Research on Moving Force Estimation of the Bridge Structure using the Adaptive Input Estimation Method

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1 Electroc Joural of Structural Egeerg (8) 28 Research o Movg Force Estmato of the Brdge Structure usg the Adaptve Iput Estmato Method Tsug-Che Che Chug Cheg Isttute of Techology Natoal Defese Uversty Ta-Hs, Tao-Yua 3359, Taa, R.O.C. E-mal: choja@cct.edu.t Mg-Hu Lee School of Defese Scece, Chug Cheg Isttute of Techology Natoal Defese Uversty, Ta-Hs, ABSTRACT: A o-le adaptve put estmato method that estmates the movg force puts of the brdge structure s preseted ths research. By usg the verse method, put forces actg o brdge structural system ca be estmated from the measured dyamc resposes. The algorthm cludes the Kalma flter (KF) ad the recursve least squares estmator (RLSE). Ths ork presets a effcet eghtg factor r of the RLSE, hch s capable of mprovg the estmato results. The capablty of the proposed algorthm s demostrated through several examples of the brdge structure system th dfferet types of the tme-varyg movg forces as the uko puts. KEYWORDS: Adaptve put estmato method, Movg force, Kalma flter. 1. INTRODUCTION The vehcle/brdge teracto movg forces are mportat for the brdge structure desg ad the relablty evaluato. Therefore, the dyamc movg forces produced by the vehcles o the brdge structure must be determed by adoptg the estmato method or measuremet techques. Drect measuremet of the vehcle movg forces usg sesors th hgher precso s expesve ad s subject to bas, hle results from the modelg errors [1~4] produced durg the computato. The Weght--Moto (WIM) systems have bee developed to solve above problems by Davs et al [5~], but these techques ca oly measure the statc vehcle axle loads. Hoever, the vehcle dyamc movg forces usually cause the damage of the road surfaces more easly tha the statc loads [8] do. Therefore, t s mportat to obta the hstores of the vehcle/brdge teracto dyamc movg forces. I recet years, may researchers have studed the hstores of the dyamc resposes of the cotuous beams due to movg vehcle heel loads. Hech et al. [9] used the fte elemet method to obta the exact mode shapes ad frequeces. Zhag et al. [1] used the assumed mode shape fucto to solve the vbrato behavor of a o-uform Beroull Euler beam. I addto, 2 Dugush ad Eseberger [11] used the fte elemet method to determe the atural frequeces ad mode shapes, ad the soluto s obtaed by usg the applcato of modal aalyss ad the drect tegrato method. All the above methods are maly cocetrated o the forard resoluto,.e. the detecto of the brdge dyamc resposes due to movg vehcle loads. Hoever, the verse problem of the vehcle movg force determato from the brdge dyamc resposes s also sgfcat ad eeded to be studed. Some researchers addressed the determato method for the above problem. Doyle [12] developed a method for determg the locato ad magtude of a mpact force by usg the phase dfferece of the sgals measured at to dfferet locatos straddlg the mpact pot. Busby ad Trujllo [13] recostructed the force hstory usg the stadg ave approach. Holladsorth ad Bushy [14] verfed ths expermet by applyg a force at a ko locato ad adoptg the accelerometers as sesors. Druz et al.[15] formulated a o-lear verse problem ad tred to fd the locato ad magtude of the exteral force. Other researchers studed the determato of the vehcle movg forces a fe years ago. The tme doma approach by La et al. [16] models the structure ad forces th a set of secod order dfferetal equatos. The forces are modeled as the

