ScienceDirect. Vibration Control of a Cable-Stayed Footbridge Using the Tension Changes of Cable

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1 Avalable onlne at ScenceDrect Proceda Engneerng 91 (2014 ) XXIII R-S-P semnar, Theoretcal Foundaton of Cvl Engneerng (23RSP) (TFoCE 2014) Vbraton Control of a Cable-Stayed Footbrdge Usng the Tenson Changes of Cable Wojcech Pakos a *, Zbgnew Wójcck a a Wroc aw Unversty of Technology, Wroc aw , Wybrze e S. Wyspa skego 27, Poland Abstract The paper presents the theoretcal analyss for actve reducng resonant vbraton n cable-stayed footbrdges. The method reles on changng the statc tenson n chosen cables. The tenson changes cause a change n those natural frequences that are close to the exctaton frequency. The paper also seeks to establsh whch combnatons of tensons n cables lead to the fastest change n the values of a chosen natural frequency. For ths purpose egenproblem senstvty analyss, formulated for the structure accordng to the second order theory were used. The effectveness of the method was tested usng an FEM model of a footbrdge The Authors. Publshed by Elsever by Elsever Ltd. Ths Ltd. s an open access artcle under the CC BY-NC-ND lcense ( Peer-revew under responsblty of organzng commttee of the XXIII R-S-P semnar, Theoretcal Foundaton of Cvl Peer-revew Engneerng under (23RSP responsblty of organzng commttee of the XXIII R-S-P semnar, Theoretcal Foundaton of Cvl Engneerng (23RSP) Keywords: a cable-stayed footbrdge; cables, actve vbraton reducton; senstvty analyss; egenproblem In the last three decades, a small number of papers have been publshed that analyze, theoretcally and expermentally, the methods of actve vbraton reducton n cable-stayed brdges. The method of actve vbraton reducton reles on addtonal forces or vbratons whch are automatcally generated, n real tme, n a gven pont of the structure [1]. Examples of references on the subject of actve vbraton reducton brdges that are worth a menton here, among others: Warntcha, Fujno et al. [2], Fujno and Susumpow [3], Susumpow and Fujno [4], Preumont and Achkre [5], Achkre [6], Schemmann and Smth [7, 8], Magaña and Rodellar [9], Bossens and Preumont [10], Rodellar and Monroy [11]. It must be hghlghted that the suggested solutons descrbed above have not yet been mplemented n real brdge structures. However, the large number of analytcal solutons and expermental studes presented by varous authors mples that the applcaton of actve methods for reducng vbraton n real cable supported brdges wll not be long n comng. The paper [12] presents the authors own method of actvely reducng resonant vbraton of cables n cable-stayed footbrdges. The method reles on changng the statc tenson n chosen cables. The term actve refers to the fact that the tenson changes would only be ntroduced when resonant vbraton wth ampltudes of a magntude dangerous to the structure occurred. A change n the cable tensons should cause a change of the natural frequences * Correspondng author. Tel.: ; ; fax: E-mal address: wojcech.pakos@pwr.wroc.pl The Authors. Publshed by Elsever Ltd. Ths s an open access artcle under the CC BY-NC-ND lcense ( Peer-revew under responsblty of organzng commttee of the XXIII R-S-P semnar, Theoretcal Foundaton of Cvl Engneerng (23RSP) do: /j.proeng

2 Wojcech Pakos and Zbgnew Wójcck / Proceda Engneerng 91 ( 2014 ) for whch resonant vbraton amplfcaton occurred. As a consequence of these changes, the resonant state would become detuned and the forced vbraton ampltudes would then reduce sgnfcantly. The effectveness of the method was earler tested on a laboratory model of a steel footbrdge [12]. The practcal effectveness of ths method s provded by the results of analyses and tests, whch show that practcally t s enough to change tenson n some selected cables to obtan a sgnfcant effect of reducng vbraton n all the cables. The choce of the cables s not ncdental and s based on the prevously performed theoretcal