Upload
duonglien
View
238
Download
4
Embed Size (px)
Citation preview
Research ArticleVortex-Induced Vibration of a Cable-Stayed Bridge
M T Song1 D Q Cao2 and W D Zhu23
1Faculty of Civil Engineering and Mechanics Jiangsu University Zhenjiang Jiangsu 212013 China2School of Astronautics Harbin Institute of Technology PO Box 137 Harbin 150001 China3Department of Mechanical Engineering University of Maryland Baltimore County Baltimore MD 21250 USA
Correspondence should be addressed to M T Song songmt2004163com
Received 21 June 2015 Revised 13 October 2015 Accepted 15 October 2015
Academic Editor Tai Thai
Copyright copy 2016 M T Song et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The dynamic response of a cable-stayed bridge that consists of a simply supported four-cable-stayed deck beam and two rigidtowers subjected to a distributed vortex shedding force on the deck beam with a uniform rectangular cross section is studiedin this work The cable-stayed bridge is modeled as a continuous system and the distributed vortex shedding force on the deckbeam is modeled using Ehsan-Scanlanrsquos model Orthogonality conditions of exact mode shapes of the linearized undamped cable-stayed bridge model are employed to convert coupled governing partial differential equations of the original cable-stayed bridgemodel with damping to a set of ordinary differential equations by using Galerkin method The dynamic response of the cable-stayed bridge is calculated using Runge-Kutta-Felhberg method in MATLAB for two cases with and without geometric nonlinearterms Convergence of the dynamic response from Galerkin method is investigated Numerical results show that the geometricnonlinearities of stay cables have significant influence on vortex-induced vibration of the cable-stayed bridge There are differentlimit cycles in the case of neglecting the geometric nonlinear terms and there are only one limit cycle and chaotic responses in thecase of considering the geometric nonlinear terms
1 Introduction
Vortex-induced vibration (VIV) of a long-span structure isof practical importance to bridge engineering after collapseof the Tacoma Narrows bridge in 1940 [1] VIV of a structureimmersed in a fluid flow results from forces generatedby alternating shedding of vortices from its surface Thestructural vibration interacts with the flow changing thefluid forces acting on the structure and strongly nonlinearstructural response with multifrequencies takes place [2]VIVmay lead to failure of a cable-stayed bridge due to fatiguedamage and affect travel safety andor comfort levels of itsoccupants [3] Hence an accurate prediction of the responseof the cable-stayed bridge to vortex shedding at an earlydesign stage is essential
To achieve this objective computational fluid dynamics(CFD) techniques are widely adopted to compute fluid forceson the structure by calculating the flow field informationMajor CFD approaches including direct numerical simula-tion [4ndash7] the time-marching scheme [8] and the vortex-in-cell method [9ndash12] mostly directly or approximately solve
the time-dependent Navier-Stokes equation however theyare limited by heavy computational requirement which isdifficult to satisfy up to now
Apart fromnumerical simulations semiempiricalmodelshave emerged as an alternative approach for predicting VIVdue to their simple forms A detailed review onVIVmodelinghas been given by Gabbai and Benaroya [13] according towhich semiempirical models can be divided into two mainclasses single-degree-of-freedom (SDOF) models and wake-oscillator models The former can be classified into negative-damping models [14ndash17] and force-coefficient data models[18ndash20] The wake-oscillator models consider two variablesa structural response variable and a fluid dynamic variable(eg the lifting force) [21ndash26]
The above semiempirical models are not able to predictthe structural response for any cross section shape of a bluffbody since theirmodel parameters rely on values of structuralmass and damping An empirical model of VIV of line-like structures with complex cross sections such as bridgedecks which requires few and relatively simple wind-tunneltests may be useful in practical applications Ehsan and
Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 1928086 14 pageshttpdxdoiorg10115520161928086
2 Shock and Vibration
Scanlan [27] proposed a SDOF model referred to as Ehsan-Scanlanrsquos model which satisfies the above requirement asingle wind-tunnel test with a relatively simple experimentalsetup called the decay-to-resonance test is needed to esti-mate its model parameters Moreover aeroelastic parametersidentified on a section model can be used to calculate theresponse of a cable-stayed bridge considering actual modelproperties of the structure Marra et al [28] applied Ehsan-Scanlanrsquos model in a realistic case study and proposed analternative identification procedure based on direct numer-ical solution of a nonlinear ordinary differential equation
Most previous studies mainly focus on VIV of cylindricalbodies and a deck-shaped body to study VIV of stay cablesand a deck beam respectively which are two main compo-nents of a cable-stayed bridge However there is interactionbetween the stay cables and deck beam when they vibrate[29] This paper presents VIV of a cable-stayed bridge thatconsists of a simply supported four-cable-stayed deck beamand two rigid towers and it aims to study effects of thegeometric nonlinearities of stay cables on the deck beamwith a uniform rectangular cross section that is subjectedto a vortex shedding force Nonlinear and linear partialdifferential equations that govern transverse and longitudinalvibrations of the stay cables and transverse vibrations ofsegments of the deck beam respectively were derived alongwith their boundary and matching conditions using a New-tonian approach Ehsan-Scanlanrsquos model is used to modelthe vortex shedding force that is considered as a distributedforce Exact natural frequencies and mode shapes of thelinearized undamped cable-stayed bridge model obtainedin [29] are used to spatially discretize coupled governingpartial differential equations of the original nonlinear cable-stayed bridge model with damping via Galerkin methodThe dynamic response of the cable-stayed bridge is obtainedby solving resulting nonlinear ordinary differential equa-tions using Runge-Kutta-Felhberg method Convergence ofGalerkin method for VIV of the cable-stayed bridge isinvestigated Numerical results show that there are significantinfluences of the stay cables on VIV of the deck beamthere are different limit cycles when one neglects geometricnonlinear terms associated with the stay cables and there areonly one limit cycle and chaotic responses in the case whenthe geometric nonlinear terms are considered
2 Problem Formulation
Consider a cable-stayed bridge that consists of a simplysupported four-cable-stayed deck beam and two towerssubjected to vortex shedding on the deck beam as shown inFigure 1The deck beam consists of seven segments separatedby its junctions with the stay cables and towersThe followingassumptions are made in this work in the formulation of thevibration problem of the cable-stayed bridgemodel subjectedto vortex shedding
(1) The cable-stayed bridge ismodeled as a planar system
(2) The towers to which the stay cables are attached arebuilt on a hard rock foundation and can be assumed to
C
Xc2
c2c1
1205791b2 b3
1205792b1 b4
c3
1205793b5 b6
1205794
c4
b7
D
A S1 S2 S3 S4 S5 S6B
Yc2 Yc1Xc1
Xb1
Yb1
Xb2Yb2
Figure 1 Schematic of a cable-stayed bridge that consists of a simplysupported four-cable-stayed deck beam and two rigid towers
be rigid [29 30] they are connected to the deck beamthrough roller supports
(3) The stay cables and deck beam have linear elasticbehaviors
(4) Each segment of the deck beam obeys the Euler-Bernoulli beam theory
21 Modeling of the Cable-Stayed Bridge A free vibrationanalysis of the planar motion of this kind of cable-stayedbridges without considering vortex shedding was presentedin [29] The four stay cables are anchored to the deck beamat junctions 119878
1 1198783 1198784 and 119878
6 and the two towers are
connected to the deck beam at junctions 1198782and 119878
5 The
junctions 1198781 1198782 119878
6divide the deck beam into seven seg-
ments 1198871 1198872 119887
7 The length mass per unit length elastic
modulus and cross-sectional area of the 119894th (119894 = 1 2 3 4)stay cable are denoted by 119871
119888119894 119898119888119894 119864119888119894 and 119860
119888119894 respectively
The length mass per unit length elastic modulus and areamoment of inertia of the 119895th (119895 = 1 2 7) segment of thedeck beam are denoted by 2119871
119887119895119898119887119895 119864119887119895 and 119868
119887119895 respectively
Let (119883119888119894 119884119888119894) be local coordinates of cable 119888
119894in the vertical
plane with the origin located at point 119862 for cables 1198881and
1198882and at point 119863 for cables 119888
3and 1198884 Let (119883
119887119895 119884119887119895) be local
coordinates of segment 119887119895of the deck beam in the vertical
plane with the origin located in the middle of segment 119887119895of
the deck beam Initial sags of the stay cables are consideredUnder the assumption of a small ratio of sag 119863
119888119894to length
119871119888119894(ie 119863
119888119894119871119888119894le 110) the static equilibrium of the
stay cable 119888119894can be approximated by a parabolic function
119884119888119894(119883119888119894) = 4119863
119888119894[119883119888119894119871119888119894minus (119883119888119894119871119888119894)2] in its domain while
the static deflection of the deck beam is assumed to benegligible The dynamic configuration of the cable-stayedbridge model is completely described by longitudinal andtransverse displacements of the stay cables 119880
119888119894(119883119888119894 119905) and
119881119888119894(119883119888119894 119905) respectively and transverse displacements of the
segments of the deck beam 119881119887119895(119883119887119895 119905) relative to the above
equilibrium configurationThe following nondimensional variables are introduced
119909119888119894=119883119888119894
119871
119909119887119895=
119883119887119895
119871
Shock and Vibration 3
119897119888119894=119871119888119894
119871
119897119887119895=
119871119887119895
119871
119889119888119894=119863119888119894
Φ
119910119888119894=119884119888119894
Φ
119906119888119894=119880119888119894
Φ
V119888119894=119881119888119894
Φ
V119887119895=
119881119887119895
Φ
120591 = 1205960119905
(1)
where 119871 = min1198711198871 1198711198872 119871
1198877 1205960= (1119871
2
1198871)radic1198641198871
11986811988711198981198871
and Φ is the diameter of stay cable 1198881 Some additional
nondimensional parameters need to be introduced to furnisha complete definition of elastodynamic properties of thecable-stayed bridge model
120583119888119894=119864119888119894119860119888119894
119867119888119894
120594119888119894=
71198712119867119888119894
sum7
119895=1119864119887119895119868119887119895
120578119887119895=
7119864119887119895119868119887119895
sum7
119895=1119864119887119895119868119887119895
120581 =Φ
119871
(2)
where 119867119888119894is the tension in the stay cable 119888
119894on which its
initial sag is dependent that is 119863119888119894= 1198981198881198941198921198712
119888119894cos 1205791198948119867119888119894
in which 119892 is the acceleration of gravity Since deck beamand cable materials of the cable-stayed bridge can generallybe assumed to have different viscous damping behaviorstransverse damping coefficients of cable 119888
119894and segment 119887
119895of
the deck beam are denoted by 119862119888119894and 119862
119887119895 respectively and
their nondimensional parameters are defined by
120585119888119894=11986211988811989412059601198712
119867119888119894
120585119887119895=
11986211988711989512059601198714
119864119887119895119868119887119895
(3)
respectively
The Newtonian method is used here to derive nonlinearequations of motion of the cable-stayed bridge model and afull set of geometric and dynamic boundary and matchingconditions Assuming that cable longitudinal inertial forces(119898119888119894119888119894) are negligible in the prevalent low-frequency trans-
verse vibration of the cable-stayed bridge the longitudinalcable displacement119880
119888119894can be statically condensed leading to
coupled nonlinear equations in terms of only the transversalcable and deck beam displacements 119881
119888119894and 119881
119887119895 respectively
The equations of motion of the cable-stayed bridge are [29]
1205732
119888119894V119888119894+ 120585119888119894V119888119894minus V10158401015840119888119894minus 120583119888119894119890119888119894(120591) (V10158401015840119888119894+ 11991010158401015840
119888119894) = 0
119909119888119894isin [0 119897
119888119894]
(4)
1205734
119887119895V119887119895+ 120585119887119895V119887119895+ V1015840101584010158401015840119887119895= 119901119887119895(119909119887119895 120591 V119887119895 V119887119895)
119909119887119895isin [minus119897119887119895 119897119887119895]
(5)
where a prime and dot denote differentiation with respect tonondimensional local abscissae 119909
119888119894and 119909
119887119895and the time 120591
respectively
120573119888119894= 1198711205960(119898119888119894
119867119888119894
)
12
120573119887119895= 119871(
1198981198871198951205962
0
119864119887119895119868119887119895
)
14
119901119887119895(119909119887119895 120591 V119887119895 V119887119895)
=
1198714119875119887119895(119883119887119895 119905 119889119881
119887119895119889119905 119889
21198811198871198951198891199052)
119864119887119895119868119887119895Φ
(6)
The uniform cable elongation 119890119888119894(120591) in (4) which results from
the static condensation procedure instantaneously dependson both the beam tip deflection and the cable transversedisplacement through the integral form [29]
119890119888119894(120591) =
120581V119888119894(119897119888119894 120591)
119897119888119894
tan 120579119894
+1205812
119897119888119894
int
119897119888119894
0
(1199101015840
119888119894V1015840119888119894+1
2V10158402119888119894)119889119909119888119894
(7)
The functions V119888119894and V
119887119895satisfy the following geometric
boundary conditions
119860 V1198871(minus1198971198871 120591) = 0
V101584010158401198871(minus1198971198871 120591) = 0
(8)
4 Shock and Vibration
119861 V1198877(1198971198877 120591) = 0
V101584010158401198877(1198971198877 120591) = 0
(9)
119862 V1198881(0 120591) = 0
V1198882(0 120591) = 0
(10)
119863 V1198883(0 120591) = 0
V1198884(0 120591) = 0
(11)
The matching conditions at the junctions 119878119896 where 119896 =
1 3 4 6 which involve cables 119888119894 where 119894 = 1 2 3 4 respect-
ively are
V119887119896(119897119887119896 120591) = V
119887119896+1(minus119897119887119896+1 120591) (12)
V1015840119887119896(119897119887119896 120591) = V1015840
119887119896+1(minus119897119887119896+1 120591) (13)
120578119896V10158401015840119887119896(119897119887119896 120591) = 120578
119896+1V10158401015840119887119896+1(minus119897119887119896+1 120591) (14)
120578119887119896V101584010158401015840119887119896(119897119887119896 120591) minus 120578
119887119896+1V101584010158401015840119887119896+1(minus119897119887119896+1 120591) =
120594119888119894120583119888119894
120581119890119888119894(120591)
sdot sin 120579119894
+ 120594119888119894[V1015840119888119894(119897119888119894 120591) + 120583
119888119894119890119888119894(120591) (V1015840119888119894(119897119888119894 120591) + 119910
1015840
119888119894(119897119888119894))]
sdot cos 120579119894
(15)
The matching conditions at the junctions 119878119896(119896 = 2 5) with
the roller supports are
V119887119896(119897119887119896 120591) = 0 (16)
V119887119896+1(minus119897119887119896+1 120591) = 0 (17)
V1015840119887119896(119897119887119896 120591) = V1015840
119887119896+1(minus119897119887119896+1 120591) (18)
120578119887119896V10158401015840119887119896(119897119887119896 120591) = 120578
119887119896+1V10158401015840119887119896+1(minus119897119887119896+1 120591) (19)
Equations (4) and (5) with the boundary and matchingconditions in (8)ndash(19) describe the nonlinear forced vibrationof the cable-stayed bridge The equations governing thesmall amplitude vibration of the cable-stayed bridge can beobtained by linearizing (8) through (19) in the neighborhoodof the equilibrium configuration An extensive analysis ofthe free vibration of the cable-stayed bridge is presented in[29]
22 Modeling of the Vortex Shedding Force The distributedvortex shedding force on the deck beamcan bemodeled usingEhsan-Scanlanrsquos model [28]
119875(119883119887119895 119905 119881119887119895
119889119881119887119895
119889119905) =
1
21205881198802(2119863)
sdot [
[
1198841(119870)(1 minus 120576
1198812
119887119895
1198632)1
119880
119889119881119887119895
119889119905
+ 1198842(119870)
119881119887119895
119863+1
2119862119871(119870) sin (120596119905 + 120579)]
]
(20)
where 120588 is the air density 119880 is the mean wind speed 119870 =
120596119863119880 is the reduced frequency during VIV in which 120596 isthe frequency of the dynamic response of the cable-stayedbridge subjected to vortex shedding 120579 is the phase angleof the harmonic force due to vortex shedding and 119884
1(119870)
120576 1198842(119870) and 119862
119871(119870) are aeroelastic parameters that can be
determined throughwind-tunnel testsTheparameters1198841(119870)
and 120576 are related to linear and nonlinear components ofthe aerodynamic damping term respectively In particular120576 takes into account the fact that VIV is self-limiting Theparameter 119884
2(119870) represents the aerodynamic stiffness term
The parameter 119862119871(119870) is related to the amplitude of the
harmonic force due to vortex shedding According to [27]the second and third terms on the right-hand side of (20) havea negligible contribution to the response of the cable-stayedbridge at lock-in Hence at lock-in (20) can be reduced tothe following form
119875(119883119887119895 119905 119881119887119895
119889119881119887119895
119889119905)
=1
21205881198802(2119863)[
[
1198841(119870)(1 minus 120576
1198812
119887119895
1198632)1
119880
119889119881119887119895
119889119905
]
]
(21)
The nondimensional force 119901119887119895(119909119887119895 120591 V119887119895 V119887119895) in (5) can be
written as
119901119887119895(119909119887119895 120591 V119887119895 V119887119895) = 1205721198871198951198841(1 minus 120576120582
2
119887119895V2119887119895) V119887119895 (22)
where
120572119887119895(119909119887119895) =
1205960120588119880119863119871
4
119864119887119895119868119887119895
120582119887119895=Φ
119863119887119895
(23)
Shock and Vibration 5
3 Solution Method
Galerkin method is used to analyze the vibration of thecable-stayed bridge The dynamic response of stay cables andsegments of the deck beam are expressed by
V119888119894(120591 119909119888119894) =
119873
sum
119904=1
119902119904(120591) 120601(119904)
119888119894(119909119888119894) (24)
V119887119895(120591 119909119887119895) =
119873
sum
119904=1
119902119904(120591) 120601(119904)
119887119895(119909119887119895) (25)
where 120601(119904)119888119894
and 120601(119904)119887119895
are corresponding components of the119904th eigenfunction of the linearized cable-stayed bridge model[29] and 119902
119904(120591) are generalized coordinates Substituting (24)
into (4) yields
1205732
119888119894(
119873
sum
119904=1
119904120601(119904)
119888119894) + 120585119888119894(
119873
sum
119904=1
119904120601(119904)
119888119894) minus (
119873
sum
119904=1
119902119904120601(119904)
119888119894
10158401015840
)
minus120583119888119894120581
119897119888119894
[
[
tan 120579119894(
119873
sum
119904=1
119902119904120601(119904)
119888119894(119897119888119894))
+ 120581int
119897119888119894
0
1199101015840
119888119894(
119873
sum
119904=1
119902119904120601(119904)
119888119894
1015840
)119889119909119888119894
+120581
2int
119897119888119894
0
(
119873
sum
119904=1
119902119904120601(119904)
119888119894
1015840
)
2
119889119909119888119894
]
]
[(
119873
sum
119904=1
119902119904120601(119904)
119888119894
10158401015840
) + 11991010158401015840
119888119894]
= 0
(26)
Substituting (25) into (5) yields
1205734
119887119895(
119873
sum
119904=1
119904120601(119904)
119887119895) + 120585119887119895(
119873
sum
119904=1
119904120601(119904)
119887119895) + (
119873
sum
119904=1
119902119904120601(119904)
119887119895
1015840101584010158401015840
)
= 1205721198871198951198841(1 minus 120576120582
2
119887119895(
119873
sum
119904=1
119902119904120601(119904)
119887119895)
2
)(
119873
sum
119904=1
119904120601(119904)
119887119895)
(27)
Multiplying (26) by 120594119888119894120601(119903)
119888119894(119909119888119894) and integrating the resulting
equation with respect to 119909119888119894from 0 to 119897
119888119894 multiplying (27)
by 120578119887119895120601(119903)
119887119895(119909119887119895) and integrating the resulting equation with
respect to 119909119887119895from minus119897
119887119895to 119897119887119895 adding all the resulting equa-
tions andusing the following orthogonality relations of eigen-functions of the linearized cable-stayed bridge model [29]
4
sum
119894=1
1205941198881198941205732
119888119894int
119897119888119894
0
120601(119903)
119888119894(119909119888119894) 120601(119904)
119888119894(119909119888119894) 119889119909119888119894+
7
sum
119895=1
1205781198871198951205734
119887119895
sdot int
119897119887119895
minus119897119887119895
120601(119903)
119887119895(119909119887119895) 120601(119904)
119887119895(119909119887119895) 119889119909119887119895= 119872119904120575119904119903
minus
4
sum
119894=1
120594119888119894[int
119897119888119894
0
120601(119903)
119888119894(119909119888119894) 120601(119904)
119888119894
10158401015840
(119909119888119894) 119889119909119888119894
minus 8120583119888119894(119904)
119888119894
119889119888119894
1198972119888119894
int
119897119888119894
0
120601(119903)
119888119894(119909119888119894) 119889119909119888119894] +
7
sum
119895=1
120578119887119895
sdot int
119897119887119895
minus119897119887119895
120601(119903)
119887119895(119909119887119895) 120601(119904)
119887119895
1015840101584010158401015840
(119909119887119895) 119889119909119887119895= 1205962
119904119872119904120575119904119903
(28)
in which (119904)119888119894= 120581(120601
(119904)
119888119894(119897119888119894)119897119888119894) tan 120579
119894+ 1205812(1119897119888119894) int119897119888119894
01199101015840
1198881198941206011015840(119904)
119888119894119889119909119888119894
120596119904is the 119904th natural frequency of the linearized cable-
stayed bridge model 119872119904are positive constants and 120575
119904119903is
the Kronecker delta one can obtain spatially discretizedequations of the cable-stayed bridge
Mq + [C + CAero] q + Kq + NQ
+ NC+ NHy
= 0 (29)
where entries of thematricesMCCAeroKNQNC andNHy
are
119872119903119904= 119872119904120575119904119903
119862119903119904=
4
sum
119894=1
120594119888119894120585119888119894int
119897119888119894
0
120601(119904)
119888119894120601(119903)
119888119894119889119909119888119894+
7
sum
119895=1
120578119887119895120585119887119895int
119897119887119895
minus119897119887119895
120601(119904)
119887119895120601(119903)
119887119895119889119909119887119895
119862Aero119903119904
= minus
7
sum
119895=1
1205781198871198951198841int
119897119887119895
minus119897119887119895
120572119887119895120601(119904)
119887119895120601(119903)
119887119895119889119909119887119895
119870119903119904= 1205962
119904119872119903119904
119873Q119903=
4
sum
119894=1
119873
sum
119898=1
119873
sum
119899=1
120594119888119894120583119888119894119902119898119902119899(41198891198881198941205812
1198973119888119894
int
119897119888119894
0
120601(119898)
119888119894
1015840
120601(119899)
119888119894
1015840
119889119909119888119894int
119897119888119894
0
120601(119903)
119888119894119889119909119888119894minus (119898)
119888119894int
119897119888119894
0
120601(119903)
119888119894120601(119899)
119888119894
10158401015840
119889119909119888119894)
119873C119903= minus1205812
4
sum
119894=1
119873
sum
119898=1
119873
sum
119899=1
119873
sum
119900=1
120594119888119894120583119888119894
2119897119888119894
119902119898119902119899119902119900int
119897119888119894
0
120601(119898)
119888119894
1015840
120601(119899)
119888119894
1015840
119889119909119888119894int
119897119888119894
0
120601(119903)
119888119894120601(119900)
119888119894
10158401015840
119889119909119888119894
119873Hy119903=
7
sum
119895=1
119873
sum
119898=1
119873
sum
119899=1
119873
sum
119900=1
12057811988711989511988411205761205822
119887119895119902119898119902119899119900int
119897119887119895
minus119897119887119895
120572119887119895120601(119898)
119887119895120601(119899)
119887119895120601(119900)
119887119895120601(119903)
119887119895119889119909119887119895
(30)
respectively
6 Shock and Vibration
615
62
625
63
Am
plitu
de (m
m)
5 10 15 20 25 300Number of truncation terms N
Figure 2 Effects of the number of Galerkin truncation terms on transverse displacements of the midpoint of the deck beam when there areno geometric nonlinear terms
It should be noted that nonlinear termsNQ andNC in (29)are induced by geometric nonlinearities of the stay cables
4 Numerical Simulation
Geometric and physical parameters of a cable-stayed bridgeand aeroelastic parameters are listed in Table 1 It shouldbe noted that a width-to-height ratio of 4 is used in thispaper not only because it is a typical ratio of bridge decksbut also because the cable-stayed bridge is supposed tohave significant response amplitudes at lock-in [28] Forall the following calculation modal damping ratio 119862
119903119904is
always equal to 001120575119903119904 The dynamic response of the cable-
stayed bridge can be calculated from (29) using Runge-Kutta-Felhbergmethod inMATLABwhere initial conditionsof generalized coordinates can be obtained from those ofphysical coordinates
119902119903(0) =
1
119872119903
[
[
4
sum
119894=1
1205941198881198941205732
119888119894int
119897119888119894
0
V119888119894(119909119888119894 0) 120601(119903)
119888119894(119909119888119894) 119889119909119888119894
+
7
sum
119895=1
1205781198871198951205734
119887119895int
119897119887119895
minus119897119887119895
V119887119895(119909119887119895 0) 120601(119903)
119887119895(119909119887119895) 119889119909119887119895
]
]
119903(0) =
1
119872119903
[
[
4
sum
119894=1
1205941198881198941205732
119888119894int
119897119888119894
0
V119888119894(119909119888119894 0) 120601(119903)
119888119894(119909119888119894) 119889119909119888119894
+
7
sum
119895=1
1205781198871198951205734
119887119895int
119897119887119895
minus119897119887119895
V119887119895(119909119887119895 0) 120601(119903)
119887119895(119909119887119895) 119889119909119887119895
]
]
(31)
For all the following calculation 119903(0) is always equal to
zero Numerical simulations for two cases are undertaken (1)neglecting the geometric nonlinear terms (ie NQ
= NC= 0
in (29)) and (2) considering the geometric nonlinear terms
41 Case Studies Neglecting the Geometric Nonlinear TermsConvergence of Galerkin method for given initial conditions119902119903(0) = 01120575
1199031 which means that the cable-stayed bridge
has its initial dynamic configuration corresponding to itsfirst mode shape and it is released from rest is shown in
Figures 2 and 3The amplitudes of the steady-state transversedisplacements of the midpoint of the deck beam withdifferent numbers of Galerkin truncation terms which areset to one through 30 are shown in Figure 2 As shown inFigure 2 the transverse displacements of the midpoint of thedeck beam initially remain stable until the truncation numberincreases to 15 and suddenly increase to their convergedvalues A truncation with 20 terms can provide accurateresults in analyzing the dynamic response of the deck beamThe difference between the amplitudes of two limit cyclesobtained by truncation with one term and 20 terms is194 which is small One can come to a conclusion that atruncation with one term is accurate enough for calculationof the steady-state response of the cable-stayed bridge whenits initial dynamic configuration corresponds to its first modeshape In other words most of the energy of the cable-stayedbridge is concentrated in its first mode in this case Phaseportraits of the response of the midpoint of the deck beamfor different numbers of Galerkin truncation terms whichis shown in Figure 3 also lead to the same conclusion Otherresults that are not shown here for the sake of brevity indicatethat Galerkin truncation with 20 terms yields accurateresults for the dynamic response of the cable-stayed bridgein the following cases in this section Hence in the followingnumerical calculations in this section the first 20 modes ofthe linearized undamped cable-stayed bridge model are usedin Galerkin method Time history responses of the midpointof the deck beam when the initial dynamic configuration ofthe cable-stayed bridge corresponds to its first mode shapebut has different amplitudes that is 119902
1(0) = 01 (the initial
displacement of themidpoint of the deck beam is 32mm) and1199021(0) = 3 (the initial displacement of themidpoint of the deck
beam is 946mm which is rather large) are shown in Figures4 and 5 respectively It can be seen that the solutions with dif-ferent 119902
1(0) converge to the same limit cycle after long-time
integrationThemagnitudes of Floquet multipliers are all lessthan unity the aforementioned limit cycle is asymptoticallystable
Solutions of a reduced-order model for a flow dynamicsystem can converge to a spurious limit cycle after long-timeintegration even if it is initializedwith a correct configuration
Shock and Vibration 7
Table 1 Geometric and physical parameters of the cable-stayed bridge and aeroelastic parameters
Parameter Unit Value
Deck beam
Mass per unit length of the deck beam (119898119887119895) kgm 16940
Elastic modulus of the deck beam (119864119887119895) Nm2 20 times 1011
Area moment of inertia of the deck beam (119868119887119895) m4 120
Length of segment 1198871of the deck beam (2119871
1198871) m 35
Length of segment 1198872of the deck beam (2119871
1198872) m 40
Length of segment 1198873of the deck beam (2119871
1198873) m 50
Length of segment 1198874of the deck beam (2119871
1198874) m 50
Length of segment 1198875of the deck beam (2119871
1198875) m 50
Length of segment 1198876of the deck beam (2119871
1198876) m 40
Length of segment 1198877of the deck beam (2119871
1198877) m 35
Stay cables
Mass per unit length of the stay cables (119898119888119894) kgm 286
Elastic modulus of the stay cables (119864119888119894) Nm2 20 times 1011
Cross-sectional area of the stay cables (119860119888119894) m2 00362
Length of stay cable 1198881(1198711198881) m 52
Length of stay cable 1198882(1198711198882) m 60
Length of stay cable 1198883(1198711198883) m 60
Length of stay cable 1198884(1198711198884) m 52
Sag-to-span ratios of the stay cables (119889119888119894= 119863119888119894119871119888119894) 001
Aeroelastic parametersAir density (120588) kgm3 1205120576 11225 [28]1198841
688 [28]
N = 1
N = 19
N = 20
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus100
minus50
0
50
100
Figure 3 Phase portraits of the response of themidpoint of the deck beam for different numbers of Galerkin truncation terms 119902119903(0) = 01120575
1119903
[31 32] Sirisup and Karniadakis [32] demonstrated that theonset of divergence from the correct limit cycle dependson the number of Galerkin truncation terms the Reynoldsnumber and the flow geometry Since the limit cycle heredoes not vanish even when the truncation number is 30which is large one can conclude that the aforementionedlimit cycle for the cable-stayed bridge is a correct one
In many cases higher mode shapes of the cable-stayedbridge would be excited one such case is that when thereare vehicles moving on the deck beam of the bridge Hence
the initial dynamic configuration of the cable-stayed bridgecan correspond to its higher mode shapes in the dynamicanalysis of the cable-stayed bridge subjected to a distributedvortex shedding force It is obvious that Galerkin truncationwith one term is not enough in these cases Through thesame method with 119873 = 20 one can find that there isthe same limit cycle as that in Figure 3 when the initialdynamic configuration corresponds to the second throughsixth mode shapes and there is a different limit cycle whenthe initial dynamic configuration corresponds to the seventh
8 Shock and Vibration
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus80
minus40
0
40
80
Vb 4(X
b 4=0)
(mm
)
Figure 4 Time response of the midpoint of the deck beam 119902119903(0) = 01120575
1119903
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus800
minus400
0
400
800
Vb 4(X
b 4=0)
(mm
)
Figure 5 Time response of the midpoint of the deck beam 119902119903(0) = 3120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 6 Time response of the midpoint of the deck beam 119902119903(0) = 01120575
7119903
mode shape (Figures 6 and 7) Two-dimensional projectionsof phase portraits onto the (119902
119903 119903) plane when 119902
119903(0) = 01120575
1119903
and 119902119903(0) = 01120575
7119903are shown in Figures 8 and 9 respectively
Results when 119902119903(0) is equal to 01120575
2119903 011205753119903 and 01120575
6119903
are the same as that when 119902119903(0) = 01120575
1119903 they are not shown
here for the sake of brevity It can be seen from Figures 8 and9 that energy of the cable-stayed bridge is concentrated inthe first mode (the seventh mode) when its initial dynamic
configuration corresponds to the first through sixth modeshapes (the seventh mode shape)
42 Case Studies Considering the Geometric Nonlinear TermsConvergence of Galerkin method for given initial conditions119902119903(0) = 01120575
1199031is shown in Figures 10 and 11 The amplitudes
of the steady-state transverse displacements of the mid-point of the deck beam with different numbers of Galerkin
Shock and Vibration 9
minus150
minus100
minus50
0
50
100
150
0 50 100minus50minus100
qr(0) = 011205751rqr(0) = 011205757r
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 7 Two different stable limit cycles corresponding to different initial configurations
q1
minus05
0
05
0 05minus05
q1
q2
minus05
0
05
0 05minus05
q2
minus05
0
05
q3
0 05minus05
q3
q4
minus05
0
05
0 05minus05
q4
q5
minus05
0
05
0 05minus05
q5
q6
minus05
0
05
0 05minus05
q6
q7
minus05
0
05
0 05minus05
q7
q8
minus05
0
05
0 05minus05
q8
q9
minus05
0
05
0 05minus05
q9
Figure 8 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903
truncation terms which are set to one through 30 are shownin Figure 10 As shown in Figure 10 a truncation with 11terms can provide accurate results in analyzing the dynamicresponse of the cable-stayed bridge and it is much lessthan the cases in Section 41 where the geometric nonlinearterms are neglected Phase portraits of the response of themidpoint of the deck beam for different numbers of Galerkin
truncation terms which are shown in Figure 11 also leadto the same conclusion It can be noted from Figure 11 thatthe limit cycle becomes more asymmetric when the numberof the truncation terms increases The reason for this isthat when the number of truncation terms increases to 7an additional asymmetric limit cycle appears on the (119902
7 7)
plane which phase portraits project onto (see Figure 12 with
10 Shock and Vibration
q1
minus1
0
1
0 03minus03
q1
q2
minus1
0
1
0 03minus03
q2
q3
minus1
0
1
0 03minus03
q3
q4
minus1
0
1
0 03minus03
q4
q5
minus1
0
1
0 03minus03
q5
q6
minus1
0
1
0 03minus03
q6
q7
minus1
0
1
0 03minus03
q7
q8
minus1
0
1
0 03minus03
q8
q9
minus1
0
1
0 03minus03
q9
Figure 9 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
7119903
5 10 15 20 25 300Number of truncation terms N
58
59
60
61
62
63
Am
plitu
de (m
m)
Figure 10 Effects of the number of Galerkin truncation terms ontransverse displacements of the midpoint of the deck beam whenthere are geometric nonlinear terms
119873 = 11) As shown in Figures 11 and 13 the dynamic responseof the midpoint of the deck beam approaches a stable limitcycle The bifurcation diagram of limit cycles with respectto 1199021(0) is shown in Figure 14 when the initial dynamic
configuration of the cable-stayed bridge corresponds to itsfirst mode shape One can find from Figure 14 that when1199021(0) is smaller than 027 a stable periodic solution exists and
minus50 0 50 100minus100minus100
minus50
0
50
100
N = 1
N = 10
N = 11
N = 30
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 11 Phase portraits of the response of themidpoint of the deckbeam for different numbers of Galerkin truncation terms 119902
119903(0) =
011205751119903
when 1199021(0) is larger than or equal to 027 chaotic responses
occur For instance when 1199021(0) = 03 and 119873 = 11 phase
portraits of the response of the midpoint of the deck beamare shown in Figure 15 which indicates occurrence of chaotic
Shock and Vibration 11
q1
minus50
0
50
0 50minus50
q1
q2
minus20
0
20
0 10minus10
q2
q3
minus20
0
20
0 10minus10
q3
q4
minus20
0
20
0 10minus10
q4
q5
minus20
0
20
0 10minus10
minus10
q5
q6
minus20
0
20
0 10minus10
q6
q7
minus20
0
20
0 10minus10
q7
q8
minus20
0
20
100
q8
q9
minus20
0
20
0 10minus10
q9
Figure 12 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903and119873 = 11
