research article numerical study of the active tendon...

Research ArticleNumerical Study of the Active Tendon Control of a Cable-StayedBridge in a Construction Phase

M. H. El Ouni1,2 and N. Ben Kahla1,2

1 Applied Mechanics and Systems Research Laboratory (AMSRL), Tunisia Polytechnic School, University of Carthage,2078 La Marsa, Tunisia

2Higher Institute of Applied Sciences and Technologies of Sousse, University of Sousse, Taffala, Ibn Khaldoun, 4003 Sousse, Tunisia

Correspondence should be addressed to N. Ben Kahla; [email protected]

Received 5 July 2012; Accepted 2 March 2013; Published 9 April 2014

Academic Editor: Sami El-Borgi

Copyright © 2014 M. H. El Ouni and N. Ben Kahla. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

This paper investigates numerically the active tendon control of a cable-stayed bridge in a construction phase. A linear FiniteElement model of small scale mock-up of the bridge is first presented. Active damping is added to the structure by using pairsof collocated force actuator-displacement sensors located on each active cable and decentralized first order positive positionfeedback (PPF) or direct velocity feedback (DVF). A comparison between these two compensators showed that each one has goodperformance for some modes and performs inadequately with the other modes. A decentralized parallel PPF-DVF is proposedto get the better of the two compensators. The proposed strategy is then compared to the one using decentralized integral forcefeedback (IFF) and showed better performance. The Finite Element model of the bridge is coupled with a nonlinear cable takinginto account sag effect, general support movements, and quadratic and cubic nonlinear couplings between in-plane and out-of-plane motions. Finally, the proposed strategy is used to control both deck and cable vibrations induced by parametric excitation.Both cable and deck vibrations are attractively damped.

1. Introduction

In the past few decades, design and construction of civilstructures showed a very deep evolution because of thetechnological progress inmaterials and devices. Cable-stayedbridges increased considerably their center span from 182.6m(Stromsund Bridge in Sweden) to 1104m (Russky Bridge inRussia). These structures are getting more slender, light, andflexible which makes them sensitive to vibrations inducedby wind, traffic, waves, or even earthquakes. Consequently,vibration control has become a major issue in civil engineer-ing.

Vibrations in cable-stayed bridges may be reduced usingpassive [1–4], semiactive [5–12], and active methods [13].Active control uses a set of actuators and sensors connectedby feedback or feed forward loops. Among the proposeddevices to control vibrations of cable-stayed bridges are theactive mass dampers [14], active aerodynamic appendages

[15], and active tendons. Several strategies have been pro-posed for the active tendon control of the global modes ofbridges, as well as for the in-plane and out-of-plane cablevibrations. Yang and Giannopolous [16] were the first topropose active tendon control to reduce vibration inducedby strong wind gusts. They studied the feedback control of asimple continuous beammodel suspended by four stay cablesusing four active tendons equipped with servohydraulicactuators. With respect to the motion of the bridge deckdetected by the sensors installed at the anchorage of eachcable, the actuators actively change the cable tension andapply time-varying forces to the deck in order to reducethe vibrations. Fujino and Susumpow [17] carried out anexperimental study on active control of planar cable vibrationby axial support motion. Using a cable-supported cantileverbeam model, Warnitchai et al. [18] performed an analyticaland experimental study on active tendon control of cable-stayed bridges. They demonstrated that the vertical global

Hindawi Publishing CorporationShock and VibrationVolume 2014, Article ID 937541, 10 pageshttp://dx.doi.org/10.1155/2014/937541

2 Shock and Vibration

mode of the bridge can be damped with a linear feedback ofthe girder velocity on the active tendon displacement and thatthe in-plane local cable vibration can be controlled efficientlyby sag induced forces. Kobayashi et al. [19] studied the tendoncontrol of cable-stayed bridges by setting active cables parallelto stay cables. They conducted an experimental study on a1/100 scale half-span model of a 410m center span cable-stayed bridge to demonstrate the effectiveness of their controlstrategy using a tendon control force proportional to thevelocity of the girder.