2 Electroc Joural of Structural Egeerg (8) 28 step fuctos a small tme terval. These equatos of moto are the expressed the modal co-ordates, ad they are solved by usg covoluto the tme doma. The forces are the determed by usg the modal superposto prcple. The frequecy ad tme doma approach by La et al. [1] performs Fourer trasformato th respect to the equatos of moto, hch are expressed modal co-ordates, ad the hstores of the forces are obtaed drectly by usg the least-squares method. The modal approach detfes the forces completely the modal co-ordates by Cha et al. [18]. Measured dsplacemets are coverted to modal dsplacemets th a assumed shape fucto. The forces are the determed by solvg the ucoupled equatos of moto the modal co-ordates. The above-metoed approaches requre the resoluto of verse matrx problems, hch are computatoally cosumg ad ot umercally ell-fuctoed dealg th the o-le real-tme sgal problem. To resolve the about-metoed problems, a o-le recursve verse method to estmate the put forces of the beam structures s preseted. The verse method s based o the Kalma flter ad the recursve least square algorthm. Tua et al. [19, 2] preseted the put estmato algorthm to cope th the 1-D ad 2-D verse heat coducto problems. Ma et al.[21] preseted a verse method to estmate the mpulsve loads o lumped-mass structural systems. The capablty of the proposed algorthm s demostrated these examples. The algorthm s a effcet o-le recursve verse method to estmate the put forces. The method s computatoally more ecoomcal tha the batch process he estmatg complex structure system put forces. I the preseted ork, the put force estmato method s appled to the brdge structure systems to cope th the movg forces. The precso of the preseted method s verfed through several examples th dfferet types of the tme-varyg movg forces as uko puts. The smulato results sho that the method s effectve determg the movg forces. the brdge structure s modeled as a smple beam th the total spa L, the flexble stffess costat EI, the mass per ut legth ρ ad the dampg coeffcet C. The beam s assumed to be a Beroull-Euler beam, hch the effects of shear deformato ad rotary erta are ot take to accout. Fgure 1. The brdge structure model uder the mult-vehcle movg force puts. Cosderg the group of vehcle forces movg from left to rght at a costat speed, the equato of moto [22] ca be expressed as: ρ 2 4 N u( x, t) u( x, t) u( x, t) A + C + EI = F ( ) ( ( )) 2 4 k t δ x xk t t t x k= 1 (1) here A s the cross secto of beam, u( x, t ) s the dsplacemet of beam, Fk ( t ) s the vehcle force matrx, δ ( t) s the Drac delta fucto, xk ( t) = vkt s the posto of the kth vehcle force ad v k s the speed of the kth vehcle. Based o modal superposto, the soluto of Equato 1 ca the be expressed as: u( x, t) = Φ ( x) Y ( t) (2) here Φ ( x ) s the th modal shape fucto ad Y ( t ) s the th modal ampltude. By substtutg Equato 2 Equato 1, multplyg each term by Φ r ( x), tegratg t over the legth of the beam ad applyg orthogoal codtos, the equato of moto terms of the modal ampltude ca be rertte as: M Y&& ( t) + C Y& ( t) + K Y ( t) = F ( t) (3) 2. PROBLEM FORMULATION To llustrate the practcablty ad precso of the preseted approach estmatg uko put movg forces, umercal smulatos of a brdge structure are vestgated here. As sho Fgure 1, 21 here L 2 ρ [ ( )] (4) L 2 [ ( )] (5) L M = A Φ x dx K = EI Φ x dx F ( t) = F ( t) δ ( x x ( t))[ Φ ( x)] dx (6) k k

