Dynamic Analysis o a Cable-S ayed Deck
S eel A ch B idge
P. Gal ´ın, J. Dom´ınguez ∗
Escuela Supe io de Ingenie os, Uni e sidad de Se illa, Camino de los
Descub imien os s/n, 41092 Se illa, Spain
Abs ac
A heo e ical and expe imen al esea ch wo k in ela ion o Ba que a cable-s ayed
b idge is desc ibed in his pape . Ba que a B idge, ac oss Guadalqui i i e , links
he ci y o Se ille wi h he Scien i ic Pa k Ca uja 93. A jam hou s ca s may co e
one hal o he b idge lanes o mo e han one hou . Full-scale es s we e ca ied ou
o measu e he b idge dynamic esponse. The expe imen al p og am included he
dynamic s udy o wo di e en li e load condi ions: he b idge wi h one hal o i
lanes ull o ca s, and he b idge emp y o ca s. Modal pa ame e s es ima ions we e
made based on he acqui ed da a. Ten ib a ion modes we e iden i ied in he e-
quency ange o 0-6 Hz by di e en echniques, being wo o hese modes e y close
o each o he . The a ic-s uc u e in e ac ion is also s udied. Expe imen al esul s
we e compa ed wi h hose ob ained om a h ee-dimensional ini e elemen model
de eloped in his wo k. Bo h se s o esul s show e y good ag eemen . Finally, a
damage iden i ica ion echnique has been applied o de e mine he in eg i y o he
s uc u e. Resul s ob ained om a es de eloped in July 2005 ha e been co ela ed
o expe imen al esul s ob ained in Oc obe 2006 using he damage index me hod.
Key wo ds: a ch b idge, damage de ec ion, ope a ional modal analysis,
ehicle-s uc u e in e ac ion
1 In oduc ion
Expe imen al es s cons i u e he mos eliable me hod o ob ain he dynamic
p ope ies (na u al equencies, mode shapes and damping a ios) o ac ual
s uc u es and o alida e, om hese esul s, nume ical models used o hei
∗Co esponding au ho . Tel.: +34 954487293; ax: +34 954487295.
Email add ess: [email p o ec ed] (J. Dom´ınguez).
P ep in submi ed o Else ie P ep in 5 No embe 2006
analysis. They pe mi also o asses he s a e o damage o a s uc u e by
compa ison wi h he dynamic p ope ies ob ained in p e ious analyses.
Dynamic iden i ica ion me hods om ambien ib a ion ha e been used in
complica ed s uc u es like dams [1], o sho e pla o ms [2], spo s s adia [3],
b idges [4], e ce e a. A heo e ical and expe imen al esea ch wo k in ela ion
o Ba que a b idge is desc ibed in his wo k. I is a s eel a ch b idge wi h
cable-s ayed deck ha links he old own o Se ille wi h he Scien i ic Pa k
Ca uja 93. A ce ain hou s, a a ic jam occu s on he b idge and ca s co e
one hal o he b idge lanes (Fig. 1) o mo e han one hou .
The expe imen al p og am de eloped in his wo k includes he dynamic cha-
ac e iza ion o he s uc u e o wo si ua ions unde no mal condi ions and
when i is co e ed by a ic. The modal p ope ies o he b idge a e iden i ied
om ambien ib a ion and he esul s ob ained o bo h si ua ions o he
b idge a e compa ed. By his analysis, he e ec ha ehicles ha e on he
dynamic beha iou o he s uc u e is s udied. The idea p esen ed in [3], whe e
changes o he modal p ope ies o a spo s adium we e de e mined a imes
when di e en ac i i ies ook place a he s adium, is explo ed in his pape
o he b idge case. The au ho s o ha pape concluded ha he dynamic
p ope ies o he s uc u e depend o some ex en on he ype o he ac i i y
celeb a ed in he s adium. The possibili y o hese changes aking place in
o he ypes o s uc u es is s udied in his wo k.
