Identification of Source Time Function for Train-induced Vibration of Underground Tunnels

Transcription

1 7th World Conference on Nondetructive Teting, 5-8 Oct 8, Shnghi, Chin Identifiction of Source Tie Function for Trin-induced Vibrtion of Underground Tunnel Juin-Fu CHAI nd Tung-Jen TENG Ntionl Center for Reerch on Erthquke Engineering, Tipei, Chinee Tiwn E-il: Abtrct The objective of thi pper i to identif the dnic ource tie function for the trin-induced vibrtion of underground tunnel. Under the conidertion of the gp between djcent lb nd the contrint fro equl-pcing ftener, the verticl force loded b the oving trin i odelled b periodicl ource tie function. Conequentl, the induced vibrtion t n pecified obervtion point on the inner wll of the ebedded wveguide cn be deterined on the bi of the trnition trix (T-trix) ethod. On the other hnd, thnk to the Tipei Rpid Trnit Corportion for the periion of n in-itu tet, the trin-induced vibrtion on the inner wll of underground tunnel w eured to vlidte the propoed periodicl ource tie function nd further, to identif the ocited loding preter. It cn be found tht the vibrtion repone deterined nuericll b the identified odel, both the wvefor in tie doin nd the Fourier pectru in frequenc doin, re in good greeent with the eured ignl. Keword: trin-induced vibrtion, underground tunnel, T-trix, dnic ource tie function. Introduction The vibrtion of underground tunnel or highw bridge cued b the trffic loding will propgte outwrdl through the urrounding oil ler, nd then, the nerb tructure will be ffected ignificntl b the induced vibrtion or noie. The nli of ground otion nd the ocited tructurl vibrtion due to underground trin i ver coplicted [, ] (Lin nd Krlov, ; Metrikine nd Vrouwenvelder, ), however, it cn be iplified b three topic: () the nli of the trin-induced vibrtion of tunnel ebedded in n infinite doin, () the nli of trnient wve cttered fro the tunnel in the urrounding oil ler, nd (3) tie hitor nli of building on the bi of the input of the trin-induced ground ccelertion [3] (Grdien nd Stuit, 3). In thi pper, the tunnel i ued to be n infinite hollow clindricl wveguide ebedded in n infinite eltic doin. Neither the interction between tunnel nor the boundr condition between oil ler i conidered. Hence, bed on the trnition trix (T-trix) ethod, the induced vibrtion of the ebedded wveguide nd the cttered wve field in the urrounding

2 oil cn be olved in ter of the externl loding function due to the oving trin. Under the conidertion of the gp between djcent lb nd the contrint fro equl-pcing ftener, the verticl force loded b the oving trin onto the tunnel cn be odeled b periodic ource tie function. In order to verif the propoed periodic ource tie function nd to identif the ocited loding preter, the trin-induced vibrtion on the inner wll of underground tunnel w eured b n in-itu tet of the Tipei Rpid Trnit te. It cn be found tht the vibrtion repone deterined nuericll b the propoed odel well the identified preter, both the wvefor in tie doin nd the Fourier pectru in frequenc doin, re in good greeent with the eured ignl. Bed on the propoed ource tie function well the identified loding preter, the trin-induced wve field cttered fro the tunnel in the urrounding oil cn be lo deterined b the T-trix ethod. Then, in the next tge, inted of the infinite urrounding ediu, the ce of oil ler will be conidered nd odelled b lered hlf-pce, nd the foreentioned wve field cn be pplied to define the perturbtion ource within the upper oil ler. Subequentl, under the boundr condition of continuit t the interfce nd trction free on the ground urfce, the induced ground vibrtion cn be deterined b en of the trniion nd reflection coefficient trice of cttered wve t the interfce nd free ground. Bed on the iulted ground ccelertion, the tie hitor nli cn be perfored to evlute the tructurl vibrtion cued b the trin oving in underground tunnel.. Source Tie Function nd Tunnel Vibrtion. Trnition Mtrix nd Trin-induced Tunnel Vibrtion A hown in Fig., the underground tunnel i odelled b n infinite hollow clindricl wveguide urrounded b infinite oil. The 3D clindricl coordinte te (r-θ- coordinte) i defined uch tht -xi i coincident with the xi of the tunnel nd the point = i defined t the iddle point of one lb. Bed on the rdition condition t infinit (r ) well the etr bout the verticl xi, the clr potentil φ, ψ nd χ with non-negtive integer correponding to P-, SV- nd SH- wve, repectivel, cn be olved nd expreed in the frequenc doin Herein, φ ψ χ () * ( r, θ, ; ω) = ε coθ H ( k r) π () * ( r, θ, ; ω) = ε in θ H ( k r) π () * ( r, θ, ; ω) = ε coθ H ( k r) π () H i the th -order Hnkel function of the econd kind, ε i the Neunn fctor (ε = for = or ε = for ), * k p nd * k re defined b p e e e ik ik ik dk dk dk ()

