APPLICATION OF GREEN S FUNCTIONS IN ANALYSIS OF THE RESPONSE OF AN INFINITE HOMOGENOUS STRUCTURE TO MOVING LOAD

Transcription

1 U.P.B. Sc. Bull., Seres A, Vol. 7, Iss., 00 ISSN APPLICATION OF GREEN S FUNCTIONS IN ANALYSIS OF THE RESPONSE OF AN INFINITE HOMOGENOUS STRUCTURE TO MOVING LOAD Traa MAZILU Scopul acestu artcol este de a prezeta o metodă umercă bazată pe fucţle Gree petru a calcula răspusul ue structur fte omogee cauzat de o forţă î mşcare. Structura are două grz rezemate cotuu pe stratur elastce ş repreztă modelul obşut al că ferate pe plăc de beto. Este dovedtă marea precze a metode. Este aalzat răspusul structur la forţe armoce cu suport fx ş cu suport mobl atât î domeul frecveţe, cât ş î domeul tmp. The am of ths paper s to preset a umercal method based o the Gree s fuctos to calculate the respose of a fte homogeous structure to movg load. The structure has two beams cotuously supported by elastc layers ad represets the commo model of the slab track. The hgh accuracy of the method s proved. The structure respose due to statoary ad movg harmoc loads s aalyzed for both frequecy ad tme domas. Keywords: Gree s fuctos, Fourer trasform, slab track PDE Classf: 35C5, 35Q7, 65M99. Itroducto The preset paper deals wth the respose of a fte homogeeous structure cosstg of two beams cotuously supported by elastc layers to movg load. Such a structure represets a commo model for the slab track that has bee employed the study of the teracto betwee a movg vehcle ad the slab track []. I addto, smlar models were used to calculate the slab track respose to a movg load [, 3]. The upper beam descrbes a ral, the lower oe models the slab, whle the two elastc layers reflect the propertes of the ral pad ad the track subsol. The ralway track structure ca be modeled by two kd of mechacal models: the cotuous models ad the perodcal models wth dscrete supports [, 5]. Oe of the oldest models for the ralway track, kow as a beam o elastc foudato, fact, a cotuous model, was preseted by Wkler 867 [6]. Reader, Depart. of Ralway Vehcles, Faculty of Trasports, Uversty POLITEHNICA of Bucharest, Romaa, e-mal: traamazlu@yahoo.com

2 0 Traa Mazlu The cotuous models are used for the slab track, but uder some codtos, ths type of model s proper for the ballasted track as well [7, 8]. I order to solve the equatos of the moto for cotuous models, may methods have bee proposed. For stace, the drect method, the Fourer trasformato method ad the couplg the wave-umber frequecy-doma method are preseted by Husse ad Hut [] for the case of the o-movg oscllatg load. I addto, they have appled the double Fourer trasform from the space-tme doma to the wave umber-frequecy doma for the track subject to a movg oscllatg load. O the other had, Wu ad Thompso [7, 8] have developed may equvalet multple degree of freedom (MDOF) models for the track dyamcs havg as a startg pot the receptace of the ral calculated for a ut statoary harmoc load. I ths paper, startg from prevous author s researches [9, 0], the Gree s fuctos method s developed to study the respose of the slab track due to the movg load. To ths ed, a umercal approach to obta the Gree s fuctos of the structure s preseted. I order to verfy the accuracy of the method, the results from the aalytcal ad umercal method are compared cosderg the steadystate behavour of the structure uder movg harmoc load. We should have to uderle that the Gree s fuctos of the structure are able to smulate the wheel/ral teracto takg to accout the oleartes of the cotact [].. Goverg equatos The structure of the slab track s composed of a massve cocrete slab, to whch the rals are embedded by meas of Corkelast. Assumg that the two rals are symmetrcally loaded, oly half-track s requred for modellg (fg. ). I fact, the slab track model may be reduced to a structure cosstg of two Euler-Beroull beams coupled by Wkler foudato as the ral ad the slab ad the ral-pad, respectvely. The track s supported by the groud that s take as

