Seled tuned liquid column dmper: cot effective olution for vibrtion dmping of lrge rch hnger W. De Corte, C. Deleie nd Ph. Vn Bogert Ghent Univerity, Deprtment of Civil Engineering, Ghent, Belgium ABSTRACT: The pper decribe the pplicbility of eled tuned liquid column dmper (STLCD) for the control of wind induced vibrtion of long nd lender rch hnger. Such intolerble trnvere hnger vibrtion were oberved on lrge rilwy rch viduct with hnger up to 35 m in the Netherlnd. A reult of thorough invetigtion, the vibrtion were ttributed to unexpectedly low mteril dmping t ζ =.11. A number of poible olution were invetigted, nd finl olution with hydrulic dmper w choen. A n lterntive olution for ce uch thi, the eled configurtion of tuned liquid dmper my be propoed. The STLCD device llow borbing the fundmentl hnger vibrtion, even when exceeding.5 Hz. For the forementioned exmple, the effect of the STLCD i nlyed numericlly nd teted t reduced cle, indicting tht the STLCD tifctorily incree the mteril dmping of the hnger, excluding the excittion type reponible for the intolerble vibrtion. 1 INTRODUCTION In their pper publihed in SEI, Vrouwenvelder nd Hoeckmn decribe ce tudy on theoreticl nd prcticl conidertion of wind induced vibrtion oberved on the tubulr hnger of rilwy viduct, the Werkpoorbridge, in the Netherlnd (Vrouwenvelder nd Hoeckmn, 4). The uthor decribe the proce of problem identifiction nd olution optimiztion, trting from the obervtion in the fll of of intolerble trnvere hnger vibrtion on the 5m pn tied rch (Fig. 1). The vibrtion were ttributed to Von Krmn excittion mde poible by n unexpectedly low mteril dmping t ζ =.11. The pper report number of rejected olution nd the nlyi method for the finl olution with vicou dmper. Although effective, thi cn be een cotly olution (Reiterer, 4), nd n intereting lterntive my be the eled configurtion of tuned liquid dmper. It i thi olution tht will be ddreed in thi pper. A tuned liquid column dmper (TLCD) i pecific type of motion dmper, relying on the motion of liquid m in rigid U hped tube (Ski et l, 1989). The externl motion of the tructurl element on which the dmper i ttched induce phe delyed motion of the liquid m. Thi motion crete internl force in the tube, countercting the externl force. Additionlly, the kinetic energy ccumulting in the tructurl element i diipted by turbulent dmping force, riing from built-in orifice plte with mll opening. The effectivene of trditionl TLCD with open verticl tube decree rpidly for frequencie over.5 Hz.
41 ARCH 7 5th Interntionl Conference on Arch Bridge Figure 1 : The Werkpoorbridge ide view Conequently, it cn only be ued for low frequency. However, if the verticl tube re eled, preure cn build up in either ide of thee ection, increing the undmped nturl circulr frequency. Such device i nmed Seled Tuned Liquid Column Dmper (STLD). In firt prt of thi pper, generl conidertion nd clcultion relted iue on the STLCD concept will be preented. In econd prt cle model tet, bed on the dimenion of the Werkpoorbridge, re dicued. Finlly, prcticl iue nd further reerch tep re ddreed. MATHEMATICAL MODELLING.1 Eqution of motion When n STLCD i ttched to ingle degree of freedom tructure (Fig. ), the et of eqution of motion cn be written Eq. (1) (Kwok et l, 1996). µγl Pw (t) (1 + µ)v && + w&& + ζ ω v& + ω v = L em m (1) n n γl ξ w& g p H H && v + w&& + α w& + w + = L ee L ee L ee ρl ee H w H + w where A = verticl ection cro-ectionl re, A 1 = horizontl ection cro-ectionl re, α = A/A 1, g = grvittionl ccelertion, H = ir pring height, L = liquid column length, L em = L B + B/α = equivlent uniform column length (equl m), L ee = L B + αb = equivlent uniform column length (equl energy), m = equivlent tructurl m, p = initil ir pring preure, n = polytropic