3 Electroc Joural of Structural Egeerg (8) 28 M, K, ad F (t) are the modal mass, the modal stffess ad the modal force of the th mode, respectvely. The modal dampg coeffcet, C = α M + β K, here α ad β are costats th proper uts. Iput estmato s a aalyss method based o the state space. The state-space model of a beam structure system eeds to be costructed before applyg the put estmato method. After covertg to the state-space model, the state varables of the secod order dyamc system th degrees of freedom are represeted by a 2 1 state vector,.e. X = Y ( t) Y& ( t) T. From Equato 1, the cotuous-tme state equato ad measuremet equato of the structure system ca be formulated as: X& ( t) = AX ( t) + BF ( t) () Z( t) = HX ( t) (8) here I A = 1 1 M K M C 1 M B = H = I 2 2 X ( t) = X ( t) X ( t) X ( t) X ( t) T A ad B are costat matrces composed of the mass, dampg rato ad stffess of the beam structure system. X ( t ) s the state vector. Z( t ) s the observato vector ad H s the measuremet matrx. The ose terferece exsts the practcal crcumstaces. The ose terferece as ot cosdered Equatos ad 8. I order to approxmate the ose to smulate the practcal codto, the ose terferece th statstcal characterstcs as added the state equato ad measuremet equato of structure system. Ths radom ose terferece as represeted by the Gaussa hte ose. The statstcal characterstc of a radom varable as descrbed detal by meas of the probablty dstrbuto ad the desty fucto, ad t ca be represeted by usg the mea ad varace values of the radom process [23]. O accout of the above reaso, by samplg Equato th the samplg terval, t, the dscrete-tme statstcal model of the system th 22 processg ose put [24] becomes: here X ( k + 1) = Φ X ( k) + Γ [ F( k) + ( k)] (9) X ( k) = X ( k) X ( k) X ( k) X ( k) T Φ = exp( A t) ( k+ 1) t { [ τ ]} Γ = exp A ( k + 1) t Bdτ k t F( k) = F ( k) F ( k) F ( k) F ( k) T ( k) = ( k) ( k) ( k) ( k) T X ( k ) represets the state vector. Φ s the state trasto matrx. Γs the put matrx. t s the samplg terval. ( k ) s the processg ose vector, hch s assumed to be the Gaussa hte ose th zero mea ad varace, E{ ( k) T ( k) } = Qδ kj, here Q = Q I 2 2. Q s the dscrete-tme processg ose covarace matrx. δ kj s the Kroecker Delta fucto. To cosder the measuremet ose, the dscrete-tme measure equato s expressed as: here Z( k) = HX ( k) + υ( k) (1) Z( k) = Z ( k) Z ( k)... Z ( k) T υ( k) = υ ( k) υ ( k)... υ ( k) T Z( k ) s the observato vector. υ ( k) represets the measuremet ose vector ad s assumed to be the Gaussa hte ose th zero mea ad varace, E{ υ( k) υ T ( k) } = Rδ, here R = R kj v I 2 2. R s the dscrete-tme measuremet ose covarace matrx. H s the measuremet matrx. 3. ADAPTIVE WEIGHTED RECURSIVE INPUT ESTIMATION METHOD Adaptve eghted put force estmato s a process of determg the appled loads from the system measuremets. The preseted adaptve eghted put estmato method cossts of to portos, the Kalmam flter ad the estmator. The Kalma flter s used to geerate the resdual ovato sequece. The resdual ovato sequece cootes bas or systematc error of the uko tme-varyg put tem ad the varace or radom error of the measuremet. The estmator s the adopted to compute the forces over tme by applyg the resdual ovato sequece to the adaptve eghted recursve least square algorthm.