The ob ained expe imen al esul s a e compa ed wi h hose om a h ee-
dimensional ini e elemen analysis. Finally, a damage iden i ica ion me hodo-
logy is applied o e i y he s uc u al in eg i y o he b idge. Expe imen al
esul s ob ained wi h a ime di e ence o one yea and a empe a u e di e ence
o 10oC a e analyzed.
2 Desc ip ion o he s uc u e
The aes he ic unc ions and symbolic alues o he s uc u es a e mo e and
mo e impo an e e y day, in pa icula when hese s uc u es a e buil in u -
ban zones [5,6]. Ba que a b idge was buil in Se ille o he 1992 In e na ional
Exhibi ion [7,8]. I is an inno a i e s uc u e, designed by JJ. A enas and M.
Pan ale´on, wi h a lying cen al a ch ising om he e ex o wo la e al ian-
gula ames. Fi een yea s a e i s cons uc ion, Ba que a is an indispensable
piece o he u ban landscape o Se ille. Ba que a is a bows ing s eel b idge.
The 168 m span s uc u e es s on wo se s o wo e ical suppo s spaced
30 m in he ans e se di ec ion loca ed a he banks o he Guadalqui i
i e (Figs. 1-3). The c oss-sec ions o he a ch and inclined legs include deep
g oo es ha p oduce enough local ine ia o a oid he need o any in e nal
2
longi udinal s i ene .
The deck c oss-sec ion is shown in Fig. 2. I is 16 m wide and 2.4 m deep. The
o al wide o he b idge is 21 m wi h wo can ile e pedes ian decks. The deck
c oss-sec ion includes wo e ical webs sepa a ed by a dis ance o 1 m (see
Fig. 2). The hange s a e ancho ed be ween hen. The hange s ha e a iable
inclina ion. The deck is suppo ed on he ex emes by ans e sal beams, wi h
a iable dep h, which es on he e ical suppo s.
3 Fini e Elemen Model
A h ee dimensional ini e elemen model (FEM) has been de eloped o he
nume ical analysis o he s uc u e using as-buil d awings o he b idge and
some double-check in-si u measu emen s. Modal analysis was ca ied ou using
ANSYS [9].
The a ch, suppo s, and he in e nal s i ene we e ep esen ed as wo-node
beam elemen s (BEAM44) wi h 6 deg ees o eedom pe node. This elemen
pe mi s he end nodes o be o se om he cen oidal axis o he beam. The
hange s we e modeled as uss elemen s (LINK10) wi h 3 deg ees o eedom
pe node. The deck slab was modeled using eigh -node shell elemen s wi h
6 deg ees o eedom pe node (SHELL93). The wo ex eme beams and he
e ical suppo s we e connec ed by sp ing elemen s (COMBINE14).
A de ailed model o all he b idge elemen s was in ended. As a consequence,
he numbe o deg ees o eedom is high. The ull model consis s o 10328
beam elemen s, 17 uss elemen s, 15672 shell elemen s and 8 sp ing elemen s,
esul ing in o 26025 elemen s and 47024 nodes. Fig. 3 shows a ull 3-D iew
o he ini e elemen model o he b idge and de ails o he deck c oss-sec ion.
4 Full-scale es ing
Dynamic p ope ies can be ob ained by measu emen o ib a ions p oduced
by ambien loads. This echnique is simple o ci il enginee ing s uc u es
han classical modal analysis because i is no necessa y o exci e he s uc u es
by shake s. In addi ion, he s uc u e can be used du ing he es ing p ocess.
The expe imen al p og am, ca ied ou du ing July 15 2005 and Oc obe 11
2006, includes dynamic cha ac e iza ion o he s uc u e in no mal condi ions
and when a hal o he b idge is co e ed by a ic. The esponse o he s uc u e
was measu ed a 16 selec ed poin s (Fig. 4) using Ende co (Model 86 and
3
Model 4370) accele ome e s. P elimina y esul s ob ained om a FE dynamic
analysis we e used o de e mine he op imum loca ion o he senso s.
