Journal of Wind Engineering
and Industrial Aerodynamics 84(2000)151}162
Identi "cation of #utter derivatives of bridge decks
Ming Gu *,Ruoxue Zhang,Haifan Xiang
Department of Bridge Engineering,Tongji Uni v ersity,Shanghai 200092,People 's Republic of China Received 5August 1998;received in revid form 8December 1998;accepted 11May 1999
Abstract
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An identi "cation method has been developed to extract all the #utter derivatives de "ned by R.H.Scanlan.In the prent work,the signals of the coupled vertical }torsional free vibration of the spring-suspended ction model are ud.The #utter derivatives of a thin plate obtained using the prent method are compared with the corresponding Theodorn theoretical values.The prent method is then ud in the identi "cation of #utter derivatives of the Jiangyin suspension Bridge over Yangtze River.The #utter critical wind speed of this bridge obtained from the full bridge aeroelastic model test in a wind tunnel shows good agreement with the estimated result from Scanlan 's #utter a
nalysis method with the #utter derivatives using the prent method. 2000Elvier Science Ltd.All rights rerved.
Keywords:Long-span bridges;Flutter derivatives;Wind tunnel test;Identi "cation method
1.Introduction
Nowadays,with the increa of spans,cable-supported bridges become more and more nsitive to wind.Wind-induced vibration is the "rst important factor in the design of long-span bridges,especially for tho located in the regions where typhoon often occurs.
The mechanisms of wind-induced vibration of long-span bridges are classi "ed as #utter,bu !eting,vortex-excited vibration and galloping [1].It has been recognized that #utter and bu !eting are the two main mechanisms.Flutter is a kind of lf-excited vibration and may destroy the bridge structures.It should be made sure that it does not occur during the lifespan of the bridge by lecting a proper deck cross ction con "guration and/or structural design [2].Bu !eting is caud by a #uctuating *Corresponding author.Tel.:#86-21-65981171;fax:#86-21-65027292.
0167-6105/00/$-e front matter 2000Elvier Science Ltd.All rights rerved.
幼儿英语游戏
PII:S 0167-6105(99)00051-3
152M.Gu et al./J.Wind Eng.Ind.Aerodyn.84(2000)151}162
component in oncoming wind and is called forced vibration.Its large amplitude which increas with wind speed and its frequent occurrence may weaken and damage the structural members of the bridge.
In the1970s,R.H.Scanlan propod a mi-experimental and mi-analytical approach for critical#utter wind speed[3]and another approach for bu!eting respon[4].The two approaches are prently widely ud.Flutter derivatives of bridge decks are parameters in the approaches esntial for the#utter and bu!eting analysis of long-span bridges.In the original technique,to extract the #utter derivatives in Scanlan's method[3],a spring-suspended ctional model was tested and the free decay vibration signals were ud.A great advantage of the free vibration technique is its simplicity,requiring no expensive and compli-cated driving machine.But Scanlan's method need three groups of test.Torsional and vertical bending motions have to be constrained,respectively,to obtain the so-called direct derivatives.Furthermore,to obtain cross derivatives,the vertical and torsion motions of the model must have the same frequency at all wind velocities.In view of this situation,ma
recognized
ny e!orts have been made to simplify the identi"cation procedure. ARMA model was ud by M.Shinozuka et al.in1982to try to identify the#utter derivatives[5].But the results emed unsatisfactory for high noi.H.Yamada et al. introduced the extended EKF method into the identi"cation procedure of the derivatives bad on the coupled vibration time histories[6].In this method,the time histories of the displacement and velocity as well as the information of the initial condition are simultaneously required.Jokobn and Hann propod a method for the determination of the aerodynamic derivatives.This method employs conversion of bu!eting respon data to the respon covariance function estimate[7].Zasso et al. tried to u the respon data of full bridge aeroelastic model to identify#utter derivatives[8].Pouln et al.ud a method which combines control theory and system identi"cation techniques to extract#utter derivatives from ction model tests for the Great Belt East Bridge[9].In1994,P.P.Sarkar and R.H.Scanlan developed Modi"ed Ibrahim Time-domain(MITD)method to extract all the direct and cross derivatives from the coupled free vibration data of2-DOF model[10].This method requires lection of the time shifts N1and N2.Sarkar and Scanlan have found a way to lect the two time shifts clo to optimal values.Besides the free vibration methods mentioned above,forced vibration method is also an e!ective one to identify the#utter derivatives[11].
An identi"cation method bad on unifying least-squares theory is prented in this paper.In this method,an uni"ed error function which is linearly compod of two error components of vertical and torsional motions is de"ned as the objective function to optimize the#utter derivatives.Besides,an initial value in the iteration procedure for the optimization of the#utter derivatives is pro-vided by MITD method.A test on a thin plate was carried out in TJ-1Boundary Layer Wind Tunnel in Tongji University,and the identi"ed#utter derivatives of the thin plate model using the prent method show good consistence with Theodorn theoretical values.The prent method is also ud in the identi-"cation of#utter derivatives of the Jiangyin suspension Bridge over the Yangtze River.