4 Electroc Joural of Structural Egeerg (8) 28 The detaled formulato of ths techque ca be foud the paper by Tua et al [25]. The equatos formulatg the Kalma flter are as follos. X ( k / k 1) = Φ X ( k 1/ k 1) (11) T T P( k / k 1) = ΦP( k 1/ k 1) Φ + ΓQΓ (12) Z( k) = Z( k) H X ( k / k 1) (13) T S( k) = HP( k / k 1) H + R (14) T 1 = (15) K ( k) P( k / k 1) H S ( k) a X ( k / k) = X ( k / k 1) + K ( k) Z( k) (16) P( k / k) = [ I Ka ( k) H ] P( k / k 1) (1) I Equatos 11 to 1, superscrpt represets the estmato value. X ( k / k 1) deotes the state estmato. P( k / k 1) s the state estmato error covarace. Z( k ) s the bas ovato caused by the measuremet ose ad the put dsturbace. S( k ) represets the ovato covarace. Ka( k ) s the Kalma Ga. X ( k / k ) s the state flter. P( k / k ) represets state flter error covarace. The state trasto matrx Φ, the measure matrx H, the dscrete-tme process ose covarace matrx Q, ad the dscrete-tme measuremet ose covarace matrx R of the Kalma flter must be obtaed to mplemet the flterg process. After the tal value, X ad P, are adopted, as the observato vector s beg putted cotuously, the output of Kalma flter ca be obtaed real-tme. The estmato value, X ( k / k 1), ad the state estmato error covarace, P( k / k 1), of the structure system ca be determed mmedately. The formulato of the adaptve eghted recursve least square algorthm s as follos: Bs ( k) = H [ ΦM s ( k 1) + I ] Γ (18) M ( k) = [ I K ( k) H ][ ΦM ( k 1) + I ] (19) a eghtg fucto s as the follog equato. 1 z( k) σ r( k) = σ (23) z( k) > σ z( k) I Equatos 18 to 23, the estmator computes the Kalma Ga Ka( k ) by applyg the ovato covarace S( k ) ad ovato matrx Z ( k ) produced by Kalma flter. By substtutg Equato 23 Equatos 2 ad 21 for eghtg factor r, the adaptve eghted recursve least square estmator ca be costructed. The procedure to estmate the uko movg force puts usg the verse method s summarzed as follos: Step1: Costruct the dscrete-tme state-space model of the system,.e. Equatos ad 9, ad measure the system resposes X ( k ). Step2: Use the Kalma flter,.e. Equatos 11 to 1, to obta the ovato matrx Z ( k ), the ovato covarace S( k ) ad the Kalma ga K ( k ). a Step3: Use the adaptve eghted recursve least square algorthm,.e. Equatos 18 to 22, to estmate the uko movg force F ˆ ( k ). 4. RESULTS AND DISCUSSION To verfy the practcablty ad precso of the preseted approach estmatg the uko movg put forces, the brdge structure s modeled as a smple beam th the total spa L = 3m, the 11 2 flexble stffess costat EI = Nm, 4 the mass per ut legth ρ = kg / m, the dampg coeffcet C = α M + β K, here s a s α =.1, ad β =. 1, ad the modal shape 1 1 T 1 T Kb ( k) = r Pb ( k 1) Bs ( k) B s( k) r Pb ( k 1) Bs ( k) + S( k) (2) fucto Φ = s( π xk / L). The tal codtos of the error covarace are gve as 1 Pb ( k) = [ I Kb ( k) Bs ( k) ] r Pb ( k 1) (21) 4 4 p( / ) = dag[1 ] for the KF ad p b () = 1 F ( k) = F ( k 1) + K ( ) ( ) ( ) b k Z k Bs k F( k 1) (22) for the adaptve eghted recursve least square estmator. The smulato codtos are set as here Z ( k ) deotes ovato value, Kb( k ) s follos. The samplg terval, t =.1 s. The the correcto ga, Bs( k ) ad M s( k ) are the sestvty matrx, M (), s ull. The eghtg sestvty matrces, P b represets the error factor s a adaptve eghtg fucto. The error covarace of the estmato, ad F ˆ ( k ) s the used to quatfy the devatos betee the estmated put vector. The eghtg factor r s estmated ad actual put movg forces s defed used to compromse betee the trackg capablty as percet root mea square dfferece (PRD) [26]: ad the degradato of estmato precso. I ths study, the adaptve eghtg fucto s preseted. The detaled dervato of ths fucto ca be foud the paper of Tua et al [2]. The adaptve 23