index ( n=1.), P w = mplitude of externl. wind force, ζ = tructurl dmping rtio, v = horizontl tructurl diplcement, v = horizontl tructurl velocity, v. = horizontl tructurl ccelertion, w = verticl reltive liquid diplcement, w = verticl reltive liquid velocity, w = verticl reltive liquid ccelertion, γ = B/L, rtio of STLCD horizontl length to totl length, µ = rtio of liquid m (m l ) to eq. tructurl m (m ), ρ =liquid denity, ω = circulr tructurl frequency, ω d = circulr STLCD frequency, ω w = circulr frequency externl force, ξ = coefficient of hed lo of liquid column. In the ce where during vibrtion, the verticl reltive liquid diplcement w remin mll, compred to the ir pring length H, the lt term of the econd eqution in Eq. 1 cn be implified to liner term, from which implified expreion for the nturl circulr frequency of the STLCD cn be deduced (Eq. ). ω g np + 1 Lee ρgh d = ()
W. De Corte, C. Deleie nd Ph. Vn Bogert 411 B H A w w A orifice A1 L Figure : STLCD fitted on SDOF tructure : nottion In the eqution of motion the fluid velocity i umed contnt cro the cro-ectionl re of the horizontl nd verticl tube. They cn be olved by direct integrtion method, nd render diplcement, velocitie, ccelertion, nd energy level in tructure nd dmper function of externl force nd initil condition, nd enble to clculte frequency repone curve. The vlue of m, ω nd P w of the SDOF nlogy cn be found by Ryleigh quotient bed method nd require clcultion prior to the poible olution of the eqution of motion.. Determintion of tructurl eigenfrequency From obervtion (Vrouwenvelder nd Hoeckmn, 4), it i tted tht the vibrtion of n rch hnger i independent of the movement of the reminder of the bridge, nd in the dmper deign, the tructure cn be limited to the hnger only. However, the hnger vibrtion till depend on the rottionl tiffnee of the connection between the hnger nd the bridge. The hnger cn be modeled m-pring ingle degree of freedom ytem, nd it firt nturl frequency cn be pproximted by Eq. (3). ω l 1 1 1 d y 1 l dy K 1α1 + K α + EI dx + N dx = [ ( )] = dx dx R y x l 1 m nly d + ρa( y) dx where K 1 = rottionl tiffne t ( being the lower boundry), K = rottionl tiffne t l (l being the upper boundry), α 1 = rottion t, α = rottion t l, y d = horizontl diplcement t STLCD poition, y(x) = vibrtion hpe, ω = circulr tructurl frequency, E = Young modulu of hnger mteril, I = moment of inerti of hnger, N = norml force in the hnger, A = cro-ectionl re of hnger, ρ = denity of hnger mteril, m nl = non-liquid m of the STLCD. In thi expreion, the vlue of α 1, α nd y d re given by y(x) or it derivtive t their repective loction..3 STLCD Optimiztion The deign nd the optimiztion of the dmper comprie the determintion of 6 chrcteritic of the TLCD: A, A1, B, L, H nd ξ. The choice of thee prmeter i n itertive proce, trting with the choice of the product γ.µ, which i the rtio of the liquid m in the horizontl tube to the equivlent tructurl m; γ.µ doe not require to be lrger then trictly necery, nd vlue of.1 to. will generlly be ufficient. With the necery liquid volume clculted, one cn chooe tube dimeter prcticl for the contruction of the dmper. In generl, n equl dimeter i ued for the verticl tube nd for the horizontl one becue in the ce of eled TLCD, the mot importnt fctor i the ir pring volume, which cn be chieved with ny dimeter. In the verticl tube, the wter level h to be low poible it m doe not ctively contribute to the dmping. However, it h to be high enough to llow the wter level to entirely fluctute in the verticl tube. Thi men tht γ (B/L) h to hve vlue (3)