3 k * p = i k k () p * ; k = i k k where k p =ω/c p nd k =ω/c re the wvenuber of the longitudinl nd her wve, repectivel. Then, bed on the clr potentil, the bi function of the diplceent vector u r, θ, ;ω cn be deterined b in the 3D doin ( ) u () (3) ( ψ e ) ; u = ( χ e ) () = φ ; u = (3) k Furtherore, the th () -order Hnkel function of the econd kind H cn be replced b the ˆ u r, θ, ;ω. It th -order Beel function J to define the regulr prt of the bi function ( ) hould be noted tht the bi function repreent the outgoing wve nd the regulr prt repreent the tnding wve. x F(, t) r S r b S b Surrounding oil ρ, λ, µ x W θ e θ z Figure. The coordinte te for the nli of trin-induced vibrtion of n underground tunnel A hown in Fig., the clindricl urfce S (r= r ) nd S b (r= r b ) re defined the inner wll nd outer wll of the tunnel, repectivel, nd S b i the interfce between the tunnel nd the urrounding oil. It hould be noted tht ll of the function nd preter correponding to the tunnel re denoted b ubcript (). Due to the trin-induced externl force loded on the inner wll S, the wve field within the tunnel (r r r b ) will propgte outwrdl nd then reflected between the inner nd outer wll. Therefore, the wve field within the tunnel (r r r b ) cn be expnded b the erie of the bi function nd the regulr prt u ˆ ˆ (4) 3 3 () = f + = = u f () = = u() On the other hnd, the wve cttered fro the tunnel in the urrounding oil (r r b ) will propgte outwrdl, nd hence, it cn be expnded b erie of the bi function u C (5) 3 = = = tunnel ρ, λ, µ Bed on the continuit condition of diplceent nd trction on the outer wll S b well the boundr condition due to the externl force loded on the inner wll S, ll of the expnion u z θ e r