3 Applcato of Gree s fuctos aalyss of the respose [...] structure to movg load a Wkler foudato as well. Oe has to emphasze that the Euler-Beroull beams model gves satsfactory results as log as the cross-sectoal dmesos are small compared to the bedg wavelegth [] ad ths hypothess s cosdered the followg les. The parameters for the track model are: the mass per legth ut m, (the dex for the upper beam ad the dex for the uder beam) ad the bedg stffess EI,. The two Wkler foudatos have the elastc costats k, per legth ut ad the vscous dampg factors c, per legth ut. It s assumed that the track s subjected to a movg load Q( depedg o the tme t, whch moves at a costat speed V. To descrbe the track evoluto uder the movg load, the absolutely tegrable C fuctos w, ( termed the ral ad slab dsplacemet are troduced so that w, : R R, w, ( t < 0) 0, () where x s the coordate alog the track. The latest codto refers to the character of the causalty. Next, the two fuctos w, ( are ecapsulated the colum vector w([w ( w (] T termed the colum vector of the dsplacemets. Accordg to the Newto s secod law, the colum vector w( verfes the PDE L tw( q( () wth the boudary ad tal codtos lm w ( 0 0, w ( 0) [ 0 0] T. (3) x Vt [ ] T where L t stads for the matrx dfferetal operator DEB + c + k c k L t t t c k DEB + ( c + c ) + ( k + k ) t t that cludes the dfferetal operator of the Euler-Beroull beams, D EB, EI, + m,, x t ad q( [Q(δ(x-V 0] T s the colum vector of the forces o the track wth δ(.) the Drac s delta fucto. Theoreme. The boudary value problem formed by the equato () ad the assocated tal codtos (3) has the soluto w ( t g( Vt', t t') Q( t')dt', () 0 where the colum vector g(t-t')[g (t-t') g (t-t')] T verfes the equato

4 Traa Mazlu [ ] T L. (5) tg( t t') δ( x x') δ( t t') 0 Proof. I order to solve the equato (), oe ca apply the D Fourer trasform wth respect to tme ad spatal coordate: L W( Q( ), (6) where ω ω W ( w( exp[ ( ωt + ξx)]dxdt, Q ( q( exp[ ( ωt + ξx)]dxdt, (where the tegrals wth respect to t verfy 0 ) ad L ω EIξ ω k m + k ωc + ωc EI ξ ω k m + k ωc + k + ω( c + c ) s the D Fourer trasform wth respect to tme ad spatal coordate of the matrx operator L t, ξ s the coordate of the wave-umber-doma, ω s the coordate of the frequecy-doma ad. The compoets of the matrx L ω are C[ω]. From Eq. (6) oe reads W ( L ξ, ωq(, (7) where the compoets of the verse matrx L ω are C(. Let the matrx Γ( M so that Γ ( L ω exp[( ωt + ξx)]dωdξ (8) π ad ω Γ L ( exp[ ( ωt + ξx)]dtdx. (9) Performg the verse Fourer trasform Eq. (7) ad accordg to the covoluto theorem, ad the causalty codto of the mechacal model, oe obtas the colum vector of the dsplacemets w( π t 0 - L ω Q( exp[( ωt + ξx)]dξdω Γ( x t t'){ q( t')}dx'dt '. (0) The matrx Γ(x-t-t') represets the soluto of the equato L Γ( x t t') δ( x x') δ( t t' E () t )