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 13 Time response of the midpoint of the deck beam 119902119903(0) =
011205751119903and119873 = 11
response On the contrary for a truncation with one termthe response always approaches a stable limit cycle no matterhow large the amplitude of the initial dynamic configurationis (see Figures 16ndash18) These mean that the true dynamicresponse of the cable-stayed bridge may not be capturedby the truncation with one term even when the initialdynamic configuration corresponds to the first mode shape
01 02 03 04 05 06 070q1(0)
minus200
minus150
minus100
minus50
0
50
Vb 4(X
b 4=0)
(mm
)
Figure 14 Bifurcation diagram of the cable-stayed bridge withrespect to 119902
1(0) 119902119903(0) = 119902
1(0)1205751119903and119873 = 11
When the initial dynamic configuration of the cable-stayedbridge corresponds to one of its higher mode shapes chaoticresponse occurs even if its amplitude is relatively small this isshown in Figure 19where the initial dynamic configuration ofthe cable-stayed bridge corresponds to its secondmode shapeand its magnitude is only 01
12 Shock and Vibration
0
minus100 0 100 200 300minus200
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus1500
minus1000
minus500
500
1000
1500
Figure 15 Chaotic response of the midpoint of the deck beam119902119903(0) = 03120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 16 Time response of themidpoint of the deck beam 1199021(0) =
01 and119873 = 1
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus1000
minus500
0
500
1000
Vb 4(X
b 4=0)
(mm
)
Figure 17 Time response of the midpoint of the deck beam 1199021(0) =
3 and119873 = 1
5 Conclusions
The dynamic behavior of a cable-stayed bridge that consistsof a simply supported four-cable-stayed deck beam andtwo rigid towers subjected to a distributed vortex shed-ding force on the deck beam has been investigated The
q1(0) = 01
q1(0) = 3
minus100
minus50
0
50
100
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 18 Phase portraits of the response of the midpoint of thedeck beam for different initial conditions119873 = 1
minus1500
minus1000
minus500
0
500
1000
1500
minus200 0 200 400minus400
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 19 Chaotic response of the midpoint of the deck beam119902119903(0) = 01120575
2119903
dynamic response of the cable-stayed bridge is calculatedusing Galerkin method in conjunction with Runge-Kutta-Felhberg method in MATLAB Convergence of Galerkinmethod for the dynamic response of the cable-stayed bridgeis studied Numerical simulations show that the geometricnonlinearities of the stay cables have significant influence onVIV of the cable-stayed bridge and further conclusions canbe summarized as follows
(1) In the case when the geometric nonlinear termsare neglected accurate calculation of the responseamplitude of the cable-stayed bridge at lock-in onlyneeds use of the first mode shape of the linearizedundamped cable-stayed bridgemodel when the initialdynamic configuration of the cable-stayed bridgecorresponds to its mode shape whose mode numberis smaller than seven There is a different limit cyclewhen the initial dynamic configuration correspondsto its mode shape whose mode number is equal to orlarger than 7
Shock and Vibration 13
(2) In the case when the geometric nonlinear termsare considered calculation of the response of thecable-stayed bridge generally needs use of multiplemode shapes of the linearized undamped cable-stayedbridge model even when the initial dynamic config-uration of the cable-stayed bridge corresponds to itsfirstmode shapeThere is a limit cycle when the initialdynamic configuration of the cable-stayed bridgecorresponds to its first mode shape and its amplitudeis smaller than 03 for the generalized coordinate andthere is chaotic response when the initial dynamicconfiguration of the cable-stayed bridge correspondsto its first mode shape with its amplitude larger than03 for the generalized coordinate or one of its highermode shapes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grant nos 11302087 and 11442006 theNatural Science Foundation of Jiangsu Province under Grantno BK20130479 the Research Foundation for AdvancedTalents of Jiangsu University under Grant no 13JDG068 andthe National Science Foundation under Grant no CMMI-1000830
References
[1] K Y R Billah and R H Scanlan ldquoResonance Tacoma NarrowsBridge failure and undergraduate physics textbooksrdquoAmericanJournal of Physics vol 59 no 2 pp 118ndash124 1991
[2] R D Blevins Flow-Induced Vibration Krieger New York NYUSA 2nd edition 1994
[3] H Zheng R E Price Y Modarres-Sadeghi and M S Tri-antafyllou ldquoOn fatigue damage of long flexible cylinders dueto the higher harmonic force components and chaotic vortex-induced vibrationsrdquo Ocean Engineering vol 88 pp 318ndash3292014
[4] C Evangelinos D Lucor and G E Karniadakis ldquoDNS-derivedforce distribution on flexible cylinders subject to vortex-induced vibrationrdquo Journal of Fluids and Structures vol 14 no3 pp 429ndash440 2000
[5] HM Blackburn R N Govardhan andCH KWilliamson ldquoAcomplementary numerical and physical investigation of vortex-inducedrdquo Journal of Fluids and Structures vol 15 no 3-4 pp481ndash488 2001
[6] R H J Willden and J M R Graham ldquoNumerical predictionof VIV on long flexible circular cylindersrdquo Journal of Fluids andStructures vol 15 no 3-4 pp 659ndash669 2001
[7] E Guilmineau and P Queutey ldquoA numerical simulation ofvortex shedding from an oscillating circular cylinderrdquo Journalof Fluids and Structures vol 16 no 6 pp 773ndash794 2002
[8] I Jadic R M C So and M P Mignolet ldquoAnalysis of fluidmdashstructure interactions using a time-marching techniquerdquo Jour-nal of Fluids and Structures vol 12 no 6 pp 631ndash654 1998
[9] C Y Zhou RM C So andK Lam ldquoVortex-induced vibrationsof an elastic circular cylinderrdquo Journal of Fluids and Structuresvol 13 no 2 pp 165ndash189 1999
[10] J R Meneghini and P W Bearman ldquoNumerical simulationof high amplitude oscillatory flow about a circular cylinderrdquoJournal of Fluids and Structures vol 9 no 4 pp 435ndash455 1995
[11] T Sarpkaya ldquoComputational methods with vorticesrdquo Journal ofFluids Engineering vol 111 no 1 pp 5ndash52 1989
[12] T Sarpkaya ldquoVortex element methods for flow simulationrdquoAdvances in Applied Mechanics vol 31 pp 113ndash247 1994
[13] R D Gabbai and H Benaroya ldquoAn overview of modeling andexperiments of vortex-induced vibration of circular cylindersrdquoJournal of Sound and Vibration vol 282 no 3ndash5 pp 575ndash6162005
[14] R H Scanlan ldquoOn the state-of-the-artmethods for calculationsof flutter vortex-induced and buffeting response of bridgestructuresrdquo Tech Rep FHWARD-80050 National TechnicalInformation Service Springfield Va USA 1981
[15] RH Scanlan ldquoBridge flutter derivatives at vortex lock-inrdquo Jour-nal of Structural Engineering vol 124 no 4 pp 450ndash458 1998
[16] B J Vickery and R I Basu ldquoAcross-wind vibrations of struc-tures of circular cross-section Part I Development of a math-ematical model for two-dimensional conditionsrdquo Journal ofWind Engineering and Industrial Aerodynamics vol 12 no 1 pp49ndash73 1983
[17] A Larsen ldquoA generalized model for assessment of vortex-induced vibrations of flexible structuresrdquo Journal of Wind Engi-neering and Industrial Aerodynamics vol 57 no 2-3 pp 281ndash294 1995
[18] T Sarpkaya ldquoFluid forces on oscillating cylindersrdquo Journal ofWaterway Port Coastal and Ocean Division (ASCE) vol 104no 3 pp 275ndash290 1978
[19] T Staubli ldquoCalculation of vibration of an elastically mountedcylinder using experimental data from a forced oscillationrdquoJournal of Fluids Engineering vol 105 no 2 pp 225ndash229 1983
[20] W D Iwan and D L R Botelho ldquoVortex-induced oscillationof structures in waterrdquo Journal of Waterway Port Coastal andOcean Engineering vol 111 no 2 pp 289ndash303 1985
[21] R T Hartlen and I G Currie ldquoLift-oscillator model of vortex-induced vibrationrdquo Journal of Engineering Mechanics Division(ASCE) vol 96 no 5 pp 577ndash591 1970
[22] R A Skop and O M Griffin ldquoA model for the vortex-excitedresonant response of bluff cylindersrdquo Journal of Sound andVibration vol 27 no 2 pp 225ndash233 1973
[23] WD Iwan andRD Blevins ldquoAmodel for vortex induced oscil-lation of structuresrdquo Journal of Applied Mechanics TransactionsASME vol 41 no 3 pp 581ndash586 1974
[24] E H Dowell ldquoNon-linear oscillator models in bluff body aero-elasticityrdquo Journal of Sound and Vibration vol 75 no 2 pp 251ndash264 1981
[25] R H M Ogink and A V Metrikine ldquoA wake oscillator withfrequency dependent coupling for the modeling of vortex-induced vibrationrdquo Journal of Sound and Vibration vol 329 no26 pp 5452ndash5473 2010
[26] X Bai and W Qin ldquoUsing vortex strength wake oscillatorin modelling of vortex induced vibrations in two degrees offreedomrdquo European Journal of Mechanics BFluids vol 48 pp165ndash173 2014
[27] F Ehsan and R H Scanlan ldquoVortexminusinduced vibrations offlexible bridgesrdquo Journal of Engineering Mechanics vol 116 no6 pp 1392ndash1411 1990
14 Shock and Vibration
[28] A M Marra C Mannini and G Bartoli ldquoVan der Pol-typeequation for modeling vortex-induced oscillations of bridgedecksrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 99 no 6-7 pp 776ndash785 2011
[29] D Q Cao M T Song W D Zhu R W Tucker and C H-T Wang ldquoModeling and analysis of the in-plane vibration ofa complex cable-stayed bridgerdquo Journal of Sound and Vibrationvol 331 no 26 pp 5685ndash5714 2012
[30] E Caetano A Cunha V Gattulli and M Lepidi ldquoCable-deckdynamic interactions at the International Guadiana Bridge on-site measurements and finite element modellingrdquo StructuralControl and Health Monitoring vol 15 no 3 pp 237ndash264 2008
[31] I Akhtar A H Nayfeh and C J Ribbens ldquoOn the stability andextension of reduced-order Galerkin models in incompressibleflowsrdquo Theoretical and Computational Fluid Dynamics vol 23no 3 pp 213ndash237 2009
[32] S Sirisup and G E Karniadakis ldquoA spectral viscosity methodfor correcting the long-term behavior of POD modelsrdquo Journalof Computational Physics vol 194 no 1 pp 92ndash116 2004
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
2 Shock and Vibration
Scanlan [27] proposed a SDOF model referred to as Ehsan-Scanlanrsquos model which satisfies the above requirement asingle wind-tunnel test with a relatively simple experimentalsetup called the decay-to-resonance test is needed to esti-mate its model parameters Moreover aeroelastic parametersidentified on a section model can be used to calculate theresponse of a cable-stayed bridge considering actual modelproperties of the structure Marra et al [28] applied Ehsan-Scanlanrsquos model in a realistic case study and proposed analternative identification procedure based on direct numer-ical solution of a nonlinear ordinary differential equation
Most previous studies mainly focus on VIV of cylindricalbodies and a deck-shaped body to study VIV of stay cablesand a deck beam respectively which are two main compo-nents of a cable-stayed bridge However there is interactionbetween the stay cables and deck beam when they vibrate[29] This paper presents VIV of a cable-stayed bridge thatconsists of a simply supported four-cable-stayed deck beamand two rigid towers and it aims to study effects of thegeometric nonlinearities of stay cables on the deck beamwith a uniform rectangular cross section that is subjectedto a vortex shedding force Nonlinear and linear partialdifferential equations that govern transverse and longitudinalvibrations of the stay cables and transverse vibrations ofsegments of the deck beam respectively were derived alongwith their boundary and matching conditions using a New-tonian approach Ehsan-Scanlanrsquos model is used to modelthe vortex shedding force that is considered as a distributedforce Exact natural frequencies and mode shapes of thelinearized undamped cable-stayed bridge model obtainedin [29] are used to spatially discretize coupled governingpartial differential equations of the original nonlinear cable-stayed bridge model with damping via Galerkin methodThe dynamic response of the cable-stayed bridge is obtainedby solving resulting nonlinear ordinary differential equa-tions using Runge-Kutta-Felhberg method Convergence ofGalerkin method for VIV of the cable-stayed bridge isinvestigated Numerical results show that there are significantinfluences of the stay cables on VIV of the deck beamthere are different limit cycles when one neglects geometricnonlinear terms associated with the stay cables and there areonly one limit cycle and chaotic responses in the case whenthe geometric nonlinear terms are considered
2 Problem Formulation
Consider a cable-stayed bridge that consists of a simplysupported four-cable-stayed deck beam and two towerssubjected to vortex shedding on the deck beam as shown inFigure 1The deck beam consists of seven segments separatedby its junctions with the stay cables and towersThe followingassumptions are made in this work in the formulation of thevibration problem of the cable-stayed bridgemodel subjectedto vortex shedding
(1) The cable-stayed bridge ismodeled as a planar system
(2) The towers to which the stay cables are attached arebuilt on a hard rock foundation and can be assumed to
C
Xc2
c2c1
1205791b2 b3
1205792b1 b4
c3
1205793b5 b6
1205794
c4
b7
D
A S1 S2 S3 S4 S5 S6B
Yc2 Yc1Xc1
Xb1
Yb1
Xb2Yb2
Figure 1 Schematic of a cable-stayed bridge that consists of a simplysupported four-cable-stayed deck beam and two rigid towers
be rigid [29 30] they are connected to the deck beamthrough roller supports
(3) The stay cables and deck beam have linear elasticbehaviors
(4) Each segment of the deck beam obeys the Euler-Bernoulli beam theory
21 Modeling of the Cable-Stayed Bridge A free vibrationanalysis of the planar motion of this kind of cable-stayedbridges without considering vortex shedding was presentedin [29] The four stay cables are anchored to the deck beamat junctions 119878
1 1198783 1198784 and 119878
6 and the two towers are
connected to the deck beam at junctions 1198782and 119878
5 The
junctions 1198781 1198782 119878
6divide the deck beam into seven seg-
ments 1198871 1198872 119887
7 The length mass per unit length elastic
modulus and cross-sectional area of the 119894th (119894 = 1 2 3 4)stay cable are denoted by 119871
119888119894 119898119888119894 119864119888119894 and 119860
119888119894 respectively
The length mass per unit length elastic modulus and areamoment of inertia of the 119895th (119895 = 1 2 7) segment of thedeck beam are denoted by 2119871
119887119895119898119887119895 119864119887119895 and 119868
119887119895 respectively
Let (119883119888119894 119884119888119894) be local coordinates of cable 119888
119894in the vertical
plane with the origin located at point 119862 for cables 1198881and
1198882and at point 119863 for cables 119888
3and 1198884 Let (119883
119887119895 119884119887119895) be local
coordinates of segment 119887119895of the deck beam in the vertical
plane with the origin located in the middle of segment 119887119895of
the deck beam Initial sags of the stay cables are consideredUnder the assumption of a small ratio of sag 119863
119888119894to length
119871119888119894(ie 119863
119888119894119871119888119894le 110) the static equilibrium of the
stay cable 119888119894can be approximated by a parabolic function
119884119888119894(119883119888119894) = 4119863
119888119894[119883119888119894119871119888119894minus (119883119888119894119871119888119894)2] in its domain while
the static deflection of the deck beam is assumed to benegligible The dynamic configuration of the cable-stayedbridge model is completely described by longitudinal andtransverse displacements of the stay cables 119880
119888119894(119883119888119894 119905) and
119881119888119894(119883119888119894 119905) respectively and transverse displacements of the
segments of the deck beam 119881119887119895(119883119887119895 119905) relative to the above
equilibrium configurationThe following nondimensional variables are introduced
119909119888119894=119883119888119894
119871
119909119887119895=
119883119887119895
119871
Shock and Vibration 3
119897119888119894=119871119888119894
119871
119897119887119895=
119871119887119895
119871
119889119888119894=119863119888119894
Φ
119910119888119894=119884119888119894
Φ
119906119888119894=119880119888119894
Φ
V119888119894=119881119888119894
Φ
V119887119895=
119881119887119895
Φ
120591 = 1205960119905
(1)
where 119871 = min1198711198871 1198711198872 119871
1198877 1205960= (1119871
2
1198871)radic1198641198871
11986811988711198981198871
and Φ is the diameter of stay cable 1198881 Some additional
nondimensional parameters need to be introduced to furnisha complete definition of elastodynamic properties of thecable-stayed bridge model
120583119888119894=119864119888119894119860119888119894
119867119888119894
120594119888119894=
71198712119867119888119894
sum7
119895=1119864119887119895119868119887119895
120578119887119895=
7119864119887119895119868119887119895
sum7
119895=1119864119887119895119868119887119895
120581 =Φ
119871
(2)
where 119867119888119894is the tension in the stay cable 119888
119894on which its
initial sag is dependent that is 119863119888119894= 1198981198881198941198921198712
119888119894cos 1205791198948119867119888119894
in which 119892 is the acceleration of gravity Since deck beamand cable materials of the cable-stayed bridge can generallybe assumed to have different viscous damping behaviorstransverse damping coefficients of cable 119888
119894and segment 119887
119895of
the deck beam are denoted by 119862119888119894and 119862
119887119895 respectively and
their nondimensional parameters are defined by
120585119888119894=11986211988811989412059601198712
119867119888119894
120585119887119895=
11986211988711989512059601198714
119864119887119895119868119887119895
(3)
respectively
The Newtonian method is used here to derive nonlinearequations of motion of the cable-stayed bridge model and afull set of geometric and dynamic boundary and matchingconditions Assuming that cable longitudinal inertial forces(119898119888119894119888119894) are negligible in the prevalent low-frequency trans-
verse vibration of the cable-stayed bridge the longitudinalcable displacement119880
119888119894can be statically condensed leading to
coupled nonlinear equations in terms of only the transversalcable and deck beam displacements 119881
119888119894and 119881
119887119895 respectively
The equations of motion of the cable-stayed bridge are [29]
1205732
119888119894V119888119894+ 120585119888119894V119888119894minus V10158401015840119888119894minus 120583119888119894119890119888119894(120591) (V10158401015840119888119894+ 11991010158401015840
119888119894) = 0
119909119888119894isin [0 119897
119888119894]
(4)
1205734
119887119895V119887119895+ 120585119887119895V119887119895+ V1015840101584010158401015840119887119895= 119901119887119895(119909119887119895 120591 V119887119895 V119887119895)
119909119887119895isin [minus119897119887119895 119897119887119895]
(5)
where a prime and dot denote differentiation with respect tonondimensional local abscissae 119909
119888119894and 119909
119887119895and the time 120591
respectively
120573119888119894= 1198711205960(119898119888119894
119867119888119894
)
12
120573119887119895= 119871(
1198981198871198951205962
0
119864119887119895119868119887119895
)
14
119901119887119895(119909119887119895 120591 V119887119895 V119887119895)
=
1198714119875119887119895(119883119887119895 119905 119889119881
119887119895119889119905 119889
21198811198871198951198891199052)
119864119887119895119868119887119895Φ
(6)
The uniform cable elongation 119890119888119894(120591) in (4) which results from
the static condensation procedure instantaneously dependson both the beam tip deflection and the cable transversedisplacement through the integral form [29]
119890119888119894(120591) =
120581V119888119894(119897119888119894 120591)
119897119888119894
tan 120579119894
+1205812
119897119888119894
int
119897119888119894
0
(1199101015840
119888119894V1015840119888119894+1
2V10158402119888119894)119889119909119888119894
(7)
The functions V119888119894and V
119887119895satisfy the following geometric
boundary conditions
119860 V1198871(minus1198971198871 120591) = 0
V101584010158401198871(minus1198971198871 120591) = 0
(8)
4 Shock and Vibration
119861 V1198877(1198971198877 120591) = 0
V101584010158401198877(1198971198877 120591) = 0
(9)
119862 V1198881(0 120591) = 0
V1198882(0 120591) = 0
(10)
119863 V1198883(0 120591) = 0
V1198884(0 120591) = 0
(11)
The matching conditions at the junctions 119878119896 where 119896 =
1 3 4 6 which involve cables 119888119894 where 119894 = 1 2 3 4 respect-
ively are
V119887119896(119897119887119896 120591) = V
119887119896+1(minus119897119887119896+1 120591) (12)
V1015840119887119896(119897119887119896 120591) = V1015840
119887119896+1(minus119897119887119896+1 120591) (13)
120578119896V10158401015840119887119896(119897119887119896 120591) = 120578
119896+1V10158401015840119887119896+1(minus119897119887119896+1 120591) (14)
120578119887119896V101584010158401015840119887119896(119897119887119896 120591) minus 120578
119887119896+1V101584010158401015840119887119896+1(minus119897119887119896+1 120591) =
120594119888119894120583119888119894
120581119890119888119894(120591)
sdot sin 120579119894
+ 120594119888119894[V1015840119888119894(119897119888119894 120591) + 120583
119888119894119890119888119894(120591) (V1015840119888119894(119897119888119894 120591) + 119910
1015840
119888119894(119897119888119894))]
sdot cos 120579119894
(15)
The matching conditions at the junctions 119878119896(119896 = 2 5) with
the roller supports are
V119887119896(119897119887119896 120591) = 0 (16)
V119887119896+1(minus119897119887119896+1 120591) = 0 (17)
V1015840119887119896(119897119887119896 120591) = V1015840
119887119896+1(minus119897119887119896+1 120591) (18)
120578119887119896V10158401015840119887119896(119897119887119896 120591) = 120578
119887119896+1V10158401015840119887119896+1(minus119897119887119896+1 120591) (19)
Equations (4) and (5) with the boundary and matchingconditions in (8)ndash(19) describe the nonlinear forced vibrationof the cable-stayed bridge The equations governing thesmall amplitude vibration of the cable-stayed bridge can beobtained by linearizing (8) through (19) in the neighborhoodof the equilibrium configuration An extensive analysis ofthe free vibration of the cable-stayed bridge is presented in[29]
22 Modeling of the Vortex Shedding Force The distributedvortex shedding force on the deck beamcan bemodeled usingEhsan-Scanlanrsquos model [28]
119875(119883119887119895 119905 119881119887119895
119889119881119887119895
119889119905) =
1
21205881198802(2119863)
sdot [
[
1198841(119870)(1 minus 120576
1198812
119887119895
1198632)1
119880
119889119881119887119895
119889119905
+ 1198842(119870)
119881119887119895
119863+1
2119862119871(119870) sin (120596119905 + 120579)]
]
(20)
where 120588 is the air density 119880 is the mean wind speed 119870 =
120596119863119880 is the reduced frequency during VIV in which 120596 isthe frequency of the dynamic response of the cable-stayedbridge subjected to vortex shedding 120579 is the phase angleof the harmonic force due to vortex shedding and 119884
1(119870)
120576 1198842(119870) and 119862
119871(119870) are aeroelastic parameters that can be
determined throughwind-tunnel testsTheparameters1198841(119870)
and 120576 are related to linear and nonlinear components ofthe aerodynamic damping term respectively In particular120576 takes into account the fact that VIV is self-limiting Theparameter 119884
2(119870) represents the aerodynamic stiffness term
The parameter 119862119871(119870) is related to the amplitude of the
harmonic force due to vortex shedding According to [27]the second and third terms on the right-hand side of (20) havea negligible contribution to the response of the cable-stayedbridge at lock-in Hence at lock-in (20) can be reduced tothe following form
119875(119883119887119895 119905 119881119887119895
119889119881119887119895
119889119905)
=1
21205881198802(2119863)[
[
1198841(119870)(1 minus 120576
1198812
119887119895
1198632)1
119880
119889119881119887119895
119889119905
]
]
(21)
The nondimensional force 119901119887119895(119909119887119895 120591 V119887119895 V119887119895) in (5) can be
written as
119901119887119895(119909119887119895 120591 V119887119895 V119887119895) = 1205721198871198951198841(1 minus 120576120582
2
119887119895V2119887119895) V119887119895 (22)
where
120572119887119895(119909119887119895) =
1205960120588119880119863119871
4
119864119887119895119868119887119895
120582119887119895=Φ
119863119887119895
(23)
Shock and Vibration 5
3 Solution Method
Galerkin method is used to analyze the vibration of thecable-stayed bridge The dynamic response of stay cables andsegments of the deck beam are expressed by
V119888119894(120591 119909119888119894) =
119873
sum
119904=1
119902119904(120591) 120601(119904)
119888119894(119909119888119894) (24)
V119887119895(120591 119909119887119895) =
119873
sum
119904=1
119902119904(120591) 120601(119904)
119887119895(119909119887119895) (25)
where 120601(119904)119888119894
and 120601(119904)119887119895
are corresponding components of the119904th eigenfunction of the linearized cable-stayed bridge model[29] and 119902
119904(120591) are generalized coordinates Substituting (24)
into (4) yields
1205732
119888119894(
119873
sum
119904=1
119904120601(119904)
119888119894) + 120585119888119894(
119873
sum
119904=1
119904120601(119904)
119888119894) minus (
119873
sum
119904=1
119902119904120601(119904)
119888119894
10158401015840
)
minus120583119888119894120581
119897119888119894
[
[
tan 120579119894(
119873
sum
119904=1
119902119904120601(119904)
119888119894(119897119888119894))
+ 120581int
119897119888119894
0
1199101015840
119888119894(
119873
sum
119904=1
119902119904120601(119904)
119888119894
1015840
)119889119909119888119894
+120581
2int
119897119888119894
0
(
119873
sum
119904=1
119902119904120601(119904)
119888119894
1015840
)
2
119889119909119888119894
]
]
[(
119873
sum
119904=1
119902119904120601(119904)
119888119894
10158401015840
) + 11991010158401015840
119888119894]
= 0
(26)
Substituting (25) into (5) yields
1205734
119887119895(
119873
sum
119904=1
119904120601(119904)
119887119895) + 120585119887119895(
119873
sum
119904=1
119904120601(119904)
119887119895) + (
119873
sum
119904=1
119902119904120601(119904)
119887119895
1015840101584010158401015840
)
= 1205721198871198951198841(1 minus 120576120582
2
119887119895(
119873
sum
119904=1
119902119904120601(119904)
119887119895)
2
)(
119873
sum
119904=1
119904120601(119904)
119887119895)
(27)
Multiplying (26) by 120594119888119894120601(119903)
119888119894(119909119888119894) and integrating the resulting
equation with respect to 119909119888119894from 0 to 119897
119888119894 multiplying (27)
by 120578119887119895120601(119903)
119887119895(119909119887119895) and integrating the resulting equation with
respect to 119909119887119895from minus119897
119887119895to 119897119887119895 adding all the resulting equa-
tions andusing the following orthogonality relations of eigen-functions of the linearized cable-stayed bridge model [29]
4
sum
119894=1
1205941198881198941205732
119888119894int
119897119888119894
0
120601(119903)
119888119894(119909119888119894) 120601(119904)
119888119894(119909119888119894) 119889119909119888119894+
7
sum
119895=1
1205781198871198951205734
119887119895
sdot int
119897119887119895
minus119897119887119895
120601(119903)
119887119895(119909119887119895) 120601(119904)
119887119895(119909119887119895) 119889119909119887119895= 119872119904120575119904119903
minus
4
sum
119894=1
120594119888119894[int
119897119888119894
0
120601(119903)
119888119894(119909119888119894) 120601(119904)
119888119894
10158401015840
(119909119888119894) 119889119909119888119894
minus 8120583119888119894(119904)
119888119894
119889119888119894
1198972119888119894
int
119897119888119894
0
120601(119903)
119888119894(119909119888119894) 119889119909119888119894] +
7
sum
119895=1
120578119887119895
sdot int
119897119887119895
minus119897119887119895
120601(119903)
119887119895(119909119887119895) 120601(119904)
119887119895
1015840101584010158401015840
(119909119887119895) 119889119909119887119895= 1205962
119904119872119904120575119904119903
(28)
in which (119904)119888119894= 120581(120601
(119904)
119888119894(119897119888119894)119897119888119894) tan 120579
119894+ 1205812(1119897119888119894) int119897119888119894
01199101015840
1198881198941206011015840(119904)
119888119894119889119909119888119894
120596119904is the 119904th natural frequency of the linearized cable-
stayed bridge model 119872119904are positive constants and 120575
119904119903is
the Kronecker delta one can obtain spatially discretizedequations of the cable-stayed bridge
Mq + [C + CAero] q + Kq + NQ
+ NC+ NHy
= 0 (29)
where entries of thematricesMCCAeroKNQNC andNHy
are
119872119903119904= 119872119904120575119904119903
119862119903119904=
4
sum
119894=1
120594119888119894120585119888119894int
119897119888119894
0
120601(119904)
119888119894120601(119903)
119888119894119889119909119888119894+
7
sum
119895=1
120578119887119895120585119887119895int
119897119887119895
minus119897119887119895
120601(119904)
119887119895120601(119903)
119887119895119889119909119887119895
119862Aero119903119904
= minus
7
sum
119895=1
1205781198871198951198841int
119897119887119895
minus119897119887119895
120572119887119895120601(119904)
119887119895120601(119903)
119887119895119889119909119887119895
119870119903119904= 1205962
119904119872119903119904
119873Q119903=
4
sum
119894=1
119873
sum
119898=1
119873
sum
119899=1
120594119888119894120583119888119894119902119898119902119899(41198891198881198941205812
1198973119888119894
int
119897119888119894
0
120601(119898)
119888119894
1015840
120601(119899)
119888119894
1015840
119889119909119888119894int
119897119888119894
0
120601(119903)
119888119894119889119909119888119894minus (119898)
119888119894int
119897119888119894
0
120601(119903)
119888119894120601(119899)
119888119894
10158401015840
119889119909119888119894)
119873C119903= minus1205812
4
sum
119894=1
119873
sum
119898=1
119873
sum
119899=1
119873
sum
119900=1
120594119888119894120583119888119894
2119897119888119894
119902119898119902119899119902119900int
119897119888119894
0
120601(119898)
119888119894
1015840
120601(119899)
119888119894
1015840
119889119909119888119894int
119897119888119894
0
120601(119903)
119888119894120601(119900)
119888119894
10158401015840
119889119909119888119894
119873Hy119903=
7
sum
119895=1
119873
sum
119898=1
119873
sum
119899=1
119873
sum
119900=1
12057811988711989511988411205761205822
119887119895119902119898119902119899119900int
119897119887119895
minus119897119887119895
120572119887119895120601(119898)
119887119895120601(119899)
119887119895120601(119900)
119887119895120601(119903)
119887119895119889119909119887119895
(30)
respectively
6 Shock and Vibration
615
62
625
63
Am
plitu
de (m
m)
5 10 15 20 25 300Number of truncation terms N
Figure 2 Effects of the number of Galerkin truncation terms on transverse displacements of the midpoint of the deck beam when there areno geometric nonlinear terms
It should be noted that nonlinear termsNQ andNC in (29)are induced by geometric nonlinearities of the stay cables
4 Numerical Simulation
Geometric and physical parameters of a cable-stayed bridgeand aeroelastic parameters are listed in Table 1 It shouldbe noted that a width-to-height ratio of 4 is used in thispaper not only because it is a typical ratio of bridge decksbut also because the cable-stayed bridge is supposed tohave significant response amplitudes at lock-in [28] Forall the following calculation modal damping ratio 119862
119903119904is
always equal to 001120575119903119904 The dynamic response of the cable-
stayed bridge can be calculated from (29) using Runge-Kutta-Felhbergmethod inMATLABwhere initial conditionsof generalized coordinates can be obtained from those ofphysical coordinates
119902119903(0) =
1
119872119903
[
[
4
sum
119894=1
1205941198881198941205732
119888119894int
119897119888119894
0
V119888119894(119909119888119894 0) 120601(119903)
119888119894(119909119888119894) 119889119909119888119894
+
7
sum
119895=1
1205781198871198951205734
119887119895int
119897119887119895
minus119897119887119895
V119887119895(119909119887119895 0) 120601(119903)
119887119895(119909119887119895) 119889119909119887119895
]
]
119903(0) =
1
119872119903
[
[
4
sum
119894=1
1205941198881198941205732
119888119894int
119897119888119894
0
V119888119894(119909119888119894 0) 120601(119903)
119888119894(119909119888119894) 119889119909119888119894
+
7
sum
119895=1
1205781198871198951205734
119887119895int
119897119887119895
minus119897119887119895
V119887119895(119909119887119895 0) 120601(119903)
119887119895(119909119887119895) 119889119909119887119895
]
]
(31)
For all the following calculation 119903(0) is always equal to
zero Numerical simulations for two cases are undertaken (1)neglecting the geometric nonlinear terms (ie NQ
= NC= 0
in (29)) and (2) considering the geometric nonlinear terms
41 Case Studies Neglecting the Geometric Nonlinear TermsConvergence of Galerkin method for given initial conditions119902119903(0) = 01120575
1199031 which means that the cable-stayed bridge
has its initial dynamic configuration corresponding to itsfirst mode shape and it is released from rest is shown in
Figures 2 and 3The amplitudes of the steady-state transversedisplacements of the midpoint of the deck beam withdifferent numbers of Galerkin truncation terms which areset to one through 30 are shown in Figure 2 As shown inFigure 2 the transverse displacements of the midpoint of thedeck beam initially remain stable until the truncation numberincreases to 15 and suddenly increase to their convergedvalues A truncation with 20 terms can provide accurateresults in analyzing the dynamic response of the deck beamThe difference between the amplitudes of two limit cyclesobtained by truncation with one term and 20 terms is194 which is small One can come to a conclusion that atruncation with one term is accurate enough for calculationof the steady-state response of the cable-stayed bridge whenits initial dynamic configuration corresponds to its first modeshape In other words most of the energy of the cable-stayedbridge is concentrated in its first mode in this case Phaseportraits of the response of the midpoint of the deck beamfor different numbers of Galerkin truncation terms whichis shown in Figure 3 also lead to the same conclusion Otherresults that are not shown here for the sake of brevity indicatethat Galerkin truncation with 20 terms yields accurateresults for the dynamic response of the cable-stayed bridgein the following cases in this section Hence in the followingnumerical calculations in this section the first 20 modes ofthe linearized undamped cable-stayed bridge model are usedin Galerkin method Time history responses of the midpointof the deck beam when the initial dynamic configuration ofthe cable-stayed bridge corresponds to its first mode shapebut has different amplitudes that is 119902
1(0) = 01 (the initial
displacement of themidpoint of the deck beam is 32mm) and1199021(0) = 3 (the initial displacement of themidpoint of the deck
beam is 946mm which is rather large) are shown in Figures4 and 5 respectively It can be seen that the solutions with dif-ferent 119902
1(0) converge to the same limit cycle after long-time
integrationThemagnitudes of Floquet multipliers are all lessthan unity the aforementioned limit cycle is asymptoticallystable
Solutions of a reduced-order model for a flow dynamicsystem can converge to a spurious limit cycle after long-timeintegration even if it is initializedwith a correct configuration
Shock and Vibration 7
Table 1 Geometric and physical parameters of the cable-stayed bridge and aeroelastic parameters
Parameter Unit Value
Deck beam
Mass per unit length of the deck beam (119898119887119895) kgm 16940
Elastic modulus of the deck beam (119864119887119895) Nm2 20 times 1011
Area moment of inertia of the deck beam (119868119887119895) m4 120
Length of segment 1198871of the deck beam (2119871
1198871) m 35
Length of segment 1198872of the deck beam (2119871
1198872) m 40
Length of segment 1198873of the deck beam (2119871
1198873) m 50
Length of segment 1198874of the deck beam (2119871
1198874) m 50
Length of segment 1198875of the deck beam (2119871
1198875) m 50
Length of segment 1198876of the deck beam (2119871
1198876) m 40
Length of segment 1198877of the deck beam (2119871
1198877) m 35
Stay cables
Mass per unit length of the stay cables (119898119888119894) kgm 286
Elastic modulus of the stay cables (119864119888119894) Nm2 20 times 1011
Cross-sectional area of the stay cables (119860119888119894) m2 00362
Length of stay cable 1198881(1198711198881) m 52
Length of stay cable 1198882(1198711198882) m 60
Length of stay cable 1198883(1198711198883) m 60
Length of stay cable 1198884(1198711198884) m 52
Sag-to-span ratios of the stay cables (119889119888119894= 119863119888119894119871119888119894) 001
Aeroelastic parametersAir density (120588) kgm3 1205120576 11225 [28]1198841
688 [28]
N = 1
N = 19
N = 20
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus100
minus50
0
50
100
Figure 3 Phase portraits of the response of themidpoint of the deck beam for different numbers of Galerkin truncation terms 119902119903(0) = 01120575
1119903