All the studies on active tendon control presented aboveused noncollocated pairs of actuator sensor which maydestabilize the structure for certain gain values and may alsocause spillover instability. Achkire and Preumont [20] solvedthis problem using a collocated displacement actuator-forcesensor configuration. They considered the active vibrationcontrol of cable-stayed bridges with an active tendon con-trolling the axial displacement of the cable anchor point. Byusing a piezoelectric actuator collocated with a force sensormeasuring the cable tension, integral force feedback (IFF) isapplied to offer an active damping control. An experimentalsetup consisting of a cable in connection with a spring-mass systemwas tested to evaluate control efficiency. Bossensand Preumont [21] proposed a simplified linear theory topredict the closed-loop poles with a root locus technique andreported an experimental study of two cable-stayed bridgemodels using active tendon control. The first one is a smallsize mock-up (3m-length) representative of a cable-stayedbridge in a construction phase. The second mock-up is a30m length cable-stayed cantilever structure, equipped withhydraulic actuators. The experimental results showed thatthe active tendon control brought a substantial reductionin the deck and cable vibration amplitudes. Using the samecontrol strategy (decentralized collocated IFF), El Ouni et al.[22, 23] studied numerically and experimentally the effect ofactive tendon control on the principal parametric resonanceof a stay cable using a small scale mock-up of a cable-stayed bridge [24].They showed that the threshold excitationamplitude of the deck, needed to trigger the parametricexcitation, increases by an increase of the active damping inthe structure. Other active control laws can be also used ina similar way as the IFF, such us first order positive positionfeedback (PPF) proposed by Baz et al. [25] and direct velocityfeedback (DVF) proposed by Balas [26].

This paper investigates numerically the active tendoncontrol of a small scale mock-up of a cable-stayed bridge in aconstruction phase. Active damping is added to the structureby using pairs of collocated force actuator-displacementsensors located on each active cable. This configuration isfirst examined with decentralized PPF and DVF. Then, aparallel PPF-DVF is proposed to get the better of the twocompensators and compared to the one using decentralizedIFF. A Finite Element model of the bridge is coupled with anonlinear cable which takes into account sag effect, generalsupport movements, and quadratic and cubic nonlinear cou-plings between in-plane and out-of-plane motions. Finally,the proposed strategy is used to control both deck and cablevibrations induced by parametric excitations.

Cable

Rectangularstiffener

Pillar

U-shaped beam

Concrete base

Masses

Figure 1: Description of the main components of the bridge.

(1)

(2)(3)

(4)

(5)(6)(7)

(8)

AB

Figure 2: FE model of the bridge created by Matlab/SDT.

2. The 3D Finite Element Model

A model of a smart cable-stayed bridge was developed inActive Structure Laboratory at ULB [24]. This mock-uprepresents a small scale model of the bridge in a constructionphase. The bridge is made of a central steel pillar resting ona concrete block and a deck supported by 8 stainless steelcables. The deck is made of two U-shaped aluminum beams,steel rectangular stiffeners, and forty additional masses (seeFigure 1). The height of the pillar is 1.6m; the total lengthand width of the deck are, respectively, 3 and 0.32m. TheMatlab/SDTools software has been used to build a 3D FiniteElement (FE) model of the bridge (see Figure 2). Shellelements are used to model the pillar, the U-shaped beams,and the stiffeners. The additional masses are modeled by 3Delements and the cables are represented by linear bars. Aclamped support condition at the lower end of the pillar isadopted. Thus, the final bridge model is composed of 29172nodes and 23743 elements and has 112980 degrees of freedom.The natural damping of all modes of the structure is equal to1%. For more details about the bridge demonstrator, the FE

Shock and Vibration 3

model, and the numerical and experimental modal analysissee [23, 24].

3. Active Tendon Control Using DecentralizedPPF and DVF

The global equation of motion of the linear cable-stayedbridge equipped with pairs of a force actuator and a displace-ment sensor in the chosen active cables (n) can be written asfollows:

𝑀�� + 𝐶�� + 𝐾𝑥 = 𝐵𝑛

𝐹cont + 𝐹excit, (1)

where 𝑀, 𝐶, and 𝐾 are, respectively, the mass, damping,and the stiffness of the bridge. ��, ��, and 𝑥 are, respectively,the acceleration, velocity, and displacement vectors. B is theinfluence matrix relating the local coordinate systems of theactive tendons to the global coordinates.𝐹excit is the excitationforce vector. 𝑛𝐹cont are the control forces.