5 Applcato of Gree s fuctos aalyss of the respose [...] structure to movg load 3 where E stads for the ut matrx. Ideed, performg the Fourer trasform, oe obtas the ecessary equalty exp[ ( ωt' +ξx')] L ξ, ωl ω exp[ ( ωt' + ξx' )] E () because ad Γ( x t t')exp[ ( ωt + ξx)]dtdx Γ( u, v)exp{ [ ω( u + t') + ξ( v + x')]}dudv exp[ ( ωt' +ξx')] L δ( x x') δ( t t')exp[ ( ωt + ξx)]dtdx exp[ ( ωt' + ξx')]. The matrx Γ(x-t-t') s termed as the Gree s matrx assocated to the operator L t through the equato () ad wll be wrte as Γ(t-t'). Takg to accout the partcular colum vector q(, oe eeds oly the frst colum of the matrx Γ(t-t'), deoted as g(t-t') [g (t-t') g (t-t')] T, whch cotas the tme-doma Gree s fuctos of the ral ad the slab. These fuctos descrbe the resposes of the ral ad the slab, the secto x at the momet t-t', f a mpulse force occurred at the momet t' the secto x' alog the ral. Therefore, g(t-t') wll be called the colum vector of the tme-doma Gree s fuctos of the track. I these crcumstaces, the colum vector of the dsplacemets ca be calculated as follows t w ( g( t t') Q( t') δ( x' Vt')dx'dt' g( Vt', t t') Q( t')dt'. (3) 0-0 From Eq. (), t follows that the colum vector g(t-t') satsfes the equato (5). Oe ca observe that the colum vector of the tme-doma Gree s fuctos of the ral ad the slab are atteuated space ad tme-domas: lm g ( t τ) lm g( t τ) 0. () ' x x t τ Usg the Fourer trasform, the colum vector G([G ( G (] T ca be assocated to the colum vector of the tme-doma Gree s fuctos t ω G ( { g( }exp( ω dt. (5) The compoets of the colum vector {G(} are the frequecydoma Gree s fuctos of the ral ad the slab. These fuctos, termed the receptaces, represet the respose of the ral ad the slab the secto provoked by a utary harmoc mpulse by a agular frequecy ω, occurrg the secto x' of the ral. Next, the colum vector {G(} s called the colum vector of the frequecy-doma Gree s fuctos of the track.

6 Traa Mazlu Theoreme. The colum vector of the frequecy-doma Gree s fuctos G( has the followg propertes: a) t verfes the equato L [ ] T ωg( δ( x x') 0, (6) b) t verfes the equalty lm G ( lm G( 0 (7) x x' ω ± c) t has the form * T GD ( L ω[ δ( x0 x') 0] d 0 G ( x, (8) where L ω stads for the Fourer trasform of the matrx operator L t, L ω s the adjot matrx of the matrx L ω ad the fucto G D ( remas to be specfed later. Proof. The frst two codtos result by applyg the Fourer trasform to the equato (5) ad the lmts (). I order to arrve to the form (8), Eq. (6) s * modfed by applyg the adjot matrx operator L ω, [ δ( x x') ] T dag( D, D){ G ( } L * x, ω 0, (9) where the dfferetal operator 8 d d D a0 + a 8 + a 8 dx dx (0) has the complex coeffcets a 0, a ad a 8 depedg o the track s parameters ad the agular frequecy a 0 EIEI, a EI[ k + k ω m + ω( c + c) ] + EI( k ω m + ωc ), 3 a8 ω mm ω [ m ( c + c ) + mc ] ω [ m ( k + k ) + mk + cc ] + () + ω( kc + kc ) + kk. Fally, the colum vector of the frequecy-doma Gree s fuctos results from Eqs. (9) ad takes the form (8), where G D ( s the Gree s fucto of the operator D. To perform the tegral from Eq. (8), the Gree s fucto of the operator D s requred. I fact, ths fucto verfes the equato DG D ( ω ) δ( x x') () ad the boudary codtos lm ( 0 (3) G D x ± due to the dampg of the track. Ths kd of equato s usually solved by applyg the Fourer trasform from the space-doma to the wave-umber doma ad the, by usg the verse