[31 32] Sirisup and Karniadakis [32] demonstrated that theonset of divergence from the correct limit cycle dependson the number of Galerkin truncation terms the Reynoldsnumber and the flow geometry Since the limit cycle heredoes not vanish even when the truncation number is 30which is large one can conclude that the aforementionedlimit cycle for the cable-stayed bridge is a correct one
In many cases higher mode shapes of the cable-stayedbridge would be excited one such case is that when thereare vehicles moving on the deck beam of the bridge Hence
the initial dynamic configuration of the cable-stayed bridgecan correspond to its higher mode shapes in the dynamicanalysis of the cable-stayed bridge subjected to a distributedvortex shedding force It is obvious that Galerkin truncationwith one term is not enough in these cases Through thesame method with 119873 = 20 one can find that there isthe same limit cycle as that in Figure 3 when the initialdynamic configuration corresponds to the second throughsixth mode shapes and there is a different limit cycle whenthe initial dynamic configuration corresponds to the seventh
8 Shock and Vibration
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus80
minus40
0
40
80
Vb 4(X
b 4=0)
(mm
)
Figure 4 Time response of the midpoint of the deck beam 119902119903(0) = 01120575
1119903
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus800
minus400
0
400
800
Vb 4(X
b 4=0)
(mm
)
Figure 5 Time response of the midpoint of the deck beam 119902119903(0) = 3120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 6 Time response of the midpoint of the deck beam 119902119903(0) = 01120575
7119903
mode shape (Figures 6 and 7) Two-dimensional projectionsof phase portraits onto the (119902
119903 119903) plane when 119902
119903(0) = 01120575
1119903
and 119902119903(0) = 01120575
7119903are shown in Figures 8 and 9 respectively
Results when 119902119903(0) is equal to 01120575
2119903 011205753119903 and 01120575
6119903
are the same as that when 119902119903(0) = 01120575
1119903 they are not shown
here for the sake of brevity It can be seen from Figures 8 and9 that energy of the cable-stayed bridge is concentrated inthe first mode (the seventh mode) when its initial dynamic
configuration corresponds to the first through sixth modeshapes (the seventh mode shape)
42 Case Studies Considering the Geometric Nonlinear TermsConvergence of Galerkin method for given initial conditions119902119903(0) = 01120575
1199031is shown in Figures 10 and 11 The amplitudes
of the steady-state transverse displacements of the mid-point of the deck beam with different numbers of Galerkin
Shock and Vibration 9
minus150
minus100
minus50
0
50
100
150
0 50 100minus50minus100
qr(0) = 011205751rqr(0) = 011205757r
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 7 Two different stable limit cycles corresponding to different initial configurations
q1
minus05
0
05
0 05minus05
q1
q2
minus05
0
05
0 05minus05
q2
minus05
0
05
q3
0 05minus05
q3
q4
minus05
0
05
0 05minus05
q4
q5
minus05
0
05
0 05minus05
q5
q6
minus05
0
05
0 05minus05
q6
q7
minus05
0
05
0 05minus05
q7
q8
minus05
0
05
0 05minus05
q8
q9
minus05
0
05
0 05minus05
q9
Figure 8 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903
truncation terms which are set to one through 30 are shownin Figure 10 As shown in Figure 10 a truncation with 11terms can provide accurate results in analyzing the dynamicresponse of the cable-stayed bridge and it is much lessthan the cases in Section 41 where the geometric nonlinearterms are neglected Phase portraits of the response of themidpoint of the deck beam for different numbers of Galerkin
truncation terms which are shown in Figure 11 also leadto the same conclusion It can be noted from Figure 11 thatthe limit cycle becomes more asymmetric when the numberof the truncation terms increases The reason for this isthat when the number of truncation terms increases to 7an additional asymmetric limit cycle appears on the (119902
7 7)
plane which phase portraits project onto (see Figure 12 with
10 Shock and Vibration
q1
minus1
0
1
0 03minus03
q1
q2
minus1
0
1
0 03minus03
q2
q3
minus1
0
1
0 03minus03
q3
q4
minus1
0
1
0 03minus03
q4
q5
minus1
0
1
0 03minus03
q5
q6
minus1
0
1
0 03minus03
q6
q7
minus1
0
1
0 03minus03
q7
q8
minus1
0
1
0 03minus03
q8
q9
minus1
0
1
0 03minus03
q9
Figure 9 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
7119903
5 10 15 20 25 300Number of truncation terms N
58
59
60
61
62
63
Am
plitu
de (m
m)
Figure 10 Effects of the number of Galerkin truncation terms ontransverse displacements of the midpoint of the deck beam whenthere are geometric nonlinear terms
119873 = 11) As shown in Figures 11 and 13 the dynamic responseof the midpoint of the deck beam approaches a stable limitcycle The bifurcation diagram of limit cycles with respectto 1199021(0) is shown in Figure 14 when the initial dynamic
configuration of the cable-stayed bridge corresponds to itsfirst mode shape One can find from Figure 14 that when1199021(0) is smaller than 027 a stable periodic solution exists and
minus50 0 50 100minus100minus100
minus50
0
50
100
N = 1
N = 10
N = 11
N = 30
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 11 Phase portraits of the response of themidpoint of the deckbeam for different numbers of Galerkin truncation terms 119902
119903(0) =
011205751119903
when 1199021(0) is larger than or equal to 027 chaotic responses
occur For instance when 1199021(0) = 03 and 119873 = 11 phase
portraits of the response of the midpoint of the deck beamare shown in Figure 15 which indicates occurrence of chaotic
Shock and Vibration 11
q1
minus50
0
50
0 50minus50
q1
q2
minus20
0
20
0 10minus10
q2
q3
minus20
0
20
0 10minus10
q3
q4
minus20
0
20
0 10minus10
q4
q5
minus20
0
20
0 10minus10
minus10
q5
q6
minus20
0
20
0 10minus10
q6
q7
minus20
0
20
0 10minus10
q7
q8
minus20
0
20
100
q8
q9
minus20
0
20
0 10minus10
q9
Figure 12 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903and119873 = 11
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 13 Time response of the midpoint of the deck beam 119902119903(0) =
011205751119903and119873 = 11
response On the contrary for a truncation with one termthe response always approaches a stable limit cycle no matterhow large the amplitude of the initial dynamic configurationis (see Figures 16ndash18) These mean that the true dynamicresponse of the cable-stayed bridge may not be capturedby the truncation with one term even when the initialdynamic configuration corresponds to the first mode shape
01 02 03 04 05 06 070q1(0)
minus200
minus150
minus100
minus50
0
50
Vb 4(X
b 4=0)
(mm
)
Figure 14 Bifurcation diagram of the cable-stayed bridge withrespect to 119902
1(0) 119902119903(0) = 119902
1(0)1205751119903and119873 = 11
When the initial dynamic configuration of the cable-stayedbridge corresponds to one of its higher mode shapes chaoticresponse occurs even if its amplitude is relatively small this isshown in Figure 19where the initial dynamic configuration ofthe cable-stayed bridge corresponds to its secondmode shapeand its magnitude is only 01
12 Shock and Vibration
0
minus100 0 100 200 300minus200
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus1500
minus1000
minus500
500
1000
1500
Figure 15 Chaotic response of the midpoint of the deck beam119902119903(0) = 03120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 16 Time response of themidpoint of the deck beam 1199021(0) =
01 and119873 = 1
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus1000
minus500
0
500
1000
Vb 4(X
b 4=0)
(mm
)
Figure 17 Time response of the midpoint of the deck beam 1199021(0) =
3 and119873 = 1
5 Conclusions
The dynamic behavior of a cable-stayed bridge that consistsof a simply supported four-cable-stayed deck beam andtwo rigid towers subjected to a distributed vortex shed-ding force on the deck beam has been investigated The
q1(0) = 01
q1(0) = 3
minus100
minus50
0
50
100
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 18 Phase portraits of the response of the midpoint of thedeck beam for different initial conditions119873 = 1
minus1500
minus1000
minus500
0
500
1000
1500
minus200 0 200 400minus400
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 19 Chaotic response of the midpoint of the deck beam119902119903(0) = 01120575
2119903
dynamic response of the cable-stayed bridge is calculatedusing Galerkin method in conjunction with Runge-Kutta-Felhberg method in MATLAB Convergence of Galerkinmethod for the dynamic response of the cable-stayed bridgeis studied Numerical simulations show that the geometricnonlinearities of the stay cables have significant influence onVIV of the cable-stayed bridge and further conclusions canbe summarized as follows
(1) In the case when the geometric nonlinear termsare neglected accurate calculation of the responseamplitude of the cable-stayed bridge at lock-in onlyneeds use of the first mode shape of the linearizedundamped cable-stayed bridgemodel when the initialdynamic configuration of the cable-stayed bridgecorresponds to its mode shape whose mode numberis smaller than seven There is a different limit cyclewhen the initial dynamic configuration correspondsto its mode shape whose mode number is equal to orlarger than 7
Shock and Vibration 13
(2) In the case when the geometric nonlinear termsare considered calculation of the response of thecable-stayed bridge generally needs use of multiplemode shapes of the linearized undamped cable-stayedbridge model even when the initial dynamic config-uration of the cable-stayed bridge corresponds to itsfirstmode shapeThere is a limit cycle when the initialdynamic configuration of the cable-stayed bridgecorresponds to its first mode shape and its amplitudeis smaller than 03 for the generalized coordinate andthere is chaotic response when the initial dynamicconfiguration of the cable-stayed bridge correspondsto its first mode shape with its amplitude larger than03 for the generalized coordinate or one of its highermode shapes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grant nos 11302087 and 11442006 theNatural Science Foundation of Jiangsu Province under Grantno BK20130479 the Research Foundation for AdvancedTalents of Jiangsu University under Grant no 13JDG068 andthe National Science Foundation under Grant no CMMI-1000830
References
[1] K Y R Billah and R H Scanlan ldquoResonance Tacoma NarrowsBridge failure and undergraduate physics textbooksrdquoAmericanJournal of Physics vol 59 no 2 pp 118ndash124 1991
[2] R D Blevins Flow-Induced Vibration Krieger New York NYUSA 2nd edition 1994
[3] H Zheng R E Price Y Modarres-Sadeghi and M S Tri-antafyllou ldquoOn fatigue damage of long flexible cylinders dueto the higher harmonic force components and chaotic vortex-induced vibrationsrdquo Ocean Engineering vol 88 pp 318ndash3292014
[4] C Evangelinos D Lucor and G E Karniadakis ldquoDNS-derivedforce distribution on flexible cylinders subject to vortex-induced vibrationrdquo Journal of Fluids and Structures vol 14 no3 pp 429ndash440 2000
[5] HM Blackburn R N Govardhan andCH KWilliamson ldquoAcomplementary numerical and physical investigation of vortex-inducedrdquo Journal of Fluids and Structures vol 15 no 3-4 pp481ndash488 2001
[6] R H J Willden and J M R Graham ldquoNumerical predictionof VIV on long flexible circular cylindersrdquo Journal of Fluids andStructures vol 15 no 3-4 pp 659ndash669 2001
[7] E Guilmineau and P Queutey ldquoA numerical simulation ofvortex shedding from an oscillating circular cylinderrdquo Journalof Fluids and Structures vol 16 no 6 pp 773ndash794 2002
[8] I Jadic R M C So and M P Mignolet ldquoAnalysis of fluidmdashstructure interactions using a time-marching techniquerdquo Jour-nal of Fluids and Structures vol 12 no 6 pp 631ndash654 1998
[9] C Y Zhou RM C So andK Lam ldquoVortex-induced vibrationsof an elastic circular cylinderrdquo Journal of Fluids and Structuresvol 13 no 2 pp 165ndash189 1999
[10] J R Meneghini and P W Bearman ldquoNumerical simulationof high amplitude oscillatory flow about a circular cylinderrdquoJournal of Fluids and Structures vol 9 no 4 pp 435ndash455 1995
[11] T Sarpkaya ldquoComputational methods with vorticesrdquo Journal ofFluids Engineering vol 111 no 1 pp 5ndash52 1989
[12] T Sarpkaya ldquoVortex element methods for flow simulationrdquoAdvances in Applied Mechanics vol 31 pp 113ndash247 1994
[13] R D Gabbai and H Benaroya ldquoAn overview of modeling andexperiments of vortex-induced vibration of circular cylindersrdquoJournal of Sound and Vibration vol 282 no 3ndash5 pp 575ndash6162005
[14] R H Scanlan ldquoOn the state-of-the-artmethods for calculationsof flutter vortex-induced and buffeting response of bridgestructuresrdquo Tech Rep FHWARD-80050 National TechnicalInformation Service Springfield Va USA 1981
[15] RH Scanlan ldquoBridge flutter derivatives at vortex lock-inrdquo Jour-nal of Structural Engineering vol 124 no 4 pp 450ndash458 1998
[16] B J Vickery and R I Basu ldquoAcross-wind vibrations of struc-tures of circular cross-section Part I Development of a math-ematical model for two-dimensional conditionsrdquo Journal ofWind Engineering and Industrial Aerodynamics vol 12 no 1 pp49ndash73 1983
[17] A Larsen ldquoA generalized model for assessment of vortex-induced vibrations of flexible structuresrdquo Journal of Wind Engi-neering and Industrial Aerodynamics vol 57 no 2-3 pp 281ndash294 1995
[18] T Sarpkaya ldquoFluid forces on oscillating cylindersrdquo Journal ofWaterway Port Coastal and Ocean Division (ASCE) vol 104no 3 pp 275ndash290 1978
[19] T Staubli ldquoCalculation of vibration of an elastically mountedcylinder using experimental data from a forced oscillationrdquoJournal of Fluids Engineering vol 105 no 2 pp 225ndash229 1983
[20] W D Iwan and D L R Botelho ldquoVortex-induced oscillationof structures in waterrdquo Journal of Waterway Port Coastal andOcean Engineering vol 111 no 2 pp 289ndash303 1985
[21] R T Hartlen and I G Currie ldquoLift-oscillator model of vortex-induced vibrationrdquo Journal of Engineering Mechanics Division(ASCE) vol 96 no 5 pp 577ndash591 1970
[22] R A Skop and O M Griffin ldquoA model for the vortex-excitedresonant response of bluff cylindersrdquo Journal of Sound andVibration vol 27 no 2 pp 225ndash233 1973
[23] WD Iwan andRD Blevins ldquoAmodel for vortex induced oscil-lation of structuresrdquo Journal of Applied Mechanics TransactionsASME vol 41 no 3 pp 581ndash586 1974
[24] E H Dowell ldquoNon-linear oscillator models in bluff body aero-elasticityrdquo Journal of Sound and Vibration vol 75 no 2 pp 251ndash264 1981
[25] R H M Ogink and A V Metrikine ldquoA wake oscillator withfrequency dependent coupling for the modeling of vortex-induced vibrationrdquo Journal of Sound and Vibration vol 329 no26 pp 5452ndash5473 2010
[26] X Bai and W Qin ldquoUsing vortex strength wake oscillatorin modelling of vortex induced vibrations in two degrees offreedomrdquo European Journal of Mechanics BFluids vol 48 pp165ndash173 2014
[27] F Ehsan and R H Scanlan ldquoVortexminusinduced vibrations offlexible bridgesrdquo Journal of Engineering Mechanics vol 116 no6 pp 1392ndash1411 1990
14 Shock and Vibration
[28] A M Marra C Mannini and G Bartoli ldquoVan der Pol-typeequation for modeling vortex-induced oscillations of bridgedecksrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 99 no 6-7 pp 776ndash785 2011
[29] D Q Cao M T Song W D Zhu R W Tucker and C H-T Wang ldquoModeling and analysis of the in-plane vibration ofa complex cable-stayed bridgerdquo Journal of Sound and Vibrationvol 331 no 26 pp 5685ndash5714 2012
[30] E Caetano A Cunha V Gattulli and M Lepidi ldquoCable-deckdynamic interactions at the International Guadiana Bridge on-site measurements and finite element modellingrdquo StructuralControl and Health Monitoring vol 15 no 3 pp 237ndash264 2008
[31] I Akhtar A H Nayfeh and C J Ribbens ldquoOn the stability andextension of reduced-order Galerkin models in incompressibleflowsrdquo Theoretical and Computational Fluid Dynamics vol 23no 3 pp 213ndash237 2009
[32] S Sirisup and G E Karniadakis ldquoA spectral viscosity methodfor correcting the long-term behavior of POD modelsrdquo Journalof Computational Physics vol 194 no 1 pp 92ndash116 2004
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 3
119897119888119894=119871119888119894
119871
119897119887119895=
119871119887119895
119871
119889119888119894=119863119888119894
Φ
119910119888119894=119884119888119894
Φ
119906119888119894=119880119888119894
Φ
V119888119894=119881119888119894
Φ
V119887119895=
119881119887119895
Φ
120591 = 1205960119905
(1)
where 119871 = min1198711198871 1198711198872 119871
1198877 1205960= (1119871
2
1198871)radic1198641198871
11986811988711198981198871
and Φ is the diameter of stay cable 1198881 Some additional
nondimensional parameters need to be introduced to furnisha complete definition of elastodynamic properties of thecable-stayed bridge model
120583119888119894=119864119888119894119860119888119894
119867119888119894
120594119888119894=
71198712119867119888119894
sum7
119895=1119864119887119895119868119887119895
120578119887119895=
7119864119887119895119868119887119895
sum7
119895=1119864119887119895119868119887119895
120581 =Φ
119871
(2)
where 119867119888119894is the tension in the stay cable 119888
119894on which its
initial sag is dependent that is 119863119888119894= 1198981198881198941198921198712
119888119894cos 1205791198948119867119888119894
in which 119892 is the acceleration of gravity Since deck beamand cable materials of the cable-stayed bridge can generallybe assumed to have different viscous damping behaviorstransverse damping coefficients of cable 119888
119894and segment 119887
119895of
the deck beam are denoted by 119862119888119894and 119862
119887119895 respectively and
their nondimensional parameters are defined by
120585119888119894=11986211988811989412059601198712
119867119888119894
120585119887119895=
11986211988711989512059601198714
119864119887119895119868119887119895
(3)
respectively
The Newtonian method is used here to derive nonlinearequations of motion of the cable-stayed bridge model and afull set of geometric and dynamic boundary and matchingconditions Assuming that cable longitudinal inertial forces(119898119888119894119888119894) are negligible in the prevalent low-frequency trans-
verse vibration of the cable-stayed bridge the longitudinalcable displacement119880
119888119894can be statically condensed leading to
coupled nonlinear equations in terms of only the transversalcable and deck beam displacements 119881
119888119894and 119881
119887119895 respectively
The equations of motion of the cable-stayed bridge are [29]
1205732
119888119894V119888119894+ 120585119888119894V119888119894minus V10158401015840119888119894minus 120583119888119894119890119888119894(120591) (V10158401015840119888119894+ 11991010158401015840
119888119894) = 0
119909119888119894isin [0 119897
119888119894]
(4)
1205734
119887119895V119887119895+ 120585119887119895V119887119895+ V1015840101584010158401015840119887119895= 119901119887119895(119909119887119895 120591 V119887119895 V119887119895)
119909119887119895isin [minus119897119887119895 119897119887119895]
(5)
where a prime and dot denote differentiation with respect tonondimensional local abscissae 119909
119888119894and 119909
119887119895and the time 120591
respectively
120573119888119894= 1198711205960(119898119888119894
119867119888119894
)
12
120573119887119895= 119871(
1198981198871198951205962
0
119864119887119895119868119887119895
)
14
119901119887119895(119909119887119895 120591 V119887119895 V119887119895)
=
1198714119875119887119895(119883119887119895 119905 119889119881
119887119895119889119905 119889
21198811198871198951198891199052)
119864119887119895119868119887119895Φ
(6)
The uniform cable elongation 119890119888119894(120591) in (4) which results from
the static condensation procedure instantaneously dependson both the beam tip deflection and the cable transversedisplacement through the integral form [29]
119890119888119894(120591) =
120581V119888119894(119897119888119894 120591)
119897119888119894
tan 120579119894
+1205812
119897119888119894
int
119897119888119894
0
(1199101015840
119888119894V1015840119888119894+1
2V10158402119888119894)119889119909119888119894
(7)
The functions V119888119894and V
119887119895satisfy the following geometric
boundary conditions
119860 V1198871(minus1198971198871 120591) = 0
V101584010158401198871(minus1198971198871 120591) = 0
(8)
4 Shock and Vibration
119861 V1198877(1198971198877 120591) = 0
V101584010158401198877(1198971198877 120591) = 0
(9)
119862 V1198881(0 120591) = 0
V1198882(0 120591) = 0
(10)
119863 V1198883(0 120591) = 0
V1198884(0 120591) = 0
(11)
The matching conditions at the junctions 119878119896 where 119896 =
1 3 4 6 which involve cables 119888119894 where 119894 = 1 2 3 4 respect-
ively are
V119887119896(119897119887119896 120591) = V
119887119896+1(minus119897119887119896+1 120591) (12)
V1015840119887119896(119897119887119896 120591) = V1015840
119887119896+1(minus119897119887119896+1 120591) (13)
120578119896V10158401015840119887119896(119897119887119896 120591) = 120578
119896+1V10158401015840119887119896+1(minus119897119887119896+1 120591) (14)
120578119887119896V101584010158401015840119887119896(119897119887119896 120591) minus 120578
119887119896+1V101584010158401015840119887119896+1(minus119897119887119896+1 120591) =
120594119888119894120583119888119894
120581119890119888119894(120591)
sdot sin 120579119894
+ 120594119888119894[V1015840119888119894(119897119888119894 120591) + 120583
119888119894119890119888119894(120591) (V1015840119888119894(119897119888119894 120591) + 119910
1015840
119888119894(119897119888119894))]
sdot cos 120579119894
(15)
The matching conditions at the junctions 119878119896(119896 = 2 5) with
the roller supports are
V119887119896(119897119887119896 120591) = 0 (16)
V119887119896+1(minus119897119887119896+1 120591) = 0 (17)
V1015840119887119896(119897119887119896 120591) = V1015840
119887119896+1(minus119897119887119896+1 120591) (18)
120578119887119896V10158401015840119887119896(119897119887119896 120591) = 120578
119887119896+1V10158401015840119887119896+1(minus119897119887119896+1 120591) (19)
Equations (4) and (5) with the boundary and matchingconditions in (8)ndash(19) describe the nonlinear forced vibrationof the cable-stayed bridge The equations governing thesmall amplitude vibration of the cable-stayed bridge can beobtained by linearizing (8) through (19) in the neighborhoodof the equilibrium configuration An extensive analysis ofthe free vibration of the cable-stayed bridge is presented in[29]
22 Modeling of the Vortex Shedding Force The distributedvortex shedding force on the deck beamcan bemodeled usingEhsan-Scanlanrsquos model [28]
119875(119883119887119895 119905 119881119887119895
119889119881119887119895
119889119905) =
1
21205881198802(2119863)
sdot [
[
1198841(119870)(1 minus 120576
1198812
119887119895
1198632)1
119880
119889119881119887119895
119889119905
+ 1198842(119870)
119881119887119895
119863+1
2119862119871(119870) sin (120596119905 + 120579)]
]
(20)
where 120588 is the air density 119880 is the mean wind speed 119870 =
120596119863119880 is the reduced frequency during VIV in which 120596 isthe frequency of the dynamic response of the cable-stayedbridge subjected to vortex shedding 120579 is the phase angleof the harmonic force due to vortex shedding and 119884
1(119870)
120576 1198842(119870) and 119862
119871(119870) are aeroelastic parameters that can be
determined throughwind-tunnel testsTheparameters1198841(119870)
and 120576 are related to linear and nonlinear components ofthe aerodynamic damping term respectively In particular120576 takes into account the fact that VIV is self-limiting Theparameter 119884
2(119870) represents the aerodynamic stiffness term
The parameter 119862119871(119870) is related to the amplitude of the
harmonic force due to vortex shedding According to [27]the second and third terms on the right-hand side of (20) havea negligible contribution to the response of the cable-stayedbridge at lock-in Hence at lock-in (20) can be reduced tothe following form
119875(119883119887119895 119905 119881119887119895
119889119881119887119895
119889119905)
=1
21205881198802(2119863)[
[
1198841(119870)(1 minus 120576
1198812
119887119895
1198632)1
119880
119889119881119887119895
119889119905
]
]
(21)
The nondimensional force 119901119887119895(119909119887119895 120591 V119887119895 V119887119895) in (5) can be
written as
119901119887119895(119909119887119895 120591 V119887119895 V119887119895) = 1205721198871198951198841(1 minus 120576120582
2
119887119895V2119887119895) V119887119895 (22)
where
120572119887119895(119909119887119895) =
1205960120588119880119863119871
4
119864119887119895119868119887119895
120582119887119895=Φ
119863119887119895
(23)
Shock and Vibration 5
3 Solution Method
Galerkin method is used to analyze the vibration of thecable-stayed bridge The dynamic response of stay cables andsegments of the deck beam are expressed by
V119888119894(120591 119909119888119894) =
119873
sum
119904=1
119902119904(120591) 120601(119904)
119888119894(119909119888119894) (24)
V119887119895(120591 119909119887119895) =
119873
sum
119904=1
119902119904(120591) 120601(119904)
119887119895(119909119887119895) (25)
where 120601(119904)119888119894
and 120601(119904)119887119895
are corresponding components of the119904th eigenfunction of the linearized cable-stayed bridge model[29] and 119902
119904(120591) are generalized coordinates Substituting (24)
into (4) yields
1205732
119888119894(
119873
sum
119904=1
119904120601(119904)
119888119894) + 120585119888119894(
119873
sum
119904=1
119904120601(119904)
119888119894) minus (
119873
sum
119904=1
119902119904120601(119904)
119888119894
10158401015840
)
minus120583119888119894120581
119897119888119894
[
[
tan 120579119894(
119873
sum
119904=1
119902119904120601(119904)
119888119894(119897119888119894))
+ 120581int
119897119888119894
0
1199101015840
119888119894(
119873
sum
119904=1
119902119904120601(119904)
119888119894
1015840
)119889119909119888119894
+120581
2int
119897119888119894
0
(
119873
sum
119904=1
119902119904120601(119904)
119888119894
1015840
)
2
119889119909119888119894
]
]
[(
119873
sum
119904=1
119902119904120601(119904)
119888119894
10158401015840
) + 11991010158401015840
119888119894]
= 0
(26)
Substituting (25) into (5) yields
1205734
119887119895(
119873
sum
119904=1
119904120601(119904)
119887119895) + 120585119887119895(
119873
sum
119904=1
119904120601(119904)
119887119895) + (
119873
sum
119904=1
119902119904120601(119904)
119887119895
1015840101584010158401015840
)
= 1205721198871198951198841(1 minus 120576120582
2
119887119895(
119873
sum
119904=1
119902119904120601(119904)
119887119895)
2
)(
119873
sum
119904=1
119904120601(119904)
119887119895)
(27)
Multiplying (26) by 120594119888119894120601(119903)
119888119894(119909119888119894) and integrating the resulting
equation with respect to 119909119888119894from 0 to 119897
119888119894 multiplying (27)
by 120578119887119895120601(119903)
119887119895(119909119887119895) and integrating the resulting equation with
respect to 119909119887119895from minus119897
119887119895to 119897119887119895 adding all the resulting equa-
tions andusing the following orthogonality relations of eigen-functions of the linearized cable-stayed bridge model [29]
4
sum
119894=1
1205941198881198941205732
119888119894int
119897119888119894
0
120601(119903)
119888119894(119909119888119894) 120601(119904)
119888119894(119909119888119894) 119889119909119888119894+
7
sum
119895=1
1205781198871198951205734
119887119895
sdot int
119897119887119895
minus119897119887119895
120601(119903)
119887119895(119909119887119895) 120601(119904)
119887119895(119909119887119895) 119889119909119887119895= 119872119904120575119904119903
minus
4
sum
119894=1
120594119888119894[int
119897119888119894
0
120601(119903)
119888119894(119909119888119894) 120601(119904)
119888119894
10158401015840
(119909119888119894) 119889119909119888119894
minus 8120583119888119894(119904)
119888119894
119889119888119894
1198972119888119894
int
119897119888119894
0
120601(119903)
119888119894(119909119888119894) 119889119909119888119894] +
7
sum
119895=1
120578119887119895
sdot int
119897119887119895
minus119897119887119895
120601(119903)
119887119895(119909119887119895) 120601(119904)
119887119895
1015840101584010158401015840
(119909119887119895) 119889119909119887119895= 1205962
119904119872119904120575119904119903
(28)
in which (119904)119888119894= 120581(120601
(119904)
119888119894(119897119888119894)119897119888119894) tan 120579
119894+ 1205812(1119897119888119894) int119897119888119894
01199101015840
1198881198941206011015840(119904)
119888119894119889119909119888119894
120596119904is the 119904th natural frequency of the linearized cable-
stayed bridge model 119872119904are positive constants and 120575
119904119903is
the Kronecker delta one can obtain spatially discretizedequations of the cable-stayed bridge
Mq + [C + CAero] q + Kq + NQ
+ NC+ NHy
= 0 (29)
where entries of thematricesMCCAeroKNQNC andNHy
are
119872119903119904= 119872119904120575119904119903
119862119903119904=
4
sum
119894=1
120594119888119894120585119888119894int
119897119888119894
0
120601(119904)
119888119894120601(119903)
119888119894119889119909119888119894+
7
sum
119895=1
120578119887119895120585119887119895int
119897119887119895
minus119897119887119895
120601(119904)
119887119895120601(119903)
119887119895119889119909119887119895
119862Aero119903119904
= minus
7
sum
119895=1
1205781198871198951198841int
119897119887119895
minus119897119887119895
120572119887119895120601(119904)
119887119895120601(119903)
119887119895119889119909119887119895
119870119903119904= 1205962
119904119872119903119904
119873Q119903=
4
sum
119894=1
119873
sum
119898=1
119873
sum
119899=1
120594119888119894120583119888119894119902119898119902119899(41198891198881198941205812
1198973119888119894
int
119897119888119894
0
120601(119898)
119888119894
1015840
120601(119899)
119888119894
1015840
119889119909119888119894int
119897119888119894
0
120601(119903)
119888119894119889119909119888119894minus (119898)
119888119894int
119897119888119894
0
120601(119903)
119888119894120601(119899)
119888119894
10158401015840
119889119909119888119894)
119873C119903= minus1205812
4
sum
119894=1
119873
sum
119898=1
119873
sum
119899=1
119873
sum
119900=1
120594119888119894120583119888119894
2119897119888119894
119902119898119902119899119902119900int
119897119888119894
0
120601(119898)
119888119894
1015840
120601(119899)
119888119894
1015840
119889119909119888119894int
119897119888119894
0
120601(119903)
119888119894120601(119900)
119888119894
10158401015840
119889119909119888119894
119873Hy119903=
7
sum
119895=1
119873
sum
119898=1
119873
sum
119899=1
119873
sum
119900=1
12057811988711989511988411205761205822
119887119895119902119898119902119899119900int
119897119887119895
minus119897119887119895
120572119887119895120601(119898)
119887119895120601(119899)
119887119895120601(119900)
119887119895120601(119903)
119887119895119889119909119887119895
(30)
respectively
6 Shock and Vibration
615
62
625
63
Am
plitu
de (m
m)
5 10 15 20 25 300Number of truncation terms N
Figure 2 Effects of the number of Galerkin truncation terms on transverse displacements of the midpoint of the deck beam when there areno geometric nonlinear terms
It should be noted that nonlinear termsNQ andNC in (29)are induced by geometric nonlinearities of the stay cables
4 Numerical Simulation
Geometric and physical parameters of a cable-stayed bridgeand aeroelastic parameters are listed in Table 1 It shouldbe noted that a width-to-height ratio of 4 is used in thispaper not only because it is a typical ratio of bridge decksbut also because the cable-stayed bridge is supposed tohave significant response amplitudes at lock-in [28] Forall the following calculation modal damping ratio 119862
119903119904is
always equal to 001120575119903119904 The dynamic response of the cable-
stayed bridge can be calculated from (29) using Runge-Kutta-Felhbergmethod inMATLABwhere initial conditionsof generalized coordinates can be obtained from those ofphysical coordinates
119902119903(0) =
1
119872119903
[
[
4
sum
119894=1
1205941198881198941205732
119888119894int
119897119888119894
0
V119888119894(119909119888119894 0) 120601(119903)
119888119894(119909119888119894) 119889119909119888119894
+
7
sum
119895=1
1205781198871198951205734
119887119895int
119897119887119895
minus119897119887119895
V119887119895(119909119887119895 0) 120601(119903)
119887119895(119909119887119895) 119889119909119887119895
]
]
119903(0) =
1
119872119903
[
[
4
sum
119894=1
1205941198881198941205732
119888119894int
119897119888119894
0
V119888119894(119909119888119894 0) 120601(119903)
119888119894(119909119888119894) 119889119909119888119894
+
7
sum
119895=1
1205781198871198951205734
119887119895int
119897119887119895
minus119897119887119895
V119887119895(119909119887119895 0) 120601(119903)
119887119895(119909119887119895) 119889119909119887119895
]
]
(31)
For all the following calculation 119903(0) is always equal to
zero Numerical simulations for two cases are undertaken (1)neglecting the geometric nonlinear terms (ie NQ
= NC= 0
in (29)) and (2) considering the geometric nonlinear terms
41 Case Studies Neglecting the Geometric Nonlinear TermsConvergence of Galerkin method for given initial conditions119902119903(0) = 01120575
1199031 which means that the cable-stayed bridge
has its initial dynamic configuration corresponding to itsfirst mode shape and it is released from rest is shown in
Figures 2 and 3The amplitudes of the steady-state transversedisplacements of the midpoint of the deck beam withdifferent numbers of Galerkin truncation terms which areset to one through 30 are shown in Figure 2 As shown inFigure 2 the transverse displacements of the midpoint of thedeck beam initially remain stable until the truncation numberincreases to 15 and suddenly increase to their convergedvalues A truncation with 20 terms can provide accurateresults in analyzing the dynamic response of the deck beamThe difference between the amplitudes of two limit cyclesobtained by truncation with one term and 20 terms is194 which is small One can come to a conclusion that atruncation with one term is accurate enough for calculationof the steady-state response of the cable-stayed bridge whenits initial dynamic configuration corresponds to its first modeshape In other words most of the energy of the cable-stayedbridge is concentrated in its first mode in this case Phaseportraits of the response of the midpoint of the deck beamfor different numbers of Galerkin truncation terms whichis shown in Figure 3 also lead to the same conclusion Otherresults that are not shown here for the sake of brevity indicatethat Galerkin truncation with 20 terms yields accurateresults for the dynamic response of the cable-stayed bridgein the following cases in this section Hence in the followingnumerical calculations in this section the first 20 modes ofthe linearized undamped cable-stayed bridge model are usedin Galerkin method Time history responses of the midpointof the deck beam when the initial dynamic configuration ofthe cable-stayed bridge corresponds to its first mode shapebut has different amplitudes that is 119902
1(0) = 01 (the initial
displacement of themidpoint of the deck beam is 32mm) and1199021(0) = 3 (the initial displacement of themidpoint of the deck
beam is 946mm which is rather large) are shown in Figures4 and 5 respectively It can be seen that the solutions with dif-ferent 119902
1(0) converge to the same limit cycle after long-time
integrationThemagnitudes of Floquet multipliers are all lessthan unity the aforementioned limit cycle is asymptoticallystable
Solutions of a reduced-order model for a flow dynamicsystem can converge to a spurious limit cycle after long-timeintegration even if it is initializedwith a correct configuration
Shock and Vibration 7
Table 1 Geometric and physical parameters of the cable-stayed bridge and aeroelastic parameters
Parameter Unit Value
Deck beam
Mass per unit length of the deck beam (119898119887119895) kgm 16940
Elastic modulus of the deck beam (119864119887119895) Nm2 20 times 1011
Area moment of inertia of the deck beam (119868119887119895) m4 120
Length of segment 1198871of the deck beam (2119871
1198871) m 35
Length of segment 1198872of the deck beam (2119871
1198872) m 40
Length of segment 1198873of the deck beam (2119871
1198873) m 50
Length of segment 1198874of the deck beam (2119871
1198874) m 50
Length of segment 1198875of the deck beam (2119871
1198875) m 50
Length of segment 1198876of the deck beam (2119871
1198876) m 40
Length of segment 1198877of the deck beam (2119871
1198877) m 35
Stay cables
Mass per unit length of the stay cables (119898119888119894) kgm 286
Elastic modulus of the stay cables (119864119888119894) Nm2 20 times 1011
Cross-sectional area of the stay cables (119860119888119894) m2 00362
Length of stay cable 1198881(1198711198881) m 52
Length of stay cable 1198882(1198711198882) m 60
Length of stay cable 1198883(1198711198883) m 60
Length of stay cable 1198884(1198711198884) m 52
Sag-to-span ratios of the stay cables (119889119888119894= 119863119888119894119871119888119894) 001
Aeroelastic parametersAir density (120588) kgm3 1205120576 11225 [28]1198841
688 [28]
N = 1
N = 19
N = 20
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus100
minus50
0
50
100
Figure 3 Phase portraits of the response of themidpoint of the deck beam for different numbers of Galerkin truncation terms 119902119903(0) = 01120575
1119903