5 Electroc Joural of Structural Egeerg (8) 28 = 1 [ F ( t ) F ( t )] ex es Error(%) = 1% = 1 [ F ( t )] ex 2 2 (24) here s total umber of estmato tme steps, Fex ( t ) ad Fes ( t ) are the actual ad estmated forces at tme t, respectvely. 4.1 Sgular-vehcle movg force put estmato The sgular-vehcle movg force put s smulated by adoptg a vehcle th the statc eght, Fk = 2KN, actg o the brdge structure, ad the costat velocty, vk = 1 m / sec, over the brdge. Accordg to Equato 6, the tme-varyg movg force put s smulated as belo: F s( π v t / L) t t t F ( t) = t t, t t k k d d (25) here t represets the tal tme he the vehcle eters the brdge. There s tme delay for.3s order to obta a ell-determed result of smulato. The termal tme he the vehcle leaves the brdge, td = L / v. The dyamc respose of the brdge s obtaed by adoptg a umercal method th the system ose ad the measuremet ose. The Kalma estmato parameters used the umercal model are gve as follos. The covarace matrx of process ose, Q = Q I 2 2. Set Q = 1. The covarace matrx of measure 2 1 v = σ = 1. -met ose, R = R I 2 2. Set R Fgure 2 shos the trackg capablty of the estmator th dfferet eghtg factors. The trackg capablty of the estmator s superor th a smaller value of eghtg factor r, the error betee the estmated ad exact movg forces put s smaller. O the cotrary, the opposte effect th larger value of eghtg factor r s preseted. Fgure 3 shos the sgular-vehcle movg force put estmato result. The dsplacemet as measured at the mddle of the brdge. The result reveals a very good estmatg ablty, that s, the estmato values coverge to the actual values rapdly. Sce p( / ) ad p b () are ormally uko, the estmator as talzed th large values of p( / ) p ad b (), such as 14, hch troduces the effect of treatg the tal error as a large value, so that the estmator ll gore the frst fe estmates. 24 Fgure 2. Hstory of the estmated ad actual sgular-vehcle movg force puts th dfferet eghtg factors. ( F = 2( KN), v = 1 m / sec, Q = 1, ad σ = 1 ) The error (PRD) of the estmated sgular-vehcle movg force put s approxmately 8.8%. The fluece of the process ad measuremet oses o the estmato results has bee cosdered. The estmato results have demostrated the avalablty of the preseted verse estmato algorthm copg th the sgular-vehcle movg force put. Fgure 3. Hstory of the estmated ad actual sgular-vehcle movg force puts ad the dsplacemets at the mddle of the brdge( F = 2( KN), v = 1 m / sec, Q = 1, σ = 1, ad error = 8.8% ) Fgure 4 depcts the correspodg hstory of the sgular-vehcle movg force put estmato result ad the dsplacemets. The sgular-vehcle movg force put s smulated by adoptg a md-sze vehcle th the statc eght, Fk = 15KN, actg o the brdge structure, ad the costat velocty, v = 2 m / sec. The error (PRD) of the estmated sgular-vehcle movg force put s approxmately 16.4%.

