Motions of Infinite Mass-Spring Systems

Transcription

1 Motio of Ifiite M-Sprig Syte KHIEM V. GO Deprtet of Aeroutic d Atroutic Stford Uiverity Stford, CA UITED STATES OF AMERICA Atrct: - I the tudy of phyicl, echicl, d electricl yte oe ofte ecouter differetildifferece equtio d recurrece reltio. The ource fro which thee equtio rie y e quite differet ut their theticl for re very iilr. For exple, there i logy etwee -prig yte d electricl yte wherey poit e correpod to iductce d prig correpod to cpcitce []. Aother re where differetil-differece equtio occur i i the uericl olutio of the wve equtio if the ptil vrile i dicretized []. It iportt to udertd how thee yte ehve tie evolve d how chge i the preter of the odel ifluece thi ehvior. I thi pper, we tudy yte coitig of poit e oied together y prig. I prticulr, we preet oe of the theticl ethod ivolved d how they re ued to olve thee prcticl prole. By otiig the olutio for thee iple -prig yte, we idirectly oti olutio for y iilr pplied prole i echic, phyic, d egieerig. Key-Word: - M-prig yte, virtio, wve, eprtio of vrile, Lplce trfor, Beel fuctio Itroductio Thi pper preet how thetic coe ito ply i the odelig proce of iple, yet fudetl, phyicl yte o tht oe c udertd how thi type of yte ehve uect to the chge i the preter. I prticulr, we re lookig t iple -prig yte with ideticl e d prig cott. We fid the olutio for thi yte y two ethod: eprtio of vrile d Lplce trfor (Sectio. The, we pertur the yte y chgig oe of the prig cott. Thi reult i the pperce of eigevlue d virtio tht h the for of tdig wve (Sectio 3. Ltly, we coider rther geeric yte with vriou e d prig tiffe uect to forcig fuctio (t. I thi ce, we oti the olutio y e of odl lyi (Sectio 4. I ot ce, the tudy of ifiite -prig yte ivolve differetildifferece equtio, which, i tur, reult i three ter recurrece reltio, whoe olutio c e quickly otied. To keep the cotet focued, legthy lger hve ee eliited. A uch, the udiece re ecourged to verify the reult preeted here. The key prt i thi work lie i the theticl odelig of thee phyicl odel, which hould e iteretig d iportt to thoe who coider workig o egieerig d ciece. I dditio, thi pper provide good ource for the techig of thetic to egieerig tudet. F A Siple M-Sprig Syte Let coider yte with ifiitely y oect of the e coected y ideticl prig. I uch yte, the diplceet of oe oect deped o the diplceet of other. So, y recurrece reltio, if we kow the diplceet of y two coecutive oect, the we c deterie the diplceet of the third dcet oect. We ue tht the diplceet of the oect t the zeroth poitio, i.e. t, i d tht it iitil velocity i zero. The other oect re uppoed to hve zero iitil diplceet d zero iitil velocity. Fro the coitio of ewto ecod lw of otio d Hooke lw, the equtio of otio for thi yte red follow: x x x Fig. A ifiite yte x && k( x x k( x x x ( x& ( x ( x& ( ( We will olve ( y two ethod: eprtio of vrile d Lplce trfor.

2 . Solutio y Seprtio of Vrile We eek olutio of the for x ( t ce, where α i cott idepedet of d t. Sutitutig ito ( we oti kc ( k α c kc. To olve thi recurrece reltio, we ue c λ o tht kλ (k α λ k whoe olutio re If α α λ ± ± ω ω α 4ω, it c e how tht λ i rel d α 4ω > λ <. Coequetly, λ, d λ λ. A the olutio of thi type i uphyicl, we dicrd it d coider the ce whe α 4ω <. λ ω α ± ± i α 4ω ω α Sice λ, d hece it uffice to let ± α coϕ ω ϕ α 4ω α iϕ ω i o tht λ e d λ e i. It follow fro tht α ω ϕ, where ϕ (,. Thu, the geerl olutio of ( i of the for x ( t αt ϕ e αt ϕ e f ( ϕ ϕ f ( ϕ e e αtϕ αtϕ Alo, for the oudry coditio, we hve f ( ϕ f ( ϕ iϕ x ( e ( f g ( where g ( ϕ f ( ϕ. I view of thi equtio, we deduce tht f. Hece, ( yield x g iϕ e f ( ϕ ( Expdig f ( i Fourier erie, we hve: ϕ iϕ f(ϕ c e where c f ( ϕ e i ϕ So, x ( 4c, i.e. c 4. Alo, c for ecue x (. f ( ϕ g ( ϕ 4. It follow tht x ( t 4 { exp[ ϕ αt ] exp[ ϕ αt ]} Therefore, for (recll tht α deped o ϕ, x ( t 4 where ( e e co(ωt i ϕ J (ω, t J ( z co( z i t dt co( z i t dt deote the Beel fuctio of order zero [3]. For, we ue J ( z co( z i u u du. x ( t x ( t Siilrly, [ co( ϕ αt co( ϕ αt ] co(ωt iϕ ϕ co(ωt i ϕ ϕ J ωt J (ω [ ] ( t i x ( t x ( t [ ϕ αt ϕ αt] i i ϕ coαt Thu, x ( t x ( t for ll t d, d the olutio i x ( t [ J (ωt],, ±, ±,... Below re the plot of the -prig otio with prig cott k, tie tep h, d ω k.5.