7 Applcato of Gree s fuctos aalyss of the respose [...] structure to movg load 5 Fourer trasform va the cotour tegrato gve by the theory of the fuctos of complex varables []. I ths paper, a dfferet soluto s cosdered ad the Gree s fucto of the D operator s obtaed usg ts outstadg features [3]. The ODE Dy0 has the followg solutos y ( x) A exp( λ x) wth 8, () where λ λ ( are egevalues of the operator D. It ca be see that f λ s oe soluto of the characterstc equato, the λ ad ± λ are solutos as well. Practcally, each quadrat cotas two solutos of the D operator characterstc equato. I fact, λ descrbes the bedg wave, whch propagates through the track structure. The bedg wave s a propagatg oe whe λ s a magary quatty. Ths wave propagates from the left to the rght whe Im λ > 0, ad vce versa whe Im λ < 0. The bedg wave atteuates whe λ s a complex umber. Accordg to the boudary codtos, the atteuated wave decreases wth dstace (Reλ > 0 for x ad Reλ < 0 for x ). Fally, the bedg wave s a evaescet oe whe λ s a real quatty. I addto, the evaescet wave has to propagate followg the same rule as the atteuated oe, accordg to the boudary codtos. Accordg to prevous cosderatos, the Gree s fucto of the operator D has the forms D( x0, A ( x0)exp( λx) 8 + D( x0, A ( x0)exp( λx) 5 G for < x < x 0, G for x 0 < x <, (5) wth Reλ > 0 for ad Reλ < 0 for 5 8. O the other had, the Gree fucto s cotuous at x x 0 ad ts frst 6 dervates are cotuous as well + + d GD d GD GD ( ω ) GD(, ( (, 6. (6) dx dx Further o, the 7 th dervatve of the Gree s fucto has a dscotuty at xx d G d G D D ( x , ( x 7 0 0, dx dx. (7) a0 All these codtos lead to the ext matrx equato

8 6 Traa Mazlu... X 0 λ λ... λ8 X 0 (8) λ λ... λ8 X 8 a0 where X A (x 0 )exp(λ x 0 ) for ad X -A (x 0 )exp(λ x 0 ) for 5 8. Obvously, the matrx from Eq. (8) has the Vadermode determat ad fact, all Cramer s determats are Vadermode determats, as well. Eq. (8) has the followg soluto X, wth, k 8 (9) a0 ( λ k λ ) k ad fally, the Gree s fucto of the D operator s obtaed exp[ λ( x x0) ] G D( for < x < x 0, a ( λ λ ) 0 k k 8 exp[ λ( x x0) ] 0 5 ( λk λ) + G D( for x 0 < x <. (30) a k Itroducg the Gree s fucto of the D operator from Eq. (30) Eq. (8) ad performg the tegral, oe obtas exp ( λ ) ( λ x x' exp x x' ) K λ + ω ( ) ( p r) G 3 ( ) λ G ω exp( λ ' ) exp( λ ' ) ( ) x x x x sk ( ), (3) 3 λ where λ, are cotaed the frst quadrat ad ( λ K EI EI λ ), p EI, r k + k ω m + ω( c + c ), s k + ωc. It s mportat to emphasze that the tme-doma Gree s fuctos of the ral ad the slab wll be computed from the correspodg frequecy-doma Gree s fuctos. Performg the verse Fourer trasform appled to the frequecy-doma Gree s fuctos, t follows that the tme-doma Gree s fuctos for the track are g( t τ) π π 0 G( exp[ω( t τ)]dω ReG( cos[ ω( t τ)]dω. The last expresso resdes from the causal character of the track model ad t s used effectvely for the computato. More precsely, the tme-doma (3)

9 Applcato of Gree s fuctos aalyss of the respose [...] structure to movg load 7 Gree s fuctos are calculated by umercal tegrato wth the help of the cubc sple fuctos vokg the followg propertes. Accordg to the atteuato tme-doma, there s a certa T for whch the orm of the Gree s fuctos s cocetrated the [0, T] terval. Therefore, the colum vector of the dsplacemets at loadg secto (x V results as w ( Vt, t g( Vt, Vτ, t τ) Q( τ)dτ, for 0 t < T (33) 0 t t T w ( Vt, g( Vt, Vτ, t τ) Q( τ) dτ, for t T. (3) It may be observed that for ay cotact secto pot x Vt, there s a correspodg colum vector of the Gree s fuctos g(vt, Vτ, t-τ) whch depeds o 0 τ t ad t s calculated from g( t-τ). For the partcular case whe t > T, oly the hstory for τ [t T, t] s ecessary, accordg to the atteuato tme-doma. Moreover, all cotact pots x Vt wth t > T have the same sequece of colum vector of the tme-doma Gree s fuctos because the track has the homogeeous structure. From the umercal tegrato pot of vew, there are two steps to follow: the frst refers to 0 t < T - the trastory perod of umerc tegrato, whle the secod assumes T t, whch meas the statoary perod of umercal tegrato. As results, the umercal smulato legth has to be hgher tha the trastory T perod. Whe s used the small tme-steps method o short Δt tme tervals order to tegrate the equatos of moto, the tme-doma Gree s fuctos wll be calculated N T/Δt + ad all the obtaed values may be ecapsulated the so-called track s Gree matrx whch depeds o the speed value V. Ths matrx cludes the requred values for the trastory perod of umercal tegrato. More specfc, a tme partto - t 0, t, t wth t 0 0, t t ad Δtt - t - where - should be cosdered. The track s Gree matrx has the form N [ g g... g g ] g t..., (36) where g g(vt N, Vt, t N -t ). The colum vector of the dsplacemets at the loadg secto may be defed as t w ( Vt, t ) g( Vt, Vτ, t τ) Q( τ)dτ. (37) t Assumg that the tme terval [t -, t ], the Gree fuctos ad the ormal cotact force Q(τ) have a lear varato, the prevous tegratos may be performed