analyss of the egenproblem senstvty wth respect to the desgn parameter p, whch s the tenson n cable. The paper presents the theoretcal (numercal) analyss of the possbltes of usng the authors own method of actvely reducng resonant vbraton of the cables n the model of a real structure a cable-stayed steel footbrdge n Le nca, a quarter of the cty Wroc aw n Poland. The man am of the analyss was to examne the possblty of actvely reducng forced resonant vbraton n the cables themselves by the tenson changes n cables. In fg. 1 the vew of the numercal model of the footbrdge s shown. The footbrdge was desgned as a two-spans and one pylon system. The length of each span s 34.0 m. Fg. 1. (a) The geometry of the system; (b) 3D-FEM Model - Cosmos/M. The deck structure conssts of two tubes 323.9/12.5 axal spacng 3.00 m connected by cross beams as I-beams HEB140 spacng 2.00 m. On the cross members le strngers HEB100 wth axal spacng from 400 to 500 mm covered wth 12 mm thck metal sheet. For numercal modellng stffness of the man grders equal I x = cm 4 was assumed (take nto account the elements handral). The heght of the steel ppe pylon s m. The pylon s attached to a renforced concrete pllar rgdly usng steel anchors. The contacts assembly and all other connectons are made as welded. A total of 24 cables are made of 2T15 (2 7 5 mm) ropes produced by Freyssnet Company. Passve anchorage of the cables were placed n the pylon, whereas actve ones are located n the deck. N c s the statc tenson n the th cable wth respect to the pre-tenson of the cable and the deadweght of the whole structure. N cr = 530 kn s the Mnmum Breakng Load of cable. The deck s supported by fxed bearngs on a pllar and by one-and two-way sldng bearngs on abutments. Addtonally, steel wshbones are used on all supports. Materal and geometry data was assumed accordng to paper [12]. Numercal analyses were done wth the help of a standard computer FEM software - Cosmos/M System. Beam elements and shell elements were used for modellng threedmensonal structures. 2. Egenproblem senstvty analyss Numercal soluton of the proposed actve vbraton reducton s based on the nonlnear matrx equaton of moton (1) of the three-dmensonal computatonal model of the cable-stayed footbrdge. The matrx equaton of moton of a 3D computer model of the structure has been created wth the help of standard computer software leanng on FEM Cosmos/M System. Bq Cq ( K K ) q F( t). E G (1) Symbol q s a Lagrange generalzed coordnates vector, B s a mass matrx, C s a dampng matrx, K E and K G are elastcty and geometrcal matrces of stffness of the system, whereas F(t) s a vector of exctng forces. Ths

3 144 Wojcech Pakos and Zbgnew Wójcck / Proceda Engneerng 91 ( 2014 ) paper focuses on the delberate forcng vbraton,.e. the exctaton of vbratons by squats people. Synchronzaton of persons performng squats leads to adjustng the frequency of the exctaton to the natural frequency (resonance), what causes excessve vbraton exctaton. In order to establsh how to steer the cable tenson n a way that wll ensure the most effcent resonant vbraton elmnaton, the problem of egenproblem senstvty analyss wth respect to change of the cable tenson n cable stayed footbrdges s consdered. The senstvty of solutons of dfferental equaton s a calculaton dervatve of the soluton wth respect to the desgn parameter p, upon whch the soluton of equaton depends (here p s a cable tenson). Intal equaton s the equaton of moton of the system wthout the dampng and the exctng forces Bq ( K K ) q 0. E G (2) For a gven tenson n cable N the geometrcal stffness matrx s gven by the relatonshp K G = N K N where K N = s the matrx of geometrcal stffness that corresponds to the solated value of tenson force n cable. The frst order logarthmc senstvty functon of the natural frequency wth respect to parameter N, whch s the value of tenson force n the th cable, s then gven by formula (3) ln N N s T 1 2 N ln N N 2 l B K w. (3) Symbol l T marks the