41 ARCH 7 5th Interntionl Conference on Arch Bridge cloe to 1 i poible. Bed in thee initil vlue of A, A1, B nd L now determined, the remining chrcteritic cn be determined from Eq. (1) in order to tune the dmper to the firt nturl frequency of the hnger. In fct, the dmper h to be tuned for omewht lower frequency, the dmper m h mll influence on the tructurl eigenfrequency. Thi influence i determined by the tuning rtio χ (ω d /ω ). The finl tep of the optimiztion conit of finding the optimum vlue for χ nd ξ, ccording to the generl method preented by Go nd Kwok (Go nd Kwok, 1997). The optimum vlue of the tuning rtio thu define the height of the ir pring H, nd the optimum vlue for the hed lo coefficient determine the dimenion of the orifice plte..4 Numericl Exmple Thi numericl exmple i bed on the dimenion of the Werkpoorbridge hnger (Hoeckmn nd Vrouwenvelder, 4). Thee dimenion re : mximum hnger length = 35m, outer dimeter = 394mm, inner dimeter = 34mm, denity = 785 kg/m³, norml force = 148kN, Young modulu = 1 GP, tructurl dmping rtio =.11. Eq. (3) clculte the firt nturl vibrtion of the hnger ω = 15.43 rd/ or f =.46 Hz. Thi vlue i cloe to the reported vlue of.5 Hz (Vrouwenvelder nd Hoeckmn, 4). The poition of the dmper i choen t /3 of the length of the hnger; thi to be out of the free pce required for rilwy trffic nd ervice. Second, choice i mde for the m rtio γ.µ =.1 nd choice for the non-liquid dmper m m nl = 1 kg. With the vlue of m nl = 1 kg nd y d = 3.33 m, the firt nturl vibrtion frequency cn be clculted ω = 15.3 rd/ or f =.43 Hz. With n equivlent m of m = 5667 kg nd the m rtio γ.µ =.1, the m of the horizontl liquid volume i equl to 56.67 kg. Bed on thi vlue, the horizontl tube dimeter cn be choen. m. With thi, the re of the horizontl tube A i equl to A =.315 m². To obtin m of 56.67 kg t ρ liquid = 1 kg/m³, the horizontl length B become B = 1.8 m. Generlly it i choen to hve A = A 1, which i done in thi exmple well. The length L hould be tken hort poible ince the verticl liquid m doe not effectively co-operte in the dmping proce. Neverthele, the length of ech verticl prt hould be ufficient to llow the wter level to remin in the verticl prt during the vibrtion. The liquid height in ech tube i tken t,1 m. The length L become L= m. The tuning rtio χ, defining the ir pring height H nd the coefficient of hed lo ξ, defining the orifice opening φ cn be found by erching the (χ,ξ) combintion generting the minimum horizontl diplcement. Thee diplcement re clculted by numericl integrtion of Eq. (1). Uing thi method, the dmper i found to be tuned t (χ =.995,ξ = 6). Thee vlue correpond to H =.55 m nd φ orifice =.86 m. Without TLCD, the mximum diplcement in the centre of the hnger with mteril dmping rtio equl to ζ =.11 i y tt /ζ = 56 mm for lod ocillting t the eigenfrequency of the originl ytem. With the dmper intlled, the mximum diplcement i reduced to 1.8 mm. A the dmping rtio i proportionte to the mximum diplcement (y mx = y tt / ζ), the dmping rtio of the hnger i increed to ζ =.7. Evidently, the propoed olution will control vibrtion in one direction only, lthough dmping will be needed in t let two orthogonl direction. The propoed olution conider dmper ttched to ingle hnger only. A olution which i more prcticl, i to ttch one dmper to two trnverely oppoite hnger by men of connecting rod. In ddition, thi configurtion i lo viully more cceptble. In thi itution, with the geometry of the dmper kept equl, the m rtio i equl to γ.µ =.5. With the dmper tuned t (χ =.997,ξ = ), the mximum diplcement in the centre of the ech hnger i reduced to.5mm. Thi i lo vlue within cceptble limit, hence even mller vlue of the m rtio γ.µ could be pplied. 3 SCALE MODEL 3.1 Generl Chrcteritic The numericl exmple preented in prgrph.4 w teted in cle model. For thi, cled model of the longet hnger of the Werkpoorbridge w fbricted. The model w