4 coefficient f, fˆ nd C cn be deterined. Furtherore, it hould be noted tht the externl trction vector loded on the inner wll S cn be expreed in frequenc doin t π ( r ),, ; ω = ik θ D e dk t with = ~ co θ D = ε in θ (6) co θ Then, ll of the expnion coefficient f, fˆ nd C cn be expreed in ter of the ~ loding vector, nd further, the tie hitor of the induced vibrtion u (r,θ,;t) on the inner t wll S cn be expreed b ( ) u r, θ, ; t = u ( r,, ; ) = θ t with ( r, θ, ; t) Herein, ~ ik + iω t u = D e dk dω T t (7) 4π T i the th -order trnition trix, ll of the eleent cn be deterined nd expreed explicitl.. Periodic Source Tie Function In thi pper, the trin i ued to ove long the poitive -xi with contnt peed c, nd the firt et of wheel will p the point = t t= coincidentl. Furtherore, hown in Fig., ech et of the wheel conit of two wheel with width of W. Therefore, the r- nd θ- t r,θ, ; t loded b the firt et of wheel onto coponent of the externl trction vector ( ) the inner wll cn be expreed b (, t) [ δ ( θ θ ) + δ ( θ + θ )] coθ ; tθ = F(, ) [ δ ( θ θ ) + δ ( θ + θ )] in θ tr = F t (8) Where θ =in - (W/) i the ngle between the contct point nd the verticl xi (ee Fig. ), nd F(, t) i the contct force. Under the conidertion of the gp between djcent lb nd the contrint fro equl-pcing ftener, the verticl contct force F(, t) cn be defined b F (, t) = Q( t) δ ( ct) with Q( t) Q ( + γ co Ω t + γ co t) Ω = (9) Herein, δ(-ct) i to define the loction of the wheel long -xi, nd Q(t) i periodic function repreenting the plitude of contct force. Q i the verged contct preure, Ω nd γ re the ngulr frequenc nd plitude rtio of the force due to the gp between djcent lb, nd Ω nd γ re correponding to the ftener. In ddition, the ngulr frequencie Ω nd Ω cn be deterined b the length of lb L nd the pcing of ftener L f well the oving peed c Ω =πc/l nd Ω =πc/l f, repectivel. It cn be found tht [δ(θ-θ )+δ(θ+θ )] nd [-δ(θ-θ )+δ(θ+θ )] re the even nd odd function of θ, repectivel, nd cn be expnded b the erie of co θ nd in θ

5 [ δ ( θ θ ) ( ) + δ θ + θ ] = [ δ ( θ θ ) + δ ( θ + θ )] = b ε coθ ε in θ with ε = coθ π ε b = in θ π () Therefore, the externl trction vector loded on the inner wll S cn be expreed in the frequenc doin t π ( r ),, ; ω = ik θ D e dk t with = ~ ~ ~ coθ t = F ( k, ω) b in θ () Fro Eq. (9), we hve ~ F ik iωt γ γ i i ( k, ω) = F(, t) e ddt = πq δ ( ω k c) + δ ( ω Ω k c) + δ ( ω + Ω k c) i = Bed on the loding vector t ~ expreed b Eq. () nd (), the vibrtion u i W i () (r,θ,;t) on the inner wll induced b the firt et of wheel cn be deterined b Eq. (7). For the n th et of wheel with ditnce L n fro the firt et of wheel, the ource tie function cn be defined b F(, t-l n /c), nd hence, the ocited repone cn be deterined fro tht cued b the firt et of wheel with the pecific tie del L n /c nd ultiplied b weighting W n. Therefore, the totl repone cn be defined b ( ) = N r, θ,, t W u ( r, θ,, t L c) n W n u (3) n= 3. In-itu Meureent nd Identifiction of Source Tie Function In order to verif the propoed periodic ource tie function nd to identif the loding preter, n in-itu tet w orgnized to eure the trin-induced vibrtion on the inner wll of n underground tunnel (Tipei Rpid Trnit). A hown in Fig., the icro-treor were etup t the iddle point of two djcent lb, nd the were fixed onto the tunnel ground between the ril with θ =. A the trin pe, the trin-induced vibrtion t the elected two eureent loction cn be eured iultneoul, nd Figure 3() how the tpicl velocit ignl (verticl coponent). Becue the elected tet point re both locted t the iddle point of one lb, the wvefor of the eured ignl V nd V re uch iilr to ech other nd tie del exit between the two ignl V nd V. Thi tie del repreent the oving tie of the trin ping through the elected two tet point, nd cn be deterined on the bi of the cro correltion function. The cro correltion function i defined b