Since nine Model 86 accele ome e s we e a ailable o he es ing and wo o
hese senso s (a loca ions 1 and 2) we e held s a iona y o e e ence du ing
he es ca ied ou in 2005 and one o hem (a loca ion 2) du ing he es
de eloped in 2006, wo se -ups we e equi ed o co e he 16 measu emen
poin s. I is wo h o men ion ha in ou pu -only modal analysis, whe e he
inpu o ce emains unknown and may a y be ween he se -ups, he di e en
measu emen s se ups can only be linked i he e a e some senso s in common.
The e e ence accele ome e s we e chosen in o de o be able e y ca e ully o
measu e all global modes o he b idge.
The hange s we e no ins umen ed in he i s es because, acco ding o p e-
limina y nume ical analysis and p eceding expe imen al s udies [10,11], sig-
ni ican in e ac ion be ween he s ay-cables and he es o he s uc u e was
no expec ed. This in e ac ion is signi ican when he lowes na u al equen-
cies o he s uc u e and he na u al equencies o he cables a e close. In he
p esen case, cables a e sho , es ima ing hei i s na u al equencies a ound
6 Hz. The e a e a leas , 10 global modes o he s uc u e below his alues.
The e o e i is no expec ed ha cables ha e an impo an pa icipa ion in he
global modes o he b idge. This assump ion was alida ed by he second es ,
whe e he cables we e ins umen ed (Figu e 5a.). The powe spec al densi y
o he cable’s esponse is shown in Figu e 6 whe e i can be obse ed ha he
i s bending mode o he cable is a ound 6 Hz.
Du ing he i s es , da a o he esponse o he s uc u e we e acqui ed
when he s uc u e was unde luid a ic condi ions, in loca ions 1 o 9,
and when ca s co e one hal o he lanes o he b idge, in poin s 1,2 and
10 o 16. In o de o ob ain he mode shapes o he s uc u e, he esponse
a all poin s ha e been used. T a ic-s uc u e in e ac ion is no expec ed o
cause changes in s uc u al mode shapes. Na u al equencies and damping
a ios, we e de e mined using each one o he se -ups independen ly. In he
second es , he esponse o he s uc u e was acqui e in all loca ions o bo h
si ua ions: when he s uc u e is unde luid a ic condi ions and when ca s
co e one hal o he lanes o he b idge.
Ambien ib a ion esponse was acqui ed du ing 1000 seconds pe channel
and pe se -up. The da a we e sampled o 64 Hz. Da a we e decima ed (o de
3) o ca y ou da a analysis in he equency ange o in e es (0 o 10 Hz).
Da a eco ds we e Hannning-windowed wi h 66.67% o e lapping o spec al
a e aging.
Acqui ed da a a e a ailable o in e es ed eade by sending an e-mail message
o he au ho s.
4
5 Da a Analysis
Di e en p ocedu es o ob ain modal pa ame e s om ambien ib a ion da a
ha e been used in his wo k. In ou pu -only modal analysis, also called ope a-
ional modal analysis, he applied o ces a e unknown and, he e o e, nei he
he equency esponse unc ion no he impulse esponse unc ion can be ob-
ained o de e mine modal pa ame e s as in classical modal analysis [12]. The
signal a one o he ixed ansduce s is used as a e e ence o de e mine he
equency esponse unc ion and he impulse esponse unc ion.
Fou complemen a y iden i ica ion me hods ha e been conside ed in he p esen
wo k: h ee o hem based on equency domain analysis and one on ime
domain analysis. The s udy has been de eloped using MATLAB [13] and
ARTEMIS [14] so wa e.