W. De Corte, C. Deleie nd Ph. Vn Bogert 413 deigned ccording to the Werkpoorbridge geometry t 1/1 cle. Fig. 3 give 3D repreenttion of the cle model, while Fig. 4 how the model built. Figure 3 : 3D repreenttion of the cle model. Figure 4 : A built cle model. The cle model differ from the rel itution on point. The firt i obviouly tht the model only repreent the hnger, nd the ret of the tructure i omitted. The econd i the bence of the norml force N, which chnge the tructurl eigenfrequency with bout 5%. It w decided tht ttempt to introduce norml force into the pecimen, till llowing for ey hndling, would led to uncceptble chnge of the boundry condition nd probbly of the tructurl dmping. The firt eigenfrequency w clculted in the bence of the norml force, nd conequently meured on the tet pecimen. Finlly, ll dmper relted vlue were reclculted bed on thee new dt. The vibrtion were regitered with reitive trin guge mounted on the hnger t the centre poition. The recorded trin ignl were trnformed into diplcement nd the frequency nlyi i done through Ft Fourier Trnform of the recording. Uing trin guge h the dvntge over ccelerometer tht they re extremely mll compred to the model dimenion. A reult, they do not chnge the frequency nd tructurl dmping chrcteritic. 3. Scling reltion The cle model cn predict the behviour of the rel hnger only if both tructure re geometriclly, kinemticlly nd dynmiclly uniform. To comply with thee condition, ll term in the eqution of motion hould be multiplicble with cle fctor. For choen liner geo-
414 ARCH 7 5th Interntionl Conference on Arch Bridge metric cle α L =.1, the geometric, kinemtic nd dynmic uniformity condition require the remining cle fctor to be : α w = 1 (frequency); α t =.1 (time); α v =1 (velocity); α = 1 (ccelertion), nd α F =.1 (force). Thi et of cle fctor determine the frequency nd time cle, mot of the other cle fctor cn lo be derived from the geometry. With well choen vlue for the pring height H nd the coefficient of hed lo ξ, n overll cle fctor cn be clculted which i pproximtely the me for ech term of the eqution of motion. The dimenion of the model, obviouly bed on commercilly vilble tube ection re : hnger length = 3.58m, outer dimeter = 4.4mm, inner dimeter = 37.mm, denity = 785 kg/m³, norml force = kn, Young modulu = 1 GP. Eq. (3) clculte the firt nturl vibrtion of the cled hnger ω = 17.48 rd/ or f =.9 Hz. A will be explined lter, thi vlue i not chieved in the tet etup, the boundry condition do not fully retrin the ngulr rottion t both end of the hnger. 3.3 Meured eigenfrequencie The eigenfrequency of the hnger in the cle model found from Fourier nlyi of the trin reult re equl to f x = 17.6 Hz, nd f y = 13.86 Hz (Fig. 3). Thi difference in vibrtion frequency cn be explined be unequl boundry condition in longitudinl nd trnvere direction of the upper hnger connection. Indeed the connection with the tubulr member of the frme i le rigid in the trnvere direction, lthough evidently neither connection i fully rigid, ince the clculted vlue of f =.9 Hz i never reched. 3.4 Scled dmper chrcteritic Bed on the model dimenion of prgrph 3., cled dmper with the following dimenion i clculted : B =.5m, L =.33m, inner dimeter =.7m, orifice opening =.95m, H =.16m. Thi dmper i intended for ue in trnvere direction only, nd in firt tet, the effect of uch dmper i teted by imple meurement of the tructurl dmping ζ, without nd with the dmper intlled. Fig. 5 diply the meured trin vlue in the undmped hnger, while Fig. 6 diply the meured trin vlue in the dmped hnger with the dmper intlled t the centre poition (x d = 1.6m). From Fig. 5, nd uing the logrithmic decrement Λ = π.ζ (for mll vlue of ζ), the tructurl dmping of the undmped cle model i clculted ζ =.17. Thi vlue prcticlly coincide with the vlue of ζ =.11 from the Werkpoorbridge. Figure 5 : Undmped trin ignl.