The control forces of the decentralized DVF [27] are𝑛

𝐹cont =𝑛

𝐻1(𝑠) (𝑛

𝑥𝑖−𝑛

𝑥𝑗) = −

𝑛𝑔1𝑠 (𝑛

𝑥𝑖−𝑛

𝑥𝑗) , (2)

where 𝑛𝐻1(𝑠) is the feedback control law of the DVF, s is the

Laplace variable, 𝑛𝑔1are the controller gains, and ( 𝑛𝑥

𝑖−𝑛

𝑥𝑗)

are the relative displacements of the extremities (𝑖 and 𝑗) ofthe cables projected on the chord lines.

The control forces of the decentralized first order PPF [27]are

𝑛

𝐹cont =𝑛

𝐻2(𝑠) (𝑛

𝑥𝑖−𝑛

𝑥𝑗) =

𝑛𝑔2

1 + 𝜏𝑠(𝑛

𝑥𝑖−𝑛

𝑥𝑗) , (3)

where 𝑛𝐻2(𝑠) is the feedback controller law of the PPF, 𝑛𝑔

2

are the controller gains, and 𝜏 is a design parameter whichdecides the damping ratio, defines the position of the poleof the first order PPF on the real axis, and fixes the stabilitymargin.

Themain idea in developing a decentralized parallel PPF-DVF strategy is as follows: can we get the better of the twocompensators in order to control the maximum of modes?

The control forces of the proposed decentralized parallelPPF-DVF strategy are

𝑛

𝐹cont =𝑛

𝐻3(𝑠) (𝑛

𝑥𝑖−𝑛

𝑥𝑗)

= [−𝑛𝑔1𝑠 +

𝑛𝑔2

1 + 𝜏 𝑠] (𝑛

𝑥𝑖−𝑛

𝑥𝑗) ,

(4)

where 𝑛𝐻3(𝑠) is the feedback controller law of the proposed

concept and the controller gains 𝑛𝑔1and 𝑛𝑔

2can be tuned to

get optimal damping on the target modes.The block diagramof the proposed control system is given in Figure 3.

4. Comparison between DifferentControl Strategies

Figure 4(a) shows the root locus of the DVF added throughthe four small tendons. This control law is unconditionally

System

Performance metric

PPF

DVF

Fexcitn

ng21 + 𝜏s

−ng1s

nxi −nxjFcont ( )

Figure 3: Block diagram of the proposed control system.

stable for all gain values, since all loops are contained inthe left side of the imaginary axis. Figure 4(b) shows theroot locus of the first order PPF added through the foursmall tendons. The PPF is conditionally stable and has thesame poles and zeros as in the DVF case, because the twocontrollers use the same actuator and sensor configuration.When the pole travelling on the real axis reaches the origin(the stability limit), the controller becomes unstable. In fact,when the pole reaches the stability limit the negative stiffnessof the controller should exceed the static stiffness of thesystem, which leads to the static collapse of the bridge. Forsome modes, only the initial part of the loop is available,because of the stability condition. Note also that in the PPFcase, the loops do not leave the open loop poles orthogonalto the imaginary axis as in the DVF case (as a result of thenegative stiffness which softens the system), which suggeststhat the control effort may be larger [27].

Figure 4(c) shows the root locus of the proposed strategy.It is also conditionally stable but the loops of the highfrequencies are wider than those of the PPF and similarly forthe low frequencies which seemwider than those of the DVF.The major advantage of the proposed strategy is that the sizeof the loops can be tuned not only through 𝜏 but also through𝑛𝑔1and 𝑛𝑔

2.

Themaximumdamping ratio for decentralizedDVF, PPF,and parallel PPF-DVF is determined for the first 17 modesusing the root locus technique and is plotted in Figure 5 asa function of mode number. The PPF seems more efficientfor modes 1, 3, 5, 6, and 9. The DVF is more efficient formodes 8, 10, 11, 13, 14, and 15. The parallel PPF-DVF hasimportant damping for many modes. With all the controlstrategies, weak controllability is observed for modes 2, 4, 12,16, and 17. Figure 6 shows the FRF between the white noiseexcitation (𝐹excit) through the cable number 2 and the verticaldisplacement of the girder (𝑈

𝑧) in point A with different

control strategies. Using the parallel PPF-DVF, the negativestiffness effect of the PPF is reduced and an average FRF isobtained between the FRF with PPF and the one with DVF.