3 Fig. Fig.3 Fig.4 Fig.5 Fig.6 A how i the lyi, x ( t x ( t for ll x d t, hece the otio of the prig i yetric. I plottig, we let the iitil tte of the -prig yte t i. Fig. repreet the prig otio with the tie tep of.5 d ω. ote tht the e otio exit if the tie tep i decreed y hlf d ω i doule, or if the tie Fig.7 tep i doule d ω i decreed y hlf. A the tie tep get ller, Fig.3 & 4, the prig otio pproche the iitil tte, which i x ( y uptio. The e ehvior lo exit whe ω. Thi i ecue ω ted to zero, the e ted to ifiity, d thi gretly reduce the virtio of the prig.. Solutio y Lplce Trfor We trt fro yte with oect d the let. We ue tht the two ed of the fiite chi re free. k( x x k x x - Fig.8 ( t ω ( x x ( t ω ( x x ω ( x x < ( t ω ( x x ω ( x x ( t ω ( x x ω ( x x < ( ( t ω ( x x (3 Applyig Lplce trfor to (3, we hve ( ( ( ( ( t ( e x ( t dt ω ω < ω (4 ω < ω The equtio (4 & (4e decrie the two free ed of the chi. Puttig µ, (4 ecoe the qudrtic equtio whoe olutio re ω µ ( ± ( ± ω µ ω µ 4ω ω,

4 ote tht µ, µ, d < <. The µ > µ olutio of (4 re of the for C µ C µ. It follow tht. Hece, it uffice to oly coider. Lettig ρ ω d ε ω, we oti for d, repectively, ( ( µ Cµ Cµ C ( Cµ ( ρ ( C C ρ C µ (5 C µ ε (6 where i the lt equtio, we ued (4c d the fct tht x x. Fro (5 d (6, we oti ohoogeeou yte for d C : µ µ ρ µ C C C ε µ µ µ ρ µ ρ ρ Uig Crer rule d the fct tht µ > d < µ <, we coclude tht C ε, C ε We hve C d C. Thu for fixed,, 4ω 4ω ω µ Tkig the ivere Lplce trfor, the olutio i foud to e x ( t J (ωt, which i the e reult otied erlier y the eprtio of vrile ethod. Thi i direct follow fro the forul L { J ( x} ( p p ν ( p. It c e foud i [5] (forul #39, p A Pertured M-Sprig Syte We ow coider the e yte i Sectio except tht we replce oe of the prig y prig of vrile tiffe k. Thi yte i govered y the equtio elow: k x x k ' x Fig.9 A pertured yte k ν k' ( x x k( x x k( x x k( x x (7 k( x x k' ( x x k( x x k( x x Auig gi tht the olutio re of the for x ( t λ e, we get x && α λ e. Settig µ α, the yte (7 ecoe k' ( λ λ k( λ λ k( λ λ k( λ λ µλ k( λ λ k' ( λ λ k( λ λ k( λ λ (8 ow, ( 4ω ( 4ω ω Uig the e lyi efore, we oti, for, λ λ, d λ ( µ ± 4µk µ k. ± ± Proceedig with the e procedure, we oti ( ( ω 4ω 4ω 4ω 4ω ω 4ω ω (. It follow tht 4ω ( ω 4ω ω 4ω 4ω 4ω λ, x ( t λe where λ ηλ, re olutio tht go to zero ±. ote tht i goig through the thetic, it i foud, for µ < 4k, k ' k. Siilr lyi how tht there i o eigevlue for µ >. ote tht the prig repoe i the rel prt of the olutio, which i Re[ x (t ] λ co[ αt]. Therefore, the virtio of the prig exhiit yetric chrcteritic.