2.Unifying least-square method
The motion equations of a bridge deck ction in smooth wind#ow in a wind tunnel can be written according to the#utter analysis theory developed by R.H. Scanlan[3]as follows:
之间英文m(h$#2 F F h# F h)"¸ ,(1) I( #2 ? ? # ? )"M ,(2) where m and I are the model mass and mass inertia moment per unit length, respectively, F and ?are the mechanical damping ratios in bending and torsion,element什么意思
respectively, F and ?are the corresponding natural mechanical frequencies,h and are the vertical be
nding displacement and torsion angle,respectively,and¸ and M are the aerodynamic lf-excited force and moment,respectively,given by
¸
" ; B KH H (K)h;#KH H (K)B ;#K H H (K) #K H H (K)h B ,(3) M " ; B KA H (K)h;#KA H (K)B ;#K A H (K) #K A H (K)h B ,(4)
where is the air density,H H G and A H G(i"1,2,3)are the#utter derivatives,B is the deck width of the model;;is mean wind velocity,K("B /;)is the reduced frequency.
Eqs.(1)and(2)can be written in matrix style as
[M]+x,#[C]+x,#[K]+x,"+F ,,(5) where[M],[C]and[K]are the matrices of mass,damping and sti!ness of the bridge, respectively,+F ,is the lf-excited force vector,where¸ and M are the compo-nents;+x,2"+h ,2.
Eq.(5)can be rewritten by moving+F ,,that is,¸ and M ,to the left side as, [M]+x,#W C C X+x,#W K C X+x,"+0,,(6) where W C C X and W K C X are damping and sti!ness matrices of the wind-bridge system, that is to say,the aerodynamic damping and aerodynamic sti!ness are included in the two
matrices,respectively.
According to the complex-mode theory,the estimated values of the m th sampling data of the respon of the ctional model with two degrees vertical bending and torsion,are constructed as
h K"
P
(A FP e H P K R#A H FP e H H P K R),(7)
sour的反义词
K"
P (A?P e H P K R#A H?P e H H P K R),(8) M.Gu et al./J.Wind Eng.Ind.Aerodyn.84(2000)151}162153
where P and A P are assumed to be
P" P#j P, H P" P#j P,(9)
A P";P#j<P,A H P";P#j<P.(10) Substituting Eqs.(9)and(10)into Eqs.(7)and(8),respectively,yields
h K"2
P
e?P K R[;FP cos( P m t)!<FP sin( P t)],(11)
K"2
P
e?P K R[;?P cos( P m t)!<?P sin( P t)].(12) Thus the error vectors between the estimated values and measured values are +e F,"+e F e F 2e F+,,(13) +e?,"+e? e? 2e?+,,(14) where
e FK"h K!h K,(15)
e?K" K! K.(16) According to the general method[12],the least-squares solutions of mode para-meters are extracted by minimizing+e F,2+e F,and+e?,2+e?,using time histories of h(t) and (t),respectively.Then is given by averaging the values of the two complex eigenvalues,and comple
x eigenvectors are given by normalizing+A,P.This method is actually a`partial a curve"t.So if two ts of values of extracted by h(t)and (t), respectively,have a relatively great di!erence,there will be a big error in identi"cation, which may result in an unacceptable error of estimated#utter derivatives eventually. Considering this point,a thought of unifying identi"cation is introduced in the prent ,using time histories of h and simultaneously to identify the#utter derivatives.A unifying error function is de"ned as
J"+e F,2+e F,#+e?,2+e?,.(17) An alternate iteration technique is adopted to the calculation,and the detailed procedure is as follows:
京翰教育(1) P(r"1,2)obtained from MITD method is treated as the initial values P. Introducing two new variables as
C PK"2e? P K cos( P m ),(18)
S PK"2e? P K sin( P m ),(19) 154M.Gu et al./J.Wind Eng.Ind.Aerodyn.84(2000)151}162
and substituting them into Eqs.(11)and (12)yields:
h K " P (;FP C PK !<FP S PK ),(20) K " P
(;?P C PK !<?P S PK ).(21)Then the objective function becomes
J "[+h ,!([C ]+;F ,![S ]+<F ,)#+ ,!([C ]+;?,![S ]+<?
])]2;[+h ,!([C ]+;F ,![S ]+<F ,)#+ ,!([C ]+;?,![S ]+<?
])],(22)
where
[C ]"[C GH ],[S ]"[S GH
](i "1,2,2M ;j "1,2).Letting
*J F "0,*J F "0,*J ?"0,and *J ?"0,(23)the coupled equations in matrix style can be obtained as follows: A
gift什么意思D D
B ;
英语演讲mp3F <F " X F >F
,(24) A
D D B ;?<? " X ?>? ,(25)
where A "[C ]2[C ],B "[S ]2[S ],D "![C ]2[S ],
X F "[C ]2+h ,,X ?"[C ]2+ ,,>F "![S ]2+h ,,>?
"![S ]2+ ,.Solving Eqs.(24)and (25)one can obtain the results,;P and <P ,which are then treated as the initial values ; P and < P .Substituting ; P and < P into Eq.(22)the initial value of the objective function,J ,can be found.
(2)Substituting ; P and < P into Eqs.(11)and (12),still taking P as the initial value and assuming:
" # ,(26)
" # ,(27) " #
,(28)lively是什么意思
" # ,(29)thus the problem of solving P and P is transferred into the problem of "nding H (j "1,2,3,4)
.M.Gu et al./J.Wind Eng.Ind.Aerodyn.84(2000)151}162155