5. Comparison between Parallel PPF-DVFand IFF

The decentralized IFF [21] uses a collocated pairs of dis-placement actuator-force sensor in each active cable. The IFF


0

100

200

300

400

5000.10.20.320.44

0.58

0.7

0.84

0.95

Im(s

)

DVF

Re(s)−250 −200 −150 −100 −50 500

(a)

0

100

200

300

400

5000.10.20.320.44

0.58

0.7

0.84

0.95

PPF

Stability limit

Re(s)−250 −200 −150 −100 −50 500

Im(s

)

(b)

0.10.20.320.44

0.58

0.7

0.84

0.95

//PPF-DVF

Stability limit

0

100

200

300

400

500

−250 −200 −150 −100 −50 500

Re(s)

Im(s

)

(c)

Figure 4: Root locus of the DVF (a), the first order PPF (b), and the parallel PPF-DVF (c) added through four small tendons.

is unconditionally stable but suffers from negative stiffnessproblem which may be solved by adding a 2nd order highpass filter (2Hz) in series with the IFF [21]. The maximumdamping ratio for decentralized IFF and parallel PPF-DVFare compared for the first 17 modes (see Figure 7). Bothstrategies successfully provide the cable-stayed bridge with

active damping but the parallel PPF-DVF shows better per-formance for all modes except mode number 7.The proposedstrategy is conditionally stable and also has a problem ofnegative stiffness which must be treated carefully for realapplications. The FRF between the force of excitation (𝐹excit)and the vertical displacement of the deck (Uz) in point A is


0 2 4 6 8 10 12 14 16 180

10

20

30

40

50

60

70

80

90

DVFPPF//PPF-DVF

100

Mode number

Dam

ping

ratio

(%)

Figure 5: Maximum damping ratio as a function of mode numberfor different active control strategies.

Frequency (Hz)

With DVFWith PPFWith //PPF-DVF

Open loop

Fex

cit

10−2

10−3

10−4

10−5

10−6

10−70 5 10 15 20 25 30 35 40 45 50

Uz/

Figure 6: FRF between the force of excitation (Fexcit) and thevertical displacement of the deck (𝑈

𝑧) in point A, for different active

control strategies.

plotted in Figure 8 for the two strategies with a maximumdamping on mode number 1.

6. Active Tendon Control ofa Nonlinear Cable-Stayed Bridgeunder Parametric Excitation

6.1. Nonlinear Modelling of an Inclined Small Sag Cable. Thenonlinear model of the inclined cable takes into accountgeneral support movement, sag effect, and quadratic andcubic nonlinear couplings between in-plane and out-of-planemotions. The cable model is presented in Figure 9. The localcoordinate system is chosen such that the x-axis is definedalong the chord line and y-axis in the horizontal plane.

0 2 4 6 8 10 12 14 16 180

10

20

30

40

50

60

70

80

IFF//PPF-DVF

90

100

Mode number

Dam

ping

ratio

(%)

Figure 7: Maximum damping ratio as a function of mode numberfor IFF and parallel PPF-DVF concepts.

With IFFWith //PPF-DVF

Open loop

Frequency (Hz)0 5 10 15 20 25 30 35 40 45 50

10−2

10−3

10−4

10−5

10−6

10−7

Fex

cit

Uz/

Figure 8: FRF between the force of excitation (Fexcit) and thevertical displacement of the deck (𝑈

𝑧) in point A for IFF and parallel

PPF-DVF concepts.

The z-axis is then taken perpendicular to the chord line,in the gravity plane. The cable displacements are separatedinto three parts: the static, the quasi-static, and the dynamiccontributions (for more details see [28]).