10 8 Traa Mazlu g Q + g Q ( g g )( Q Q ) w ( Vt, t ) Δt +, (38) 3 where Q Q(t ) ad g s take from the track s Gree matrx. For t > T, the ral ad slab dsplacemets wll be calculated by summg N terms oly, accordg to Eq. (3). For the colum vector of the dsplacemets at a partcular secto of the track, the smlar formulae have to be computed. 3. Numercal results Further o, both frequecy ad tme-doma umercal aalyss of a partcular slab track uder a movg load s preseted. Physcal parameters of the track model used these computatos are as follows: m 60 kg/m, EI 6. MNm, m 750 kg/m, EI 7 MNm, k 5 MN/m, c 7 kns/m, k 60 MN/m ad c 0 kns/m. Fg. presets the frequecy-doma Gree s fuctos of the track,.e. the ral ad slab receptaces, whch have bee computed at the pot of a utary harmoc mpulse force. As t ca be observed, the track respose has two peaks at 9 ad 50 Hz because the ral ad the slab vbrate as a dscrete system wth two degrees of freedom. At the frst resoace frequecy, the ral ad the slab are phase ad the, they vbrate at-phase. The frst peak belogs to the slab resoace, ad the secod oe s the effect of the ral s resoace. The two frequeces of resoace may be approxmately calculated usg the formula from the sgle-degree-of freedom system cosstg of a equvalet mass ad a sprg wth equvalet stffess k, f,. π m,

11 Applcato of Gree s fuctos aalyss of the respose [...] structure to movg load 9 The ral receptace s sgfcatly hgher tha the slab receptace due to ts low erta ad the elastcty of the ral-pad. The dagram of the track s receptaces s smlar wth the results from the precedg related researches [3, 5]. Fg. 3 shows the fluece of the speed of the movg harmoc load o the receptace of the ral. I order to calculate the receptace of the ral due to movg harmoc load, the equatos of moto have bee solved troducg the movg frame ad the, followg the same method. It ca be see that by creasg the speed of movg harmoc load, the resoace frequeces of the track decrease. I addto, the receptace of the ral lows aroud resoace frequeces due to the speed of the harmoc exctato. The Gree s fuctos of the track for the tme-doma aalyss may be calculated by the umercal tegrato of Eq. (3) from 0 to 5 khz wth the step tegrato of Hz. Fg. presets the dsplacemet of the loadg pot for the frequecy of the utary harmoc load of 30 Hz ad for two values of the speed, 60 ad 0 m/s. To ths ed, Eq. (38) has bee used. The results of the steady-state behavour