[31 32] Sirisup and Karniadakis [32] demonstrated that theonset of divergence from the correct limit cycle dependson the number of Galerkin truncation terms the Reynoldsnumber and the flow geometry Since the limit cycle heredoes not vanish even when the truncation number is 30which is large one can conclude that the aforementionedlimit cycle for the cable-stayed bridge is a correct one
In many cases higher mode shapes of the cable-stayedbridge would be excited one such case is that when thereare vehicles moving on the deck beam of the bridge Hence
the initial dynamic configuration of the cable-stayed bridgecan correspond to its higher mode shapes in the dynamicanalysis of the cable-stayed bridge subjected to a distributedvortex shedding force It is obvious that Galerkin truncationwith one term is not enough in these cases Through thesame method with 119873 = 20 one can find that there isthe same limit cycle as that in Figure 3 when the initialdynamic configuration corresponds to the second throughsixth mode shapes and there is a different limit cycle whenthe initial dynamic configuration corresponds to the seventh
8 Shock and Vibration
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus80
minus40
0
40
80
Vb 4(X
b 4=0)
(mm
)
Figure 4 Time response of the midpoint of the deck beam 119902119903(0) = 01120575
1119903
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus800
minus400
0
400
800
Vb 4(X
b 4=0)
(mm
)
Figure 5 Time response of the midpoint of the deck beam 119902119903(0) = 3120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 6 Time response of the midpoint of the deck beam 119902119903(0) = 01120575
7119903
mode shape (Figures 6 and 7) Two-dimensional projectionsof phase portraits onto the (119902
119903 119903) plane when 119902
119903(0) = 01120575
1119903
and 119902119903(0) = 01120575
7119903are shown in Figures 8 and 9 respectively
Results when 119902119903(0) is equal to 01120575
2119903 011205753119903 and 01120575
6119903
are the same as that when 119902119903(0) = 01120575
1119903 they are not shown
here for the sake of brevity It can be seen from Figures 8 and9 that energy of the cable-stayed bridge is concentrated inthe first mode (the seventh mode) when its initial dynamic
configuration corresponds to the first through sixth modeshapes (the seventh mode shape)
42 Case Studies Considering the Geometric Nonlinear TermsConvergence of Galerkin method for given initial conditions119902119903(0) = 01120575
1199031is shown in Figures 10 and 11 The amplitudes
of the steady-state transverse displacements of the mid-point of the deck beam with different numbers of Galerkin
Shock and Vibration 9
minus150
minus100
minus50
0
50
100
150
0 50 100minus50minus100
qr(0) = 011205751rqr(0) = 011205757r
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 7 Two different stable limit cycles corresponding to different initial configurations
q1
minus05
0
05
0 05minus05
q1
q2
minus05
0
05
0 05minus05
q2
minus05
0
05
q3
0 05minus05
q3
q4
minus05
0
05
0 05minus05
q4
q5
minus05
0
05
0 05minus05
q5
q6
minus05
0
05
0 05minus05
q6
q7
minus05
0
05
0 05minus05
q7
q8
minus05
0
05
0 05minus05
q8
q9
minus05
0
05
0 05minus05
q9
Figure 8 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903
truncation terms which are set to one through 30 are shownin Figure 10 As shown in Figure 10 a truncation with 11terms can provide accurate results in analyzing the dynamicresponse of the cable-stayed bridge and it is much lessthan the cases in Section 41 where the geometric nonlinearterms are neglected Phase portraits of the response of themidpoint of the deck beam for different numbers of Galerkin
truncation terms which are shown in Figure 11 also leadto the same conclusion It can be noted from Figure 11 thatthe limit cycle becomes more asymmetric when the numberof the truncation terms increases The reason for this isthat when the number of truncation terms increases to 7an additional asymmetric limit cycle appears on the (119902
7 7)
plane which phase portraits project onto (see Figure 12 with
10 Shock and Vibration
q1
minus1
0
1
0 03minus03
q1
q2
minus1
0
1
0 03minus03
q2
q3
minus1
0
1
0 03minus03
q3
q4
minus1
0
1
0 03minus03
q4
q5
minus1
0
1
0 03minus03
q5
q6
minus1
0
1
0 03minus03
q6
q7
minus1
0
1
0 03minus03
q7
q8
minus1
0
1
0 03minus03
q8
q9
minus1
0
1
0 03minus03
q9
Figure 9 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
7119903
5 10 15 20 25 300Number of truncation terms N
58
59
60
61
62
63
Am
plitu
de (m
m)
Figure 10 Effects of the number of Galerkin truncation terms ontransverse displacements of the midpoint of the deck beam whenthere are geometric nonlinear terms
119873 = 11) As shown in Figures 11 and 13 the dynamic responseof the midpoint of the deck beam approaches a stable limitcycle The bifurcation diagram of limit cycles with respectto 1199021(0) is shown in Figure 14 when the initial dynamic
configuration of the cable-stayed bridge corresponds to itsfirst mode shape One can find from Figure 14 that when1199021(0) is smaller than 027 a stable periodic solution exists and
minus50 0 50 100minus100minus100
minus50
0
50
100
N = 1
N = 10
N = 11
N = 30
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 11 Phase portraits of the response of themidpoint of the deckbeam for different numbers of Galerkin truncation terms 119902
119903(0) =
011205751119903
when 1199021(0) is larger than or equal to 027 chaotic responses
occur For instance when 1199021(0) = 03 and 119873 = 11 phase
portraits of the response of the midpoint of the deck beamare shown in Figure 15 which indicates occurrence of chaotic
Shock and Vibration 11
q1
minus50
0
50
0 50minus50
q1
q2
minus20
0
20
0 10minus10
q2
q3
minus20
0
20
0 10minus10
q3
q4
minus20
0
20
0 10minus10
q4
q5
minus20
0
20
0 10minus10
minus10
q5
q6
minus20
0
20
0 10minus10
q6
q7
minus20
0
20
0 10minus10
q7
q8
minus20
0
20
100
q8
q9
minus20
0
20
0 10minus10
q9
Figure 12 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903and119873 = 11
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 13 Time response of the midpoint of the deck beam 119902119903(0) =
011205751119903and119873 = 11
response On the contrary for a truncation with one termthe response always approaches a stable limit cycle no matterhow large the amplitude of the initial dynamic configurationis (see Figures 16ndash18) These mean that the true dynamicresponse of the cable-stayed bridge may not be capturedby the truncation with one term even when the initialdynamic configuration corresponds to the first mode shape
01 02 03 04 05 06 070q1(0)
minus200
minus150
minus100
minus50
0
50
Vb 4(X
b 4=0)
(mm
)
Figure 14 Bifurcation diagram of the cable-stayed bridge withrespect to 119902
1(0) 119902119903(0) = 119902
1(0)1205751119903and119873 = 11
When the initial dynamic configuration of the cable-stayedbridge corresponds to one of its higher mode shapes chaoticresponse occurs even if its amplitude is relatively small this isshown in Figure 19where the initial dynamic configuration ofthe cable-stayed bridge corresponds to its secondmode shapeand its magnitude is only 01
12 Shock and Vibration
0
minus100 0 100 200 300minus200
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus1500
minus1000
minus500
500
1000
1500
Figure 15 Chaotic response of the midpoint of the deck beam119902119903(0) = 03120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 16 Time response of themidpoint of the deck beam 1199021(0) =
01 and119873 = 1
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus1000
minus500
0
500
1000
Vb 4(X
b 4=0)
(mm
)
Figure 17 Time response of the midpoint of the deck beam 1199021(0) =
3 and119873 = 1
5 Conclusions
The dynamic behavior of a cable-stayed bridge that consistsof a simply supported four-cable-stayed deck beam andtwo rigid towers subjected to a distributed vortex shed-ding force on the deck beam has been investigated The
q1(0) = 01
q1(0) = 3
minus100
minus50
0
50
100
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 18 Phase portraits of the response of the midpoint of thedeck beam for different initial conditions119873 = 1
minus1500
minus1000
minus500
0
500
1000
1500
minus200 0 200 400minus400
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 19 Chaotic response of the midpoint of the deck beam119902119903(0) = 01120575
2119903
dynamic response of the cable-stayed bridge is calculatedusing Galerkin method in conjunction with Runge-Kutta-Felhberg method in MATLAB Convergence of Galerkinmethod for the dynamic response of the cable-stayed bridgeis studied Numerical simulations show that the geometricnonlinearities of the stay cables have significant influence onVIV of the cable-stayed bridge and further conclusions canbe summarized as follows
(1) In the case when the geometric nonlinear termsare neglected accurate calculation of the responseamplitude of the cable-stayed bridge at lock-in onlyneeds use of the first mode shape of the linearizedundamped cable-stayed bridgemodel when the initialdynamic configuration of the cable-stayed bridgecorresponds to its mode shape whose mode numberis smaller than seven There is a different limit cyclewhen the initial dynamic configuration correspondsto its mode shape whose mode number is equal to orlarger than 7
Shock and Vibration 13
(2) In the case when the geometric nonlinear termsare considered calculation of the response of thecable-stayed bridge generally needs use of multiplemode shapes of the linearized undamped cable-stayedbridge model even when the initial dynamic config-uration of the cable-stayed bridge corresponds to itsfirstmode shapeThere is a limit cycle when the initialdynamic configuration of the cable-stayed bridgecorresponds to its first mode shape and its amplitudeis smaller than 03 for the generalized coordinate andthere is chaotic response when the initial dynamicconfiguration of the cable-stayed bridge correspondsto its first mode shape with its amplitude larger than03 for the generalized coordinate or one of its highermode shapes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grant nos 11302087 and 11442006 theNatural Science Foundation of Jiangsu Province under Grantno BK20130479 the Research Foundation for AdvancedTalents of Jiangsu University under Grant no 13JDG068 andthe National Science Foundation under Grant no CMMI-1000830
References
[1] K Y R Billah and R H Scanlan ldquoResonance Tacoma NarrowsBridge failure and undergraduate physics textbooksrdquoAmericanJournal of Physics vol 59 no 2 pp 118ndash124 1991
[2] R D Blevins Flow-Induced Vibration Krieger New York NYUSA 2nd edition 1994
[3] H Zheng R E Price Y Modarres-Sadeghi and M S Tri-antafyllou ldquoOn fatigue damage of long flexible cylinders dueto the higher harmonic force components and chaotic vortex-induced vibrationsrdquo Ocean Engineering vol 88 pp 318ndash3292014
[4] C Evangelinos D Lucor and G E Karniadakis ldquoDNS-derivedforce distribution on flexible cylinders subject to vortex-induced vibrationrdquo Journal of Fluids and Structures vol 14 no3 pp 429ndash440 2000
[5] HM Blackburn R N Govardhan andCH KWilliamson ldquoAcomplementary numerical and physical investigation of vortex-inducedrdquo Journal of Fluids and Structures vol 15 no 3-4 pp481ndash488 2001
[6] R H J Willden and J M R Graham ldquoNumerical predictionof VIV on long flexible circular cylindersrdquo Journal of Fluids andStructures vol 15 no 3-4 pp 659ndash669 2001
[7] E Guilmineau and P Queutey ldquoA numerical simulation ofvortex shedding from an oscillating circular cylinderrdquo Journalof Fluids and Structures vol 16 no 6 pp 773ndash794 2002
[8] I Jadic R M C So and M P Mignolet ldquoAnalysis of fluidmdashstructure interactions using a time-marching techniquerdquo Jour-nal of Fluids and Structures vol 12 no 6 pp 631ndash654 1998
[9] C Y Zhou RM C So andK Lam ldquoVortex-induced vibrationsof an elastic circular cylinderrdquo Journal of Fluids and Structuresvol 13 no 2 pp 165ndash189 1999
[10] J R Meneghini and P W Bearman ldquoNumerical simulationof high amplitude oscillatory flow about a circular cylinderrdquoJournal of Fluids and Structures vol 9 no 4 pp 435ndash455 1995
[11] T Sarpkaya ldquoComputational methods with vorticesrdquo Journal ofFluids Engineering vol 111 no 1 pp 5ndash52 1989
[12] T Sarpkaya ldquoVortex element methods for flow simulationrdquoAdvances in Applied Mechanics vol 31 pp 113ndash247 1994
[13] R D Gabbai and H Benaroya ldquoAn overview of modeling andexperiments of vortex-induced vibration of circular cylindersrdquoJournal of Sound and Vibration vol 282 no 3ndash5 pp 575ndash6162005
[14] R H Scanlan ldquoOn the state-of-the-artmethods for calculationsof flutter vortex-induced and buffeting response of bridgestructuresrdquo Tech Rep FHWARD-80050 National TechnicalInformation Service Springfield Va USA 1981
[15] RH Scanlan ldquoBridge flutter derivatives at vortex lock-inrdquo Jour-nal of Structural Engineering vol 124 no 4 pp 450ndash458 1998
[16] B J Vickery and R I Basu ldquoAcross-wind vibrations of struc-tures of circular cross-section Part I Development of a math-ematical model for two-dimensional conditionsrdquo Journal ofWind Engineering and Industrial Aerodynamics vol 12 no 1 pp49ndash73 1983
[17] A Larsen ldquoA generalized model for assessment of vortex-induced vibrations of flexible structuresrdquo Journal of Wind Engi-neering and Industrial Aerodynamics vol 57 no 2-3 pp 281ndash294 1995
[18] T Sarpkaya ldquoFluid forces on oscillating cylindersrdquo Journal ofWaterway Port Coastal and Ocean Division (ASCE) vol 104no 3 pp 275ndash290 1978
[19] T Staubli ldquoCalculation of vibration of an elastically mountedcylinder using experimental data from a forced oscillationrdquoJournal of Fluids Engineering vol 105 no 2 pp 225ndash229 1983
[20] W D Iwan and D L R Botelho ldquoVortex-induced oscillationof structures in waterrdquo Journal of Waterway Port Coastal andOcean Engineering vol 111 no 2 pp 289ndash303 1985
[21] R T Hartlen and I G Currie ldquoLift-oscillator model of vortex-induced vibrationrdquo Journal of Engineering Mechanics Division(ASCE) vol 96 no 5 pp 577ndash591 1970
[22] R A Skop and O M Griffin ldquoA model for the vortex-excitedresonant response of bluff cylindersrdquo Journal of Sound andVibration vol 27 no 2 pp 225ndash233 1973
[23] WD Iwan andRD Blevins ldquoAmodel for vortex induced oscil-lation of structuresrdquo Journal of Applied Mechanics TransactionsASME vol 41 no 3 pp 581ndash586 1974
[24] E H Dowell ldquoNon-linear oscillator models in bluff body aero-elasticityrdquo Journal of Sound and Vibration vol 75 no 2 pp 251ndash264 1981
[25] R H M Ogink and A V Metrikine ldquoA wake oscillator withfrequency dependent coupling for the modeling of vortex-induced vibrationrdquo Journal of Sound and Vibration vol 329 no26 pp 5452ndash5473 2010
[26] X Bai and W Qin ldquoUsing vortex strength wake oscillatorin modelling of vortex induced vibrations in two degrees offreedomrdquo European Journal of Mechanics BFluids vol 48 pp165ndash173 2014
[27] F Ehsan and R H Scanlan ldquoVortexminusinduced vibrations offlexible bridgesrdquo Journal of Engineering Mechanics vol 116 no6 pp 1392ndash1411 1990
14 Shock and Vibration
[28] A M Marra C Mannini and G Bartoli ldquoVan der Pol-typeequation for modeling vortex-induced oscillations of bridgedecksrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 99 no 6-7 pp 776ndash785 2011
[29] D Q Cao M T Song W D Zhu R W Tucker and C H-T Wang ldquoModeling and analysis of the in-plane vibration ofa complex cable-stayed bridgerdquo Journal of Sound and Vibrationvol 331 no 26 pp 5685ndash5714 2012
[30] E Caetano A Cunha V Gattulli and M Lepidi ldquoCable-deckdynamic interactions at the International Guadiana Bridge on-site measurements and finite element modellingrdquo StructuralControl and Health Monitoring vol 15 no 3 pp 237ndash264 2008
[31] I Akhtar A H Nayfeh and C J Ribbens ldquoOn the stability andextension of reduced-order Galerkin models in incompressibleflowsrdquo Theoretical and Computational Fluid Dynamics vol 23no 3 pp 213ndash237 2009
[32] S Sirisup and G E Karniadakis ldquoA spectral viscosity methodfor correcting the long-term behavior of POD modelsrdquo Journalof Computational Physics vol 194 no 1 pp 92ndash116 2004
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 Shock and Vibration
119861 V1198877(1198971198877 120591) = 0
V101584010158401198877(1198971198877 120591) = 0
(9)
119862 V1198881(0 120591) = 0
V1198882(0 120591) = 0
(10)
119863 V1198883(0 120591) = 0
V1198884(0 120591) = 0
(11)
The matching conditions at the junctions 119878119896 where 119896 =
1 3 4 6 which involve cables 119888119894 where 119894 = 1 2 3 4 respect-
ively are
V119887119896(119897119887119896 120591) = V
119887119896+1(minus119897119887119896+1 120591) (12)
V1015840119887119896(119897119887119896 120591) = V1015840
119887119896+1(minus119897119887119896+1 120591) (13)
120578119896V10158401015840119887119896(119897119887119896 120591) = 120578
119896+1V10158401015840119887119896+1(minus119897119887119896+1 120591) (14)
120578119887119896V101584010158401015840119887119896(119897119887119896 120591) minus 120578
119887119896+1V101584010158401015840119887119896+1(minus119897119887119896+1 120591) =
120594119888119894120583119888119894
120581119890119888119894(120591)
sdot sin 120579119894
+ 120594119888119894[V1015840119888119894(119897119888119894 120591) + 120583
119888119894119890119888119894(120591) (V1015840119888119894(119897119888119894 120591) + 119910
1015840
119888119894(119897119888119894))]
sdot cos 120579119894
(15)
The matching conditions at the junctions 119878119896(119896 = 2 5) with
the roller supports are
V119887119896(119897119887119896 120591) = 0 (16)
V119887119896+1(minus119897119887119896+1 120591) = 0 (17)
V1015840119887119896(119897119887119896 120591) = V1015840
119887119896+1(minus119897119887119896+1 120591) (18)
120578119887119896V10158401015840119887119896(119897119887119896 120591) = 120578
119887119896+1V10158401015840119887119896+1(minus119897119887119896+1 120591) (19)
Equations (4) and (5) with the boundary and matchingconditions in (8)ndash(19) describe the nonlinear forced vibrationof the cable-stayed bridge The equations governing thesmall amplitude vibration of the cable-stayed bridge can beobtained by linearizing (8) through (19) in the neighborhoodof the equilibrium configuration An extensive analysis ofthe free vibration of the cable-stayed bridge is presented in[29]
22 Modeling of the Vortex Shedding Force The distributedvortex shedding force on the deck beamcan bemodeled usingEhsan-Scanlanrsquos model [28]
119875(119883119887119895 119905 119881119887119895
119889119881119887119895
119889119905) =
1
21205881198802(2119863)
sdot [
[
1198841(119870)(1 minus 120576
1198812
119887119895
1198632)1
119880
119889119881119887119895
119889119905
+ 1198842(119870)
119881119887119895
119863+1
2119862119871(119870) sin (120596119905 + 120579)]
]
(20)
where 120588 is the air density 119880 is the mean wind speed 119870 =
120596119863119880 is the reduced frequency during VIV in which 120596 isthe frequency of the dynamic response of the cable-stayedbridge subjected to vortex shedding 120579 is the phase angleof the harmonic force due to vortex shedding and 119884
1(119870)
120576 1198842(119870) and 119862
119871(119870) are aeroelastic parameters that can be
determined throughwind-tunnel testsTheparameters1198841(119870)
and 120576 are related to linear and nonlinear components ofthe aerodynamic damping term respectively In particular120576 takes into account the fact that VIV is self-limiting Theparameter 119884
2(119870) represents the aerodynamic stiffness term
The parameter 119862119871(119870) is related to the amplitude of the
harmonic force due to vortex shedding According to [27]the second and third terms on the right-hand side of (20) havea negligible contribution to the response of the cable-stayedbridge at lock-in Hence at lock-in (20) can be reduced tothe following form
119875(119883119887119895 119905 119881119887119895
119889119881119887119895
119889119905)
=1
21205881198802(2119863)[
[
1198841(119870)(1 minus 120576
1198812
119887119895
1198632)1
119880
119889119881119887119895
119889119905
]
]
(21)
The nondimensional force 119901119887119895(119909119887119895 120591 V119887119895 V119887119895) in (5) can be
written as
119901119887119895(119909119887119895 120591 V119887119895 V119887119895) = 1205721198871198951198841(1 minus 120576120582
2
119887119895V2119887119895) V119887119895 (22)
where
120572119887119895(119909119887119895) =
1205960120588119880119863119871
4
119864119887119895119868119887119895
120582119887119895=Φ
119863119887119895
(23)
Shock and Vibration 5
3 Solution Method
Galerkin method is used to analyze the vibration of thecable-stayed bridge The dynamic response of stay cables andsegments of the deck beam are expressed by
V119888119894(120591 119909119888119894) =
119873
sum
119904=1
119902119904(120591) 120601(119904)
119888119894(119909119888119894) (24)
V119887119895(120591 119909119887119895) =
119873
sum
119904=1
119902119904(120591) 120601(119904)
119887119895(119909119887119895) (25)
where 120601(119904)119888119894
and 120601(119904)119887119895
are corresponding components of the119904th eigenfunction of the linearized cable-stayed bridge model[29] and 119902
119904(120591) are generalized coordinates Substituting (24)
into (4) yields
1205732
119888119894(
119873
sum
119904=1
119904120601(119904)
119888119894) + 120585119888119894(
119873
sum
119904=1
119904120601(119904)
119888119894) minus (
119873
sum
119904=1
119902119904120601(119904)
119888119894
10158401015840
)
minus120583119888119894120581
119897119888119894
[
[
tan 120579119894(
119873
sum
119904=1
119902119904120601(119904)
119888119894(119897119888119894))
+ 120581int
119897119888119894
0
1199101015840
119888119894(
119873
sum
119904=1
119902119904120601(119904)
119888119894
1015840
)119889119909119888119894
+120581
2int
119897119888119894
0
(
119873
sum
119904=1
119902119904120601(119904)
119888119894
1015840
)
2
119889119909119888119894
]
]
[(
119873
sum
119904=1
119902119904120601(119904)
119888119894
10158401015840
) + 11991010158401015840
119888119894]
= 0
(26)
Substituting (25) into (5) yields
1205734
119887119895(
119873
sum
119904=1
119904120601(119904)
119887119895) + 120585119887119895(
119873
sum
119904=1
119904120601(119904)
119887119895) + (
119873
sum
119904=1
119902119904120601(119904)
119887119895
1015840101584010158401015840
)
= 1205721198871198951198841(1 minus 120576120582
2
119887119895(
119873
sum
119904=1
119902119904120601(119904)
119887119895)
2
)(
119873
sum
119904=1
119904120601(119904)
119887119895)
(27)
Multiplying (26) by 120594119888119894120601(119903)
119888119894(119909119888119894) and integrating the resulting
equation with respect to 119909119888119894from 0 to 119897
119888119894 multiplying (27)
by 120578119887119895120601(119903)
119887119895(119909119887119895) and integrating the resulting equation with
respect to 119909119887119895from minus119897
119887119895to 119897119887119895 adding all the resulting equa-
tions andusing the following orthogonality relations of eigen-functions of the linearized cable-stayed bridge model [29]
4
sum
119894=1
1205941198881198941205732
119888119894int
119897119888119894
0
120601(119903)
119888119894(119909119888119894) 120601(119904)
119888119894(119909119888119894) 119889119909119888119894+
7
sum
119895=1
1205781198871198951205734
119887119895
sdot int
119897119887119895
minus119897119887119895
120601(119903)
119887119895(119909119887119895) 120601(119904)
119887119895(119909119887119895) 119889119909119887119895= 119872119904120575119904119903
minus
4
sum
119894=1
120594119888119894[int
119897119888119894
0
120601(119903)
119888119894(119909119888119894) 120601(119904)
119888119894
10158401015840
(119909119888119894) 119889119909119888119894
minus 8120583119888119894(119904)
119888119894
119889119888119894
1198972119888119894
int
119897119888119894
0
120601(119903)
119888119894(119909119888119894) 119889119909119888119894] +
7
sum
119895=1
120578119887119895
sdot int
119897119887119895
minus119897119887119895
120601(119903)
119887119895(119909119887119895) 120601(119904)
119887119895
1015840101584010158401015840
(119909119887119895) 119889119909119887119895= 1205962
119904119872119904120575119904119903
(28)
in which (119904)119888119894= 120581(120601
(119904)
119888119894(119897119888119894)119897119888119894) tan 120579
119894+ 1205812(1119897119888119894) int119897119888119894
01199101015840
1198881198941206011015840(119904)
119888119894119889119909119888119894
120596119904is the 119904th natural frequency of the linearized cable-
stayed bridge model 119872119904are positive constants and 120575
119904119903is
the Kronecker delta one can obtain spatially discretizedequations of the cable-stayed bridge
Mq + [C + CAero] q + Kq + NQ
+ NC+ NHy
= 0 (29)
where entries of thematricesMCCAeroKNQNC andNHy
are
119872119903119904= 119872119904120575119904119903
119862119903119904=
4
sum
119894=1
120594119888119894120585119888119894int
119897119888119894
0
120601(119904)
119888119894120601(119903)
119888119894119889119909119888119894+
7
sum
119895=1
120578119887119895120585119887119895int
119897119887119895
minus119897119887119895
120601(119904)
119887119895120601(119903)
119887119895119889119909119887119895
119862Aero119903119904
= minus
7
sum
119895=1
1205781198871198951198841int
119897119887119895
minus119897119887119895
120572119887119895120601(119904)
119887119895120601(119903)
119887119895119889119909119887119895
119870119903119904= 1205962
119904119872119903119904
119873Q119903=
4
sum
119894=1
119873
sum
119898=1
119873
sum
119899=1
120594119888119894120583119888119894119902119898119902119899(41198891198881198941205812
1198973119888119894
int
119897119888119894
0
120601(119898)
119888119894
1015840
120601(119899)
119888119894
1015840
119889119909119888119894int
119897119888119894
0
120601(119903)
119888119894119889119909119888119894minus (119898)
119888119894int
119897119888119894
0
120601(119903)
119888119894120601(119899)
119888119894
10158401015840
119889119909119888119894)
119873C119903= minus1205812
4
sum
119894=1
119873
sum
119898=1
119873
sum
119899=1
119873
sum
119900=1
120594119888119894120583119888119894
2119897119888119894
119902119898119902119899119902119900int
119897119888119894
0
120601(119898)
119888119894
1015840
120601(119899)
119888119894
1015840
119889119909119888119894int
119897119888119894
0
120601(119903)
119888119894120601(119900)
119888119894
10158401015840
119889119909119888119894
119873Hy119903=
7
sum
119895=1
119873
sum
119898=1
119873
sum
119899=1
119873
sum
119900=1
12057811988711989511988411205761205822
119887119895119902119898119902119899119900int
119897119887119895
minus119897119887119895
120572119887119895120601(119898)
119887119895120601(119899)
119887119895120601(119900)
119887119895120601(119903)
119887119895119889119909119887119895
(30)
respectively
6 Shock and Vibration
615
62
625
63
Am
plitu
de (m
m)
5 10 15 20 25 300Number of truncation terms N
Figure 2 Effects of the number of Galerkin truncation terms on transverse displacements of the midpoint of the deck beam when there areno geometric nonlinear terms
It should be noted that nonlinear termsNQ andNC in (29)are induced by geometric nonlinearities of the stay cables
4 Numerical Simulation
Geometric and physical parameters of a cable-stayed bridgeand aeroelastic parameters are listed in Table 1 It shouldbe noted that a width-to-height ratio of 4 is used in thispaper not only because it is a typical ratio of bridge decksbut also because the cable-stayed bridge is supposed tohave significant response amplitudes at lock-in [28] Forall the following calculation modal damping ratio 119862
119903119904is
always equal to 001120575119903119904 The dynamic response of the cable-
stayed bridge can be calculated from (29) using Runge-Kutta-Felhbergmethod inMATLABwhere initial conditionsof generalized coordinates can be obtained from those ofphysical coordinates
119902119903(0) =
1
119872119903
[
[
4
sum
119894=1
1205941198881198941205732
119888119894int
119897119888119894
0
V119888119894(119909119888119894 0) 120601(119903)
119888119894(119909119888119894) 119889119909119888119894
+
7
sum
119895=1
1205781198871198951205734
119887119895int
119897119887119895
minus119897119887119895
V119887119895(119909119887119895 0) 120601(119903)
119887119895(119909119887119895) 119889119909119887119895
]
]
119903(0) =
1
119872119903
[
[
4
sum
119894=1
1205941198881198941205732
119888119894int
119897119888119894
0
V119888119894(119909119888119894 0) 120601(119903)
119888119894(119909119888119894) 119889119909119888119894
+
7
sum
119895=1
1205781198871198951205734
119887119895int
119897119887119895
minus119897119887119895
V119887119895(119909119887119895 0) 120601(119903)
119887119895(119909119887119895) 119889119909119887119895
]
]
(31)
For all the following calculation 119903(0) is always equal to
zero Numerical simulations for two cases are undertaken (1)neglecting the geometric nonlinear terms (ie NQ
= NC= 0
in (29)) and (2) considering the geometric nonlinear terms
41 Case Studies Neglecting the Geometric Nonlinear TermsConvergence of Galerkin method for given initial conditions119902119903(0) = 01120575
1199031 which means that the cable-stayed bridge
has its initial dynamic configuration corresponding to itsfirst mode shape and it is released from rest is shown in
Figures 2 and 3The amplitudes of the steady-state transversedisplacements of the midpoint of the deck beam withdifferent numbers of Galerkin truncation terms which areset to one through 30 are shown in Figure 2 As shown inFigure 2 the transverse displacements of the midpoint of thedeck beam initially remain stable until the truncation numberincreases to 15 and suddenly increase to their convergedvalues A truncation with 20 terms can provide accurateresults in analyzing the dynamic response of the deck beamThe difference between the amplitudes of two limit cyclesobtained by truncation with one term and 20 terms is194 which is small One can come to a conclusion that atruncation with one term is accurate enough for calculationof the steady-state response of the cable-stayed bridge whenits initial dynamic configuration corresponds to its first modeshape In other words most of the energy of the cable-stayedbridge is concentrated in its first mode in this case Phaseportraits of the response of the midpoint of the deck beamfor different numbers of Galerkin truncation terms whichis shown in Figure 3 also lead to the same conclusion Otherresults that are not shown here for the sake of brevity indicatethat Galerkin truncation with 20 terms yields accurateresults for the dynamic response of the cable-stayed bridgein the following cases in this section Hence in the followingnumerical calculations in this section the first 20 modes ofthe linearized undamped cable-stayed bridge model are usedin Galerkin method Time history responses of the midpointof the deck beam when the initial dynamic configuration ofthe cable-stayed bridge corresponds to its first mode shapebut has different amplitudes that is 119902
1(0) = 01 (the initial
displacement of themidpoint of the deck beam is 32mm) and1199021(0) = 3 (the initial displacement of themidpoint of the deck
beam is 946mm which is rather large) are shown in Figures4 and 5 respectively It can be seen that the solutions with dif-ferent 119902
1(0) converge to the same limit cycle after long-time
integrationThemagnitudes of Floquet multipliers are all lessthan unity the aforementioned limit cycle is asymptoticallystable
Solutions of a reduced-order model for a flow dynamicsystem can converge to a spurious limit cycle after long-timeintegration even if it is initializedwith a correct configuration
Shock and Vibration 7
Table 1 Geometric and physical parameters of the cable-stayed bridge and aeroelastic parameters
Parameter Unit Value
Deck beam
Mass per unit length of the deck beam (119898119887119895) kgm 16940
Elastic modulus of the deck beam (119864119887119895) Nm2 20 times 1011
Area moment of inertia of the deck beam (119868119887119895) m4 120
Length of segment 1198871of the deck beam (2119871
1198871) m 35
Length of segment 1198872of the deck beam (2119871
1198872) m 40
Length of segment 1198873of the deck beam (2119871
1198873) m 50
Length of segment 1198874of the deck beam (2119871
1198874) m 50
Length of segment 1198875of the deck beam (2119871
1198875) m 50
Length of segment 1198876of the deck beam (2119871
1198876) m 40
Length of segment 1198877of the deck beam (2119871
1198877) m 35
Stay cables
Mass per unit length of the stay cables (119898119888119894) kgm 286
Elastic modulus of the stay cables (119864119888119894) Nm2 20 times 1011
Cross-sectional area of the stay cables (119860119888119894) m2 00362
Length of stay cable 1198881(1198711198881) m 52
Length of stay cable 1198882(1198711198882) m 60
Length of stay cable 1198883(1198711198883) m 60
Length of stay cable 1198884(1198711198884) m 52
Sag-to-span ratios of the stay cables (119889119888119894= 119863119888119894119871119888119894) 001
Aeroelastic parametersAir density (120588) kgm3 1205120576 11225 [28]1198841
688 [28]
N = 1
N = 19
N = 20
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus100
minus50
0
50
100
Figure 3 Phase portraits of the response of themidpoint of the deck beam for different numbers of Galerkin truncation terms 119902119903(0) = 01120575
1119903
[31 32] Sirisup and Karniadakis [32] demonstrated that theonset of divergence from the correct limit cycle dependson the number of Galerkin truncation terms the Reynoldsnumber and the flow geometry Since the limit cycle heredoes not vanish even when the truncation number is 30which is large one can conclude that the aforementionedlimit cycle for the cable-stayed bridge is a correct one
In many cases higher mode shapes of the cable-stayedbridge would be excited one such case is that when thereare vehicles moving on the deck beam of the bridge Hence
the initial dynamic configuration of the cable-stayed bridgecan correspond to its higher mode shapes in the dynamicanalysis of the cable-stayed bridge subjected to a distributedvortex shedding force It is obvious that Galerkin truncationwith one term is not enough in these cases Through thesame method with 119873 = 20 one can find that there isthe same limit cycle as that in Figure 3 when the initialdynamic configuration corresponds to the second throughsixth mode shapes and there is a different limit cycle whenthe initial dynamic configuration corresponds to the seventh
8 Shock and Vibration
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus80
minus40
0
40
80
Vb 4(X
b 4=0)
(mm
)
Figure 4 Time response of the midpoint of the deck beam 119902119903(0) = 01120575
1119903
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus800
minus400
0
400
800
Vb 4(X
b 4=0)
(mm
)
Figure 5 Time response of the midpoint of the deck beam 119902119903(0) = 3120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 6 Time response of the midpoint of the deck beam 119902119903(0) = 01120575
7119903
mode shape (Figures 6 and 7) Two-dimensional projectionsof phase portraits onto the (119902
119903 119903) plane when 119902
119903(0) = 01120575
1119903
and 119902119903(0) = 01120575
7119903are shown in Figures 8 and 9 respectively
Results when 119902119903(0) is equal to 01120575
2119903 011205753119903 and 01120575
6119903
are the same as that when 119902119903(0) = 01120575
1119903 they are not shown
here for the sake of brevity It can be seen from Figures 8 and9 that energy of the cable-stayed bridge is concentrated inthe first mode (the seventh mode) when its initial dynamic
configuration corresponds to the first through sixth modeshapes (the seventh mode shape)
42 Case Studies Considering the Geometric Nonlinear TermsConvergence of Galerkin method for given initial conditions119902119903(0) = 01120575
1199031is shown in Figures 10 and 11 The amplitudes
of the steady-state transverse displacements of the mid-point of the deck beam with different numbers of Galerkin
Shock and Vibration 9
minus150
minus100
minus50
0
50
100
150
0 50 100minus50minus100
qr(0) = 011205751rqr(0) = 011205757r
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 7 Two different stable limit cycles corresponding to different initial configurations
q1
minus05
0
05
0 05minus05
q1
q2
minus05
0
05
0 05minus05
q2
minus05
0
05
q3
0 05minus05
q3
q4
minus05
0
05
0 05minus05
q4
q5
minus05
0
05
0 05minus05
q5
q6
minus05
0
05
0 05minus05
q6
q7
minus05
0
05
0 05minus05
q7
q8
minus05
0
05
0 05minus05
q8
q9
minus05
0
05
0 05minus05
q9
Figure 8 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903
truncation terms which are set to one through 30 are shownin Figure 10 As shown in Figure 10 a truncation with 11terms can provide accurate results in analyzing the dynamicresponse of the cable-stayed bridge and it is much lessthan the cases in Section 41 where the geometric nonlinearterms are neglected Phase portraits of the response of themidpoint of the deck beam for different numbers of Galerkin
truncation terms which are shown in Figure 11 also leadto the same conclusion It can be noted from Figure 11 thatthe limit cycle becomes more asymmetric when the numberof the truncation terms increases The reason for this isthat when the number of truncation terms increases to 7an additional asymmetric limit cycle appears on the (119902
7 7)
plane which phase portraits project onto (see Figure 12 with
10 Shock and Vibration
q1
minus1
0
1
0 03minus03
q1
q2
minus1
0
1
0 03minus03
q2
q3
minus1
0
1
0 03minus03
q3
q4
minus1
0
1
0 03minus03
q4
q5
minus1
0
1
0 03minus03
q5
q6
minus1
0
1
0 03minus03
q6
q7
minus1
0
1
0 03minus03
q7
q8
minus1