6 Electroc Joural of Structural Egeerg (8) 28 Kalma estmato parameters are adjusted that Q = 1, ad Rv = σ = 1. Fgure 6 shos the hstory of the sgular-vehcle movg force put estmato result ad the dsplacemets. The result reveals a very good estmatg ablty, that s, the error (PRD) of the estmated sgular-vehcle movg force put s apparetly reduced (1.9%). Fgure 4. Hstory of the estmated ad actual sgular-vehcle movg force puts ad the dsplacemets at the mddle of the brdge. ( F = 15( KN ), v = 2 m / sec, Q = 1, σ = 1, ad error = 16.4% ) Fgure 5 shos the hstory of the sgularvehcle movg force put estmato result ad the dsplacemets. The movg force put s smulated by adoptg a compact vehcle th the statc eght, Fk = 1KN, actg o the brdge structure, ad the costat velocty, v = 3 m / sec. The error (PRD) of the estmated sgular-vehcle movg force put s approxmately 2.42%. Fgure 6. Hstory of the estmated ad actual sgular-vehcle movg force puts ad the dsplacemets at the mddle of the 8 brdge. ( F = 1( KN), v = 3 m / sec, Q = 1, σ = 1, ad error = 1.9% ) 4.2 Mult-vehcle movg force put estmato Fgure 5. Hstory of the estmated ad actual sgular-vehcle movg force puts ad the dsplacemets at the mddle of the brdge. ( F = 1( KN), v = 3 m / sec, Q = 1, σ = 1, ad error = 2.42% ) Accordg to Fgures 3 to 6, the performace of the estmator s ot satsfyg he adoptg the statc lght-eght ad the hgh-velocty vehcle, the error (PRD) s relatvely hgher. Hoever, the preseted verse method stll has the trackg capablty to detfy the movg force puts. To obta better estmato results, the values of Three movg force puts are smulated by adoptg multple vehcles th the statc eghts, Fk = 1~3 = 1KN, actg o the brdge structure, ad the costat veloctes, vk = 1~3 = 1 m / sec. The tal tme t as the frst vehcle eters the brdge s delayed for.3s to obtaed a more ell-determed result of smulato. The tme terval betee ay to vehcles etres s.5s. Accordg to Equato 6, the tme-varyg movg force puts are smulated as follos: Fk s( πvkt / L) t t td F ( t) = (26) t t, t td here td = L / v k = 1~3, hch represets the termal tme as the vehcles leave the brdge. The Kalma estmato parameters used the umercal model are adopted as follos: The covarace matrx of the processg ose, Q = 1. The covarace matrx 2 1 of the measuremet ose, Rv = σ = 1. The dyamc respose of the brdge s obtaed by usg a umercal method th system ose ad measuremet ose take to accout. The hstory of the mult-vehcle movg force put estmato result ad the dsplacemets at the mddle of the 25

7 Electroc Joural of Structural Egeerg (8) 28 brdge are sho Fgure. The error (PRD) of the estmated mult-vehcle movg force put s approxmately 1.66%. The estmato result demostrates the avalablty of the preseted verse estmato algorthm use of estmatg the mult-vehcle movg force puts. vehcle/brdge teracto flueces the estmato resoluto, but the result s stll acceptable. To obta better estmato results, the values of Kalma estmato parameters ca be adjusted that 2 12 Q = 1, ad Rv = σ = 1. Fgure 9 shos the hstory of the mult-vehcle movg force put estmato result ad the dsplacemets. The result reveals a better estmatg performace, ad the error (PRD) of the estmated sgular-vehcle movg force put s apparetly reduced (5.3%). The performace of the estmator s flueced by the Kalma estmato parameter, R. Fgure. Hstory of the estmated ad actual mult-vehcle movg force puts ad the dsplacemets at the mddle of the brdge. ( F 1 = F 2 = F 3 = 1( KN), v 1 = v 2 = v 3 = 1 m / sec ). Fgure 9. Hstory of the estmated ad actual mult-vehcle movg force puts ad the dsplacemets at the mddle of the brdge. ( F 1 = F 2 = F 3 = 1( KN), v 1 = 3, v 2 = 2, v 3 = 1 m / sec) Fgure 1 shos the hstory of the mult-vehcle movg force put estmato results ad the dsplacemets. The statc eght of the frst vehcle puts movg force, F1 = 2( KN), to the brdge structure. The secod force, F2 = 15( KN), ad the thrd force, F3 = 1( KN). The velocty of the frst vehcle, v 1 = 3 m / sec, the secod oe, v2 = 2 m / sec, ad the thrd oe, v3 = 1 m / sec. The Kalma estmato parameters are set that, Q = 1, ad 2 1 Rv = σ = 1. The error (PRD) of the estmated mult-vehcle Fgure 8. Hstory of the estmated ad actual mult-vehcle movg force puts ad the dsplacemets at the mddle of the movg force put s about 15.