W. De Corte, C. Deleie nd Ph. Vn Bogert 415 Figure 6 : Dmped trin ignl. In the me wy, Fig. 6 revel tht the tructurl dmping i increed to ζ =.15 with the dmper intlled. Thi i more thn 1 fold incree of the tructurl dmping. The effect of the dmper cn lo be clculted with the eqution of motion of the SDOF nlogy. Although the clculted logrithmic decrement i not contnt, given the non liner dmping, nd the effective dmping depend on the initil deformtion, tructurl dmping vlue of ζ =.9 w clculted. Thi clculted vlue more thn double the meured vlue, which cn be ttributed to the fct tht in the model vibrtion i conidered in one direction only while in the cle model vibrtion energy i ditributed over two orthogonl eigenmode. Hence, in the cle model the dmper will ct in both direction, lthough intended for unixil dmping only. A reult, the logrithmic decrement i pproximtely hlf of wht w expected initilly. In ddition, when building reduced cle dmper, difference of up to 3 % between clculted nd chieved eigenfrequency i poible. Thi, however, trnlte to proportionlly lrger difference in logrithmic decrement, w indirectly noticed in the field experiment decribed in prgrph 3.5. 3.5 Field experiment Meurement were done on the Zeebrugge hrbour brekwter. The wind peed t thi loction re of ufficient mgnitude to obtin ueful reult. Following the model reembly with lightly tiffer connection, the eigenfrequency in the dmper direction w meured t 15.5 Hz, nd the dmper w reconfigured following thi frequency chnge. Meurement were done by recording the trin ignl over period up to 1 econd, nd for conequent meurement, the dmper i tuned to lightly different frequencie, nd i plced on vriou height. f d = 14.3 Hz ; χ =.97 ; ξ = 58,6 Diplcement (m),5,4,3, Amplitude (µstrin),5,,15,1,1,5, 13 13,5 14 14,5 15 15,5 16 Frequency (Hz) 13 14 15 Frequency (Hz) 16 Figure 7 : Predicted nd meured frequency repone curve for ner - optimum dmper.
416 ARCH 7 5th Interntionl Conference on Arch Bridge f d = 13.7 ; χ =.933 ; ξ = 58,6,5 Diplcement (m),5,4,3,,1, 13 13,5 14 14,5 15 15,5 16 Frequency (Hz) Amplitude (µstrin),,15,1,5 13 14 15 Frequency (Hz) 16 Figure 8 : Predicted nd meured frequency repone curve for non - optimum dmper. Becue of the turbulent nture of the wind, the model i excited t brod rnge of frequencie. Conequently, frequency repone curve i derived directly from Ft Fourier Trnform of the dt et. Becue of the fct tht the wind peed i likely not to be contnt for period of time long enough to chieve mximum diplcement nd becue of the reltively long meuring time, the obtined frequency-repone curve h lower mplitude thn the theoreticl curve. In pite of tht, the hpe of the clculted nd meured frequency repone curve correpond very well, lthough the meurement reveled n pproximtely.5 % (.4 Hz) higher thn clculted dmper frequency. Thi tken into ccount, the curve gree very well, cn be een in Fig. 7 nd 8. 4 CONCLUSIONS Wind induced vibrtion of teel rch hnger cn effectively be controlled by the ue of eled tuned liquid column dmper (STLCD). It w implied tht the vibrting hnger cn be identified to SDOF ytem. After clcultion of the relevnt prmeter of uch ytem, nd by dequte choice of rtio of dmper m to hnger m, the dmper cn be optimized. Scle model tet indicte tht the STLCD i effective lthough it doe not completely function predicted by the mthemticl eqution. Severl reon including the unixil nture of the clcultion model, nd the indirect nture of the frequency meurement cn be put forwrd. Further prcticl reerch, including direct frequency meurement re necery before implementtion. REFERENCES Kwok, K.C.S., Smli, B. nd Xu, Y.L., 1991, Control of wind induced vibrtion of tll tructure by optimized tuned liquid column dmper, Proc. Ai-Pcific Conf. on Computtionl Mechnic, Hong Kong, p. 569-574. Reiterer, M., 4, Control of pedetrin-induced bridge vibrtion by tuned liquid column dmper, Proc. Third Europen Conf. on Structurl Control, Vienn, Autri, 4. Ski, F., Tked, S. nd Tmki, T., 1989, Tuned liquid column dmper-new type device for uppreion of building vibrtion, Proc. Int. Conf. on Highrie Building, Nnjing, Chin, p. 96-931. Vrouwenvelder, A.C.W.M. nd Hoeckmn, W., 4, Wind-induced vibrtion of tubulr digonl of the Werkpoorbridge, Structurl Engineering Interntionl, 4/4, p. 314-31.