0
1
0 03minus03
q8
q9
minus1
0
1
0 03minus03
q9
Figure 9 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
7119903
5 10 15 20 25 300Number of truncation terms N
58
59
60
61
62
63
Am
plitu
de (m
m)
Figure 10 Effects of the number of Galerkin truncation terms ontransverse displacements of the midpoint of the deck beam whenthere are geometric nonlinear terms
119873 = 11) As shown in Figures 11 and 13 the dynamic responseof the midpoint of the deck beam approaches a stable limitcycle The bifurcation diagram of limit cycles with respectto 1199021(0) is shown in Figure 14 when the initial dynamic
configuration of the cable-stayed bridge corresponds to itsfirst mode shape One can find from Figure 14 that when1199021(0) is smaller than 027 a stable periodic solution exists and
minus50 0 50 100minus100minus100
minus50
0
50
100
N = 1
N = 10
N = 11
N = 30
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 11 Phase portraits of the response of themidpoint of the deckbeam for different numbers of Galerkin truncation terms 119902
119903(0) =
011205751119903
when 1199021(0) is larger than or equal to 027 chaotic responses
occur For instance when 1199021(0) = 03 and 119873 = 11 phase
portraits of the response of the midpoint of the deck beamare shown in Figure 15 which indicates occurrence of chaotic
Shock and Vibration 11
q1
minus50
0
50
0 50minus50
q1
q2
minus20
0
20
0 10minus10
q2
q3
minus20
0
20
0 10minus10
q3
q4
minus20
0
20
0 10minus10
q4
q5
minus20
0
20
0 10minus10
minus10
q5
q6
minus20
0
20
0 10minus10
q6
q7
minus20
0
20
0 10minus10
q7
q8
minus20
0
20
100
q8
q9
minus20
0
20
0 10minus10
q9
Figure 12 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903and119873 = 11
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 13 Time response of the midpoint of the deck beam 119902119903(0) =
011205751119903and119873 = 11
response On the contrary for a truncation with one termthe response always approaches a stable limit cycle no matterhow large the amplitude of the initial dynamic configurationis (see Figures 16ndash18) These mean that the true dynamicresponse of the cable-stayed bridge may not be capturedby the truncation with one term even when the initialdynamic configuration corresponds to the first mode shape
01 02 03 04 05 06 070q1(0)
minus200
minus150
minus100
minus50
0
50
Vb 4(X
b 4=0)
(mm
)
Figure 14 Bifurcation diagram of the cable-stayed bridge withrespect to 119902
1(0) 119902119903(0) = 119902
1(0)1205751119903and119873 = 11
When the initial dynamic configuration of the cable-stayedbridge corresponds to one of its higher mode shapes chaoticresponse occurs even if its amplitude is relatively small this isshown in Figure 19where the initial dynamic configuration ofthe cable-stayed bridge corresponds to its secondmode shapeand its magnitude is only 01
12 Shock and Vibration
0
minus100 0 100 200 300minus200
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus1500
minus1000
minus500
500
1000
1500
Figure 15 Chaotic response of the midpoint of the deck beam119902119903(0) = 03120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 16 Time response of themidpoint of the deck beam 1199021(0) =
01 and119873 = 1
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus1000
minus500
0
500
1000
Vb 4(X
b 4=0)
(mm
)
Figure 17 Time response of the midpoint of the deck beam 1199021(0) =
3 and119873 = 1
5 Conclusions
The dynamic behavior of a cable-stayed bridge that consistsof a simply supported four-cable-stayed deck beam andtwo rigid towers subjected to a distributed vortex shed-ding force on the deck beam has been investigated The
q1(0) = 01
q1(0) = 3
minus100
minus50
0
50
100
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 18 Phase portraits of the response of the midpoint of thedeck beam for different initial conditions119873 = 1
minus1500
minus1000
minus500
0
500
1000
1500
minus200 0 200 400minus400
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 19 Chaotic response of the midpoint of the deck beam119902119903(0) = 01120575
2119903
dynamic response of the cable-stayed bridge is calculatedusing Galerkin method in conjunction with Runge-Kutta-Felhberg method in MATLAB Convergence of Galerkinmethod for the dynamic response of the cable-stayed bridgeis studied Numerical simulations show that the geometricnonlinearities of the stay cables have significant influence onVIV of the cable-stayed bridge and further conclusions canbe summarized as follows
(1) In the case when the geometric nonlinear termsare neglected accurate calculation of the responseamplitude of the cable-stayed bridge at lock-in onlyneeds use of the first mode shape of the linearizedundamped cable-stayed bridgemodel when the initialdynamic configuration of the cable-stayed bridgecorresponds to its mode shape whose mode numberis smaller than seven There is a different limit cyclewhen the initial dynamic configuration correspondsto its mode shape whose mode number is equal to orlarger than 7
Shock and Vibration 13
(2) In the case when the geometric nonlinear termsare considered calculation of the response of thecable-stayed bridge generally needs use of multiplemode shapes of the linearized undamped cable-stayedbridge model even when the initial dynamic config-uration of the cable-stayed bridge corresponds to itsfirstmode shapeThere is a limit cycle when the initialdynamic configuration of the cable-stayed bridgecorresponds to its first mode shape and its amplitudeis smaller than 03 for the generalized coordinate andthere is chaotic response when the initial dynamicconfiguration of the cable-stayed bridge correspondsto its first mode shape with its amplitude larger than03 for the generalized coordinate or one of its highermode shapes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grant nos 11302087 and 11442006 theNatural Science Foundation of Jiangsu Province under Grantno BK20130479 the Research Foundation for AdvancedTalents of Jiangsu University under Grant no 13JDG068 andthe National Science Foundation under Grant no CMMI-1000830
References
[1] K Y R Billah and R H Scanlan ldquoResonance Tacoma NarrowsBridge failure and undergraduate physics textbooksrdquoAmericanJournal of Physics vol 59 no 2 pp 118ndash124 1991
[2] R D Blevins Flow-Induced Vibration Krieger New York NYUSA 2nd edition 1994
[3] H Zheng R E Price Y Modarres-Sadeghi and M S Tri-antafyllou ldquoOn fatigue damage of long flexible cylinders dueto the higher harmonic force components and chaotic vortex-induced vibrationsrdquo Ocean Engineering vol 88 pp 318ndash3292014
[4] C Evangelinos D Lucor and G E Karniadakis ldquoDNS-derivedforce distribution on flexible cylinders subject to vortex-induced vibrationrdquo Journal of Fluids and Structures vol 14 no3 pp 429ndash440 2000
[5] HM Blackburn R N Govardhan andCH KWilliamson ldquoAcomplementary numerical and physical investigation of vortex-inducedrdquo Journal of Fluids and Structures vol 15 no 3-4 pp481ndash488 2001
[6] R H J Willden and J M R Graham ldquoNumerical predictionof VIV on long flexible circular cylindersrdquo Journal of Fluids andStructures vol 15 no 3-4 pp 659ndash669 2001
[7] E Guilmineau and P Queutey ldquoA numerical simulation ofvortex shedding from an oscillating circular cylinderrdquo Journalof Fluids and Structures vol 16 no 6 pp 773ndash794 2002
[8] I Jadic R M C So and M P Mignolet ldquoAnalysis of fluidmdashstructure interactions using a time-marching techniquerdquo Jour-nal of Fluids and Structures vol 12 no 6 pp 631ndash654 1998
[9] C Y Zhou RM C So andK Lam ldquoVortex-induced vibrationsof an elastic circular cylinderrdquo Journal of Fluids and Structuresvol 13 no 2 pp 165ndash189 1999
[10] J R Meneghini and P W Bearman ldquoNumerical simulationof high amplitude oscillatory flow about a circular cylinderrdquoJournal of Fluids and Structures vol 9 no 4 pp 435ndash455 1995
[11] T Sarpkaya ldquoComputational methods with vorticesrdquo Journal ofFluids Engineering vol 111 no 1 pp 5ndash52 1989
[12] T Sarpkaya ldquoVortex element methods for flow simulationrdquoAdvances in Applied Mechanics vol 31 pp 113ndash247 1994
[13] R D Gabbai and H Benaroya ldquoAn overview of modeling andexperiments of vortex-induced vibration of circular cylindersrdquoJournal of Sound and Vibration vol 282 no 3ndash5 pp 575ndash6162005
[14] R H Scanlan ldquoOn the state-of-the-artmethods for calculationsof flutter vortex-induced and buffeting response of bridgestructuresrdquo Tech Rep FHWARD-80050 National TechnicalInformation Service Springfield Va USA 1981
[15] RH Scanlan ldquoBridge flutter derivatives at vortex lock-inrdquo Jour-nal of Structural Engineering vol 124 no 4 pp 450ndash458 1998
[16] B J Vickery and R I Basu ldquoAcross-wind vibrations of struc-tures of circular cross-section Part I Development of a math-ematical model for two-dimensional conditionsrdquo Journal ofWind Engineering and Industrial Aerodynamics vol 12 no 1 pp49ndash73 1983
[17] A Larsen ldquoA generalized model for assessment of vortex-induced vibrations of flexible structuresrdquo Journal of Wind Engi-neering and Industrial Aerodynamics vol 57 no 2-3 pp 281ndash294 1995
[18] T Sarpkaya ldquoFluid forces on oscillating cylindersrdquo Journal ofWaterway Port Coastal and Ocean Division (ASCE) vol 104no 3 pp 275ndash290 1978
[19] T Staubli ldquoCalculation of vibration of an elastically mountedcylinder using experimental data from a forced oscillationrdquoJournal of Fluids Engineering vol 105 no 2 pp 225ndash229 1983
[20] W D Iwan and D L R Botelho ldquoVortex-induced oscillationof structures in waterrdquo Journal of Waterway Port Coastal andOcean Engineering vol 111 no 2 pp 289ndash303 1985
[21] R T Hartlen and I G Currie ldquoLift-oscillator model of vortex-induced vibrationrdquo Journal of Engineering Mechanics Division(ASCE) vol 96 no 5 pp 577ndash591 1970
[22] R A Skop and O M Griffin ldquoA model for the vortex-excitedresonant response of bluff cylindersrdquo Journal of Sound andVibration vol 27 no 2 pp 225ndash233 1973
[23] WD Iwan andRD Blevins ldquoAmodel for vortex induced oscil-lation of structuresrdquo Journal of Applied Mechanics TransactionsASME vol 41 no 3 pp 581ndash586 1974
[24] E H Dowell ldquoNon-linear oscillator models in bluff body aero-elasticityrdquo Journal of Sound and Vibration vol 75 no 2 pp 251ndash264 1981
[25] R H M Ogink and A V Metrikine ldquoA wake oscillator withfrequency dependent coupling for the modeling of vortex-induced vibrationrdquo Journal of Sound and Vibration vol 329 no26 pp 5452ndash5473 2010
[26] X Bai and W Qin ldquoUsing vortex strength wake oscillatorin modelling of vortex induced vibrations in two degrees offreedomrdquo European Journal of Mechanics BFluids vol 48 pp165ndash173 2014
[27] F Ehsan and R H Scanlan ldquoVortexminusinduced vibrations offlexible bridgesrdquo Journal of Engineering Mechanics vol 116 no6 pp 1392ndash1411 1990
14 Shock and Vibration
[28] A M Marra C Mannini and G Bartoli ldquoVan der Pol-typeequation for modeling vortex-induced oscillations of bridgedecksrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 99 no 6-7 pp 776ndash785 2011
[29] D Q Cao M T Song W D Zhu R W Tucker and C H-T Wang ldquoModeling and analysis of the in-plane vibration ofa complex cable-stayed bridgerdquo Journal of Sound and Vibrationvol 331 no 26 pp 5685ndash5714 2012
[30] E Caetano A Cunha V Gattulli and M Lepidi ldquoCable-deckdynamic interactions at the International Guadiana Bridge on-site measurements and finite element modellingrdquo StructuralControl and Health Monitoring vol 15 no 3 pp 237ndash264 2008
[31] I Akhtar A H Nayfeh and C J Ribbens ldquoOn the stability andextension of reduced-order Galerkin models in incompressibleflowsrdquo Theoretical and Computational Fluid Dynamics vol 23no 3 pp 213ndash237 2009
[32] S Sirisup and G E Karniadakis ldquoA spectral viscosity methodfor correcting the long-term behavior of POD modelsrdquo Journalof Computational Physics vol 194 no 1 pp 92ndash116 2004
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 5
3 Solution Method
Galerkin method is used to analyze the vibration of thecable-stayed bridge The dynamic response of stay cables andsegments of the deck beam are expressed by
V119888119894(120591 119909119888119894) =
119873
sum
119904=1
119902119904(120591) 120601(119904)
119888119894(119909119888119894) (24)
V119887119895(120591 119909119887119895) =
119873
sum
119904=1
119902119904(120591) 120601(119904)
119887119895(119909119887119895) (25)
where 120601(119904)119888119894
and 120601(119904)119887119895
are corresponding components of the119904th eigenfunction of the linearized cable-stayed bridge model[29] and 119902
119904(120591) are generalized coordinates Substituting (24)
into (4) yields
1205732
119888119894(
119873
sum
119904=1
119904120601(119904)
119888119894) + 120585119888119894(
119873
sum
119904=1
119904120601(119904)
119888119894) minus (
119873
sum
119904=1
119902119904120601(119904)
119888119894
10158401015840
)
minus120583119888119894120581
119897119888119894
[
[
tan 120579119894(
119873
sum
119904=1
119902119904120601(119904)
119888119894(119897119888119894))
+ 120581int
119897119888119894
0
1199101015840
119888119894(
119873
sum
119904=1
119902119904120601(119904)
119888119894
1015840
)119889119909119888119894
+120581
2int
119897119888119894
0
(
119873
sum
119904=1
119902119904120601(119904)
119888119894
1015840
)
2
119889119909119888119894
]
]
[(
119873
sum
119904=1
119902119904120601(119904)
119888119894
10158401015840
) + 11991010158401015840
119888119894]
= 0
(26)
Substituting (25) into (5) yields
1205734
119887119895(
119873
sum
119904=1
119904120601(119904)
119887119895) + 120585119887119895(
119873
sum
119904=1
119904120601(119904)
119887119895) + (
119873
sum
119904=1
119902119904120601(119904)
119887119895
1015840101584010158401015840
)
= 1205721198871198951198841(1 minus 120576120582
2
119887119895(
119873
sum
119904=1
119902119904120601(119904)
119887119895)
2
)(
119873
sum
119904=1
119904120601(119904)
119887119895)
(27)
Multiplying (26) by 120594119888119894120601(119903)
119888119894(119909119888119894) and integrating the resulting
equation with respect to 119909119888119894from 0 to 119897
119888119894 multiplying (27)
by 120578119887119895120601(119903)
119887119895(119909119887119895) and integrating the resulting equation with
respect to 119909119887119895from minus119897
119887119895to 119897119887119895 adding all the resulting equa-
tions andusing the following orthogonality relations of eigen-functions of the linearized cable-stayed bridge model [29]
4
sum
119894=1
1205941198881198941205732
119888119894int
119897119888119894
0
120601(119903)
119888119894(119909119888119894) 120601(119904)
119888119894(119909119888119894) 119889119909119888119894+
7
sum
119895=1
1205781198871198951205734
119887119895
sdot int
119897119887119895
minus119897119887119895
120601(119903)
119887119895(119909119887119895) 120601(119904)
119887119895(119909119887119895) 119889119909119887119895= 119872119904120575119904119903
minus
4
sum
119894=1
120594119888119894[int
119897119888119894
0
120601(119903)
119888119894(119909119888119894) 120601(119904)
119888119894
10158401015840
(119909119888119894) 119889119909119888119894
minus 8120583119888119894(119904)
119888119894
119889119888119894
1198972119888119894
int
119897119888119894
0
120601(119903)
119888119894(119909119888119894) 119889119909119888119894] +
7
sum
119895=1
120578119887119895
sdot int
119897119887119895
minus119897119887119895
120601(119903)
119887119895(119909119887119895) 120601(119904)
119887119895
1015840101584010158401015840
(119909119887119895) 119889119909119887119895= 1205962
119904119872119904120575119904119903
(28)
in which (119904)119888119894= 120581(120601
(119904)
119888119894(119897119888119894)119897119888119894) tan 120579
119894+ 1205812(1119897119888119894) int119897119888119894
01199101015840
1198881198941206011015840(119904)
119888119894119889119909119888119894
120596119904is the 119904th natural frequency of the linearized cable-
stayed bridge model 119872119904are positive constants and 120575
119904119903is
the Kronecker delta one can obtain spatially discretizedequations of the cable-stayed bridge
Mq + [C + CAero] q + Kq + NQ
+ NC+ NHy
= 0 (29)
where entries of thematricesMCCAeroKNQNC andNHy
are
119872119903119904= 119872119904120575119904119903
119862119903119904=
4
sum
119894=1
120594119888119894120585119888119894int
119897119888119894
0
120601(119904)
119888119894120601(119903)
119888119894119889119909119888119894+
7
sum
119895=1
120578119887119895120585119887119895int
119897119887119895
minus119897119887119895
120601(119904)
119887119895120601(119903)
119887119895119889119909119887119895
119862Aero119903119904
= minus
7
sum
119895=1
1205781198871198951198841int
119897119887119895
minus119897119887119895
120572119887119895120601(119904)
119887119895120601(119903)
119887119895119889119909119887119895
119870119903119904= 1205962
119904119872119903119904
119873Q119903=
4
sum
119894=1
119873
sum
119898=1
119873
sum
119899=1
120594119888119894120583119888119894119902119898119902119899(41198891198881198941205812
1198973119888119894
int
119897119888119894
0
120601(119898)
119888119894
1015840
120601(119899)
119888119894
1015840
119889119909119888119894int
119897119888119894
0
120601(119903)
119888119894119889119909119888119894minus (119898)
119888119894int
119897119888119894
0
120601(119903)
119888119894120601(119899)
119888119894
10158401015840
119889119909119888119894)
119873C119903= minus1205812
4
sum
119894=1
119873
sum
119898=1
119873
sum
119899=1
119873
sum
119900=1
120594119888119894120583119888119894
2119897119888119894
119902119898119902119899119902119900int
119897119888119894
0
120601(119898)
119888119894
1015840
120601(119899)
119888119894
1015840
119889119909119888119894int
119897119888119894
0
120601(119903)
119888119894120601(119900)
119888119894
10158401015840
119889119909119888119894
119873Hy119903=
7
sum
119895=1
119873
sum
119898=1
119873
sum
119899=1
119873
sum
119900=1
12057811988711989511988411205761205822
119887119895119902119898119902119899119900int
119897119887119895
minus119897119887119895
120572119887119895120601(119898)
119887119895120601(119899)
119887119895120601(119900)
119887119895120601(119903)
119887119895119889119909119887119895
(30)
respectively
6 Shock and Vibration
615
62
625
63
Am
plitu
de (m
m)
5 10 15 20 25 300Number of truncation terms N
Figure 2 Effects of the number of Galerkin truncation terms on transverse displacements of the midpoint of the deck beam when there areno geometric nonlinear terms
It should be noted that nonlinear termsNQ andNC in (29)are induced by geometric nonlinearities of the stay cables
4 Numerical Simulation
Geometric and physical parameters of a cable-stayed bridgeand aeroelastic parameters are listed in Table 1 It shouldbe noted that a width-to-height ratio of 4 is used in thispaper not only because it is a typical ratio of bridge decksbut also because the cable-stayed bridge is supposed tohave significant response amplitudes at lock-in [28] Forall the following calculation modal damping ratio 119862
119903119904is
always equal to 001120575119903119904 The dynamic response of the cable-
stayed bridge can be calculated from (29) using Runge-Kutta-Felhbergmethod inMATLABwhere initial conditionsof generalized coordinates can be obtained from those ofphysical coordinates
119902119903(0) =
1
119872119903
[
[
4
sum
119894=1
1205941198881198941205732
119888119894int
119897119888119894
0
V119888119894(119909119888119894 0) 120601(119903)
119888119894(119909119888119894) 119889119909119888119894
+
7
sum
119895=1
1205781198871198951205734
119887119895int
119897119887119895
minus119897119887119895
V119887119895(119909119887119895 0) 120601(119903)
119887119895(119909119887119895) 119889119909119887119895
]
]
119903(0) =
1
119872119903
[
[
4
sum
119894=1
1205941198881198941205732
119888119894int
119897119888119894
0
V119888119894(119909119888119894 0) 120601(119903)
119888119894(119909119888119894) 119889119909119888119894
+
7
sum
119895=1
1205781198871198951205734
119887119895int
119897119887119895
minus119897119887119895
V119887119895(119909119887119895 0) 120601(119903)
119887119895(119909119887119895) 119889119909119887119895
]
]
(31)
For all the following calculation 119903(0) is always equal to
zero Numerical simulations for two cases are undertaken (1)neglecting the geometric nonlinear terms (ie NQ
= NC= 0
in (29)) and (2) considering the geometric nonlinear terms
41 Case Studies Neglecting the Geometric Nonlinear TermsConvergence of Galerkin method for given initial conditions119902119903(0) = 01120575
1199031 which means that the cable-stayed bridge
has its initial dynamic configuration corresponding to itsfirst mode shape and it is released from rest is shown in
Figures 2 and 3The amplitudes of the steady-state transversedisplacements of the midpoint of the deck beam withdifferent numbers of Galerkin truncation terms which areset to one through 30 are shown in Figure 2 As shown inFigure 2 the transverse displacements of the midpoint of thedeck beam initially remain stable until the truncation numberincreases to 15 and suddenly increase to their convergedvalues A truncation with 20 terms can provide accurateresults in analyzing the dynamic response of the deck beamThe difference between the amplitudes of two limit cyclesobtained by truncation with one term and 20 terms is194 which is small One can come to a conclusion that atruncation with one term is accurate enough for calculationof the steady-state response of the cable-stayed bridge whenits initial dynamic configuration corresponds to its first modeshape In other words most of the energy of the cable-stayedbridge is concentrated in its first mode in this case Phaseportraits of the response of the midpoint of the deck beamfor different numbers of Galerkin truncation terms whichis shown in Figure 3 also lead to the same conclusion Otherresults that are not shown here for the sake of brevity indicatethat Galerkin truncation with 20 terms yields accurateresults for the dynamic response of the cable-stayed bridgein the following cases in this section Hence in the followingnumerical calculations in this section the first 20 modes ofthe linearized undamped cable-stayed bridge model are usedin Galerkin method Time history responses of the midpointof the deck beam when the initial dynamic configuration ofthe cable-stayed bridge corresponds to its first mode shapebut has different amplitudes that is 119902
1(0) = 01 (the initial
displacement of themidpoint of the deck beam is 32mm) and1199021(0) = 3 (the initial displacement of themidpoint of the deck
beam is 946mm which is rather large) are shown in Figures4 and 5 respectively It can be seen that the solutions with dif-ferent 119902
1(0) converge to the same limit cycle after long-time
integrationThemagnitudes of Floquet multipliers are all lessthan unity the aforementioned limit cycle is asymptoticallystable
Solutions of a reduced-order model for a flow dynamicsystem can converge to a spurious limit cycle after long-timeintegration even if it is initializedwith a correct configuration
Shock and Vibration 7
Table 1 Geometric and physical parameters of the cable-stayed bridge and aeroelastic parameters
Parameter Unit Value
Deck beam
Mass per unit length of the deck beam (119898119887119895) kgm 16940
Elastic modulus of the deck beam (119864119887119895) Nm2 20 times 1011
Area moment of inertia of the deck beam (119868119887119895) m4 120
Length of segment 1198871of the deck beam (2119871
1198871) m 35
Length of segment 1198872of the deck beam (2119871
1198872) m 40
Length of segment 1198873of the deck beam (2119871
1198873) m 50
Length of segment 1198874of the deck beam (2119871
1198874) m 50
Length of segment 1198875of the deck beam (2119871
1198875) m 50
Length of segment 1198876of the deck beam (2119871
1198876) m 40
Length of segment 1198877of the deck beam (2119871
1198877) m 35
Stay cables
Mass per unit length of the stay cables (119898119888119894) kgm 286
Elastic modulus of the stay cables (119864119888119894) Nm2 20 times 1011
Cross-sectional area of the stay cables (119860119888119894) m2 00362
Length of stay cable 1198881(1198711198881) m 52
Length of stay cable 1198882(1198711198882) m 60
Length of stay cable 1198883(1198711198883) m 60
Length of stay cable 1198884(1198711198884) m 52
Sag-to-span ratios of the stay cables (119889119888119894= 119863119888119894119871119888119894) 001
Aeroelastic parametersAir density (120588) kgm3 1205120576 11225 [28]1198841
688 [28]
N = 1
N = 19
N = 20
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus100
minus50
0
50
100
Figure 3 Phase portraits of the response of themidpoint of the deck beam for different numbers of Galerkin truncation terms 119902119903(0) = 01120575
1119903
[31 32] Sirisup and Karniadakis [32] demonstrated that theonset of divergence from the correct limit cycle dependson the number of Galerkin truncation terms the Reynoldsnumber and the flow geometry Since the limit cycle heredoes not vanish even when the truncation number is 30which is large one can conclude that the aforementionedlimit cycle for the cable-stayed bridge is a correct one
In many cases higher mode shapes of the cable-stayedbridge would be excited one such case is that when thereare vehicles moving on the deck beam of the bridge Hence
the initial dynamic configuration of the cable-stayed bridgecan correspond to its higher mode shapes in the dynamicanalysis of the cable-stayed bridge subjected to a distributedvortex shedding force It is obvious that Galerkin truncationwith one term is not enough in these cases Through thesame method with 119873 = 20 one can find that there isthe same limit cycle as that in Figure 3 when the initialdynamic configuration corresponds to the second throughsixth mode shapes and there is a different limit cycle whenthe initial dynamic configuration corresponds to the seventh
8 Shock and Vibration
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus80
minus40
0
40
80
Vb 4(X
b 4=0)
(mm
)
Figure 4 Time response of the midpoint of the deck beam 119902119903(0) = 01120575
1119903
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus800
minus400
0
400
800
Vb 4(X
b 4=0)
(mm
)
Figure 5 Time response of the midpoint of the deck beam 119902119903(0) = 3120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 6 Time response of the midpoint of the deck beam 119902119903(0) = 01120575
7119903
mode shape (Figures 6 and 7) Two-dimensional projectionsof phase portraits onto the (119902
119903 119903) plane when 119902
119903(0) = 01120575
1119903
and 119902119903(0) = 01120575
7119903are shown in Figures 8 and 9 respectively
Results when 119902119903(0) is equal to 01120575
2119903 011205753119903 and 01120575
6119903
are the same as that when 119902119903(0) = 01120575
1119903 they are not shown
here for the sake of brevity It can be seen from Figures 8 and9 that energy of the cable-stayed bridge is concentrated inthe first mode (the seventh mode) when its initial dynamic
configuration corresponds to the first through sixth modeshapes (the seventh mode shape)
42 Case Studies Considering the Geometric Nonlinear TermsConvergence of Galerkin method for given initial conditions119902119903(0) = 01120575
1199031is shown in Figures 10 and 11 The amplitudes
of the steady-state transverse displacements of the mid-point of the deck beam with different numbers of Galerkin
Shock and Vibration 9
minus150
minus100
minus50
0
50
100
150
0 50 100minus50minus100
qr(0) = 011205751rqr(0) = 011205757r
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 7 Two different stable limit cycles corresponding to different initial configurations
q1
minus05
0
05
0 05minus05
q1
q2
minus05
0
05
0 05minus05
q2
minus05
0
05
q3
0 05minus05
q3
q4
minus05
0
05
0 05minus05
q4
q5
minus05
0
05
0 05minus05
q5
q6
minus05
0
05
0 05minus05
q6
q7
minus05
0
05
0 05minus05
q7
q8
minus05
0
05
0 05minus05
q8
q9
minus05
0
05
0 05minus05
q9
Figure 8 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903
truncation terms which are set to one through 30 are shownin Figure 10 As shown in Figure 10 a truncation with 11terms can provide accurate results in analyzing the dynamicresponse of the cable-stayed bridge and it is much lessthan the cases in Section 41 where the geometric nonlinearterms are neglected Phase portraits of the response of themidpoint of the deck beam for different numbers of Galerkin
truncation terms which are shown in Figure 11 also leadto the same conclusion It can be noted from Figure 11 thatthe limit cycle becomes more asymmetric when the numberof the truncation terms increases The reason for this isthat when the number of truncation terms increases to 7an additional asymmetric limit cycle appears on the (119902
7 7)
plane which phase portraits project onto (see Figure 12 with
10 Shock and Vibration
q1
minus1
0
1
0 03minus03
q1
q2
minus1
0
1
0 03minus03
q2
q3
minus1
0
1
0 03minus03
q3
q4
minus1
0
1
0 03minus03
q4
q5
minus1
0
1
0 03minus03
q5
q6
minus1
0
1
0 03minus03
q6
q7
minus1
0
1
0 03minus03
q7
q8
minus1
0
1
0 03minus03
q8
q9
minus1
0
1
0 03minus03
q9
Figure 9 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
7119903
5 10 15 20 25 300Number of truncation terms N
58
59
60
61
62
63
Am
plitu
de (m
m)
Figure 10 Effects of the number of Galerkin truncation terms ontransverse displacements of the midpoint of the deck beam whenthere are geometric nonlinear terms
119873 = 11) As shown in Figures 11 and 13 the dynamic responseof the midpoint of the deck beam approaches a stable limitcycle The bifurcation diagram of limit cycles with respectto 1199021(0) is shown in Figure 14 when the initial dynamic
configuration of the cable-stayed bridge corresponds to itsfirst mode shape One can find from Figure 14 that when1199021(0) is smaller than 027 a stable periodic solution exists and
minus50 0 50 100minus100minus100
minus50
0
50
100
N = 1
N = 10
N = 11
N = 30
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 11 Phase portraits of the response of themidpoint of the deckbeam for different numbers of Galerkin truncation terms 119902
119903(0) =
011205751119903
when 1199021(0) is larger than or equal to 027 chaotic responses
occur For instance when 1199021(0) = 03 and 119873 = 11 phase
portraits of the response of the midpoint of the deck beamare shown in Figure 15 which indicates occurrence of chaotic
Shock and Vibration 11
q1
minus50
0
50
0 50minus50
q1
q2
minus20
0
20
0 10minus10
q2
q3
minus20
0
20
0 10minus10
q3
q4
minus20
0
20
0 10minus10
q4
q5
minus20
0
20
0 10minus10
minus10
q5
q6
minus20
0
20
0 10minus10
q6
q7
minus20
0
20
0 10minus10
q7
q8
minus20
0
20
100
q8
q9
minus20
0
20
0 10minus10
q9
Figure 12 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903and119873 = 11
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 13 Time response of the midpoint of the deck beam 119902119903(0) =
011205751119903and119873 = 11
response On the contrary for a truncation with one termthe response always approaches a stable limit cycle no matterhow large the amplitude of the initial dynamic configurationis (see Figures 16ndash18) These mean that the true dynamicresponse of the cable-stayed bridge may not be capturedby the truncation with one term even when the initialdynamic configuration corresponds to the first mode shape
01 02 03 04 05 06 070q1(0)
minus200
minus150
minus100
minus50
0
50
Vb 4(X
b 4=0)
(mm
)
Figure 14 Bifurcation diagram of the cable-stayed bridge withrespect to 119902
1(0) 119902119903(0) = 119902
1(0)1205751119903and119873 = 11
When the initial dynamic configuration of the cable-stayedbridge corresponds to one of its higher mode shapes chaoticresponse occurs even if its amplitude is relatively small this isshown in Figure 19where the initial dynamic configuration ofthe cable-stayed bridge corresponds to its secondmode shapeand its magnitude is only 01
12 Shock and Vibration
0
minus100 0 100 200 300minus200
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus1500
minus1000
minus500
500
1000
1500
Figure 15 Chaotic response of the midpoint of the deck beam119902119903(0) = 03120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 16 Time response of themidpoint of the deck beam 1199021(0) =
01 and119873 = 1
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus1000
minus500
0
500
1000
Vb 4(X
b 4=0)
(mm
)
Figure 17 Time response of the midpoint of the deck beam 1199021(0) =
3 and119873 = 1
5 Conclusions
The dynamic behavior of a cable-stayed bridge that consistsof a simply supported four-cable-stayed deck beam andtwo rigid towers subjected to a distributed vortex shed-ding force on the deck beam has been investigated The
q1(0) = 01
q1(0) = 3
minus100
minus50
0
50
100
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 18 Phase portraits of the response of the midpoint of thedeck beam for different initial conditions119873 = 1
minus1500
minus1000
minus500
0
500
1000
1500
minus200 0 200 400minus400
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 19 Chaotic response of the midpoint of the deck beam119902119903(0) = 01120575
2119903
dynamic response of the cable-stayed bridge is calculatedusing Galerkin method in conjunction with Runge-Kutta-Felhberg method in MATLAB Convergence of Galerkinmethod for the dynamic response of the cable-stayed bridgeis studied Numerical simulations show that the geometricnonlinearities of the stay cables have significant influence onVIV of the cable-stayed bridge and further conclusions canbe summarized as follows
(1) In the case when the geometric nonlinear termsare neglected accurate calculation of the responseamplitude of the cable-stayed bridge at lock-in onlyneeds use of the first mode shape of the linearizedundamped cable-stayed bridgemodel when the initialdynamic configuration of the cable-stayed bridgecorresponds to its mode shape whose mode numberis smaller than seven There is a different limit cyclewhen the initial dynamic configuration correspondsto its mode shape whose mode number is equal to orlarger than 7
Shock and Vibration 13
(2) In the case when the geometric nonlinear termsare considered calculation of the response of thecable-stayed bridge generally needs use of multiplemode shapes of the linearized undamped cable-stayedbridge model even when the initial dynamic config-uration of the cable-stayed bridge corresponds to itsfirstmode shapeThere is a limit cycle when the initialdynamic configuration of the cable-stayed bridgecorresponds to its first mode shape and its amplitudeis smaller than 03 for the generalized coordinate andthere is chaotic response when the initial dynamicconfiguration of the cable-stayed bridge correspondsto its first mode shape with its amplitude larger than03 for the generalized coordinate or one of its highermode shapes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grant nos 11302087 and 11442006 theNatural Science Foundation of Jiangsu Province under Grantno BK20130479 the Research Foundation for AdvancedTalents of Jiangsu University under Grant no 13JDG068 andthe National Science Foundation under Grant no CMMI-1000830
References
[1] K Y R Billah and R H Scanlan ldquoResonance Tacoma NarrowsBridge failure and undergraduate physics textbooksrdquoAmericanJournal of Physics vol 59 no 2 pp 118ndash124 1991
[2] R D Blevins Flow-Induced Vibration Krieger New York NYUSA 2nd edition 1994
[3] H Zheng R E Price Y Modarres-Sadeghi and M S Tri-antafyllou ldquoOn fatigue damage of long flexible cylinders dueto the higher harmonic force components and chaotic vortex-induced vibrationsrdquo Ocean Engineering vol 88 pp 318ndash3292014
[4] C Evangelinos D Lucor and G E Karniadakis ldquoDNS-derivedforce distribution on flexible cylinders subject to vortex-induced vibrationrdquo Journal of Fluids and Structures vol 14 no3 pp 429ndash440 2000
[5] HM Blackburn R N Govardhan andCH KWilliamson ldquoAcomplementary numerical and physical investigation of vortex-inducedrdquo Journal of Fluids and Structures vol 15 no 3-4 pp481ndash488 2001
[6] R H J Willden and J M R Graham ldquoNumerical predictionof VIV on long flexible circular cylindersrdquo Journal of Fluids andStructures vol 15 no 3-4 pp 659ndash669 2001
[7] E Guilmineau and P Queutey ldquoA numerical simulation ofvortex shedding from an oscillating circular cylinderrdquo Journalof Fluids and Structures vol 16 no 6 pp 773ndash794 2002
[8] I Jadic R M C So and M P Mignolet ldquoAnalysis of fluidmdashstructure interactions using a time-marching techniquerdquo Jour-nal of Fluids and Structures vol 12 no 6 pp 631ndash654 1998
[9] C Y Zhou RM C So andK Lam ldquoVortex-induced vibrationsof an elastic circular cylinderrdquo Journal of Fluids and Structuresvol 13 no 2 pp 165ndash189 1999
[10] J R Meneghini and P W Bearman ldquoNumerical simulationof high amplitude oscillatory flow about a circular cylinderrdquoJournal of Fluids and Structures vol 9 no 4 pp 435ndash455 1995
[11] T Sarpkaya ldquoComputational methods with vorticesrdquo Journal ofFluids Engineering vol 111 no 1 pp 5ndash52 1989
[12] T Sarpkaya ldquoVortex element methods for flow simulationrdquoAdvances in Applied Mechanics vol 31 pp 113ndash247 1994
[13] R D Gabbai and H Benaroya ldquoAn overview of modeling andexperiments of vortex-induced vibration of circular cylindersrdquoJournal of Sound and Vibration vol 282 no 3ndash5 pp 575ndash6162005
[14] R H Scanlan ldquoOn the state-of-the-artmethods for calculationsof flutter vortex-induced and buffeting response of bridgestructuresrdquo Tech Rep FHWARD-80050 National TechnicalInformation Service Springfield Va USA 1981
[15] RH Scanlan ldquoBridge flutter derivatives at vortex lock-inrdquo Jour-nal of Structural Engineering vol 124 no 4 pp 450ndash458 1998