The i s iden i ica ion me hod employ is Peak-Picking (PP), which has been
used wi h success in many o he applica ions [5,11,15,16]. This me hod is
based on he ac ha when he equency esponse unc ion eaches a peak
a a ce ain equency, i can be associa ed o he o ce o o a esonance
equency o he s uc u e [17]. Na u al equencies a e iden i ied om peaks
o spec al densi ies unc ion:
ωdi =q1−ξ2
i·ωni (1)
This p ocedu e p oduce a good es ima ions o na u al equencies o weakly
damped s uc u es. To dis inguish be ween peaks associa ed o he exci a ion
and hose associa ed o esonance equencies o he s uc u e, mode shapes
can be used [17]. The esponse alues a all poin s o a weakly damped s uc-
u e o one o i s esonance equencies, a e in phase o ou o phase by 180o,
depending on he mode shape. Peaks o spec a densi y unc ion associa ed o
he exci a ion no mally p esen a phase di e ence o he c oss-spec a unc ion
be ween wo measu emen s poin s di e en om 0oo 180o. In addi ion, he co-
he ence unc ion be ween wo signals has a alue close o one o he esonance
equencies o he s uc u e, due o he high ela ion esponse-noise a hose
equencies. This ac helps o decide which o he equencies eally a e na u-
al equencies o he s uc u e. The PP me hod is based on he assump ion
ha he dynamic esponse o he s uc u e a esonance peaks is de e mined
o each mode. This is alid when modes a e well sepa a ed. The e o e, i is
di icul o iden i y modes e y close o each o he using his me hod. The
au o-spec a, c oss-spec a and cohe ence unc ions ob ained om he i s
es , a e shown in Fig. 7. The na u al equencies we e iden i ied om eso-
nance peaks in au o-spec a and c oss-spec a unc ions. Cohe ence unc ion
peaks a e he same ha he peaks o he p e ious unc ions, and he phase
alue o he c oss-spec a unc ion o hose peaks a e 0oo 180o, p o iding
5

addi ional e idence ha hese peaks co espond o na u al equencies o he
s uc u e.
The second iden i ica ion echnique used in he p esen s udy is he so called
A e aged No malized Powe Spec al Densi ies (ANPSDs) which is a p ac ical
implemen a ion, de eloped by Felbe [18], o he PP me hod. In his case, au o-
spec a unc ions a e no malized and a e aged o ob ain an a e age spec a
densi y unc ion ha , no mally, shows all esonance equencies o he sys em.
ANPSD(ω) = 1
l
l
X
i=1
PSD(ω)
Pn
j=1 PSD(ωj)(2)
whe e lis he numbe o measu emen loca ions. The na u al equencies o he
s uc u e a e ob ained om he simple obse a ion o he peak in he ANPSDs
diag am. This me hod was used success ully in he dynamic cha ac e iza ion
o an a ch b idge in e e ence [19]. The ANPSDs diag ams o bo h si ua ions
o he b idge ob ained om he i s es a e shown in Fig. 8.
The hi d iden i ica ion echnique used, also in he equency domain, is called
Enhanced F equency Domain Decomposi ion (EFDD) [20] which, om a sim-
ple o m, in oduces signi ican imp o emen s o Peak Picking Technique. This
me hod is based on a modal decomposi ion ealiza ion o he spec al densi y
ma ix, being one o he ad an ages o he me hod he possibili y o iden i y
e y close modes.
The Powe Spec al Densi y ma ix (PSD) o he mmeasu ed esponses can
be exp essed, o a ligh ly damped s uc u e, as:
Gyy(jω) = X
k²Sub(ω)
dkφkφT
k
jω −λk
+dkφkφT
k
jω −λk
(3)
whe e dkis a scala cons an , φkis he mode shape ec o , λkis he pole o he
ou pu PSD, and Sub(ω) is he limi ed numbe o modes ha will con ibu e
o he esponse a equency ω.