6.1.1. Out-of-Plane Cable Motion. The transverse out-of-plane displacements of the cable are described by the follow-ing equation ofmotion governing the generalized coordinatesyn of the nth out-of-plane mode of vibration:

1

2𝑚𝑙 { 𝑦

𝑛+ 2𝜉𝑦𝑛𝜔𝑦𝑛

𝑦𝑛+𝑛2

𝜋2

𝑚𝑙2(𝑇0+ 𝑇𝑞+ 𝑇𝑑) 𝑦𝑛}

= 𝐹𝑦𝑛

−𝑚𝑙

𝑛𝜋(V𝑎+ (−1)

𝑛+1V𝑏) ,

(5)


Chord line

a

by

z

x

g

Dynamic state

Static sag profile

ub(t)

u(x, t)

wa(t)

ua(t)

va(t)

ws(x)

w(x, t)

v(x, t)

vb(t)

wb(t)

𝜃

Figure 9: 3D model of an inclined cable with general supportmovements.

wherem is the mass per unit length; is the chord length of thecable; 𝜉

𝑦𝑛

, 𝜔𝑦𝑛

, and 𝐹𝑦𝑛

are, respectively, the modal damping,the frequency, and the modal component of the externalforces applied to the cable, associated with the generalizedcoordinates 𝑦

𝑛of the cable mode n; 𝑇

0, 𝑇𝑞, and 𝑇

𝑑are,

respectively, the static tension in the cable at its equilibrium,the tension increment induced by the supportmovement, andthe tension increment induced by the dynamic motion of thecable; 𝑇

𝑑is responsible for the quadratic and cubic nonlinear

couplings between in-plane and out-of-plane motions; V𝑎

and V𝑏are, respectively, the transverse acceleration of the

anchorage points a and b according to the y-axis.The expressions of 𝑇

𝑞and 𝑇

𝑑are given in the Appendix.

6.1.2. In-Plane Cable Motion. The in-plane displacements ofthe cable (perpendicular to its chord line) is described bythe following equation of motion governing the generalizedcoordinates 𝑧

𝑛of the nth in-plane mode of vibration and

accounting for the gravity effect (𝛾 = 𝜌𝑔 cos 𝜃):

1

2𝑚𝑙 {��

𝑛+ 2𝜉𝑧𝑛

𝜔𝑧𝑛

��𝑛+𝑛2

𝜋2

𝑚𝑙2(𝑇0+ 𝑇𝑞+ 𝑇𝑑) 𝑧𝑛}

= 𝐹𝑧𝑛

−𝑚𝑙

𝑛𝜋(��𝑎+ (−1)

𝑛+1

��𝑏)

+

𝑚𝑙2

𝐸𝑞𝛾

(𝜎𝑠)2

(1 + (−1)𝑛+1

)

(𝑛𝜋)3

(��𝑏− ��𝑎)

−𝛾𝐴𝑙

𝑇0

(1 + (−1)𝑛+1

)

𝑛𝜋𝑇𝑑,

(6)

where𝐸𝑞is the effectivemodulus of elasticity (see Appendix),

𝛾 is the component of distributed weight along the cable,𝜌 is the cable density, 𝑔 is the gravity, 𝜃 is the angle ofthe chord line with respect to the horizontal, A is the crosssection of the cable and 𝜎𝑠 is the static stress, 𝜉

𝑧𝑛

, 𝜔𝑧𝑛

, and 𝐹𝑧𝑛

are, respectively, the modal damping, the frequency, and the

Nonlinear cables

FE model of thebridge without cables

Reaction forces applied by the cables to the pylon and the deck

Anchorage displacementsapplied by the pylon and the deck

Figure 10: Principle of coupling between the FEmodel of the bridgeand the nonlinear cables.

modal component of the external forces applied to the cable,associated with the generalized coordinates 𝑧

𝑛of the cable

mode n. ��𝑎and ��

𝑏are, respectively, the in-plane acceleration

of the anchorage points a and b according to the z-axis. ��𝑎

and ��𝑏are, respectively, the longitudinal acceleration of the

anchorage points a and b according to the y-axis.