[16] B J Vickery and R I Basu ldquoAcross-wind vibrations of struc-tures of circular cross-section Part I Development of a math-ematical model for two-dimensional conditionsrdquo Journal ofWind Engineering and Industrial Aerodynamics vol 12 no 1 pp49ndash73 1983
[17] A Larsen ldquoA generalized model for assessment of vortex-induced vibrations of flexible structuresrdquo Journal of Wind Engi-neering and Industrial Aerodynamics vol 57 no 2-3 pp 281ndash294 1995
[18] T Sarpkaya ldquoFluid forces on oscillating cylindersrdquo Journal ofWaterway Port Coastal and Ocean Division (ASCE) vol 104no 3 pp 275ndash290 1978
[19] T Staubli ldquoCalculation of vibration of an elastically mountedcylinder using experimental data from a forced oscillationrdquoJournal of Fluids Engineering vol 105 no 2 pp 225ndash229 1983
[20] W D Iwan and D L R Botelho ldquoVortex-induced oscillationof structures in waterrdquo Journal of Waterway Port Coastal andOcean Engineering vol 111 no 2 pp 289ndash303 1985
[21] R T Hartlen and I G Currie ldquoLift-oscillator model of vortex-induced vibrationrdquo Journal of Engineering Mechanics Division(ASCE) vol 96 no 5 pp 577ndash591 1970
[22] R A Skop and O M Griffin ldquoA model for the vortex-excitedresonant response of bluff cylindersrdquo Journal of Sound andVibration vol 27 no 2 pp 225ndash233 1973
[23] WD Iwan andRD Blevins ldquoAmodel for vortex induced oscil-lation of structuresrdquo Journal of Applied Mechanics TransactionsASME vol 41 no 3 pp 581ndash586 1974
[24] E H Dowell ldquoNon-linear oscillator models in bluff body aero-elasticityrdquo Journal of Sound and Vibration vol 75 no 2 pp 251ndash264 1981
[25] R H M Ogink and A V Metrikine ldquoA wake oscillator withfrequency dependent coupling for the modeling of vortex-induced vibrationrdquo Journal of Sound and Vibration vol 329 no26 pp 5452ndash5473 2010
[26] X Bai and W Qin ldquoUsing vortex strength wake oscillatorin modelling of vortex induced vibrations in two degrees offreedomrdquo European Journal of Mechanics BFluids vol 48 pp165ndash173 2014
[27] F Ehsan and R H Scanlan ldquoVortexminusinduced vibrations offlexible bridgesrdquo Journal of Engineering Mechanics vol 116 no6 pp 1392ndash1411 1990
14 Shock and Vibration
[28] A M Marra C Mannini and G Bartoli ldquoVan der Pol-typeequation for modeling vortex-induced oscillations of bridgedecksrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 99 no 6-7 pp 776ndash785 2011
[29] D Q Cao M T Song W D Zhu R W Tucker and C H-T Wang ldquoModeling and analysis of the in-plane vibration ofa complex cable-stayed bridgerdquo Journal of Sound and Vibrationvol 331 no 26 pp 5685ndash5714 2012
[30] E Caetano A Cunha V Gattulli and M Lepidi ldquoCable-deckdynamic interactions at the International Guadiana Bridge on-site measurements and finite element modellingrdquo StructuralControl and Health Monitoring vol 15 no 3 pp 237ndash264 2008
[31] I Akhtar A H Nayfeh and C J Ribbens ldquoOn the stability andextension of reduced-order Galerkin models in incompressibleflowsrdquo Theoretical and Computational Fluid Dynamics vol 23no 3 pp 213ndash237 2009
[32] S Sirisup and G E Karniadakis ldquoA spectral viscosity methodfor correcting the long-term behavior of POD modelsrdquo Journalof Computational Physics vol 194 no 1 pp 92ndash116 2004
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 Shock and Vibration
615
62
625
63
Am
plitu
de (m
m)
5 10 15 20 25 300Number of truncation terms N
Figure 2 Effects of the number of Galerkin truncation terms on transverse displacements of the midpoint of the deck beam when there areno geometric nonlinear terms
It should be noted that nonlinear termsNQ andNC in (29)are induced by geometric nonlinearities of the stay cables
4 Numerical Simulation
Geometric and physical parameters of a cable-stayed bridgeand aeroelastic parameters are listed in Table 1 It shouldbe noted that a width-to-height ratio of 4 is used in thispaper not only because it is a typical ratio of bridge decksbut also because the cable-stayed bridge is supposed tohave significant response amplitudes at lock-in [28] Forall the following calculation modal damping ratio 119862
119903119904is
always equal to 001120575119903119904 The dynamic response of the cable-
stayed bridge can be calculated from (29) using Runge-Kutta-Felhbergmethod inMATLABwhere initial conditionsof generalized coordinates can be obtained from those ofphysical coordinates
119902119903(0) =
1
119872119903
[
[
4
sum
119894=1
1205941198881198941205732
119888119894int
119897119888119894
0
V119888119894(119909119888119894 0) 120601(119903)
119888119894(119909119888119894) 119889119909119888119894
+
7
sum
119895=1
1205781198871198951205734
119887119895int
119897119887119895
minus119897119887119895
V119887119895(119909119887119895 0) 120601(119903)
119887119895(119909119887119895) 119889119909119887119895
]
]
119903(0) =
1
119872119903
[
[
4
sum
119894=1
1205941198881198941205732
119888119894int
119897119888119894
0
V119888119894(119909119888119894 0) 120601(119903)
119888119894(119909119888119894) 119889119909119888119894
+
7
sum
119895=1
1205781198871198951205734
119887119895int
119897119887119895
minus119897119887119895
V119887119895(119909119887119895 0) 120601(119903)
119887119895(119909119887119895) 119889119909119887119895
]
]
(31)
For all the following calculation 119903(0) is always equal to
zero Numerical simulations for two cases are undertaken (1)neglecting the geometric nonlinear terms (ie NQ
= NC= 0
in (29)) and (2) considering the geometric nonlinear terms
41 Case Studies Neglecting the Geometric Nonlinear TermsConvergence of Galerkin method for given initial conditions119902119903(0) = 01120575
1199031 which means that the cable-stayed bridge
has its initial dynamic configuration corresponding to itsfirst mode shape and it is released from rest is shown in
Figures 2 and 3The amplitudes of the steady-state transversedisplacements of the midpoint of the deck beam withdifferent numbers of Galerkin truncation terms which areset to one through 30 are shown in Figure 2 As shown inFigure 2 the transverse displacements of the midpoint of thedeck beam initially remain stable until the truncation numberincreases to 15 and suddenly increase to their convergedvalues A truncation with 20 terms can provide accurateresults in analyzing the dynamic response of the deck beamThe difference between the amplitudes of two limit cyclesobtained by truncation with one term and 20 terms is194 which is small One can come to a conclusion that atruncation with one term is accurate enough for calculationof the steady-state response of the cable-stayed bridge whenits initial dynamic configuration corresponds to its first modeshape In other words most of the energy of the cable-stayedbridge is concentrated in its first mode in this case Phaseportraits of the response of the midpoint of the deck beamfor different numbers of Galerkin truncation terms whichis shown in Figure 3 also lead to the same conclusion Otherresults that are not shown here for the sake of brevity indicatethat Galerkin truncation with 20 terms yields accurateresults for the dynamic response of the cable-stayed bridgein the following cases in this section Hence in the followingnumerical calculations in this section the first 20 modes ofthe linearized undamped cable-stayed bridge model are usedin Galerkin method Time history responses of the midpointof the deck beam when the initial dynamic configuration ofthe cable-stayed bridge corresponds to its first mode shapebut has different amplitudes that is 119902
1(0) = 01 (the initial
displacement of themidpoint of the deck beam is 32mm) and1199021(0) = 3 (the initial displacement of themidpoint of the deck
beam is 946mm which is rather large) are shown in Figures4 and 5 respectively It can be seen that the solutions with dif-ferent 119902
1(0) converge to the same limit cycle after long-time
integrationThemagnitudes of Floquet multipliers are all lessthan unity the aforementioned limit cycle is asymptoticallystable
Solutions of a reduced-order model for a flow dynamicsystem can converge to a spurious limit cycle after long-timeintegration even if it is initializedwith a correct configuration
Shock and Vibration 7
Table 1 Geometric and physical parameters of the cable-stayed bridge and aeroelastic parameters
Parameter Unit Value
Deck beam
Mass per unit length of the deck beam (119898119887119895) kgm 16940
Elastic modulus of the deck beam (119864119887119895) Nm2 20 times 1011
Area moment of inertia of the deck beam (119868119887119895) m4 120
Length of segment 1198871of the deck beam (2119871
1198871) m 35
Length of segment 1198872of the deck beam (2119871
1198872) m 40
Length of segment 1198873of the deck beam (2119871
1198873) m 50
Length of segment 1198874of the deck beam (2119871
1198874) m 50
Length of segment 1198875of the deck beam (2119871
1198875) m 50
Length of segment 1198876of the deck beam (2119871
1198876) m 40
Length of segment 1198877of the deck beam (2119871
1198877) m 35
Stay cables
Mass per unit length of the stay cables (119898119888119894) kgm 286
Elastic modulus of the stay cables (119864119888119894) Nm2 20 times 1011
Cross-sectional area of the stay cables (119860119888119894) m2 00362
Length of stay cable 1198881(1198711198881) m 52
Length of stay cable 1198882(1198711198882) m 60
Length of stay cable 1198883(1198711198883) m 60
Length of stay cable 1198884(1198711198884) m 52
Sag-to-span ratios of the stay cables (119889119888119894= 119863119888119894119871119888119894) 001
Aeroelastic parametersAir density (120588) kgm3 1205120576 11225 [28]1198841
688 [28]
N = 1
N = 19
N = 20
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus100
minus50
0
50
100
Figure 3 Phase portraits of the response of themidpoint of the deck beam for different numbers of Galerkin truncation terms 119902119903(0) = 01120575
1119903
[31 32] Sirisup and Karniadakis [32] demonstrated that theonset of divergence from the correct limit cycle dependson the number of Galerkin truncation terms the Reynoldsnumber and the flow geometry Since the limit cycle heredoes not vanish even when the truncation number is 30which is large one can conclude that the aforementionedlimit cycle for the cable-stayed bridge is a correct one
In many cases higher mode shapes of the cable-stayedbridge would be excited one such case is that when thereare vehicles moving on the deck beam of the bridge Hence
the initial dynamic configuration of the cable-stayed bridgecan correspond to its higher mode shapes in the dynamicanalysis of the cable-stayed bridge subjected to a distributedvortex shedding force It is obvious that Galerkin truncationwith one term is not enough in these cases Through thesame method with 119873 = 20 one can find that there isthe same limit cycle as that in Figure 3 when the initialdynamic configuration corresponds to the second throughsixth mode shapes and there is a different limit cycle whenthe initial dynamic configuration corresponds to the seventh
8 Shock and Vibration
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus80
minus40
0
40
80
Vb 4(X
b 4=0)
(mm
)
Figure 4 Time response of the midpoint of the deck beam 119902119903(0) = 01120575
1119903
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus800
minus400
0
400
800
Vb 4(X
b 4=0)
(mm
)
Figure 5 Time response of the midpoint of the deck beam 119902119903(0) = 3120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 6 Time response of the midpoint of the deck beam 119902119903(0) = 01120575
7119903
mode shape (Figures 6 and 7) Two-dimensional projectionsof phase portraits onto the (119902
119903 119903) plane when 119902
119903(0) = 01120575
1119903
and 119902119903(0) = 01120575
7119903are shown in Figures 8 and 9 respectively
Results when 119902119903(0) is equal to 01120575
2119903 011205753119903 and 01120575
6119903
are the same as that when 119902119903(0) = 01120575
1119903 they are not shown
here for the sake of brevity It can be seen from Figures 8 and9 that energy of the cable-stayed bridge is concentrated inthe first mode (the seventh mode) when its initial dynamic
configuration corresponds to the first through sixth modeshapes (the seventh mode shape)
42 Case Studies Considering the Geometric Nonlinear TermsConvergence of Galerkin method for given initial conditions119902119903(0) = 01120575
1199031is shown in Figures 10 and 11 The amplitudes
of the steady-state transverse displacements of the mid-point of the deck beam with different numbers of Galerkin
Shock and Vibration 9
minus150
minus100
minus50
0
50
100
150
0 50 100minus50minus100
qr(0) = 011205751rqr(0) = 011205757r
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 7 Two different stable limit cycles corresponding to different initial configurations
q1
minus05
0
05
0 05minus05
q1
q2
minus05
0
05
0 05minus05
q2
minus05
0
05
q3
0 05minus05
q3
q4
minus05
0
05
0 05minus05
q4
q5
minus05
0
05
0 05minus05
q5
q6
minus05
0
05
0 05minus05
q6
q7
minus05
0
05
0 05minus05
q7
q8
minus05
0
05
0 05minus05
q8
q9
minus05
0
05
0 05minus05
q9
Figure 8 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903
truncation terms which are set to one through 30 are shownin Figure 10 As shown in Figure 10 a truncation with 11terms can provide accurate results in analyzing the dynamicresponse of the cable-stayed bridge and it is much lessthan the cases in Section 41 where the geometric nonlinearterms are neglected Phase portraits of the response of themidpoint of the deck beam for different numbers of Galerkin
truncation terms which are shown in Figure 11 also leadto the same conclusion It can be noted from Figure 11 thatthe limit cycle becomes more asymmetric when the numberof the truncation terms increases The reason for this isthat when the number of truncation terms increases to 7an additional asymmetric limit cycle appears on the (119902
7 7)
plane which phase portraits project onto (see Figure 12 with
10 Shock and Vibration
q1
minus1
0
1
0 03minus03
q1
q2
minus1
0
1
0 03minus03
q2
q3
minus1
0
1
0 03minus03
q3
q4
minus1
0
1
0 03minus03
q4
q5
minus1
0
1
0 03minus03
q5
q6
minus1
0
1
0 03minus03
q6
q7
minus1
0
1
0 03minus03
q7
q8
minus1
0
1
0 03minus03
q8
q9
minus1
0
1
0 03minus03
q9
Figure 9 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
7119903
5 10 15 20 25 300Number of truncation terms N
58
59
60
61
62
63
Am
plitu
de (m
m)
Figure 10 Effects of the number of Galerkin truncation terms ontransverse displacements of the midpoint of the deck beam whenthere are geometric nonlinear terms
119873 = 11) As shown in Figures 11 and 13 the dynamic responseof the midpoint of the deck beam approaches a stable limitcycle The bifurcation diagram of limit cycles with respectto 1199021(0) is shown in Figure 14 when the initial dynamic
configuration of the cable-stayed bridge corresponds to itsfirst mode shape One can find from Figure 14 that when1199021(0) is smaller than 027 a stable periodic solution exists and
minus50 0 50 100minus100minus100
minus50
0
50
100
N = 1
N = 10
N = 11
N = 30
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 11 Phase portraits of the response of themidpoint of the deckbeam for different numbers of Galerkin truncation terms 119902
119903(0) =
011205751119903
when 1199021(0) is larger than or equal to 027 chaotic responses
occur For instance when 1199021(0) = 03 and 119873 = 11 phase
portraits of the response of the midpoint of the deck beamare shown in Figure 15 which indicates occurrence of chaotic
Shock and Vibration 11
q1
minus50
0
50
0 50minus50
q1
q2
minus20
0
20
0 10minus10
q2
q3
minus20
0
20
0 10minus10
q3
q4
minus20
0
20
0 10minus10
q4
q5
minus20
0
20
0 10minus10
minus10
q5
q6
minus20
0
20
0 10minus10
q6
q7
minus20
0
20
0 10minus10
q7
q8
minus20
0
20
100
q8
q9
minus20
0
20
0 10minus10
q9
Figure 12 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903and119873 = 11
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 13 Time response of the midpoint of the deck beam 119902119903(0) =
011205751119903and119873 = 11
response On the contrary for a truncation with one termthe response always approaches a stable limit cycle no matterhow large the amplitude of the initial dynamic configurationis (see Figures 16ndash18) These mean that the true dynamicresponse of the cable-stayed bridge may not be capturedby the truncation with one term even when the initialdynamic configuration corresponds to the first mode shape
01 02 03 04 05 06 070q1(0)
minus200
minus150
minus100
minus50
0
50
Vb 4(X
b 4=0)
(mm
)
Figure 14 Bifurcation diagram of the cable-stayed bridge withrespect to 119902
1(0) 119902119903(0) = 119902
1(0)1205751119903and119873 = 11
When the initial dynamic configuration of the cable-stayedbridge corresponds to one of its higher mode shapes chaoticresponse occurs even if its amplitude is relatively small this isshown in Figure 19where the initial dynamic configuration ofthe cable-stayed bridge corresponds to its secondmode shapeand its magnitude is only 01
12 Shock and Vibration
0
minus100 0 100 200 300minus200
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus1500
minus1000
minus500
500
1000
1500
Figure 15 Chaotic response of the midpoint of the deck beam119902119903(0) = 03120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 16 Time response of themidpoint of the deck beam 1199021(0) =
01 and119873 = 1
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus1000
minus500
0
500
1000
Vb 4(X
b 4=0)
(mm
)
Figure 17 Time response of the midpoint of the deck beam 1199021(0) =
3 and119873 = 1
5 Conclusions
The dynamic behavior of a cable-stayed bridge that consistsof a simply supported four-cable-stayed deck beam andtwo rigid towers subjected to a distributed vortex shed-ding force on the deck beam has been investigated The
q1(0) = 01
q1(0) = 3
minus100
minus50
0
50
100
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 18 Phase portraits of the response of the midpoint of thedeck beam for different initial conditions119873 = 1
minus1500
minus1000
minus500
0
500
1000
1500
minus200 0 200 400minus400
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 19 Chaotic response of the midpoint of the deck beam119902119903(0) = 01120575
2119903
dynamic response of the cable-stayed bridge is calculatedusing Galerkin method in conjunction with Runge-Kutta-Felhberg method in MATLAB Convergence of Galerkinmethod for the dynamic response of the cable-stayed bridgeis studied Numerical simulations show that the geometricnonlinearities of the stay cables have significant influence onVIV of the cable-stayed bridge and further conclusions canbe summarized as follows
(1) In the case when the geometric nonlinear termsare neglected accurate calculation of the responseamplitude of the cable-stayed bridge at lock-in onlyneeds use of the first mode shape of the linearizedundamped cable-stayed bridgemodel when the initialdynamic configuration of the cable-stayed bridgecorresponds to its mode shape whose mode numberis smaller than seven There is a different limit cyclewhen the initial dynamic configuration correspondsto its mode shape whose mode number is equal to orlarger than 7
Shock and Vibration 13
(2) In the case when the geometric nonlinear termsare considered calculation of the response of thecable-stayed bridge generally needs use of multiplemode shapes of the linearized undamped cable-stayedbridge model even when the initial dynamic config-uration of the cable-stayed bridge corresponds to itsfirstmode shapeThere is a limit cycle when the initialdynamic configuration of the cable-stayed bridgecorresponds to its first mode shape and its amplitudeis smaller than 03 for the generalized coordinate andthere is chaotic response when the initial dynamicconfiguration of the cable-stayed bridge correspondsto its first mode shape with its amplitude larger than03 for the generalized coordinate or one of its highermode shapes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grant nos 11302087 and 11442006 theNatural Science Foundation of Jiangsu Province under Grantno BK20130479 the Research Foundation for AdvancedTalents of Jiangsu University under Grant no 13JDG068 andthe National Science Foundation under Grant no CMMI-1000830
References
[1] K Y R Billah and R H Scanlan ldquoResonance Tacoma NarrowsBridge failure and undergraduate physics textbooksrdquoAmericanJournal of Physics vol 59 no 2 pp 118ndash124 1991
[2] R D Blevins Flow-Induced Vibration Krieger New York NYUSA 2nd edition 1994
[3] H Zheng R E Price Y Modarres-Sadeghi and M S Tri-antafyllou ldquoOn fatigue damage of long flexible cylinders dueto the higher harmonic force components and chaotic vortex-induced vibrationsrdquo Ocean Engineering vol 88 pp 318ndash3292014
[4] C Evangelinos D Lucor and G E Karniadakis ldquoDNS-derivedforce distribution on flexible cylinders subject to vortex-induced vibrationrdquo Journal of Fluids and Structures vol 14 no3 pp 429ndash440 2000
[5] HM Blackburn R N Govardhan andCH KWilliamson ldquoAcomplementary numerical and physical investigation of vortex-inducedrdquo Journal of Fluids and Structures vol 15 no 3-4 pp481ndash488 2001
[6] R H J Willden and J M R Graham ldquoNumerical predictionof VIV on long flexible circular cylindersrdquo Journal of Fluids andStructures vol 15 no 3-4 pp 659ndash669 2001
[7] E Guilmineau and P Queutey ldquoA numerical simulation ofvortex shedding from an oscillating circular cylinderrdquo Journalof Fluids and Structures vol 16 no 6 pp 773ndash794 2002
[8] I Jadic R M C So and M P Mignolet ldquoAnalysis of fluidmdashstructure interactions using a time-marching techniquerdquo Jour-nal of Fluids and Structures vol 12 no 6 pp 631ndash654 1998
[9] C Y Zhou RM C So andK Lam ldquoVortex-induced vibrationsof an elastic circular cylinderrdquo Journal of Fluids and Structuresvol 13 no 2 pp 165ndash189 1999
[10] J R Meneghini and P W Bearman ldquoNumerical simulationof high amplitude oscillatory flow about a circular cylinderrdquoJournal of Fluids and Structures vol 9 no 4 pp 435ndash455 1995
[11] T Sarpkaya ldquoComputational methods with vorticesrdquo Journal ofFluids Engineering vol 111 no 1 pp 5ndash52 1989
[12] T Sarpkaya ldquoVortex element methods for flow simulationrdquoAdvances in Applied Mechanics vol 31 pp 113ndash247 1994
[13] R D Gabbai and H Benaroya ldquoAn overview of modeling andexperiments of vortex-induced vibration of circular cylindersrdquoJournal of Sound and Vibration vol 282 no 3ndash5 pp 575ndash6162005
[14] R H Scanlan ldquoOn the state-of-the-artmethods for calculationsof flutter vortex-induced and buffeting response of bridgestructuresrdquo Tech Rep FHWARD-80050 National TechnicalInformation Service Springfield Va USA 1981
[15] RH Scanlan ldquoBridge flutter derivatives at vortex lock-inrdquo Jour-nal of Structural Engineering vol 124 no 4 pp 450ndash458 1998
[16] B J Vickery and R I Basu ldquoAcross-wind vibrations of struc-tures of circular cross-section Part I Development of a math-ematical model for two-dimensional conditionsrdquo Journal ofWind Engineering and Industrial Aerodynamics vol 12 no 1 pp49ndash73 1983
[17] A Larsen ldquoA generalized model for assessment of vortex-induced vibrations of flexible structuresrdquo Journal of Wind Engi-neering and Industrial Aerodynamics vol 57 no 2-3 pp 281ndash294 1995
[18] T Sarpkaya ldquoFluid forces on oscillating cylindersrdquo Journal ofWaterway Port Coastal and Ocean Division (ASCE) vol 104no 3 pp 275ndash290 1978
[19] T Staubli ldquoCalculation of vibration of an elastically mountedcylinder using experimental data from a forced oscillationrdquoJournal of Fluids Engineering vol 105 no 2 pp 225ndash229 1983
[20] W D Iwan and D L R Botelho ldquoVortex-induced oscillationof structures in waterrdquo Journal of Waterway Port Coastal andOcean Engineering vol 111 no 2 pp 289ndash303 1985
[21] R T Hartlen and I G Currie ldquoLift-oscillator model of vortex-induced vibrationrdquo Journal of Engineering Mechanics Division(ASCE) vol 96 no 5 pp 577ndash591 1970
[22] R A Skop and O M Griffin ldquoA model for the vortex-excitedresonant response of bluff cylindersrdquo Journal of Sound andVibration vol 27 no 2 pp 225ndash233 1973
[23] WD Iwan andRD Blevins ldquoAmodel for vortex induced oscil-lation of structuresrdquo Journal of Applied Mechanics TransactionsASME vol 41 no 3 pp 581ndash586 1974
[24] E H Dowell ldquoNon-linear oscillator models in bluff body aero-elasticityrdquo Journal of Sound and Vibration vol 75 no 2 pp 251ndash264 1981
[25] R H M Ogink and A V Metrikine ldquoA wake oscillator withfrequency dependent coupling for the modeling of vortex-induced vibrationrdquo Journal of Sound and Vibration vol 329 no26 pp 5452ndash5473 2010
[26] X Bai and W Qin ldquoUsing vortex strength wake oscillatorin modelling of vortex induced vibrations in two degrees offreedomrdquo European Journal of Mechanics BFluids vol 48 pp165ndash173 2014
[27] F Ehsan and R H Scanlan ldquoVortexminusinduced vibrations offlexible bridgesrdquo Journal of Engineering Mechanics vol 116 no6 pp 1392ndash1411 1990
14 Shock and Vibration
[28] A M Marra C Mannini and G Bartoli ldquoVan der Pol-typeequation for modeling vortex-induced oscillations of bridgedecksrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 99 no 6-7 pp 776ndash785 2011
[29] D Q Cao M T Song W D Zhu R W Tucker and C H-T Wang ldquoModeling and analysis of the in-plane vibration ofa complex cable-stayed bridgerdquo Journal of Sound and Vibrationvol 331 no 26 pp 5685ndash5714 2012
[30] E Caetano A Cunha V Gattulli and M Lepidi ldquoCable-deckdynamic interactions at the International Guadiana Bridge on-site measurements and finite element modellingrdquo StructuralControl and Health Monitoring vol 15 no 3 pp 237ndash264 2008
[31] I Akhtar A H Nayfeh and C J Ribbens ldquoOn the stability andextension of reduced-order Galerkin models in incompressibleflowsrdquo Theoretical and Computational Fluid Dynamics vol 23no 3 pp 213ndash237 2009
[32] S Sirisup and G E Karniadakis ldquoA spectral viscosity methodfor correcting the long-term behavior of POD modelsrdquo Journalof Computational Physics vol 194 no 1 pp 92ndash116 2004
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 7
Table 1 Geometric and physical parameters of the cable-stayed bridge and aeroelastic parameters
Parameter Unit Value
Deck beam
Mass per unit length of the deck beam (119898119887119895) kgm 16940
Elastic modulus of the deck beam (119864119887119895) Nm2 20 times 1011
Area moment of inertia of the deck beam (119868119887119895) m4 120
Length of segment 1198871of the deck beam (2119871
1198871) m 35
Length of segment 1198872of the deck beam (2119871
1198872) m 40
Length of segment 1198873of the deck beam (2119871
1198873) m 50
Length of segment 1198874of the deck beam (2119871
1198874) m 50
Length of segment 1198875of the deck beam (2119871
1198875) m 50
Length of segment 1198876of the deck beam (2119871
1198876) m 40
Length of segment 1198877of the deck beam (2119871
1198877) m 35
Stay cables
Mass per unit length of the stay cables (119898119888119894) kgm 286
Elastic modulus of the stay cables (119864119888119894) Nm2 20 times 1011
Cross-sectional area of the stay cables (119860119888119894) m2 00362
Length of stay cable 1198881(1198711198881) m 52
Length of stay cable 1198882(1198711198882) m 60
Length of stay cable 1198883(1198711198883) m 60
Length of stay cable 1198884(1198711198884) m 52
Sag-to-span ratios of the stay cables (119889119888119894= 119863119888119894119871119888119894) 001
Aeroelastic parametersAir density (120588) kgm3 1205120576 11225 [28]1198841
688 [28]
N = 1
N = 19
N = 20
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus100
minus50
0
50
100
Figure 3 Phase portraits of the response of themidpoint of the deck beam for different numbers of Galerkin truncation terms 119902119903(0) = 01120575
1119903
[31 32] Sirisup and Karniadakis [32] demonstrated that theonset of divergence from the correct limit cycle dependson the number of Galerkin truncation terms the Reynoldsnumber and the flow geometry Since the limit cycle heredoes not vanish even when the truncation number is 30which is large one can conclude that the aforementionedlimit cycle for the cable-stayed bridge is a correct one
In many cases higher mode shapes of the cable-stayedbridge would be excited one such case is that when thereare vehicles moving on the deck beam of the bridge Hence
the initial dynamic configuration of the cable-stayed bridgecan correspond to its higher mode shapes in the dynamicanalysis of the cable-stayed bridge subjected to a distributedvortex shedding force It is obvious that Galerkin truncationwith one term is not enough in these cases Through thesame method with 119873 = 20 one can find that there isthe same limit cycle as that in Figure 3 when the initialdynamic configuration corresponds to the second throughsixth mode shapes and there is a different limit cycle whenthe initial dynamic configuration corresponds to the seventh
8 Shock and Vibration
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus80
minus40
0
40
80
Vb 4(X
b 4=0)
(mm
)
Figure 4 Time response of the midpoint of the deck beam 119902119903(0) = 01120575
1119903
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus800
minus400
0
400
800
Vb 4(X
b 4=0)
(mm
)
Figure 5 Time response of the midpoint of the deck beam 119902119903(0) = 3120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 6 Time response of the midpoint of the deck beam 119902119903(0) = 01120575
7119903
mode shape (Figures 6 and 7) Two-dimensional projectionsof phase portraits onto the (119902
119903 119903) plane when 119902
119903(0) = 01120575
1119903
and 119902119903(0) = 01120575
7119903are shown in Figures 8 and 9 respectively
Results when 119902119903(0) is equal to 01120575
2119903 011205753119903 and 01120575
6119903
are the same as that when 119902119903(0) = 01120575
1119903 they are not shown
here for the sake of brevity It can be seen from Figures 8 and9 that energy of the cable-stayed bridge is concentrated inthe first mode (the seventh mode) when its initial dynamic
configuration corresponds to the first through sixth modeshapes (the seventh mode shape)
42 Case Studies Considering the Geometric Nonlinear TermsConvergence of Galerkin method for given initial conditions119902119903(0) = 01120575
1199031is shown in Figures 10 and 11 The amplitudes
of the steady-state transverse displacements of the mid-point of the deck beam with different numbers of Galerkin
Shock and Vibration 9
minus150
minus100
minus50
0
50
100
150
0 50 100minus50minus100
qr(0) = 011205751rqr(0) = 011205757r
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 7 Two different stable limit cycles corresponding to different initial configurations
q1
minus05
0
05
0 05minus05
q1
q2
minus05
0
05
0 05minus05
q2
minus05
0
05
q3
0 05minus05
q3
q4
minus05
0
05
0 05minus05
q4
q5
minus05
0
05
0 05minus05
q5
q6
minus05
0
05
0 05minus05
q6
q7
minus05
0
05
0 05minus05
q7
q8
minus05
0
05
0 05minus05
q8
q9
minus05
0
05
0 05minus05
q9
Figure 8 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903
truncation terms which are set to one through 30 are shownin Figure 10 As shown in Figure 10 a truncation with 11terms can provide accurate results in analyzing the dynamicresponse of the cable-stayed bridge and it is much lessthan the cases in Section 41 where the geometric nonlinearterms are neglected Phase portraits of the response of themidpoint of the deck beam for different numbers of Galerkin
truncation terms which are shown in Figure 11 also leadto the same conclusion It can be noted from Figure 11 thatthe limit cycle becomes more asymmetric when the numberof the truncation terms increases The reason for this isthat when the number of truncation terms increases to 7an additional asymmetric limit cycle appears on the (119902
7 7)
plane which phase portraits project onto (see Figure 12 with
10 Shock and Vibration
q1
minus1
0
1
0 03minus03
q1
q2
minus1
0
1
0 03minus03
q2
q3
minus1
0
1
0 03minus03
q3
q4
minus1
0
1
0 03minus03
q4
q5
minus1
0
1
0 03minus03
q5
q6
minus1
0
1
0 03minus03
q6
q7
minus1
0
1
0 03minus03
q7
q8
minus1
0
1
0 03minus03
q8
q9
minus1
0
1
0 03minus03
q9
Figure 9 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
7119903
5 10 15 20 25 300Number of truncation terms N
58
59
60
61
62
63
Am
plitu
de (m
m)
Figure 10 Effects of the number of Galerkin truncation terms ontransverse displacements of the midpoint of the deck beam whenthere are geometric nonlinear terms
119873 = 11) As shown in Figures 11 and 13 the dynamic responseof the midpoint of the deck beam approaches a stable limitcycle The bifurcation diagram of limit cycles with respectto 1199021(0) is shown in Figure 14 when the initial dynamic
configuration of the cable-stayed bridge corresponds to itsfirst mode shape One can find from Figure 14 that when1199021(0) is smaller than 027 a stable periodic solution exists and
minus50 0 50 100minus100minus100
minus50
0
50
100
N = 1
N = 10
N = 11
N = 30
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 11 Phase portraits of the response of themidpoint of the deckbeam for different numbers of Galerkin truncation terms 119902
119903(0) =
011205751119903
when 1199021(0) is larger than or equal to 027 chaotic responses
occur For instance when 1199021(0) = 03 and 119873 = 11 phase
portraits of the response of the midpoint of the deck beamare shown in Figure 15 which indicates occurrence of chaotic
Shock and Vibration 11
q1
minus50
0
50
0 50minus50
q1
q2
minus20
0
20
0 10minus10
q2
q3
minus20
0
20
0 10minus10
q3
q4
minus20
0
20
0 10minus10
q4
q5
minus20
0
20
0 10minus10
minus10
q5
q6
minus20
0
20
0 10minus10
q6
q7
minus20
0
20
0 10minus10
q7
q8
minus20
0
20
100
q8
q9
minus20
0
20
0 10minus10
q9
Figure 12 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903and119873 = 11
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 13 Time response of the midpoint of the deck beam 119902119903(0) =
011205751119903and119873 = 11
response On the contrary for a truncation with one termthe response always approaches a stable limit cycle no matterhow large the amplitude of the initial dynamic configurationis (see Figures 16ndash18) These mean that the true dynamicresponse of the cable-stayed bridge may not be capturedby the truncation with one term even when the initialdynamic configuration corresponds to the first mode shape
01 02 03 04 05 06 070q1(0)
minus200
minus150
minus100
minus50
0
50
Vb 4(X
b 4=0)
(mm
)
Figure 14 Bifurcation diagram of the cable-stayed bridge withrespect to 119902
1(0) 119902119903(0) = 119902
1(0)1205751119903and119873 = 11
When the initial dynamic configuration of the cable-stayedbridge corresponds to one of its higher mode shapes chaoticresponse occurs even if its amplitude is relatively small this isshown in Figure 19where the initial dynamic configuration ofthe cable-stayed bridge corresponds to its secondmode shapeand its magnitude is only 01
12 Shock and Vibration
0
minus100 0 100 200 300minus200
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus1500
minus1000
minus500
500
1000
1500
Figure 15 Chaotic response of the midpoint of the deck beam119902119903(0) = 03120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 16 Time response of themidpoint of the deck beam 1199021(0) =
01 and119873 = 1
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus1000
minus500
0
500
1000
Vb 4(X
b 4=0)
(mm
)
Figure 17 Time response of the midpoint of the deck beam 1199021(0) =
3 and119873 = 1
5 Conclusions
The dynamic behavior of a cable-stayed bridge that consistsof a simply supported four-cable-stayed deck beam andtwo rigid towers subjected to a distributed vortex shed-ding force on the deck beam has been investigated The
q1(0) = 01
q1(0) = 3
minus100
minus50
0
50
100
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 18 Phase portraits of the response of the midpoint of thedeck beam for different initial conditions119873 = 1
minus1500
minus1000
minus500
0
500
1000
1500
minus200 0 200 400minus400
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 19 Chaotic response of the midpoint of the deck beam119902119903(0) = 01120575
2119903
dynamic response of the cable-stayed bridge is calculatedusing Galerkin method in conjunction with Runge-Kutta-Felhberg method in MATLAB Convergence of Galerkinmethod for the dynamic response of the cable-stayed bridgeis studied Numerical simulations show that the geometricnonlinearities of the stay cables have significant influence onVIV of the cable-stayed bridge and further conclusions canbe summarized as follows
(1) In the case when the geometric nonlinear termsare neglected accurate calculation of the responseamplitude of the cable-stayed bridge at lock-in onlyneeds use of the first mode shape of the linearizedundamped cable-stayed bridgemodel when the initialdynamic configuration of the cable-stayed bridgecorresponds to its mode shape whose mode numberis smaller than seven There is a different limit cyclewhen the initial dynamic configuration correspondsto its mode shape whose mode number is equal to orlarger than 7
Shock and Vibration 13
(2) In the case when the geometric nonlinear termsare considered calculation of the response of thecable-stayed bridge generally needs use of multiplemode shapes of the linearized undamped cable-stayedbridge model even when the initial dynamic config-uration of the cable-stayed bridge corresponds to itsfirstmode shapeThere is a limit cycle when the initialdynamic configuration of the cable-stayed bridgecorresponds to its first mode shape and its amplitudeis smaller than 03 for the generalized coordinate andthere is chaotic response when the initial dynamicconfiguration of the cable-stayed bridge correspondsto its first mode shape with its amplitude larger than03 for the generalized coordinate or one of its highermode shapes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grant nos 11302087 and 11442006 theNatural Science Foundation of Jiangsu Province under Grantno BK20130479 the Research Foundation for AdvancedTalents of Jiangsu University under Grant no 13JDG068 andthe National Science Foundation under Grant no CMMI-1000830