The i s s ep o he EFDD is o es ima e he ou pu PSD ma ix a disc e e
equencies, and o ca y ou he Singula Value Decomposi ion (SVD) o he
ma ix ˆ
Gyy(jωi) = UiSiUH
i(4)
whe e he ma ix Uicon ains he singula ec o s uij and Siis a diagonal
ma ix wi h he scala singula alues sij. Close o a peak, whe e he kmode
is dominan , he e will be only one mode in Sub(ω) and, he e o e, he i s
singula ec o uk1is an es ima e o he mode shape; i.e., ˆ
φk=uk1, and he
co esponding singula alue is he au o powe spec al densi y unc ion o
he singula deg ee o eedom sys em. This powe spec al densi y unc ion is
iden i ied a ound he peak compa ing he es ima ed mode shape ˆ
φkwi h he
6
singula ec o s om he equency lines a ound he peak. I he MAC alue
ob ained om he singula ec o and ˆ
φkis highe han a e e ence alue close
o one, he singula alue belongs o he au o powe spec al densi y unc ion.
Once he au o powe spec al densi y unc ion has been ob ained a ound he
peak, he na u al equency and he damping a io a e es ima ed by In e se
Fas Fou ie T ans o m. The singula alues decomposi ion o he spec al
densi y ma ix is shown in Fig. 9. Peaks ep esen ing ib a ion modes ha e
been selec ed.
I can be obse ed ha wo modes e y close o 2 Hz exis o he b idge unde
s udy, one o hem no been de e mined by he p e ious me hods. This ype
o modes can be easily iden i ied wi h EFDD, by obse a ion no only o he
highes singula alue bu also o he nex one.
The las echnique is a mo e elabo a ed ma hema ical p ocedu e ha wo ks
di ec ly wi h ime domain acqui ed da a. I is called S ochas ic Subspace
Iden i ica ion (SSI). The in e es ed eade can ind de ails o he ma hema i-
cal app oach in e e ences [21,22]. SSI is a powe ul ool (pe haps he mos
ad anced iden i ica ion me hod ha exis s up o day) ha has been used o
dynamic cha ac e iza ion o many s uc u es [21,23,24]. A b ie e iew o he
main ideas o his me hod is p esen ed in he ollowing.
The dynamic beha iou o a s uc u e is desc ibed by:
M¨
U( ) + C˙
U( ) + KU( ) = F( ) (5)
whe e M,C, and Ka e he mass, damping, and s i ness ma ices o he
s uc u e, espec i ely. U( ) and F( ) a e he displacemen and inpu o ce
ec o s espec i ely. This equa ion can be w i en as a s a e space equa ion:
˙x( ) = Acx( ) + Bcu( ) (6)
whe e he s a e ec o x( )=[U( ),˙
U( )]Tand he s a e ma ix Ac, and he
sys em con ol in luence coe icien ma ix Bc, a e de ined by
Ac=


0I
−M−1K−M−1C


Bc=


0
−M−1


whe e F( ) = B2u( ) (7)
The ou pu ec o y( ) can be exp essed as
y( ) = Ccx( ) + Dcu( ) (8)
whe e Ccis he ou pu in luence coe icien ma ix and Dcis he ou pu con-
ol in luence coe icien ma ix. These equa ions cons i u e a con inuous- ime
s a e space model o a dynamic sys em.
7
Since expe imen al da a a e disc e e, a disc e e- ime s a e space model can be
ob ained by sampling he con inuous- ime s a e space model:
xk+1 =Axk+Buk(9)
yk=Cxk+Duk(10)
whe e xk=x(k∆ ) is he disc e e- ime s a e ec o con aining he sampled
displacemen s and eloci ies; uk,yka e he sampled inpu and ou pu , espec-
i ely; A= exp(Ac∆ ) is he disc e e s a e ma ix;and B= [A−I]A−1
cBc
is he disc e e inpu ma ix.
Including he s ochas ic componen s; i.e., he noise due o dis u bances and
modeling inaccu acies (wk) and he noise due o senso inaccu acy ( k), he
disc e e s a e space model can be w i en as:
xk+1 =Axk+Buk+wk(11)
yk=Cxk+Duk+ k(12)
The p ocess noise wkand measu emen noise ka e assumed o be ze o-mean,
whi e noise, s a is ically independen , and wi h co a iance ma ices
E



wp
p


µwT
p T
p¶
=


Q S
STR


δpq (13)
whe e Eis he expec ed alue ope a o and δpq is he K onecke del a. Q,R
and Sa e p ocess and measu emen noise co a iance ma ices.