6.2. Coupling between the Nonlinear Cables and the FEModel of the Bridge. As an alternative to a general nonlinearFinite Element approach which would be extremely timeconsuming, we had developed, using SDTools [29] andMatlab/Simulink, software which combines a Finite Elementmodel of the linear structure with a nonlinear analyticalmodel of the cables accounting for general supportmovementand cubic and quadratic couplings between in-plane and out-of-plane motions of the cable. Figure 10 shows the principleof coupling between the FE model of the bridge and the non-linear cables: the structure motion imposes displacements tothe cables supports and the reactions of the cables supportsact like external forces to the structure. Using SDTools, itcan be achieved numerically by creating pairs of collocatedforce actuator-displacement sensors in the anchorage pointsand coupling the cables to the rest of the structure throughSimulink (for more details about the coupling see [23]).

Taking into account the nonlinear dynamics of the nccables and active damping, the global equation of motion ofthe cable-stayed bridge can be expressed in modal coordi-nates as follows:

𝜇𝑖{ 𝑒𝑖+ 2𝜉𝑖𝜔𝑖𝑒𝑖+ 𝜔2

𝑖𝑒𝑖}

= 𝐹𝑖+ 𝜙𝑇

𝑖𝐵𝑛

𝐹cont

− 𝜙𝑇

𝑖

𝑛𝑐

∑

𝑘

[(𝐿𝑘𝑇

𝑎, 𝐿𝑘𝑇

𝑏) (𝐹𝑘

𝑢𝑎

𝐹𝑘

V𝑎

𝐹𝑘

𝑤𝑎

𝐹𝑘

𝑢𝑏

𝐹𝑘

V𝑏

𝐹𝑘

𝑤𝑏

)𝑇

] ,

(7)

where 𝜇𝑖, 𝜉𝑖, 𝜔𝑖, and 𝐹

𝑖are, respectively, the modal mass, the

modal damping, the frequency, and the modal componentof the external forces applied to the bridge without cables,associated with the generalized coordinates 𝑒

𝑖of the bridge

mode i. 𝜙𝑖represents the mode shapes of the bridge without

cables. 𝐿𝑘𝑇𝑎

and 𝐿𝑘𝑇

𝑏are the transformation matrices allowing

the transformation from the global coordinates of the bridgeto the local coordinates of the cable k.𝐹𝑘

𝑢𝑎

, 𝐹𝑘

V𝑎

, 𝐹𝑘

𝑤𝑎

, 𝐹𝑘

𝑢𝑏

, 𝐹𝑘

V𝑏

,


(1)(2)

(3)(4)

(5)(6)

(7) (8)

Active tendons(3, 4, 5 and 6)

Sine wave(f = 13.55Hz)

f/2

Figure 11: Description of the numerical experience.

and 𝐹𝑘

𝑤𝑏

are the reaction forces on the anchorage points (aand b) written in the local coordinates of the cable k (seeAppendix). 𝑛𝐹cont are the control forces of the n active cablesand are given in (4).

The equations of motions of the cables and the bridge aresolved simultaneously and interactively using the fourth andfifth order Dormand-Prince Runge-Kutta method.

6.3. Parametric Excitation. In cable-stayed bridges, the pres-ence of many low frequencies in the deck or tower and inthe stay cables may give rise to parametric excitation. Thecoupling between a local cable and a global structure makesthe bridge sensitive to very small motion of the deck ortower which may cause dynamic instabilities and very largeoscillations of the stay cables (see Figure 11). This may occurwhen the frequency of the anchorage motion is close to thefundamental frequency or twice the first natural frequency ofthe cable.

In order to produce a principal (first order) parametricexcitation corresponding to a fundamental natural frequencyof the in-plane mode (6.77Hz) equal to the half of thefrequency of the first symmetric flexural mode shape of thebridge (13.55Hz), the tension of cable number 2 is tuned.Active damping is added through the four short active cablesusing decentralized parallel PPF-DVF strategy. Then, theglobal flexural mode had been harmonically excited by afrequency equal to 13.55Hz and force amplitude of 2Nthrough the actuator of cable number 1. Finally, the in-planemidspan motion of cable number 2 and the deck vibrationin the anchorage point A in the vertical direction had beenrecorded. Figure 11 describes the principle of the numericalexperience. In order to produce a fundamental (secondorder) parametric excitation, the same numerical experiencedescribed above is repeated but the cable tension is tuned toobtain a fundamental natural frequency of the in-planemodeequal to the frequency of the first symmetric flexural modeshape of the bridge. The evolution in time of the in-planemotion of cable number 2 at midspan (L/2) and the deckvibration in the anchorage point A in the vertical direction,under principal and fundamental parametric excitations, areplotted in Figures 12(a) and 13(a) for both cases, with andwithout active control. The amplitude of the deck is well

damped and the parametric resonance is cancelled. Figures12(b) and 13(b) show the trajectory of the cable at midspanbefore control triggering and then during the first 10 secondsafter switching on the control and finally during the last 10seconds. The cable is attractively damped for both in-planeand out-of-plane motions.