References
[1] K Y R Billah and R H Scanlan ldquoResonance Tacoma NarrowsBridge failure and undergraduate physics textbooksrdquoAmericanJournal of Physics vol 59 no 2 pp 118ndash124 1991
[2] R D Blevins Flow-Induced Vibration Krieger New York NYUSA 2nd edition 1994
[3] H Zheng R E Price Y Modarres-Sadeghi and M S Tri-antafyllou ldquoOn fatigue damage of long flexible cylinders dueto the higher harmonic force components and chaotic vortex-induced vibrationsrdquo Ocean Engineering vol 88 pp 318ndash3292014
[4] C Evangelinos D Lucor and G E Karniadakis ldquoDNS-derivedforce distribution on flexible cylinders subject to vortex-induced vibrationrdquo Journal of Fluids and Structures vol 14 no3 pp 429ndash440 2000
[5] HM Blackburn R N Govardhan andCH KWilliamson ldquoAcomplementary numerical and physical investigation of vortex-inducedrdquo Journal of Fluids and Structures vol 15 no 3-4 pp481ndash488 2001
[6] R H J Willden and J M R Graham ldquoNumerical predictionof VIV on long flexible circular cylindersrdquo Journal of Fluids andStructures vol 15 no 3-4 pp 659ndash669 2001
[7] E Guilmineau and P Queutey ldquoA numerical simulation ofvortex shedding from an oscillating circular cylinderrdquo Journalof Fluids and Structures vol 16 no 6 pp 773ndash794 2002
[8] I Jadic R M C So and M P Mignolet ldquoAnalysis of fluidmdashstructure interactions using a time-marching techniquerdquo Jour-nal of Fluids and Structures vol 12 no 6 pp 631ndash654 1998
[9] C Y Zhou RM C So andK Lam ldquoVortex-induced vibrationsof an elastic circular cylinderrdquo Journal of Fluids and Structuresvol 13 no 2 pp 165ndash189 1999
[10] J R Meneghini and P W Bearman ldquoNumerical simulationof high amplitude oscillatory flow about a circular cylinderrdquoJournal of Fluids and Structures vol 9 no 4 pp 435ndash455 1995
[11] T Sarpkaya ldquoComputational methods with vorticesrdquo Journal ofFluids Engineering vol 111 no 1 pp 5ndash52 1989
[12] T Sarpkaya ldquoVortex element methods for flow simulationrdquoAdvances in Applied Mechanics vol 31 pp 113ndash247 1994
[13] R D Gabbai and H Benaroya ldquoAn overview of modeling andexperiments of vortex-induced vibration of circular cylindersrdquoJournal of Sound and Vibration vol 282 no 3ndash5 pp 575ndash6162005
[14] R H Scanlan ldquoOn the state-of-the-artmethods for calculationsof flutter vortex-induced and buffeting response of bridgestructuresrdquo Tech Rep FHWARD-80050 National TechnicalInformation Service Springfield Va USA 1981
[15] RH Scanlan ldquoBridge flutter derivatives at vortex lock-inrdquo Jour-nal of Structural Engineering vol 124 no 4 pp 450ndash458 1998
[16] B J Vickery and R I Basu ldquoAcross-wind vibrations of struc-tures of circular cross-section Part I Development of a math-ematical model for two-dimensional conditionsrdquo Journal ofWind Engineering and Industrial Aerodynamics vol 12 no 1 pp49ndash73 1983
[17] A Larsen ldquoA generalized model for assessment of vortex-induced vibrations of flexible structuresrdquo Journal of Wind Engi-neering and Industrial Aerodynamics vol 57 no 2-3 pp 281ndash294 1995
[18] T Sarpkaya ldquoFluid forces on oscillating cylindersrdquo Journal ofWaterway Port Coastal and Ocean Division (ASCE) vol 104no 3 pp 275ndash290 1978
[19] T Staubli ldquoCalculation of vibration of an elastically mountedcylinder using experimental data from a forced oscillationrdquoJournal of Fluids Engineering vol 105 no 2 pp 225ndash229 1983
[20] W D Iwan and D L R Botelho ldquoVortex-induced oscillationof structures in waterrdquo Journal of Waterway Port Coastal andOcean Engineering vol 111 no 2 pp 289ndash303 1985
[21] R T Hartlen and I G Currie ldquoLift-oscillator model of vortex-induced vibrationrdquo Journal of Engineering Mechanics Division(ASCE) vol 96 no 5 pp 577ndash591 1970
[22] R A Skop and O M Griffin ldquoA model for the vortex-excitedresonant response of bluff cylindersrdquo Journal of Sound andVibration vol 27 no 2 pp 225ndash233 1973
[23] WD Iwan andRD Blevins ldquoAmodel for vortex induced oscil-lation of structuresrdquo Journal of Applied Mechanics TransactionsASME vol 41 no 3 pp 581ndash586 1974
[24] E H Dowell ldquoNon-linear oscillator models in bluff body aero-elasticityrdquo Journal of Sound and Vibration vol 75 no 2 pp 251ndash264 1981
[25] R H M Ogink and A V Metrikine ldquoA wake oscillator withfrequency dependent coupling for the modeling of vortex-induced vibrationrdquo Journal of Sound and Vibration vol 329 no26 pp 5452ndash5473 2010
[26] X Bai and W Qin ldquoUsing vortex strength wake oscillatorin modelling of vortex induced vibrations in two degrees offreedomrdquo European Journal of Mechanics BFluids vol 48 pp165ndash173 2014
[27] F Ehsan and R H Scanlan ldquoVortexminusinduced vibrations offlexible bridgesrdquo Journal of Engineering Mechanics vol 116 no6 pp 1392ndash1411 1990
14 Shock and Vibration
[28] A M Marra C Mannini and G Bartoli ldquoVan der Pol-typeequation for modeling vortex-induced oscillations of bridgedecksrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 99 no 6-7 pp 776ndash785 2011
[29] D Q Cao M T Song W D Zhu R W Tucker and C H-T Wang ldquoModeling and analysis of the in-plane vibration ofa complex cable-stayed bridgerdquo Journal of Sound and Vibrationvol 331 no 26 pp 5685ndash5714 2012
[30] E Caetano A Cunha V Gattulli and M Lepidi ldquoCable-deckdynamic interactions at the International Guadiana Bridge on-site measurements and finite element modellingrdquo StructuralControl and Health Monitoring vol 15 no 3 pp 237ndash264 2008
[31] I Akhtar A H Nayfeh and C J Ribbens ldquoOn the stability andextension of reduced-order Galerkin models in incompressibleflowsrdquo Theoretical and Computational Fluid Dynamics vol 23no 3 pp 213ndash237 2009
[32] S Sirisup and G E Karniadakis ldquoA spectral viscosity methodfor correcting the long-term behavior of POD modelsrdquo Journalof Computational Physics vol 194 no 1 pp 92ndash116 2004
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 Shock and Vibration
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus80
minus40
0
40
80
Vb 4(X
b 4=0)
(mm
)
Figure 4 Time response of the midpoint of the deck beam 119902119903(0) = 01120575
1119903
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus800
minus400
0
400
800
Vb 4(X
b 4=0)
(mm
)
Figure 5 Time response of the midpoint of the deck beam 119902119903(0) = 3120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 6 Time response of the midpoint of the deck beam 119902119903(0) = 01120575
7119903
mode shape (Figures 6 and 7) Two-dimensional projectionsof phase portraits onto the (119902
119903 119903) plane when 119902
119903(0) = 01120575
1119903
and 119902119903(0) = 01120575
7119903are shown in Figures 8 and 9 respectively
Results when 119902119903(0) is equal to 01120575
2119903 011205753119903 and 01120575
6119903
are the same as that when 119902119903(0) = 01120575
1119903 they are not shown
here for the sake of brevity It can be seen from Figures 8 and9 that energy of the cable-stayed bridge is concentrated inthe first mode (the seventh mode) when its initial dynamic
configuration corresponds to the first through sixth modeshapes (the seventh mode shape)
42 Case Studies Considering the Geometric Nonlinear TermsConvergence of Galerkin method for given initial conditions119902119903(0) = 01120575
1199031is shown in Figures 10 and 11 The amplitudes
of the steady-state transverse displacements of the mid-point of the deck beam with different numbers of Galerkin
Shock and Vibration 9
minus150
minus100
minus50
0
50
100
150
0 50 100minus50minus100
qr(0) = 011205751rqr(0) = 011205757r
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 7 Two different stable limit cycles corresponding to different initial configurations
q1
minus05
0
05
0 05minus05
q1
q2
minus05
0
05
0 05minus05
q2
minus05
0
05
q3
0 05minus05
q3
q4
minus05
0
05
0 05minus05
q4
q5
minus05
0
05
0 05minus05
q5
q6
minus05
0
05
0 05minus05
q6
q7
minus05
0
05
0 05minus05
q7
q8
minus05
0
05
0 05minus05
q8
q9
minus05
0
05
0 05minus05
q9
Figure 8 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903
truncation terms which are set to one through 30 are shownin Figure 10 As shown in Figure 10 a truncation with 11terms can provide accurate results in analyzing the dynamicresponse of the cable-stayed bridge and it is much lessthan the cases in Section 41 where the geometric nonlinearterms are neglected Phase portraits of the response of themidpoint of the deck beam for different numbers of Galerkin
truncation terms which are shown in Figure 11 also leadto the same conclusion It can be noted from Figure 11 thatthe limit cycle becomes more asymmetric when the numberof the truncation terms increases The reason for this isthat when the number of truncation terms increases to 7an additional asymmetric limit cycle appears on the (119902
7 7)
plane which phase portraits project onto (see Figure 12 with
10 Shock and Vibration
q1
minus1
0
1
0 03minus03
q1
q2
minus1
0
1
0 03minus03
q2
q3
minus1
0
1
0 03minus03
q3
q4
minus1
0
1
0 03minus03
q4
q5
minus1
0
1
0 03minus03
q5
q6
minus1
0
1
0 03minus03
q6
q7
minus1
0
1
0 03minus03
q7
q8
minus1
0
1
0 03minus03
q8
q9
minus1
0
1
0 03minus03
q9
Figure 9 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
7119903
5 10 15 20 25 300Number of truncation terms N
58
59
60
61
62
63
Am
plitu
de (m
m)
Figure 10 Effects of the number of Galerkin truncation terms ontransverse displacements of the midpoint of the deck beam whenthere are geometric nonlinear terms
119873 = 11) As shown in Figures 11 and 13 the dynamic responseof the midpoint of the deck beam approaches a stable limitcycle The bifurcation diagram of limit cycles with respectto 1199021(0) is shown in Figure 14 when the initial dynamic
configuration of the cable-stayed bridge corresponds to itsfirst mode shape One can find from Figure 14 that when1199021(0) is smaller than 027 a stable periodic solution exists and
minus50 0 50 100minus100minus100
minus50
0
50
100
N = 1
N = 10
N = 11
N = 30
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 11 Phase portraits of the response of themidpoint of the deckbeam for different numbers of Galerkin truncation terms 119902
119903(0) =
011205751119903
when 1199021(0) is larger than or equal to 027 chaotic responses
occur For instance when 1199021(0) = 03 and 119873 = 11 phase
portraits of the response of the midpoint of the deck beamare shown in Figure 15 which indicates occurrence of chaotic
Shock and Vibration 11
q1
minus50
0
50
0 50minus50
q1
q2
minus20
0
20
0 10minus10
q2
q3
minus20
0
20
0 10minus10
q3
q4
minus20
0
20
0 10minus10
q4
q5
minus20
0
20
0 10minus10
minus10
q5
q6
minus20
0
20
0 10minus10
q6
q7
minus20
0
20
0 10minus10
q7
q8
minus20
0
20
100
q8
q9
minus20
0
20
0 10minus10
q9
Figure 12 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903and119873 = 11
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 13 Time response of the midpoint of the deck beam 119902119903(0) =
011205751119903and119873 = 11
response On the contrary for a truncation with one termthe response always approaches a stable limit cycle no matterhow large the amplitude of the initial dynamic configurationis (see Figures 16ndash18) These mean that the true dynamicresponse of the cable-stayed bridge may not be capturedby the truncation with one term even when the initialdynamic configuration corresponds to the first mode shape
01 02 03 04 05 06 070q1(0)
minus200
minus150
minus100
minus50
0
50
Vb 4(X
b 4=0)
(mm
)
Figure 14 Bifurcation diagram of the cable-stayed bridge withrespect to 119902
1(0) 119902119903(0) = 119902
1(0)1205751119903and119873 = 11
When the initial dynamic configuration of the cable-stayedbridge corresponds to one of its higher mode shapes chaoticresponse occurs even if its amplitude is relatively small this isshown in Figure 19where the initial dynamic configuration ofthe cable-stayed bridge corresponds to its secondmode shapeand its magnitude is only 01
12 Shock and Vibration
0
minus100 0 100 200 300minus200
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus1500
minus1000
minus500
500
1000
1500
Figure 15 Chaotic response of the midpoint of the deck beam119902119903(0) = 03120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 16 Time response of themidpoint of the deck beam 1199021(0) =
01 and119873 = 1
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus1000
minus500
0
500
1000
Vb 4(X
b 4=0)
(mm
)
Figure 17 Time response of the midpoint of the deck beam 1199021(0) =
3 and119873 = 1
5 Conclusions
The dynamic behavior of a cable-stayed bridge that consistsof a simply supported four-cable-stayed deck beam andtwo rigid towers subjected to a distributed vortex shed-ding force on the deck beam has been investigated The
q1(0) = 01
q1(0) = 3
minus100
minus50
0
50
100
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 18 Phase portraits of the response of the midpoint of thedeck beam for different initial conditions119873 = 1
minus1500
minus1000
minus500
0
500
1000
1500
minus200 0 200 400minus400
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 19 Chaotic response of the midpoint of the deck beam119902119903(0) = 01120575
2119903
dynamic response of the cable-stayed bridge is calculatedusing Galerkin method in conjunction with Runge-Kutta-Felhberg method in MATLAB Convergence of Galerkinmethod for the dynamic response of the cable-stayed bridgeis studied Numerical simulations show that the geometricnonlinearities of the stay cables have significant influence onVIV of the cable-stayed bridge and further conclusions canbe summarized as follows
(1) In the case when the geometric nonlinear termsare neglected accurate calculation of the responseamplitude of the cable-stayed bridge at lock-in onlyneeds use of the first mode shape of the linearizedundamped cable-stayed bridgemodel when the initialdynamic configuration of the cable-stayed bridgecorresponds to its mode shape whose mode numberis smaller than seven There is a different limit cyclewhen the initial dynamic configuration correspondsto its mode shape whose mode number is equal to orlarger than 7
Shock and Vibration 13
(2) In the case when the geometric nonlinear termsare considered calculation of the response of thecable-stayed bridge generally needs use of multiplemode shapes of the linearized undamped cable-stayedbridge model even when the initial dynamic config-uration of the cable-stayed bridge corresponds to itsfirstmode shapeThere is a limit cycle when the initialdynamic configuration of the cable-stayed bridgecorresponds to its first mode shape and its amplitudeis smaller than 03 for the generalized coordinate andthere is chaotic response when the initial dynamicconfiguration of the cable-stayed bridge correspondsto its first mode shape with its amplitude larger than03 for the generalized coordinate or one of its highermode shapes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grant nos 11302087 and 11442006 theNatural Science Foundation of Jiangsu Province under Grantno BK20130479 the Research Foundation for AdvancedTalents of Jiangsu University under Grant no 13JDG068 andthe National Science Foundation under Grant no CMMI-1000830
References
[1] K Y R Billah and R H Scanlan ldquoResonance Tacoma NarrowsBridge failure and undergraduate physics textbooksrdquoAmericanJournal of Physics vol 59 no 2 pp 118ndash124 1991
[2] R D Blevins Flow-Induced Vibration Krieger New York NYUSA 2nd edition 1994
[3] H Zheng R E Price Y Modarres-Sadeghi and M S Tri-antafyllou ldquoOn fatigue damage of long flexible cylinders dueto the higher harmonic force components and chaotic vortex-induced vibrationsrdquo Ocean Engineering vol 88 pp 318ndash3292014
[4] C Evangelinos D Lucor and G E Karniadakis ldquoDNS-derivedforce distribution on flexible cylinders subject to vortex-induced vibrationrdquo Journal of Fluids and Structures vol 14 no3 pp 429ndash440 2000
[5] HM Blackburn R N Govardhan andCH KWilliamson ldquoAcomplementary numerical and physical investigation of vortex-inducedrdquo Journal of Fluids and Structures vol 15 no 3-4 pp481ndash488 2001
[6] R H J Willden and J M R Graham ldquoNumerical predictionof VIV on long flexible circular cylindersrdquo Journal of Fluids andStructures vol 15 no 3-4 pp 659ndash669 2001
[7] E Guilmineau and P Queutey ldquoA numerical simulation ofvortex shedding from an oscillating circular cylinderrdquo Journalof Fluids and Structures vol 16 no 6 pp 773ndash794 2002
[8] I Jadic R M C So and M P Mignolet ldquoAnalysis of fluidmdashstructure interactions using a time-marching techniquerdquo Jour-nal of Fluids and Structures vol 12 no 6 pp 631ndash654 1998
[9] C Y Zhou RM C So andK Lam ldquoVortex-induced vibrationsof an elastic circular cylinderrdquo Journal of Fluids and Structuresvol 13 no 2 pp 165ndash189 1999
[10] J R Meneghini and P W Bearman ldquoNumerical simulationof high amplitude oscillatory flow about a circular cylinderrdquoJournal of Fluids and Structures vol 9 no 4 pp 435ndash455 1995
[11] T Sarpkaya ldquoComputational methods with vorticesrdquo Journal ofFluids Engineering vol 111 no 1 pp 5ndash52 1989
[12] T Sarpkaya ldquoVortex element methods for flow simulationrdquoAdvances in Applied Mechanics vol 31 pp 113ndash247 1994
[13] R D Gabbai and H Benaroya ldquoAn overview of modeling andexperiments of vortex-induced vibration of circular cylindersrdquoJournal of Sound and Vibration vol 282 no 3ndash5 pp 575ndash6162005
[14] R H Scanlan ldquoOn the state-of-the-artmethods for calculationsof flutter vortex-induced and buffeting response of bridgestructuresrdquo Tech Rep FHWARD-80050 National TechnicalInformation Service Springfield Va USA 1981
[15] RH Scanlan ldquoBridge flutter derivatives at vortex lock-inrdquo Jour-nal of Structural Engineering vol 124 no 4 pp 450ndash458 1998
[16] B J Vickery and R I Basu ldquoAcross-wind vibrations of struc-tures of circular cross-section Part I Development of a math-ematical model for two-dimensional conditionsrdquo Journal ofWind Engineering and Industrial Aerodynamics vol 12 no 1 pp49ndash73 1983
[17] A Larsen ldquoA generalized model for assessment of vortex-induced vibrations of flexible structuresrdquo Journal of Wind Engi-neering and Industrial Aerodynamics vol 57 no 2-3 pp 281ndash294 1995
[18] T Sarpkaya ldquoFluid forces on oscillating cylindersrdquo Journal ofWaterway Port Coastal and Ocean Division (ASCE) vol 104no 3 pp 275ndash290 1978
[19] T Staubli ldquoCalculation of vibration of an elastically mountedcylinder using experimental data from a forced oscillationrdquoJournal of Fluids Engineering vol 105 no 2 pp 225ndash229 1983
[20] W D Iwan and D L R Botelho ldquoVortex-induced oscillationof structures in waterrdquo Journal of Waterway Port Coastal andOcean Engineering vol 111 no 2 pp 289ndash303 1985
[21] R T Hartlen and I G Currie ldquoLift-oscillator model of vortex-induced vibrationrdquo Journal of Engineering Mechanics Division(ASCE) vol 96 no 5 pp 577ndash591 1970
[22] R A Skop and O M Griffin ldquoA model for the vortex-excitedresonant response of bluff cylindersrdquo Journal of Sound andVibration vol 27 no 2 pp 225ndash233 1973
[23] WD Iwan andRD Blevins ldquoAmodel for vortex induced oscil-lation of structuresrdquo Journal of Applied Mechanics TransactionsASME vol 41 no 3 pp 581ndash586 1974
[24] E H Dowell ldquoNon-linear oscillator models in bluff body aero-elasticityrdquo Journal of Sound and Vibration vol 75 no 2 pp 251ndash264 1981
[25] R H M Ogink and A V Metrikine ldquoA wake oscillator withfrequency dependent coupling for the modeling of vortex-induced vibrationrdquo Journal of Sound and Vibration vol 329 no26 pp 5452ndash5473 2010
[26] X Bai and W Qin ldquoUsing vortex strength wake oscillatorin modelling of vortex induced vibrations in two degrees offreedomrdquo European Journal of Mechanics BFluids vol 48 pp165ndash173 2014
[27] F Ehsan and R H Scanlan ldquoVortexminusinduced vibrations offlexible bridgesrdquo Journal of Engineering Mechanics vol 116 no6 pp 1392ndash1411 1990
14 Shock and Vibration
[28] A M Marra C Mannini and G Bartoli ldquoVan der Pol-typeequation for modeling vortex-induced oscillations of bridgedecksrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 99 no 6-7 pp 776ndash785 2011
[29] D Q Cao M T Song W D Zhu R W Tucker and C H-T Wang ldquoModeling and analysis of the in-plane vibration ofa complex cable-stayed bridgerdquo Journal of Sound and Vibrationvol 331 no 26 pp 5685ndash5714 2012
[30] E Caetano A Cunha V Gattulli and M Lepidi ldquoCable-deckdynamic interactions at the International Guadiana Bridge on-site measurements and finite element modellingrdquo StructuralControl and Health Monitoring vol 15 no 3 pp 237ndash264 2008
[31] I Akhtar A H Nayfeh and C J Ribbens ldquoOn the stability andextension of reduced-order Galerkin models in incompressibleflowsrdquo Theoretical and Computational Fluid Dynamics vol 23no 3 pp 213ndash237 2009
[32] S Sirisup and G E Karniadakis ldquoA spectral viscosity methodfor correcting the long-term behavior of POD modelsrdquo Journalof Computational Physics vol 194 no 1 pp 92ndash116 2004
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 9
minus150
minus100
minus50
0
50
100
150
0 50 100minus50minus100
qr(0) = 011205751rqr(0) = 011205757r
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 7 Two different stable limit cycles corresponding to different initial configurations
q1
minus05
0
05
0 05minus05
q1
q2
minus05
0
05
0 05minus05
q2
minus05
0
05
q3
0 05minus05
q3
q4
minus05
0
05
0 05minus05
q4
q5
minus05
0
05
0 05minus05
q5
q6
minus05
0
05
0 05minus05
q6
q7
minus05
0
05
0 05minus05
q7
q8
minus05
0
05
0 05minus05
q8
q9
minus05
0
05
0 05minus05
q9
Figure 8 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903
truncation terms which are set to one through 30 are shownin Figure 10 As shown in Figure 10 a truncation with 11terms can provide accurate results in analyzing the dynamicresponse of the cable-stayed bridge and it is much lessthan the cases in Section 41 where the geometric nonlinearterms are neglected Phase portraits of the response of themidpoint of the deck beam for different numbers of Galerkin
truncation terms which are shown in Figure 11 also leadto the same conclusion It can be noted from Figure 11 thatthe limit cycle becomes more asymmetric when the numberof the truncation terms increases The reason for this isthat when the number of truncation terms increases to 7an additional asymmetric limit cycle appears on the (119902
7 7)
plane which phase portraits project onto (see Figure 12 with
10 Shock and Vibration
q1
minus1
0
1
0 03minus03
q1
q2
minus1
0
1
0 03minus03
q2
q3
minus1
0
1
0 03minus03
q3
q4
minus1
0
1
0 03minus03
q4
q5
minus1
0
1
0 03minus03
q5
q6
minus1
0
1
0 03minus03
q6
q7
minus1
0
1
0 03minus03
q7
q8
minus1
0
1
0 03minus03
q8
q9
minus1
0
1
0 03minus03
q9
Figure 9 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
7119903
5 10 15 20 25 300Number of truncation terms N
58
59
60
61
62
63
Am
plitu
de (m
m)
Figure 10 Effects of the number of Galerkin truncation terms ontransverse displacements of the midpoint of the deck beam whenthere are geometric nonlinear terms
119873 = 11) As shown in Figures 11 and 13 the dynamic responseof the midpoint of the deck beam approaches a stable limitcycle The bifurcation diagram of limit cycles with respectto 1199021(0) is shown in Figure 14 when the initial dynamic
configuration of the cable-stayed bridge corresponds to itsfirst mode shape One can find from Figure 14 that when1199021(0) is smaller than 027 a stable periodic solution exists and
minus50 0 50 100minus100minus100
minus50
0
50
100
N = 1
N = 10
N = 11
N = 30
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 11 Phase portraits of the response of themidpoint of the deckbeam for different numbers of Galerkin truncation terms 119902
119903(0) =
011205751119903
when 1199021(0) is larger than or equal to 027 chaotic responses
occur For instance when 1199021(0) = 03 and 119873 = 11 phase
portraits of the response of the midpoint of the deck beamare shown in Figure 15 which indicates occurrence of chaotic
Shock and Vibration 11
q1
minus50
0
50
0 50minus50
q1
q2
minus20
0
20
0 10minus10
q2
q3
minus20
0
20
0 10minus10
q3
q4
minus20
0
20
0 10minus10
q4
q5
minus20
0
20
0 10minus10
minus10
q5
q6
minus20
0
20
0 10minus10
q6
q7
minus20
0
20
0 10minus10
q7
q8
minus20
0
20
100
q8
q9
minus20
0
20
0 10minus10
q9
Figure 12 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903and119873 = 11
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 13 Time response of the midpoint of the deck beam 119902119903(0) =
011205751119903and119873 = 11
response On the contrary for a truncation with one termthe response always approaches a stable limit cycle no matterhow large the amplitude of the initial dynamic configurationis (see Figures 16ndash18) These mean that the true dynamicresponse of the cable-stayed bridge may not be capturedby the truncation with one term even when the initialdynamic configuration corresponds to the first mode shape
01 02 03 04 05 06 070q1(0)
minus200
minus150
minus100
minus50
0
50
Vb 4(X
b 4=0)
(mm
)
Figure 14 Bifurcation diagram of the cable-stayed bridge withrespect to 119902
1(0) 119902119903(0) = 119902
1(0)1205751119903and119873 = 11
When the initial dynamic configuration of the cable-stayedbridge corresponds to one of its higher mode shapes chaoticresponse occurs even if its amplitude is relatively small this isshown in Figure 19where the initial dynamic configuration ofthe cable-stayed bridge corresponds to its secondmode shapeand its magnitude is only 01
12 Shock and Vibration
0
minus100 0 100 200 300minus200
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus1500
minus1000
minus500
500
1000
1500
Figure 15 Chaotic response of the midpoint of the deck beam119902119903(0) = 03120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 16 Time response of themidpoint of the deck beam 1199021(0) =
01 and119873 = 1
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus1000
minus500
0
500
1000
Vb 4(X
b 4=0)
(mm
)
Figure 17 Time response of the midpoint of the deck beam 1199021(0) =
3 and119873 = 1
5 Conclusions
The dynamic behavior of a cable-stayed bridge that consistsof a simply supported four-cable-stayed deck beam andtwo rigid towers subjected to a distributed vortex shed-ding force on the deck beam has been investigated The
q1(0) = 01
q1(0) = 3
minus100
minus50
0
50
100
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 18 Phase portraits of the response of the midpoint of thedeck beam for different initial conditions119873 = 1
minus1500
minus1000
minus500
0
500
1000
1500
minus200 0 200 400minus400
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 19 Chaotic response of the midpoint of the deck beam119902119903(0) = 01120575
2119903
dynamic response of the cable-stayed bridge is calculatedusing Galerkin method in conjunction with Runge-Kutta-Felhberg method in MATLAB Convergence of Galerkinmethod for the dynamic response of the cable-stayed bridgeis studied Numerical simulations show that the geometricnonlinearities of the stay cables have significant influence onVIV of the cable-stayed bridge and further conclusions canbe summarized as follows
(1) In the case when the geometric nonlinear termsare neglected accurate calculation of the responseamplitude of the cable-stayed bridge at lock-in onlyneeds use of the first mode shape of the linearizedundamped cable-stayed bridgemodel when the initialdynamic configuration of the cable-stayed bridgecorresponds to its mode shape whose mode numberis smaller than seven There is a different limit cyclewhen the initial dynamic configuration correspondsto its mode shape whose mode number is equal to orlarger than 7
Shock and Vibration 13
(2) In the case when the geometric nonlinear termsare considered calculation of the response of thecable-stayed bridge generally needs use of multiplemode shapes of the linearized undamped cable-stayedbridge model even when the initial dynamic config-uration of the cable-stayed bridge corresponds to itsfirstmode shapeThere is a limit cycle when the initialdynamic configuration of the cable-stayed bridgecorresponds to its first mode shape and its amplitudeis smaller than 03 for the generalized coordinate andthere is chaotic response when the initial dynamicconfiguration of the cable-stayed bridge correspondsto its first mode shape with its amplitude larger than03 for the generalized coordinate or one of its highermode shapes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grant nos 11302087 and 11442006 theNatural Science Foundation of Jiangsu Province under Grantno BK20130479 the Research Foundation for AdvancedTalents of Jiangsu University under Grant no 13JDG068 andthe National Science Foundation under Grant no CMMI-1000830
References
[1] K Y R Billah and R H Scanlan ldquoResonance Tacoma NarrowsBridge failure and undergraduate physics textbooksrdquoAmericanJournal of Physics vol 59 no 2 pp 118ndash124 1991
[2] R D Blevins Flow-Induced Vibration Krieger New York NYUSA 2nd edition 1994
[3] H Zheng R E Price Y Modarres-Sadeghi and M S Tri-antafyllou ldquoOn fatigue damage of long flexible cylinders dueto the higher harmonic force components and chaotic vortex-induced vibrationsrdquo Ocean Engineering vol 88 pp 318ndash3292014
[4] C Evangelinos D Lucor and G E Karniadakis ldquoDNS-derivedforce distribution on flexible cylinders subject to vortex-induced vibrationrdquo Journal of Fluids and Structures vol 14 no3 pp 429ndash440 2000
[5] HM Blackburn R N Govardhan andCH KWilliamson ldquoAcomplementary numerical and physical investigation of vortex-inducedrdquo Journal of Fluids and Structures vol 15 no 3-4 pp481ndash488 2001
[6] R H J Willden and J M R Graham ldquoNumerical predictionof VIV on long flexible circular cylindersrdquo Journal of Fluids andStructures vol 15 no 3-4 pp 659ndash669 2001
[7] E Guilmineau and P Queutey ldquoA numerical simulation ofvortex shedding from an oscillating circular cylinderrdquo Journalof Fluids and Structures vol 16 no 6 pp 773ndash794 2002
[8] I Jadic R M C So and M P Mignolet ldquoAnalysis of fluidmdashstructure interactions using a time-marching techniquerdquo Jour-nal of Fluids and Structures vol 12 no 6 pp 631ndash654 1998
[9] C Y Zhou RM C So andK Lam ldquoVortex-induced vibrationsof an elastic circular cylinderrdquo Journal of Fluids and Structuresvol 13 no 2 pp 165ndash189 1999
[10] J R Meneghini and P W Bearman ldquoNumerical simulationof high amplitude oscillatory flow about a circular cylinderrdquoJournal of Fluids and Structures vol 9 no 4 pp 435ndash455 1995
[11] T Sarpkaya ldquoComputational methods with vorticesrdquo Journal ofFluids Engineering vol 111 no 1 pp 5ndash52 1989
[12] T Sarpkaya ldquoVortex element methods for flow simulationrdquoAdvances in Applied Mechanics vol 31 pp 113ndash247 1994
[13] R D Gabbai and H Benaroya ldquoAn overview of modeling andexperiments of vortex-induced vibration of circular cylindersrdquoJournal of Sound and Vibration vol 282 no 3ndash5 pp 575ndash6162005
[14] R H Scanlan ldquoOn the state-of-the-artmethods for calculationsof flutter vortex-induced and buffeting response of bridgestructuresrdquo Tech Rep FHWARD-80050 National TechnicalInformation Service Springfield Va USA 1981
[15] RH Scanlan ldquoBridge flutter derivatives at vortex lock-inrdquo Jour-nal of Structural Engineering vol 124 no 4 pp 450ndash458 1998
[16] B J Vickery and R I Basu ldquoAcross-wind vibrations of struc-tures of circular cross-section Part I Development of a math-ematical model for two-dimensional conditionsrdquo Journal ofWind Engineering and Industrial Aerodynamics vol 12 no 1 pp49ndash73 1983
[17] A Larsen ldquoA generalized model for assessment of vortex-induced vibrations of flexible structuresrdquo Journal of Wind Engi-neering and Industrial Aerodynamics vol 57 no 2-3 pp 281ndash294 1995
[18] T Sarpkaya ldquoFluid forces on oscillating cylindersrdquo Journal ofWaterway Port Coastal and Ocean Division (ASCE) vol 104no 3 pp 275ndash290 1978
[19] T Staubli ldquoCalculation of vibration of an elastically mountedcylinder using experimental data from a forced oscillationrdquoJournal of Fluids Engineering vol 105 no 2 pp 225ndash229 1983
[20] W D Iwan and D L R Botelho ldquoVortex-induced oscillationof structures in waterrdquo Journal of Waterway Port Coastal andOcean Engineering vol 111 no 2 pp 289ndash303 1985
[21] R T Hartlen and I G Currie ldquoLift-oscillator model of vortex-induced vibrationrdquo Journal of Engineering Mechanics Division(ASCE) vol 96 no 5 pp 577ndash591 1970
[22] R A Skop and O M Griffin ldquoA model for the vortex-excitedresonant response of bluff cylindersrdquo Journal of Sound andVibration vol 27 no 2 pp 225ndash233 1973
[23] WD Iwan andRD Blevins ldquoAmodel for vortex induced oscil-lation of structuresrdquo Journal of Applied Mechanics TransactionsASME vol 41 no 3 pp 581ndash586 1974
[24] E H Dowell ldquoNon-linear oscillator models in bluff body aero-elasticityrdquo Journal of Sound and Vibration vol 75 no 2 pp 251ndash264 1981
[25] R H M Ogink and A V Metrikine ldquoA wake oscillator withfrequency dependent coupling for the modeling of vortex-induced vibrationrdquo Journal of Sound and Vibration vol 329 no26 pp 5452ndash5473 2010
[26] X Bai and W Qin ldquoUsing vortex strength wake oscillatorin modelling of vortex induced vibrations in two degrees offreedomrdquo European Journal of Mechanics BFluids vol 48 pp165ndash173 2014
[27] F Ehsan and R H Scanlan ldquoVortexminusinduced vibrations offlexible bridgesrdquo Journal of Engineering Mechanics vol 116 no6 pp 1392ndash1411 1990
14 Shock and Vibration
[28] A M Marra C Mannini and G Bartoli ldquoVan der Pol-typeequation for modeling vortex-induced oscillations of bridgedecksrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 99 no 6-7 pp 776ndash785 2011
[29] D Q Cao M T Song W D Zhu R W Tucker and C H-T Wang ldquoModeling and analysis of the in-plane vibration ofa complex cable-stayed bridgerdquo Journal of Sound and Vibrationvol 331 no 26 pp 5685ndash5714 2012
[30] E Caetano A Cunha V Gattulli and M Lepidi ldquoCable-deckdynamic interactions at the International Guadiana Bridge on-site measurements and finite element modellingrdquo StructuralControl and Health Monitoring vol 15 no 3 pp 237ndash264 2008
[31] I Akhtar A H Nayfeh and C J Ribbens ldquoOn the stability andextension of reduced-order Galerkin models in incompressibleflowsrdquo Theoretical and Computational Fluid Dynamics vol 23no 3 pp 213ndash237 2009
[32] S Sirisup and G E Karniadakis ldquoA spectral viscosity methodfor correcting the long-term behavior of POD modelsrdquo Journalof Computational Physics vol 194 no 1 pp 92ndash116 2004
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 Shock and Vibration
q1
minus1
0
1
0 03minus03
q1
q2
minus1
0
1
0 03minus03
q2
q3
minus1
0
1
0 03minus03
q3
q4
minus1
0
1
0 03minus03
q4
q5
minus1
0
1
0 03minus03
q5
q6
minus1
0
1
0 03minus03
q6
q7
minus1
0
1
0 03minus03
q7
q8
minus1
0
1
0 03minus03
q8
q9
minus1
0
1
0 03minus03
q9
Figure 9 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
7119903
5 10 15 20 25 300Number of truncation terms N
58
59
60
61
62
63
Am
plitu
de (m
m)
Figure 10 Effects of the number of Galerkin truncation terms ontransverse displacements of the midpoint of the deck beam whenthere are geometric nonlinear terms
119873 = 11) As shown in Figures 11 and 13 the dynamic responseof the midpoint of the deck beam approaches a stable limitcycle The bifurcation diagram of limit cycles with respectto 1199021(0) is shown in Figure 14 when the initial dynamic
configuration of the cable-stayed bridge corresponds to itsfirst mode shape One can find from Figure 14 that when1199021(0) is smaller than 027 a stable periodic solution exists and
minus50 0 50 100minus100minus100
minus50
0
50
100
N = 1
N = 10
N = 11
N = 30
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 11 Phase portraits of the response of themidpoint of the deckbeam for different numbers of Galerkin truncation terms 119902
119903(0) =
011205751119903
when 1199021(0) is larger than or equal to 027 chaotic responses
occur For instance when 1199021(0) = 03 and 119873 = 11 phase
portraits of the response of the midpoint of the deck beamare shown in Figure 15 which indicates occurrence of chaotic
Shock and Vibration 11
q1
minus50
0
50
0 50minus50
q1
q2
minus20
0
20
0 10minus10
q2
q3
minus20
0
20
0 10minus10
q3
q4
minus20
0
20
0 10minus10
q4
q5
minus20
0
20
0 10minus10
minus10
q5
q6
minus20
0
20
0 10minus10
q6
q7
minus20
0
20
0 10minus10
q7
q8
minus20
0
20
100
q8
q9
minus20
0
20
0 10minus10
q9
Figure 12 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903and119873 = 11
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 13 Time response of the midpoint of the deck beam 119902119903(0) =
011205751119903and119873 = 11
response On the contrary for a truncation with one termthe response always approaches a stable limit cycle no matterhow large the amplitude of the initial dynamic configurationis (see Figures 16ndash18) These mean that the true dynamicresponse of the cable-stayed bridge may not be capturedby the truncation with one term even when the initialdynamic configuration corresponds to the first mode shape
01 02 03 04 05 06 070q1(0)
minus200
minus150
minus100
minus50
0
50
Vb 4(X
b 4=0)
(mm
)
Figure 14 Bifurcation diagram of the cable-stayed bridge withrespect to 119902
1(0) 119902119903(0) = 119902
1(0)1205751119903and119873 = 11