6 ( ) V ( t) V ( t ) R τ τ = dt (4) where V nd V re the eured ignl, nd the vlue tht cue the function R(τ) to rech it xiu i tken for the trveling tie between the two elected point. Therefore, the oving peed of the trin cn be deterined b the trveling ditnce between the two tet point nd the clculted oving tie. In ddition to the tie ignl, the Fourier plitude of V nd V re deterined nd copred in Fig. 3(b), nd Figure 3(c) how the zooed ignl for the low frequenc rnge ( f 4 Hz). It cn be found tht the Fourier pectr of V nd V for the two elected tet point re lot the e. It hould be noted tht, bed on ll of the eured ignl, the tunnel vibrtion re lot the e nd the oving peed of the trin ping through the tet loction i contnt of 6 k/h. dt cquiition te lb ignl cble ril ftener icro-treor Figure. Setup of the icro-treor in the in-itu tet in the underground tunnel Bed on the pecil ce with oving peed (c=6 k/h), length of lb (L =4.5 ) nd the pcing of ftener (L f =.75 ), the ngulr frequencie of the propoed periodic ource tie function cn be deterined Ω =7. rd/ (f =.5 Hz) nd Ω =39.6 rd/ (f =. Hz), nd the ocited plitude Q Q =Q γ nd Q =Q γ will be identified b the eured ignl. Bed on Eq. (7) well the regrded preter of Tipei Rpid Trnit te hown in Tble, the r-coponent of velocit repone cued b the firt et of wheel on the inner wll with = (iddle point of lb) nd θ = cn de deterined nuericll. A trin conit of 6 crrige, nd the length of ech crrige i 3.5. There re 4 et of wheel on ech crrige, nd the ditnce of the three rer et fro the firt one re 4, 6, nd, repectivel. Therefore, the totl repone due to the totl 4 et of wheel of oving trin cn be crried out b Eq. (3). Bed on the coprion of the nuericl reult

7 with the eured ignl, the plitude of the periodic loding cn be identified Q =.8, Q =.4 nd Q =., nd the weighting of the four et of wheel on ech crrige cn be lo identified W =., W =.4, W 3 =.8 nd W 4 =.. Figure 4 how the nuericl reult which re deterined on the bi of the propoed periodic ource tie function well the identified loding preter. It cn be found fro Fig. 4 tht the vibrtion repone deterined nuericll b the identified odel, both the wvefor in tie doin nd the Fourier pectru in frequenc doin, re in good greeent with the eured ignl. Tble : Preter for the nli of velocit repone of underground tunnel Underground tunnel Inner dieter: =8 Outer dieter: b=35 S-wve velocit: C =5 / P-wve velocit: C p =45 / M denit: ρ =4 kg/3 Surrounding oil S-wve velocit: C =8 / P-wve velocit: C p =38 / M denit: ρ= kg/3 Figure 3. Vibrtion ignl eured on the inner wll of n underground tunnel in the in-itu tet: () tie hitorie of velocit, (b) Fourier trnfortion nd (c) zoo in low frequenc rnge.

8 Figure 4. Coprion of the ignl deterined nuericll b the propoed odel nd the identified loding preter nd the eured ignl: () tie hitorie of velocit, nd (b) Fourier trnfortion in low frequenc rnge. 4. Concluion In thi pper, the vibrtion of n underground tunnel nd the ocited cttered wve field in the urrounding oil cn be olved in ter of the externl loding function due to oving trin. Under the conidertion of the gp between djcent lb nd the contrint fro equlpcing ftener, the verticl force loded b the oving trin onto the tunnel cn be odeled b periodic ource tie function. In ddition, the propoed periodic ource tie function nd the ocited loding preter cn be verified nd identified b the eureent of n in-itu tet of the Tipei Rpid Trnit te. It cn be found tht the vibrtion repone deterined nuericll b the identified odel, both the wvefor in tie doin nd the Fourier pectru in frequenc doin, re in good greeent with the eured ignl. Reference [] Lin, Q. nd Krlov, V. V.,, Effect of tunnel dieter on ground vibrtion generted b underground trin, J. Low Frequenc Noie nd Vibrtion, 9: 7-5. [] Metrikine, A. V. nd Vrouwenvelder, A.C.W.M.,, Surfce ground vibrtion due to oving trin in tunnel: two-dienionl odel, J. Sound nd Vibrtion, 34: [3] Grdien, W. nd Stuit, H. G., 3, Modeling of oil vibrtion fro rilw tunnel, J. Sound nd Vibrtion, 67 (3):