In ambien ib a ion es ing, only he esponses o he s uc u e a e measu ed,
ob aining:
xk+1 =Axk+wk(14)
yk=Cxk+ k(15)
whe e he inpu ukis modeled by he noise e ms.
This equa ion cons i u es he basis o ime domain modal iden i ica ion om
ambien ib a ion es ing. The e a e se e al echniques o ca y ou modal
iden i ica ion based on his equa ion.
Fig. 10 shows a s abiliza ion diag am ob ained by applying SSI. Sys em o de
and s able poles can be ound in hese diag ams which p o ide modes o he
s uc u e.
8
6 Dynamic beha iou o he b idge
The dynamic beha iou o Ba que a B idge is go e ned by e ical bending
and o sional modes, in he equency ange o 0−6 Hz. Ten modes ha e been
iden i ied in his equency ange. Table 1 shows he ob ained na u al equen-
cies o he b idge unde luid a ic condi ions whe eas Table 2 co esponds
o he si ua ion when he b idge is jammed. Damping a ios ob ained o bo h
b idge condi ions a e shown in Tables 3 and 4.
I is obse ed in bo h es s ha he na u al equencies o he s uc u e change
e y li le due o he a ic condi ions on he b idge. The i s es was ca ied
ou on July 2005, while ha he second one ook place on Oc obe 2006, wi h
an ambien empe a u e di e ence o 10oC. Ve y li le changes appea in he
na u al equencies ob ained om he wo expe imen al es s as can be seen
om he alues shown in Tables 1 and 2. Damping a ios may inc ease up
o 200 pe cen when he b idge is jammed wi h ehicles, as compa ed o
he emp y si ua ion. Inc emen s in damping a ios o he same o de we e
obse ed in bo h se ies o es s. A e y simila conclusion was eached in [3],
whe e i was concluded ha he damping a ios in a spo s adium inc eases
o a signi ican ex en when i is c owded.
Mode shapes o he s uc u e iden i ied om expe imen al modal analysis and
om nume ical analysis a e shown in Figs. 11 and 12.A comple e ag eemen
be ween bo h se s o mode shapes can be obse ed.
In o de o quan i y his ag eemen , expe imen al esul s ha e been compa ed
wi h he ini e elemen esul s by using he Modal Assu ance C i e ion (MAC)
[25]. MAC alues a y om 0 o 1; a alue o one implies pe ec co ela ion o
he wo modes ec o s (one ec o is p opo ional o he o he ), while a alue
close o ze o indica es no co ela ed modes (o hogonal modes). The MAC
alue is de ined as:
MAC(φA,k, φB,j) = (φT
A,kφB,j)2
(φT
A,kφA,k)(φT
B,jφB,j)(16)
whe e φC,k is he k-mode o da a se C, and Tmeans anspose ma ix.
The ob ained MAC ma ix be ween he esul s ob ained om he es ca ied
ou in 2005 and hose ob ained om he es ca ied ou in 2006, a e shown in
Fig. 13. The mode ec o co ela ion using MAC seems qui e good, inding he
highes di e ence in he i h mode shape. The MAC ma ix be ween he nu-
me ical and expe imen al esul s ha e also ob ained. Fig. 14 shows ha he e
is a qui e good ag eemen be ween expe imen al and nume ical esul s. The e-
o e, he nume ical model can be used o ep esen he dynamic beha iou o
he b idge.
9
(a)
(b)
Fig. 3. The 3-D FE model o Ba que a B idge (a) and de ail o he deck c oss-sec ion
model (b).
16

Fig. 4. Measu emen loca ions.