5 Oviouly, λ ( cotrol the plitude of the prig otio, which decree ±. Therefore, the virtio die out t oth ed. The e pheoeo hppe the differece i the prig tiffe k d k icree. Thi c e ee i Fig. d Fig.3 d whe copred with Fig.. O the other hd, thi differece decree, k' k, we re coig ck to the origil iple -prig yte dicued i Sectio. d.. A uch, the prig virte with plitude, which re ecoig cott, Fig.5. I dditio, the e re doule, Fig., the otio o the right of.5 i ivere yetric to tht o the left. Below re the plot of the otio of the prig yte i the itervl 5 5 d t. Fig. Fig. Fig. Fig.3 Fig.4 Fig.5 4 M-Sprig Syte with Vriou Me d Sprig Stiffe Suppoe ow we coider fiite dicrete yte F ( t F ( t F 3 ( t x ( x ( x ( t 3 k k k 3 k4 Fig.6 A fiite yte To get the ifiite -prig yte, we ut eed to let the uer of e go to ifiity t oth ed. I relity, the two ed of either yte, fiite or ifiite, ut e coected to oethig to hold it till efore virtio c exit. Thu it uffice to look t fiite dicrete yte, Fig.6. We re to eek olutio for thi yte y the odl lyi t 3 t ethod. By ewto ecod lw of otio, [ ]{ ( t} [ k]{ x( t] { F( t} (9 For illutrtio, we let 3, 4, d ue ll the prig hve the e tiffe k. Thu, the d tiffe trice re [ ] 4, k [ k] k where x ( t} [ x ( t x ( t x ( t ] T k k k k k { 3 i diplceet vector. Uig the ethod of odl lyi, we firt olve the free virtio prole or hoogeeou yte. I doig o, we gi ue olutio of r r the for x( t u exp( iωt d utitute it ito (9, we oti the eigevlue prole

6 ([ k] ω [ ]{ u} {} To udertd the reltiohip etwee the fiite d ifiite yte, oe c oerve the ehvior of the eigefrequecy ω the dieio of [k] d [] get lrger. The ove equtio h otrivil olutio if d oly if det ([ k] ω [ ]. Upo olvig thi chrcteritic equtio, we get o tht & ξ ( t ω ξ ( t ( t for,, 3, where ~ T ( t [ u ] { F( t}. Uig the iitil coditio we olve thi ecod order differetil equtio for ξ. ω ω Ω ω Ω, where Ω k. 5Ω Coequetly, the turl ode of the yte re 5 { u }, { u}, { u } 3 ( 5 I order to exploit the uefule of the eigeode towrd otiig the olutio, we orlize the with repect to the trix []. A uch, we ue the orlized odl vector re of the for { u ~ } i ci{ u} i for i,,3. By the orthogolity reltio of odl vector, we hve T { u ~ } [ ]{ u ~ } δ. It follow tht i ~.3 { u i u } { }, { } { u} [ u ~ ] ~.5.7 u, { u ~ } 3 { u} To llow iterctio og the ode, we expre the diplceet x r (t lier uperpoitio of the orl ode o tht { x( t} [ u ~ ]{ ξ ( t}, where T { ξ ( t } { ξ( t ξ ( t ξ 3 ( t} for oe coefficiet ξ, ξ, ξ 3. Upo utitutio ito (9, we get [ ][ u ~ ]{ & ξ ( t} [ k][ u ~ ]{ ξ ( t} { F( t} Applyig the orthogolity reltio of eigevector [ ~ T u ] [ ][ u ~ ]{ && [ ~ T ( t} u ] [ k][ u ~ ]{ ( t} [ u ~ T ξ ξ ] { F ( t} ω Fig.7 turl ode of virtio 5 Cocluio I hort, thi pper preet how thetic coe ito ply i the odelig proce of iple, yet fudetl, phyicl yte o tht oe c udertd how thi type of yte ehve uect to the chge i the preter; e d prig tiffe. I prticulr, we look t everl iple -prig yte d preet oe of the well kow theticl ethod ivolved i otiig the olutio for thee yte. By otiig the olutio for thee iple -prig yte, we idirectly oti olutio for y iilr pplied prole i echic, phyic, d egieerig. Referece: [] J. Keeer, Priciple of Applied Mthetic, Trfor. & Approx., A. Weley, Y, 988 [] G. Bldock d T. Bridge, Mtheticl Theory of Wve Motio, J. Wiley & So, 98 [3] C. Beder d S. Orzg, Advced Mtheticl Method for Scietit d Egieer, McGrw-Hill, 978 [4] L. Meirovitch, Priciple d Techique of Virtio, Pretice Hll, 996 [5] I. Grdhtey d I. Ryzhik, Tle of Itegrl, Serie, d Product, Acdeic Pre, 98