%. I ths case, brdge. ( F1 = F2 = F3 = 1( KN), v1 = 3, v2 = 2, v3 = 1 m / sec Q = 1, although the complex smulato stuato flueces σ = 1, ad error = 19.54% ) the estmato resoluto, the result s stll Fgure 8 shos the hstory of the mult-vehcle acceptable. movg force put estmato result ad the The mproved estmato result by usg a dsplacemets. The vehcle movg force puts are smaller value of the Kalma estmato parameter smulated by adoptg the same statc eghts, R as sho Fgure 11. The error (PRD) of the Fk = 1~3 = 1KN, actg o the brdge structure. The estmated mutl-vehcle movg force put s velocty of the frst vehcle, v1 = 3 m / sec, the secod apparetly reduced (5.31%). The capablty of the oe, v2 = 2 m / sec, ad the thrd, v3 = 1 m / sec. The estmato are demostrated through ths turg Kalma estmato parameters are set that Q = 1, Kalma parameter ( Q = 1, ad Rv = σ = 1 ) ad Rv = σ = 1. The error (PRD) of the example th complex mult-vehcle movg force estmated mult-vehcle movg force put s puts appled. approxmately 19.54%. I ths case, the complex 26

8 Electroc Joural of Structural Egeerg (8) CONCLUSIONS Fgure 1. Hstory of the estmated ad actual mult-vehcle movg force puts ad the dsplacemets at the mddle of the brdge. ( F 1 = 2, F 2 = 15, F 3 = 1( KN), v 1 = 3, v 2 = 2, v3 = 1 m / sec ) The above smulato results demostrate that the system modal ose ad measuremet ose ll fluece the estmato resoluto. The trackg capablty of the estmator s degraded as sho Fgures 8 ad 1 he applyg a larger value of the measuremet ose covarace, R v. O the cotrary, the opposte effect he applyg a smaller measuremet ose covarace s preseted Fgures 9 ad 11. The smulato results sho that the proposed method has good performace trackg uko movg forces mposed o the brdge structure system. Fgure 11. Hstory of the estmated ad actual mult-vehcle movg force puts ad the dsplacemets at the mddle of the brdge. ( F 1 = 2, F 2 = 15, F 3 = 1( KN), v 1 = 3, v 2 = 2, v3 = 1 m / sec ) I ths paper, a verse adaptve eghted put estmato methodology s proposed to estmate the uko tme-varyg movg forces produced by the vehcle the brdge structure system. Ths algorthm cludes the Kalma flter (KF) ad the adaptve eghted recursve least square estmator (RLSE), hch recursvely estmates the uko puts uder a stuato that the system volves the measuremet ad modelg errors. The algorthm s a effcet o-le recursve verse method to estmate the force puts. The capabltes of the proposed algorthm are demostrated by usg to smulato examples. The method has fast adaptve capablty ad good performace trackg the movg forces, by adequately choosg the smaller Kalma parameter R v, alog th the adaptve eghtg factor r. The estmato method proposed ths paper ca be appled to further research extesvely. Future orks of ths study ould address the problems of the force put estmato the to or three dmesoal structural system ad the applcatos the optmal cotrol scope. 6. REFERENCES [1] R. Cate, "Dyamc behavor of hghay brdges uder the passage of heavy vehcles", Sss Federal Laboratores for Materals Testg ad Research (EMPA) Report, 1992, pp [2] R. J. Heyood, "Ifluece of truck suspesos o the dyamc respose of a short spa brdge", Iteratoal Joural of Vehcle Desg, 1994, pp [3] M. F. Gree ad D. Cebo, "Dyamc respose of hghay brdges to heavy vehcle loads: theory ad expermetal valdato", Joural of Soud ad Vbrato, Vol. 1, 1994, pp [4] Y. B. Yag ad J. D. Yau, "Vehcle-brdge teracto elemet for dyamc aalyss", Joural of Structural Egeerg, ASCE 123, 199, pp ,. [5] P. Davs ad F. Sommervlle, "Lo-Cost Axle Load determato", Proceedgs of the 13 th ARRB ad Ffth REAAA Combed Coferece, part 6, 1986, pp [6] R. J. Peters, "AXWAY-a system to obta 2