17
(a) (b)
Fig. 5. Accele ome e s a cables and a ch membe
18
0 5 10 15 20
0
1
2
3
4
5
6
7x 10-3
F equency [Hz]
PSD [m/s2/Hz]
Accele ome e a cable
Fig. 6. Powe spec al densi y o he cable’s esponse
19
0 1 2 3 4 5 6
0
0.5
1
1.5
2
2.5 x 10-5
F equency [Hz]
Au o-spec a magni ude [m/s
2]
T ansduce 1
T ansduce 2
T ansduce 5
T ansduce 8
T ansduce 9
(a) Au o-spec a unc ion
0 1 2 3 4 5 6
0
0.5
1
1.5 x 10-5
F equency [Hz]
C oss-spec a Magni ude [m/s
2]
Re e ence T ansduce 2
T ansduce 4
T ansduce 7
T ansduce 12
T ansduce 14
T ansduce 16
(b) C oss-spec a unc ion (magni ude)
0 1 2 3 4 5
-150
-100
-50
0
50
100
150
F equency [Hz]
C oss-spec a Phase [º]
Re e ence T ansduce 2
T ansduce 4
T ansduce 7
T ansduce 12
T ansduce 14
T ansduce 16
(c) C oss-spec a unc ion (phase)
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F equency [Hz]
Cohe ence
Re e ence T ansduce 2
T ansduce 4
T ansduce 7
T ansduce 12
T ansduce 14
T ansduce 16
(d) Cohe ence unc ion
Fig. 7. Au o-spec a unc ion, c oss-spec a unc ion (magni ude and phase) and
cohe ence unc ion.
20
0 1 2 3 4 5 6
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
F equency [Hz]
ANPSDs
No mal
T a ic jam
Fig. 8. A e age No malized Powe Spec al Densi ies.
21

d B | ( 1 . 0 s ¯ ² ) ² / H z
F e q u e n c y [ H z ]
0 2 4 6
- 8 0
- 6 0
- 4 0
- 2 0
0
2 0
F e q u e n c y D o m a i n D e c o m p o s i i o n - P e a k P i c k i n g
A e a g e o h e N o m a l i z e d S i n g u l a V a l u e s o
S p e c a l D e n s i y M a i c e s o a l l D a a S e s .
Fig. 9. Singula Value Decomposi ion o he Spec al Densi ies Ma ices.
22
S a e Space
Dimension
F equency [Hz]
0 2 4 6
S abiliza ion Diag am
Da a Se : Measu emen 2
CVA [Da a D i en]
10
20
30
40
50
60
70
80
Ma ke s
S able Modes
Uns able Modes
Noise Modes
Fig. 10. S abiliza ion Diag am.
23
(a) Mode 1 (b) Mode 2
(c) Mode 3 (d) Mode 4
(e) Mode 5 ( ) Mode 6
(g) Mode 7 (h) Mode 8
(i) Mode 9 (j) Mode 10
Fig. 11. Mode shapes om expe imen al modal analysis.
24
(a) Mode 1 (b) Mode 2
(c) Mode 3 (d) Mode 4
(e) Mode 5 ( ) Mode 6
(g) Mode 7 (h) Mode 8
(i) Mode 9 (j) Mode 10
Fig. 12. Mode shapes om nume ical modal analysis.
25
Table 4: Damping a ios. T a ic jam on he b idge.
ξEF DD[%] 13 ξEF DD[%] 14 ξSSI [%] 15 ξSSI [%] 16
1.178 2.196 2.26 3.527
1.748 2.266 4.119 2.054
1.663 2.629 1.665 2.601
2.674 2.843 2.053 3.132
0.809 1.434 1.296 1.117
1.161 1.183 1.703 2.667
1.272 2.8 1.48 2.58
1.208 1.583 1.76 2.459
1.186 1.515 1.185 1.503
1.462 1.302 2.048 1.498
13 Tes ca ied ou in 2005
14 Tes ca ied ou in 2006
15 Tes ca ied ou in 2005
16 Tes ca ied ou in 2006
32
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