12 50 Traa Mazlu gve by the aalytcal soluto are preseted as well for comparso. It may observe the trastory behavour the case of the umercal soluto. The two solutos are very smlar. Ideed, the dfferece betwee the ampltudes obtaed by the two methods s % at 60 m/s ad 0.0% at 0 m/s. Fg. 5 presets the dsplacemet of the fxg pot belogg to the secto of the ral stuated at m from the frame, whe a utary harmoc load wth the frequecy of 30 Hz travels alog the ral at 60 m/s. I ths case, Eq. (6) s appled. The smulato of the steady-state behavour s show usg a aalytcal method. Oce aga, the results from the umercal ad aalytcal method are matched. The relatve error betwee the results gve by the two methods s 0.83 % for the dsplacemet obtaed at t 0, s, whe the harmoc load passes over the fxg pot. It has uderle that the fxg pot vbrates more tese the trace of the movg load. Fg. 6 dsplays the dsplacemet of the loadg pot ad the dsplacemet of the fxed pot prevously preseted. The dsplacemet of the loadg pot ad the dsplacemet of the fxed pot have the same value oly whe the movg load passes over the secto of the fxed pot. At 30 Hz, the bedg wave of the ral s a propagatg oe that propagates from the movg load to the fxg pot. Due to ths, the hstory of the fxed pot s later tha the loadg pot s except the jog momet. For stace, oe ca observe that all peaks of the fxed pot are later tha the correspodg oes of the loadg pot. Cosequetly, the fxed pot has a frequecy hgher tha the loadg pot s oe before the jog ad the, after jog, ths tred reverses. I other words, ths s the Doppler effect.

13 Applcato of Gree s fuctos aalyss of the respose [...] structure to movg load 5. Coclusos The respose of the slab track uder a movg load has bee studed, cosderg the model of a fte homogeous structure cosstg of two Euler-Beroull beams cotuously supported by elastc layers. To ths am, the method of Gree s fuctos has bee appled followg the umercal approach. The hgh accuracy of the preseted method has bee verfed. The proposed method s mportat because allows us to smulate the wheel/ral teracto uder olear codtos. The structure respose due to statoary ad movg harmoc loads has bee vestgated for both frequecy ad tme domas. Ackowledgemets Ths paper has bee partally supported by the Uversty Poltehca of Bucharest. The author would lke to express hs scere thaks to Prof. Dr. Costat Udrşte, Uversty Poltehca of Bucharest, for kd ecouragemet ad support. R E F E R E N C E S []. J. Btzebauer, J. Dkel, Dyamc teracto betwee a movg vehcle ad a fte structure excted by rregulartes Fourer trasforms soluto, Archve of Appled Mechacs 73, 3-, 003. []. M. F. M. Husse, H. E. M. Hut, Modellg of floatg-slab tracks wth cotuous slabs uder oscllatg movg loads, Joural of Soud ad Vbrato, 97, 37-5, 006. [3]. M. Shamalta, A. V. Metrke, Aalytcal study of the dyamc respose of a embedded ralway track to a movg load, Archve of Appled Mechacs 73, 3-6, 003. []. Kothe, K., Grasse, S., Modellg of ralway track ad vehcle/track teracto at hgh frequeces, Vehcle System Dyamcs, 09-63, 993. [5]. K. Popp, H. Kruse, I. Kaser, Vehcle-Track Dyamcs the Md-Frequecy Rage, Vehcle System Dyamcs, 3, 3-6, 999. [6]. L. Fryba, Hstory of Wkler foudato, Vehcle System Dyamcs Supplemet, 7-, 995. [7]. T. X. Wu, D. J. Thompso, Theoretcal Ivestgato of Wheel/Ral Nolear Iteracto due to Roughess Exctato, Vehcle System Dyamcs 3, 6-8, 000. [8]. T. X. Wu, D. J. Thompso, A hybrd model for the ose geerato due to ralway wheel flats, Joural of Soud ad Vbrato 5, 5-39, 00. [9]. Tr. Mazlu, Gree s fuctos for aalyss of dyamc respose of wheel/ral to vertcal exctato, Joural of Soud ad Vbrato, 306, 3-58, 007. [0]. Tr. Mazlu, Wheel/Ral Vbratos ( Romaa), MatrxRom, Bucharest, 008.

14 5 Traa Mazlu []. Tr. Mazlu, Aalyss of fte structure respose due to movg mass the presece of rregulartes va Gree s fuctos method, Proceedgs of Romaa Academy, Seres A, vol. 0, No., 009. []. L. Cremer, M. Heckl, Structure-Bore Soud Structural Vbratos ad Soud Radato at Audo Frequeces, Spger-Verlag Berl, Heldelberg, 973 ad 988. [3]. Gh. Moccă, The Laplace Trasform ad Operatoal Methods ( Romaa), UPB, Bucharest, 993.