When the initial dynamic configuration of the cable-stayedbridge corresponds to one of its higher mode shapes chaoticresponse occurs even if its amplitude is relatively small this isshown in Figure 19where the initial dynamic configuration ofthe cable-stayed bridge corresponds to its secondmode shapeand its magnitude is only 01
12 Shock and Vibration
0
minus100 0 100 200 300minus200
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus1500
minus1000
minus500
500
1000
1500
Figure 15 Chaotic response of the midpoint of the deck beam119902119903(0) = 03120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 16 Time response of themidpoint of the deck beam 1199021(0) =
01 and119873 = 1
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus1000
minus500
0
500
1000
Vb 4(X
b 4=0)
(mm
)
Figure 17 Time response of the midpoint of the deck beam 1199021(0) =
3 and119873 = 1
5 Conclusions
The dynamic behavior of a cable-stayed bridge that consistsof a simply supported four-cable-stayed deck beam andtwo rigid towers subjected to a distributed vortex shed-ding force on the deck beam has been investigated The
q1(0) = 01
q1(0) = 3
minus100
minus50
0
50
100
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 18 Phase portraits of the response of the midpoint of thedeck beam for different initial conditions119873 = 1
minus1500
minus1000
minus500
0
500
1000
1500
minus200 0 200 400minus400
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 19 Chaotic response of the midpoint of the deck beam119902119903(0) = 01120575
2119903
dynamic response of the cable-stayed bridge is calculatedusing Galerkin method in conjunction with Runge-Kutta-Felhberg method in MATLAB Convergence of Galerkinmethod for the dynamic response of the cable-stayed bridgeis studied Numerical simulations show that the geometricnonlinearities of the stay cables have significant influence onVIV of the cable-stayed bridge and further conclusions canbe summarized as follows
(1) In the case when the geometric nonlinear termsare neglected accurate calculation of the responseamplitude of the cable-stayed bridge at lock-in onlyneeds use of the first mode shape of the linearizedundamped cable-stayed bridgemodel when the initialdynamic configuration of the cable-stayed bridgecorresponds to its mode shape whose mode numberis smaller than seven There is a different limit cyclewhen the initial dynamic configuration correspondsto its mode shape whose mode number is equal to orlarger than 7
Shock and Vibration 13
(2) In the case when the geometric nonlinear termsare considered calculation of the response of thecable-stayed bridge generally needs use of multiplemode shapes of the linearized undamped cable-stayedbridge model even when the initial dynamic config-uration of the cable-stayed bridge corresponds to itsfirstmode shapeThere is a limit cycle when the initialdynamic configuration of the cable-stayed bridgecorresponds to its first mode shape and its amplitudeis smaller than 03 for the generalized coordinate andthere is chaotic response when the initial dynamicconfiguration of the cable-stayed bridge correspondsto its first mode shape with its amplitude larger than03 for the generalized coordinate or one of its highermode shapes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grant nos 11302087 and 11442006 theNatural Science Foundation of Jiangsu Province under Grantno BK20130479 the Research Foundation for AdvancedTalents of Jiangsu University under Grant no 13JDG068 andthe National Science Foundation under Grant no CMMI-1000830
References
[1] K Y R Billah and R H Scanlan ldquoResonance Tacoma NarrowsBridge failure and undergraduate physics textbooksrdquoAmericanJournal of Physics vol 59 no 2 pp 118ndash124 1991
[2] R D Blevins Flow-Induced Vibration Krieger New York NYUSA 2nd edition 1994
[3] H Zheng R E Price Y Modarres-Sadeghi and M S Tri-antafyllou ldquoOn fatigue damage of long flexible cylinders dueto the higher harmonic force components and chaotic vortex-induced vibrationsrdquo Ocean Engineering vol 88 pp 318ndash3292014
[4] C Evangelinos D Lucor and G E Karniadakis ldquoDNS-derivedforce distribution on flexible cylinders subject to vortex-induced vibrationrdquo Journal of Fluids and Structures vol 14 no3 pp 429ndash440 2000
[5] HM Blackburn R N Govardhan andCH KWilliamson ldquoAcomplementary numerical and physical investigation of vortex-inducedrdquo Journal of Fluids and Structures vol 15 no 3-4 pp481ndash488 2001
[6] R H J Willden and J M R Graham ldquoNumerical predictionof VIV on long flexible circular cylindersrdquo Journal of Fluids andStructures vol 15 no 3-4 pp 659ndash669 2001
[7] E Guilmineau and P Queutey ldquoA numerical simulation ofvortex shedding from an oscillating circular cylinderrdquo Journalof Fluids and Structures vol 16 no 6 pp 773ndash794 2002
[8] I Jadic R M C So and M P Mignolet ldquoAnalysis of fluidmdashstructure interactions using a time-marching techniquerdquo Jour-nal of Fluids and Structures vol 12 no 6 pp 631ndash654 1998
[9] C Y Zhou RM C So andK Lam ldquoVortex-induced vibrationsof an elastic circular cylinderrdquo Journal of Fluids and Structuresvol 13 no 2 pp 165ndash189 1999
[10] J R Meneghini and P W Bearman ldquoNumerical simulationof high amplitude oscillatory flow about a circular cylinderrdquoJournal of Fluids and Structures vol 9 no 4 pp 435ndash455 1995
[11] T Sarpkaya ldquoComputational methods with vorticesrdquo Journal ofFluids Engineering vol 111 no 1 pp 5ndash52 1989
[12] T Sarpkaya ldquoVortex element methods for flow simulationrdquoAdvances in Applied Mechanics vol 31 pp 113ndash247 1994
[13] R D Gabbai and H Benaroya ldquoAn overview of modeling andexperiments of vortex-induced vibration of circular cylindersrdquoJournal of Sound and Vibration vol 282 no 3ndash5 pp 575ndash6162005
[14] R H Scanlan ldquoOn the state-of-the-artmethods for calculationsof flutter vortex-induced and buffeting response of bridgestructuresrdquo Tech Rep FHWARD-80050 National TechnicalInformation Service Springfield Va USA 1981
[15] RH Scanlan ldquoBridge flutter derivatives at vortex lock-inrdquo Jour-nal of Structural Engineering vol 124 no 4 pp 450ndash458 1998
[16] B J Vickery and R I Basu ldquoAcross-wind vibrations of struc-tures of circular cross-section Part I Development of a math-ematical model for two-dimensional conditionsrdquo Journal ofWind Engineering and Industrial Aerodynamics vol 12 no 1 pp49ndash73 1983
[17] A Larsen ldquoA generalized model for assessment of vortex-induced vibrations of flexible structuresrdquo Journal of Wind Engi-neering and Industrial Aerodynamics vol 57 no 2-3 pp 281ndash294 1995
[18] T Sarpkaya ldquoFluid forces on oscillating cylindersrdquo Journal ofWaterway Port Coastal and Ocean Division (ASCE) vol 104no 3 pp 275ndash290 1978
[19] T Staubli ldquoCalculation of vibration of an elastically mountedcylinder using experimental data from a forced oscillationrdquoJournal of Fluids Engineering vol 105 no 2 pp 225ndash229 1983
[20] W D Iwan and D L R Botelho ldquoVortex-induced oscillationof structures in waterrdquo Journal of Waterway Port Coastal andOcean Engineering vol 111 no 2 pp 289ndash303 1985
[21] R T Hartlen and I G Currie ldquoLift-oscillator model of vortex-induced vibrationrdquo Journal of Engineering Mechanics Division(ASCE) vol 96 no 5 pp 577ndash591 1970
[22] R A Skop and O M Griffin ldquoA model for the vortex-excitedresonant response of bluff cylindersrdquo Journal of Sound andVibration vol 27 no 2 pp 225ndash233 1973
[23] WD Iwan andRD Blevins ldquoAmodel for vortex induced oscil-lation of structuresrdquo Journal of Applied Mechanics TransactionsASME vol 41 no 3 pp 581ndash586 1974
[24] E H Dowell ldquoNon-linear oscillator models in bluff body aero-elasticityrdquo Journal of Sound and Vibration vol 75 no 2 pp 251ndash264 1981
[25] R H M Ogink and A V Metrikine ldquoA wake oscillator withfrequency dependent coupling for the modeling of vortex-induced vibrationrdquo Journal of Sound and Vibration vol 329 no26 pp 5452ndash5473 2010
[26] X Bai and W Qin ldquoUsing vortex strength wake oscillatorin modelling of vortex induced vibrations in two degrees offreedomrdquo European Journal of Mechanics BFluids vol 48 pp165ndash173 2014
[27] F Ehsan and R H Scanlan ldquoVortexminusinduced vibrations offlexible bridgesrdquo Journal of Engineering Mechanics vol 116 no6 pp 1392ndash1411 1990
14 Shock and Vibration
[28] A M Marra C Mannini and G Bartoli ldquoVan der Pol-typeequation for modeling vortex-induced oscillations of bridgedecksrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 99 no 6-7 pp 776ndash785 2011
[29] D Q Cao M T Song W D Zhu R W Tucker and C H-T Wang ldquoModeling and analysis of the in-plane vibration ofa complex cable-stayed bridgerdquo Journal of Sound and Vibrationvol 331 no 26 pp 5685ndash5714 2012
[30] E Caetano A Cunha V Gattulli and M Lepidi ldquoCable-deckdynamic interactions at the International Guadiana Bridge on-site measurements and finite element modellingrdquo StructuralControl and Health Monitoring vol 15 no 3 pp 237ndash264 2008
[31] I Akhtar A H Nayfeh and C J Ribbens ldquoOn the stability andextension of reduced-order Galerkin models in incompressibleflowsrdquo Theoretical and Computational Fluid Dynamics vol 23no 3 pp 213ndash237 2009
[32] S Sirisup and G E Karniadakis ldquoA spectral viscosity methodfor correcting the long-term behavior of POD modelsrdquo Journalof Computational Physics vol 194 no 1 pp 92ndash116 2004
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 11
q1
minus50
0
50
0 50minus50
q1
q2
minus20
0
20
0 10minus10
q2
q3
minus20
0
20
0 10minus10
q3
q4
minus20
0
20
0 10minus10
q4
q5
minus20
0
20
0 10minus10
minus10
q5
q6
minus20
0
20
0 10minus10
q6
q7
minus20
0
20
0 10minus10
q7
q8
minus20
0
20
100
q8
q9
minus20
0
20
0 10minus10
q9
Figure 12 Two-dimensional projections of phase portraits onto the (119902119903 119903) plane 119902
119903(0) = 01120575
1119903and119873 = 11
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 13 Time response of the midpoint of the deck beam 119902119903(0) =
011205751119903and119873 = 11
response On the contrary for a truncation with one termthe response always approaches a stable limit cycle no matterhow large the amplitude of the initial dynamic configurationis (see Figures 16ndash18) These mean that the true dynamicresponse of the cable-stayed bridge may not be capturedby the truncation with one term even when the initialdynamic configuration corresponds to the first mode shape
01 02 03 04 05 06 070q1(0)
minus200
minus150
minus100
minus50
0
50
Vb 4(X
b 4=0)
(mm
)
Figure 14 Bifurcation diagram of the cable-stayed bridge withrespect to 119902
1(0) 119902119903(0) = 119902
1(0)1205751119903and119873 = 11
When the initial dynamic configuration of the cable-stayedbridge corresponds to one of its higher mode shapes chaoticresponse occurs even if its amplitude is relatively small this isshown in Figure 19where the initial dynamic configuration ofthe cable-stayed bridge corresponds to its secondmode shapeand its magnitude is only 01
12 Shock and Vibration
0
minus100 0 100 200 300minus200
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus1500
minus1000
minus500
500
1000
1500
Figure 15 Chaotic response of the midpoint of the deck beam119902119903(0) = 03120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 16 Time response of themidpoint of the deck beam 1199021(0) =
01 and119873 = 1
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus1000
minus500
0
500
1000
Vb 4(X
b 4=0)
(mm
)
Figure 17 Time response of the midpoint of the deck beam 1199021(0) =
3 and119873 = 1
5 Conclusions
The dynamic behavior of a cable-stayed bridge that consistsof a simply supported four-cable-stayed deck beam andtwo rigid towers subjected to a distributed vortex shed-ding force on the deck beam has been investigated The
q1(0) = 01
q1(0) = 3
minus100
minus50
0
50
100
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 18 Phase portraits of the response of the midpoint of thedeck beam for different initial conditions119873 = 1
minus1500
minus1000
minus500
0
500
1000
1500
minus200 0 200 400minus400
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 19 Chaotic response of the midpoint of the deck beam119902119903(0) = 01120575
2119903
dynamic response of the cable-stayed bridge is calculatedusing Galerkin method in conjunction with Runge-Kutta-Felhberg method in MATLAB Convergence of Galerkinmethod for the dynamic response of the cable-stayed bridgeis studied Numerical simulations show that the geometricnonlinearities of the stay cables have significant influence onVIV of the cable-stayed bridge and further conclusions canbe summarized as follows
(1) In the case when the geometric nonlinear termsare neglected accurate calculation of the responseamplitude of the cable-stayed bridge at lock-in onlyneeds use of the first mode shape of the linearizedundamped cable-stayed bridgemodel when the initialdynamic configuration of the cable-stayed bridgecorresponds to its mode shape whose mode numberis smaller than seven There is a different limit cyclewhen the initial dynamic configuration correspondsto its mode shape whose mode number is equal to orlarger than 7
Shock and Vibration 13
(2) In the case when the geometric nonlinear termsare considered calculation of the response of thecable-stayed bridge generally needs use of multiplemode shapes of the linearized undamped cable-stayedbridge model even when the initial dynamic config-uration of the cable-stayed bridge corresponds to itsfirstmode shapeThere is a limit cycle when the initialdynamic configuration of the cable-stayed bridgecorresponds to its first mode shape and its amplitudeis smaller than 03 for the generalized coordinate andthere is chaotic response when the initial dynamicconfiguration of the cable-stayed bridge correspondsto its first mode shape with its amplitude larger than03 for the generalized coordinate or one of its highermode shapes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grant nos 11302087 and 11442006 theNatural Science Foundation of Jiangsu Province under Grantno BK20130479 the Research Foundation for AdvancedTalents of Jiangsu University under Grant no 13JDG068 andthe National Science Foundation under Grant no CMMI-1000830
References
[1] K Y R Billah and R H Scanlan ldquoResonance Tacoma NarrowsBridge failure and undergraduate physics textbooksrdquoAmericanJournal of Physics vol 59 no 2 pp 118ndash124 1991
[2] R D Blevins Flow-Induced Vibration Krieger New York NYUSA 2nd edition 1994
[3] H Zheng R E Price Y Modarres-Sadeghi and M S Tri-antafyllou ldquoOn fatigue damage of long flexible cylinders dueto the higher harmonic force components and chaotic vortex-induced vibrationsrdquo Ocean Engineering vol 88 pp 318ndash3292014
[4] C Evangelinos D Lucor and G E Karniadakis ldquoDNS-derivedforce distribution on flexible cylinders subject to vortex-induced vibrationrdquo Journal of Fluids and Structures vol 14 no3 pp 429ndash440 2000
[5] HM Blackburn R N Govardhan andCH KWilliamson ldquoAcomplementary numerical and physical investigation of vortex-inducedrdquo Journal of Fluids and Structures vol 15 no 3-4 pp481ndash488 2001
[6] R H J Willden and J M R Graham ldquoNumerical predictionof VIV on long flexible circular cylindersrdquo Journal of Fluids andStructures vol 15 no 3-4 pp 659ndash669 2001
[7] E Guilmineau and P Queutey ldquoA numerical simulation ofvortex shedding from an oscillating circular cylinderrdquo Journalof Fluids and Structures vol 16 no 6 pp 773ndash794 2002
[8] I Jadic R M C So and M P Mignolet ldquoAnalysis of fluidmdashstructure interactions using a time-marching techniquerdquo Jour-nal of Fluids and Structures vol 12 no 6 pp 631ndash654 1998
[9] C Y Zhou RM C So andK Lam ldquoVortex-induced vibrationsof an elastic circular cylinderrdquo Journal of Fluids and Structuresvol 13 no 2 pp 165ndash189 1999
[10] J R Meneghini and P W Bearman ldquoNumerical simulationof high amplitude oscillatory flow about a circular cylinderrdquoJournal of Fluids and Structures vol 9 no 4 pp 435ndash455 1995
[11] T Sarpkaya ldquoComputational methods with vorticesrdquo Journal ofFluids Engineering vol 111 no 1 pp 5ndash52 1989
[12] T Sarpkaya ldquoVortex element methods for flow simulationrdquoAdvances in Applied Mechanics vol 31 pp 113ndash247 1994
[13] R D Gabbai and H Benaroya ldquoAn overview of modeling andexperiments of vortex-induced vibration of circular cylindersrdquoJournal of Sound and Vibration vol 282 no 3ndash5 pp 575ndash6162005
[14] R H Scanlan ldquoOn the state-of-the-artmethods for calculationsof flutter vortex-induced and buffeting response of bridgestructuresrdquo Tech Rep FHWARD-80050 National TechnicalInformation Service Springfield Va USA 1981
[15] RH Scanlan ldquoBridge flutter derivatives at vortex lock-inrdquo Jour-nal of Structural Engineering vol 124 no 4 pp 450ndash458 1998
[16] B J Vickery and R I Basu ldquoAcross-wind vibrations of struc-tures of circular cross-section Part I Development of a math-ematical model for two-dimensional conditionsrdquo Journal ofWind Engineering and Industrial Aerodynamics vol 12 no 1 pp49ndash73 1983
[17] A Larsen ldquoA generalized model for assessment of vortex-induced vibrations of flexible structuresrdquo Journal of Wind Engi-neering and Industrial Aerodynamics vol 57 no 2-3 pp 281ndash294 1995
[18] T Sarpkaya ldquoFluid forces on oscillating cylindersrdquo Journal ofWaterway Port Coastal and Ocean Division (ASCE) vol 104no 3 pp 275ndash290 1978
[19] T Staubli ldquoCalculation of vibration of an elastically mountedcylinder using experimental data from a forced oscillationrdquoJournal of Fluids Engineering vol 105 no 2 pp 225ndash229 1983
[20] W D Iwan and D L R Botelho ldquoVortex-induced oscillationof structures in waterrdquo Journal of Waterway Port Coastal andOcean Engineering vol 111 no 2 pp 289ndash303 1985
[21] R T Hartlen and I G Currie ldquoLift-oscillator model of vortex-induced vibrationrdquo Journal of Engineering Mechanics Division(ASCE) vol 96 no 5 pp 577ndash591 1970
[22] R A Skop and O M Griffin ldquoA model for the vortex-excitedresonant response of bluff cylindersrdquo Journal of Sound andVibration vol 27 no 2 pp 225ndash233 1973
[23] WD Iwan andRD Blevins ldquoAmodel for vortex induced oscil-lation of structuresrdquo Journal of Applied Mechanics TransactionsASME vol 41 no 3 pp 581ndash586 1974
[24] E H Dowell ldquoNon-linear oscillator models in bluff body aero-elasticityrdquo Journal of Sound and Vibration vol 75 no 2 pp 251ndash264 1981
[25] R H M Ogink and A V Metrikine ldquoA wake oscillator withfrequency dependent coupling for the modeling of vortex-induced vibrationrdquo Journal of Sound and Vibration vol 329 no26 pp 5452ndash5473 2010
[26] X Bai and W Qin ldquoUsing vortex strength wake oscillatorin modelling of vortex induced vibrations in two degrees offreedomrdquo European Journal of Mechanics BFluids vol 48 pp165ndash173 2014
[27] F Ehsan and R H Scanlan ldquoVortexminusinduced vibrations offlexible bridgesrdquo Journal of Engineering Mechanics vol 116 no6 pp 1392ndash1411 1990
14 Shock and Vibration
[28] A M Marra C Mannini and G Bartoli ldquoVan der Pol-typeequation for modeling vortex-induced oscillations of bridgedecksrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 99 no 6-7 pp 776ndash785 2011
[29] D Q Cao M T Song W D Zhu R W Tucker and C H-T Wang ldquoModeling and analysis of the in-plane vibration ofa complex cable-stayed bridgerdquo Journal of Sound and Vibrationvol 331 no 26 pp 5685ndash5714 2012
[30] E Caetano A Cunha V Gattulli and M Lepidi ldquoCable-deckdynamic interactions at the International Guadiana Bridge on-site measurements and finite element modellingrdquo StructuralControl and Health Monitoring vol 15 no 3 pp 237ndash264 2008
[31] I Akhtar A H Nayfeh and C J Ribbens ldquoOn the stability andextension of reduced-order Galerkin models in incompressibleflowsrdquo Theoretical and Computational Fluid Dynamics vol 23no 3 pp 213ndash237 2009
[32] S Sirisup and G E Karniadakis ldquoA spectral viscosity methodfor correcting the long-term behavior of POD modelsrdquo Journalof Computational Physics vol 194 no 1 pp 92ndash116 2004
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
12 Shock and Vibration
0
minus100 0 100 200 300minus200
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
minus1500
minus1000
minus500
500
1000
1500
Figure 15 Chaotic response of the midpoint of the deck beam119902119903(0) = 03120575
1119903
minus80
minus40
0
40
80
1000 2000 3000 4000 5000 5005 5010 50150t (s)
Vb 4(X
b 4=0)
(mm
)
Figure 16 Time response of themidpoint of the deck beam 1199021(0) =
01 and119873 = 1
1000 2000 3000 4000 5000 5005 5010 50150t (s)
minus1000
minus500
0
500
1000
Vb 4(X
b 4=0)
(mm
)
Figure 17 Time response of the midpoint of the deck beam 1199021(0) =
3 and119873 = 1
5 Conclusions
The dynamic behavior of a cable-stayed bridge that consistsof a simply supported four-cable-stayed deck beam andtwo rigid towers subjected to a distributed vortex shed-ding force on the deck beam has been investigated The
q1(0) = 01
q1(0) = 3
minus100
minus50
0
50
100
0 50 100minus50minus100
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 18 Phase portraits of the response of the midpoint of thedeck beam for different initial conditions119873 = 1
minus1500
minus1000
minus500
0
500
1000
1500
minus200 0 200 400minus400
(ddt)Vb 4(X
b 4=0)
(mm
s)
Vb4 (Xb4= 0) (mm)
Figure 19 Chaotic response of the midpoint of the deck beam119902119903(0) = 01120575
2119903
dynamic response of the cable-stayed bridge is calculatedusing Galerkin method in conjunction with Runge-Kutta-Felhberg method in MATLAB Convergence of Galerkinmethod for the dynamic response of the cable-stayed bridgeis studied Numerical simulations show that the geometricnonlinearities of the stay cables have significant influence onVIV of the cable-stayed bridge and further conclusions canbe summarized as follows
(1) In the case when the geometric nonlinear termsare neglected accurate calculation of the responseamplitude of the cable-stayed bridge at lock-in onlyneeds use of the first mode shape of the linearizedundamped cable-stayed bridgemodel when the initialdynamic configuration of the cable-stayed bridgecorresponds to its mode shape whose mode numberis smaller than seven There is a different limit cyclewhen the initial dynamic configuration correspondsto its mode shape whose mode number is equal to orlarger than 7
Shock and Vibration 13
(2) In the case when the geometric nonlinear termsare considered calculation of the response of thecable-stayed bridge generally needs use of multiplemode shapes of the linearized undamped cable-stayedbridge model even when the initial dynamic config-uration of the cable-stayed bridge corresponds to itsfirstmode shapeThere is a limit cycle when the initialdynamic configuration of the cable-stayed bridgecorresponds to its first mode shape and its amplitudeis smaller than 03 for the generalized coordinate andthere is chaotic response when the initial dynamicconfiguration of the cable-stayed bridge correspondsto its first mode shape with its amplitude larger than03 for the generalized coordinate or one of its highermode shapes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grant nos 11302087 and 11442006 theNatural Science Foundation of Jiangsu Province under Grantno BK20130479 the Research Foundation for AdvancedTalents of Jiangsu University under Grant no 13JDG068 andthe National Science Foundation under Grant no CMMI-1000830
References
[1] K Y R Billah and R H Scanlan ldquoResonance Tacoma NarrowsBridge failure and undergraduate physics textbooksrdquoAmericanJournal of Physics vol 59 no 2 pp 118ndash124 1991
[2] R D Blevins Flow-Induced Vibration Krieger New York NYUSA 2nd edition 1994
[3] H Zheng R E Price Y Modarres-Sadeghi and M S Tri-antafyllou ldquoOn fatigue damage of long flexible cylinders dueto the higher harmonic force components and chaotic vortex-induced vibrationsrdquo Ocean Engineering vol 88 pp 318ndash3292014
[4] C Evangelinos D Lucor and G E Karniadakis ldquoDNS-derivedforce distribution on flexible cylinders subject to vortex-induced vibrationrdquo Journal of Fluids and Structures vol 14 no3 pp 429ndash440 2000
[5] HM Blackburn R N Govardhan andCH KWilliamson ldquoAcomplementary numerical and physical investigation of vortex-inducedrdquo Journal of Fluids and Structures vol 15 no 3-4 pp481ndash488 2001
[6] R H J Willden and J M R Graham ldquoNumerical predictionof VIV on long flexible circular cylindersrdquo Journal of Fluids andStructures vol 15 no 3-4 pp 659ndash669 2001
[7] E Guilmineau and P Queutey ldquoA numerical simulation ofvortex shedding from an oscillating circular cylinderrdquo Journalof Fluids and Structures vol 16 no 6 pp 773ndash794 2002
[8] I Jadic R M C So and M P Mignolet ldquoAnalysis of fluidmdashstructure interactions using a time-marching techniquerdquo Jour-nal of Fluids and Structures vol 12 no 6 pp 631ndash654 1998
[9] C Y Zhou RM C So andK Lam ldquoVortex-induced vibrationsof an elastic circular cylinderrdquo Journal of Fluids and Structuresvol 13 no 2 pp 165ndash189 1999
[10] J R Meneghini and P W Bearman ldquoNumerical simulationof high amplitude oscillatory flow about a circular cylinderrdquoJournal of Fluids and Structures vol 9 no 4 pp 435ndash455 1995
[11] T Sarpkaya ldquoComputational methods with vorticesrdquo Journal ofFluids Engineering vol 111 no 1 pp 5ndash52 1989
[12] T Sarpkaya ldquoVortex element methods for flow simulationrdquoAdvances in Applied Mechanics vol 31 pp 113ndash247 1994
[13] R D Gabbai and H Benaroya ldquoAn overview of modeling andexperiments of vortex-induced vibration of circular cylindersrdquoJournal of Sound and Vibration vol 282 no 3ndash5 pp 575ndash6162005
[14] R H Scanlan ldquoOn the state-of-the-artmethods for calculationsof flutter vortex-induced and buffeting response of bridgestructuresrdquo Tech Rep FHWARD-80050 National TechnicalInformation Service Springfield Va USA 1981
[15] RH Scanlan ldquoBridge flutter derivatives at vortex lock-inrdquo Jour-nal of Structural Engineering vol 124 no 4 pp 450ndash458 1998
[16] B J Vickery and R I Basu ldquoAcross-wind vibrations of struc-tures of circular cross-section Part I Development of a math-ematical model for two-dimensional conditionsrdquo Journal ofWind Engineering and Industrial Aerodynamics vol 12 no 1 pp49ndash73 1983
[17] A Larsen ldquoA generalized model for assessment of vortex-induced vibrations of flexible structuresrdquo Journal of Wind Engi-neering and Industrial Aerodynamics vol 57 no 2-3 pp 281ndash294 1995
[18] T Sarpkaya ldquoFluid forces on oscillating cylindersrdquo Journal ofWaterway Port Coastal and Ocean Division (ASCE) vol 104no 3 pp 275ndash290 1978
[19] T Staubli ldquoCalculation of vibration of an elastically mountedcylinder using experimental data from a forced oscillationrdquoJournal of Fluids Engineering vol 105 no 2 pp 225ndash229 1983
[20] W D Iwan and D L R Botelho ldquoVortex-induced oscillationof structures in waterrdquo Journal of Waterway Port Coastal andOcean Engineering vol 111 no 2 pp 289ndash303 1985
[21] R T Hartlen and I G Currie ldquoLift-oscillator model of vortex-induced vibrationrdquo Journal of Engineering Mechanics Division(ASCE) vol 96 no 5 pp 577ndash591 1970
[22] R A Skop and O M Griffin ldquoA model for the vortex-excitedresonant response of bluff cylindersrdquo Journal of Sound andVibration vol 27 no 2 pp 225ndash233 1973
[23] WD Iwan andRD Blevins ldquoAmodel for vortex induced oscil-lation of structuresrdquo Journal of Applied Mechanics TransactionsASME vol 41 no 3 pp 581ndash586 1974
[24] E H Dowell ldquoNon-linear oscillator models in bluff body aero-elasticityrdquo Journal of Sound and Vibration vol 75 no 2 pp 251ndash264 1981
[25] R H M Ogink and A V Metrikine ldquoA wake oscillator withfrequency dependent coupling for the modeling of vortex-induced vibrationrdquo Journal of Sound and Vibration vol 329 no26 pp 5452ndash5473 2010
[26] X Bai and W Qin ldquoUsing vortex strength wake oscillatorin modelling of vortex induced vibrations in two degrees offreedomrdquo European Journal of Mechanics BFluids vol 48 pp165ndash173 2014
[27] F Ehsan and R H Scanlan ldquoVortexminusinduced vibrations offlexible bridgesrdquo Journal of Engineering Mechanics vol 116 no6 pp 1392ndash1411 1990
14 Shock and Vibration
[28] A M Marra C Mannini and G Bartoli ldquoVan der Pol-typeequation for modeling vortex-induced oscillations of bridgedecksrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 99 no 6-7 pp 776ndash785 2011
[29] D Q Cao M T Song W D Zhu R W Tucker and C H-T Wang ldquoModeling and analysis of the in-plane vibration ofa complex cable-stayed bridgerdquo Journal of Sound and Vibrationvol 331 no 26 pp 5685ndash5714 2012
[30] E Caetano A Cunha V Gattulli and M Lepidi ldquoCable-deckdynamic interactions at the International Guadiana Bridge on-site measurements and finite element modellingrdquo StructuralControl and Health Monitoring vol 15 no 3 pp 237ndash264 2008
[31] I Akhtar A H Nayfeh and C J Ribbens ldquoOn the stability andextension of reduced-order Galerkin models in incompressibleflowsrdquo Theoretical and Computational Fluid Dynamics vol 23no 3 pp 213ndash237 2009
[32] S Sirisup and G E Karniadakis ldquoA spectral viscosity methodfor correcting the long-term behavior of POD modelsrdquo Journalof Computational Physics vol 194 no 1 pp 92ndash116 2004
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 13
(2) In the case when the geometric nonlinear termsare considered calculation of the response of thecable-stayed bridge generally needs use of multiplemode shapes of the linearized undamped cable-stayedbridge model even when the initial dynamic config-uration of the cable-stayed bridge corresponds to itsfirstmode shapeThere is a limit cycle when the initialdynamic configuration of the cable-stayed bridgecorresponds to its first mode shape and its amplitudeis smaller than 03 for the generalized coordinate andthere is chaotic response when the initial dynamicconfiguration of the cable-stayed bridge correspondsto its first mode shape with its amplitude larger than03 for the generalized coordinate or one of its highermode shapes
Conflict of Interests
The authors declare that there is no conflict of interestsregarding publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grant nos 11302087 and 11442006 theNatural Science Foundation of Jiangsu Province under Grantno BK20130479 the Research Foundation for AdvancedTalents of Jiangsu University under Grant no 13JDG068 andthe National Science Foundation under Grant no CMMI-1000830
References
[1] K Y R Billah and R H Scanlan ldquoResonance Tacoma NarrowsBridge failure and undergraduate physics textbooksrdquoAmericanJournal of Physics vol 59 no 2 pp 118ndash124 1991
[2] R D Blevins Flow-Induced Vibration Krieger New York NYUSA 2nd edition 1994
[3] H Zheng R E Price Y Modarres-Sadeghi and M S Tri-antafyllou ldquoOn fatigue damage of long flexible cylinders dueto the higher harmonic force components and chaotic vortex-induced vibrationsrdquo Ocean Engineering vol 88 pp 318ndash3292014
[4] C Evangelinos D Lucor and G E Karniadakis ldquoDNS-derivedforce distribution on flexible cylinders subject to vortex-induced vibrationrdquo Journal of Fluids and Structures vol 14 no3 pp 429ndash440 2000
[5] HM Blackburn R N Govardhan andCH KWilliamson ldquoAcomplementary numerical and physical investigation of vortex-inducedrdquo Journal of Fluids and Structures vol 15 no 3-4 pp481ndash488 2001
[6] R H J Willden and J M R Graham ldquoNumerical predictionof VIV on long flexible circular cylindersrdquo Journal of Fluids andStructures vol 15 no 3-4 pp 659ndash669 2001
[7] E Guilmineau and P Queutey ldquoA numerical simulation ofvortex shedding from an oscillating circular cylinderrdquo Journalof Fluids and Structures vol 16 no 6 pp 773ndash794 2002
[8] I Jadic R M C So and M P Mignolet ldquoAnalysis of fluidmdashstructure interactions using a time-marching techniquerdquo Jour-nal of Fluids and Structures vol 12 no 6 pp 631ndash654 1998
[9] C Y Zhou RM C So andK Lam ldquoVortex-induced vibrationsof an elastic circular cylinderrdquo Journal of Fluids and Structuresvol 13 no 2 pp 165ndash189 1999
[10] J R Meneghini and P W Bearman ldquoNumerical simulationof high amplitude oscillatory flow about a circular cylinderrdquoJournal of Fluids and Structures vol 9 no 4 pp 435ndash455 1995
[11] T Sarpkaya ldquoComputational methods with vorticesrdquo Journal ofFluids Engineering vol 111 no 1 pp 5ndash52 1989
[12] T Sarpkaya ldquoVortex element methods for flow simulationrdquoAdvances in Applied Mechanics vol 31 pp 113ndash247 1994
[13] R D Gabbai and H Benaroya ldquoAn overview of modeling andexperiments of vortex-induced vibration of circular cylindersrdquoJournal of Sound and Vibration vol 282 no 3ndash5 pp 575ndash6162005
[14] R H Scanlan ldquoOn the state-of-the-artmethods for calculationsof flutter vortex-induced and buffeting response of bridgestructuresrdquo Tech Rep FHWARD-80050 National TechnicalInformation Service Springfield Va USA 1981
[15] RH Scanlan ldquoBridge flutter derivatives at vortex lock-inrdquo Jour-nal of Structural Engineering vol 124 no 4 pp 450ndash458 1998
[16] B J Vickery and R I Basu ldquoAcross-wind vibrations of struc-tures of circular cross-section Part I Development of a math-ematical model for two-dimensional conditionsrdquo Journal ofWind Engineering and Industrial Aerodynamics vol 12 no 1 pp49ndash73 1983
[17] A Larsen ldquoA generalized model for assessment of vortex-induced vibrations of flexible structuresrdquo Journal of Wind Engi-neering and Industrial Aerodynamics vol 57 no 2-3 pp 281ndash294 1995
[18] T Sarpkaya ldquoFluid forces on oscillating cylindersrdquo Journal ofWaterway Port Coastal and Ocean Division (ASCE) vol 104no 3 pp 275ndash290 1978
[19] T Staubli ldquoCalculation of vibration of an elastically mountedcylinder using experimental data from a forced oscillationrdquoJournal of Fluids Engineering vol 105 no 2 pp 225ndash229 1983
[20] W D Iwan and D L R Botelho ldquoVortex-induced oscillationof structures in waterrdquo Journal of Waterway Port Coastal andOcean Engineering vol 111 no 2 pp 289ndash303 1985
[21] R T Hartlen and I G Currie ldquoLift-oscillator model of vortex-induced vibrationrdquo Journal of Engineering Mechanics Division(ASCE) vol 96 no 5 pp 577ndash591 1970
[22] R A Skop and O M Griffin ldquoA model for the vortex-excitedresonant response of bluff cylindersrdquo Journal of Sound andVibration vol 27 no 2 pp 225ndash233 1973
[23] WD Iwan andRD Blevins ldquoAmodel for vortex induced oscil-lation of structuresrdquo Journal of Applied Mechanics TransactionsASME vol 41 no 3 pp 581ndash586 1974
[24] E H Dowell ldquoNon-linear oscillator models in bluff body aero-elasticityrdquo Journal of Sound and Vibration vol 75 no 2 pp 251ndash264 1981
[25] R H M Ogink and A V Metrikine ldquoA wake oscillator withfrequency dependent coupling for the modeling of vortex-induced vibrationrdquo Journal of Sound and Vibration vol 329 no26 pp 5452ndash5473 2010
[26] X Bai and W Qin ldquoUsing vortex strength wake oscillatorin modelling of vortex induced vibrations in two degrees offreedomrdquo European Journal of Mechanics BFluids vol 48 pp165ndash173 2014
[27] F Ehsan and R H Scanlan ldquoVortexminusinduced vibrations offlexible bridgesrdquo Journal of Engineering Mechanics vol 116 no6 pp 1392ndash1411 1990
14 Shock and Vibration
[28] A M Marra C Mannini and G Bartoli ldquoVan der Pol-typeequation for modeling vortex-induced oscillations of bridgedecksrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 99 no 6-7 pp 776ndash785 2011
[29] D Q Cao M T Song W D Zhu R W Tucker and C H-T Wang ldquoModeling and analysis of the in-plane vibration ofa complex cable-stayed bridgerdquo Journal of Sound and Vibrationvol 331 no 26 pp 5685ndash5714 2012
[30] E Caetano A Cunha V Gattulli and M Lepidi ldquoCable-deckdynamic interactions at the International Guadiana Bridge on-site measurements and finite element modellingrdquo StructuralControl and Health Monitoring vol 15 no 3 pp 237ndash264 2008
[31] I Akhtar A H Nayfeh and C J Ribbens ldquoOn the stability andextension of reduced-order Galerkin models in incompressibleflowsrdquo Theoretical and Computational Fluid Dynamics vol 23no 3 pp 213ndash237 2009
[32] S Sirisup and G E Karniadakis ldquoA spectral viscosity methodfor correcting the long-term behavior of POD modelsrdquo Journalof Computational Physics vol 194 no 1 pp 92ndash116 2004
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
14 Shock and Vibration
[28] A M Marra C Mannini and G Bartoli ldquoVan der Pol-typeequation for modeling vortex-induced oscillations of bridgedecksrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 99 no 6-7 pp 776ndash785 2011
[29] D Q Cao M T Song W D Zhu R W Tucker and C H-T Wang ldquoModeling and analysis of the in-plane vibration ofa complex cable-stayed bridgerdquo Journal of Sound and Vibrationvol 331 no 26 pp 5685ndash5714 2012
[30] E Caetano A Cunha V Gattulli and M Lepidi ldquoCable-deckdynamic interactions at the International Guadiana Bridge on-site measurements and finite element modellingrdquo StructuralControl and Health Monitoring vol 15 no 3 pp 237ndash264 2008
[31] I Akhtar A H Nayfeh and C J Ribbens ldquoOn the stability andextension of reduced-order Galerkin models in incompressibleflowsrdquo Theoretical and Computational Fluid Dynamics vol 23no 3 pp 213ndash237 2009
[32] S Sirisup and G E Karniadakis ldquoA spectral viscosity methodfor correcting the long-term behavior of POD modelsrdquo Journalof Computational Physics vol 194 no 1 pp 92ndash116 2004
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of