9 Electroc Joural of Structural Egeerg (8) 28 vehcle axle eghts", Proceedgs of the 12 th ARRB Coferece, Vol. 12(2), 1984, pp [] R. J. Peters, "CULWAY-a umaed ad udetectable hghay speed vehcle eghtg system", Proceedgs of the 13 th ARRB ad Ffth REAAA Combed Coferece, part 6, 1986, pp -83. [8] D. Cebo, "Assessmet of the dyamc heel forces geerated by heavy road vehcles", Proceedgs of the Symposum o Heavy Vehcle Suspeso ad Characterstcs, Australa Road Research Board, 198, pp [9] K. Hech, M. Fafard, G. Dhatt ad M. Talbot, "Dyamc behavor of mult-spa beams uder movg loads", Joural of Soud ad Vbrato, Vol. 199(1), 199, pp [1] D. Y. Zheg, Y. K. Cheug, F. T. K. Au ad Y. S. Cheg,"Vbrato of mult-spa o-uform beams due to movg loads", Joural of Soud ad Vbrato, Vol. 212(3), 1998, pp [11] Y. A. Dugush ad M. Eseberger, "Vbratos of o-uform cotuous beams uder movg loads", Joural of Soud ad Vbrato, Vol. 254(5), 22, pp [12] J. F. Doyle ad M. Eseberger, "A expermetal method for determg the locato ad tme of tato of a uko dspersg pulsev, Expermetal Mechacs, Vol. 2, 198, pp [13] H. R. Busby ad D. M. Trujllo, "Soluto of a verse dyamcs problem usg a egevalue reducto techque", Computers ad Structures, Vol. 25, 198, pp [14] P. E. Holladsorth ad H. R. Busby, "Impact force detfcato usg the geeral verse techque", Iteratoal Joural of Impact Egeerg, Vol. 8, 1989, pp [15] J. Druz, J. D. Crsp ad T. Ryall, "Determg a dyamc force o a plate-a verse problem", AIAA Joural, Vol. 29, 1991, pp [16] S. S. La, T. H. T. Cha ad Q. H. Zeg, "Movg force detfcato: a tme doma method", Joural of Soud ad Vbrato, Vol. 21, 199, pp [1] S. S. La, T. H. T. Cha ad Q. H. Zeg, "Movg force detfcato: a frequecy ad tme domas aalyss", Joural of Dyamc Systems, Measuremet ad Cotrol ASME, Vol. 12, 1999, pp [18] T. H. T. Cha, S. S. La, T. H. Yug ad X. R. Yug, "A terpretve method for movg force detfcato", Joural of Soud ad Vbrato, Vol. 219, 1999, pp [19] P. C. Tua, S. C. Lee ad W. T. Hou, "A effcet o-le thermal put estmato method usg Kalma flter ad recursve least square algorthm". Iverse Problem Eg 5, 199, pp [2] P. C. Tua ad W. T. Hou, "Adaptve robust eghtg put estmato method for the 1-D verse heat coducto problem". Numeral Heat Trasfer 34, 1998, pp [21] C. K. Ma, P. C. Tua, D. C. L ad C. S. Lu, "A study of a verse method for the estmato of mpulsve loads". It J Syst Sc 29, 1998, pp [22] H. T. C. Tommy ad B. A. Demeke, "Theoretcal study of movg force detfcato o cotuous brdges", Joural of Soud ad Vbrato, Vol. 295, 26, pp [23] T. Fu, "Radom vbrato of egeerg", Defese dustres publcato, Pekg, [24] P. L. Bogler, "Trackg a maeuverg target usg put estmato", IEEE Trasactos o Aerospace ad Electroc Systems, Vol. AES-23, No. 3, 198, pp [25] P. C. Tua, C. C. J, L. W. Fog ad W. T. Huag, "A put estmato approach to o-le to dmesoal verse heat coducto problems", Numercal Heat Trasfer B, 29, 1996, pp [26] C. K. Ma, J. M. Chag ad D. C. L, "Iput forces estmato of beam structures by a verse method", Joural of Soud ad Vbrato, 259(2), 23, pp