transformed left egenvector, whch corresponds to the rght vector w, whch fulflls the T condton l w 1. Comparson of values of logarthmc senstvty functon s s shown n Table 1. Table 1. Comparson of values of logarthmc senstvty functon s Frequency f [Hz] Descrpton of egenform N 1 N 2 N 3 N 4 N 5 N 6 f 1c 2,4112 Cables no. 1: 1-st bendng n XZ plane f 2c 2,6982 Cables no. 2: 1-st bendng n XZ plane f 3c 2,9782 Cables no. 3: 1-st bendng n XZ plane f 4c 3,6338 Cables no. 4: 1-st bendng n XZ plane f 5c 4,3854 Cables no. 5: 1-st bendng n XZ plane f 6c 4,9434 Cables no. 6: 1-st bendng n XZ plane f 1 1,7499 Platform: 1-st bendng, ant-symmetrc f 2 2,8374 Platform: 1-st bendng, symmetrc f 3 2,8854 Pylon: 1-st bendng f 4 6,2393 Platform: 1-st torsonal The nformaton ganed from the senstvty analyss may be used to calculate how to steer the cable statc tenson n a way that wll ensure the most effcent resonant vbraton elmnaton. In Table 1, the red colour s used to mark the coeffcents of a logarthmc senstvty functon that ndcate the most effectve changes n the frequency of all tendons, whle mnmzng the number of cables, whch should be drect change the tenson. Choce of two pars of cables Nos. 1 and 4 provde an effectve change n the natural frequency of all sx pars of cables,.e. change of the statc tenson n cables no. 1 wll ensure the most effcent natural frequency changes n cables No. 1 and No. 2, whereas change of the statc tenson n cables No. 4 wll ensure the most effcent natural frequency changes n cables No. 3, 4, 5 and No. 6. The effectveness of the method was numercally tested on the exctaton of resonant vbraton of cables and selected results of these analyses are presented n the next secton.

4 Wojcech Pakos and Zbgnew Wójcck / Proceda Engneerng 91 ( 2014 ) Examples of resonant vbraton reducton of cables 3.1. The frst example actve resonant vbraton reducton of cables No. 1 by changng tenson n cables No. 1. In the frst example, the nfluence of actve steerng of tenson n cable No. 1 on the ampltude of the vbratons of all the cables was examned. The am of the task was to reduce the resonant vbraton caused by harmonc forcng F 1 (t) wth frequency f 1c = Hz. Ths frequency corresponds to the frst n-plane natural modes of vbratons of cables No. 1. In fg. 2 the ampltudes of the vbratons of all cables are shown. The ampltudes of the vbratons were measured at the same tme. In fg. 1 the red colour marks the nfluence of actve steerng of tenson n cable 1 on the vertcal ampltude of vbraton q z (t) n mllmetres, whle the blue colour shows vbratons wthout actve control. Changng the tenson n cables No. 1 was started at 4 seconds and was complete after 4 seconds. The ntal statc effort n cables 1.e. N c1 /N cr was about 8.92% and after the change t was about 13.04%. A reducton of the resonant vbraton n cable 1 by about 96% was obtaned fg. 2a. Moreover, there were no sgnfcant changes n the vbraton n other cables fg. 2b-f. Fg. 2. Vertcal dsplacements q z (t)[mm] for: (a) cables no. 1; (b) cables no.2; (c) cables no.3; (d) cables no.4; (e) cables no.5; (f) cables no The second example actve resonant vbraton reducton of cables No. 2 by changng tenson n cables No. 2. In the second example the am of the task was to reduce the resonant vbraton caused by harmonc forcng F 1 (t) wth frequency f 2c = Hz. Ths frequency corresponds to the frst n-plane natural modes of vbratons of cables No. 2. Fgure 3 shows the correspondng tmng dagram of ths analyss. Changng the tenson n cables No. 1 was started at 4 seconds and was complete after 4 seconds. The ntal statc effort n cables 1.e. N c1 /N cr was about 8.92% and after the change t was about 13.04%. A reducton of the resonant vbraton n cable 2 by about 94% was obtaned fg. 3b. Moreover, a reducton of vbratons n cable No. 3 was obtaned fg. 3c. A slght ncrease n the

