F.Vaais ad O.Gedelma, Eergy pumpig i oliear mechaical oscillaors ii: resoace capure, J. ppl. Mech., vol. 68, o.,, pp [6].F.Vaais, L.I.Maevich,.I. Musieo, G.Kersche, L..Bergma, Trasie dyamics of a dispersive elasic wave guide wealy coupled o a esseially oliear ed aachme, Wave Moio, vol. 4, 5, pp.9 3. [7] R.H.Goodma, P.J.Holmes ad M.I.Weisei, Ieracio of sie-gordo is wih defecs: phase space raspor i a wo-mode model, Physica D., vol.6,, pp. 44. [8].R.Osbore, Solio physics ad he periodic iverse scaerig rasform, Physica D., vol.86, 995, pp [9] M.J.blowiz ad P..Clarso, Solios, Noliear Evoluio Equaios ad Iverse Scaerig, Cambridge Uiv. Press, Cambridge, 99.