5 146 Wojcech Pakos and Zbgnew Wójcck / Proceda Engneerng 91 ( 2014 ) ampltudes of vbraton n the cables No. 1 occurred fg. 3a. Moreover, no sgnfcant changes n vbraton n other cables was observed fg. 3 d-f Fg. 3. Vertcal dsplacements q z (t) [mm] for: (a) cables no. 1; (b) cables no. 2; (c) cables no. 3; (d) cables no. 4; (e) cables no. 5; (f) cables no The thrd example actve resonant vbraton reducton of cables No. 6 by changng tenson n cables No. 4. In the thrd example the nfluence of actve steerng of tenson n cable No.4 on the ampltude of the vbratons of all the cables was examned. The am of the task was to reduce the resonant vbraton caused by harmonc forcng F 2 (t) wth frequency f 6c = Hz. Ths frequency corresponds to the frst n-plane natural modes of vbratons of cables No. 6. In Fg. 4 the correspondng tmng dagram of ths analyss s shown. Changng the tenson n cables No. 4 was started at 4 seconds and was complete after 4 seconds. The ntal statc effort n cables 4.e. N c4 /N cr was about 7.72% and after the change t was about 11.0%. A reducton of the resonant vbraton n cable 6 by about 72% was obtaned fg. 4f. Moreover, no sgnfcant changes n the vbraton n other cables was observed fg. 4 a-e. 4. Summary and conclusons In ths paper, the theoretcal (numerc) analyss of the possbltes of usng the authors own method of actvely reducng forced resonant vbraton of the cables n the cable-stayed steel footbrdge by tenson changes n cables was presented. It was shown that changng tenson n selected cables leads to a decrease of excessve resonant oscllatons only n the cable where resonant vbratons occurred. At the same tme, there was no exctaton of excessve vbraton n all the other cables. It must be emphaszed that after the change of statc tensons n the cables, the stresses n the cables are much smaller than the permssble stresses.e % < perm. Permssble stress perm n the cables n the cable-stayed brdge should be n range 30% to 40% characterstc resstance f yk= of cables when the useful load s taken nto account.

6 Wojcech Pakos and Zbgnew Wójcck / Proceda Engneerng 91 ( 2014 ) Fg. 4. Vertcal dsplacements q z (t) [mm] for: (a) cables no. 1; (b) cables no. 2; (c) cables no. 3; (d) cables no. 4; (e) cables no. 5; (f) cables no. 6. References [1] L. T. Wrotny, The dynamcs of dynamcal systems. Repetytorum theoretcal and tasks. (In Polsh) Warszawa: Ofcyna Wydawncza Poltechnk Warszawskej, [2] P. Warntcha, Y. Fujno, B. M. Pacheco, and R. Agret, An expermental study on actve tendon control of cable-stayed brdges, Earthquake Engneerng & Structural Dynamcs, No. 22, 1993, pp [3] Y. Fujno, T. Susumpow, An Expermental Study on Actve Control of In-Plane Cable Vbraton by Axal Support Moton, Earthquake Engneerng and Structural Dynamcs, vol. Vol. 23, 1994, pp [4] T. Susumpow, Y. Fujno, Actve Control of Multmodal Cable Vbratons by Axal Support Moton, Journal of Engneerng Mechancs, 1995, pp [5] A. Preumont, Y. Achkre, Actve dampng of structures wth guy cables, Journal of Gudance Control and Dynamcs, vol. Vol. 20, no. No. 2, 1997, pp [6] Y. Achkre, Actve Tendon Control of Cable Stayed Brdges. Belgum: Actve Structures Laboratory, Unverste Lbre de Bruxelles, PhD Thess, [7] A. G. Schemmann, H. A. Smth, Vbraton control of cable-stayed brdges Part 1: Modellng ssues, Earthquake Eng. Struct. Dyn., vol. Vol. 27, 1998, pp [8] A. G. Schemmann, H. A. Smth, Vbraton control of cable cable-stayed brdges Part 2: control analyses, Earthquake Eng. Struct. Dyn., vol. Vol. 27, 1998, pp [9] M. E. Magaña, J. Rodellar, Nonlnear decentralzed actve tendon control of cable-stayed brdges, Journal of Structural Control, vol. Vol. 5, no. No. 1, 1998, pp [10] F. Bossens, A. Preumont, Actve tendon control of cable-stayed brdges: A large-scale demonstraton, Earthquake Engneerng and Structural Dynamcs, vol. Vol. 30, 2001, pp [11] J. Rodellar, C. Monroy, An actve tendon control scheme for cable-stayed brdges wth model uncertantes and sesmc exctaton, Journal of Structural Control, vol. Vol. 9, no. No. 1, 2002, pp [12] W. Pakos, PhD thess, [n Polsh], Rap. Inst. In.L d. P.Wroc. 2012, Ser. PRE; nr 3.