7. Conclusions

The active tendon control of a cable-stayed bridge in aconstruction phase had been investigated numerically. Activedamping is added to the structure by using pairs of collocatedforce actuator-displacement sensor located on each activecable and decentralized first order positive position feedback(PPF) or direct velocity feedback (DVF). A comparisonbetween these two compensators showed that each one hasgood performance for some modes and performs inade-quately with the other modes. A parallel PPF-DVF is pro-posed to get the better of the two compensators.Theproposedstrategy is then compared to the one using decentralizedintegral force feedback and showed better performance.Finally, the proposed strategy is applied to a nonlinear modelof a cable-stayed bridge in order to control both deck andcable vibrations induced by parametric excitation. Both cableand deck vibrations are attractively damped. As a futurework,a modal analysis of the cable-stayed bridge will be carriedout during all the construction phases. The proposed controlstrategy will be improved to be adaptive to different phasesof construction and semiactive tendon control of the cable-stayed bridge using MR dampers will also be investigated.

Appendix

The Irvine parameter is

𝜆2

= (𝜌𝑙𝑔 cos 𝜃

𝜎𝑠)

2

𝐸

𝜎𝑠, (A.1)

where 𝜌 is the cable density, 𝑔 is the gravity, 𝜃 is the angle ofthe chord line with respect to the horizontal,𝐸 is themodulusof elasticity, and 𝜎

𝑠 is the static stress.The effective modulus of elasticity is

𝐸𝑞=

1

1 + (𝜆2/12)𝐸. (A.2)

The tension increment induced by the supportmovementis

𝑇𝑞= 𝑇(1)

𝑞+ 𝑇(2)

𝑞, (A.3)

where

𝑇(1)

𝑞= 𝐸𝑞𝐴

𝑢𝑏− 𝑢𝑎

𝑙,

𝑇(2)

𝑞= 𝐸𝐴(1 +

𝐸𝑞

𝜎𝑠

𝜆2

12 + 𝜆2)(𝑢𝑏− 𝑢𝑎)2

2𝑙2+ 𝐸𝐴

(V𝑏− V𝑎)2

2𝑙2

+ 𝐸𝐴(1 +𝜆2

𝜎𝑠

12𝐸)(𝑤𝑏− 𝑤𝑎)2

2𝑙2,

(A.4)


Cab

le v

ibra

tion

(mm

)D

eck

vibr

atio

n (m

m)

Time (s)

Time (s)

0

2

4

6

Control on

Control on

4

3

2

1

0

−1

−2

−2

−4

−6

−3

−4

×10−1

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

(a)

Out-of-plane cable vibration (mm)0 1 2 3

0

2

4

6

Control off

In-p

lane

cabl

e vib

ratio

n (m

m)

−2

−4

−6−3 −2 −1

×10−1

20 to 30 s)Control on (30 to 40 s)Control on (

(b)

Figure 12: (a) Evolution in time of the vertical deck vibration and the in-plane cable vibration at L/2 (control on at 20 s); (b) cable trajectoryat L/2 before and after control (case of principal parametric excitation).

Dec

k vi

brat

ion

(mm

)

4

3

2

1

0

−1

−2

−3

−4

×10−1

0 5 10 15 20 25 30 35 40

Control on

Control on

Time (s)

Time (s)

0 5 10 15 20 25 30 35 40

Cabl

e vib

ratio

n (m

m)

0

2

4

6

−2

−4

−6

(a)

0

2

4

6

In-p

lane

cabl

e vib

ratio

n (m

m)

−2

−4

−6

Out-of-plane cable vibration (mm)0 1 2 3−3 −2 −1

×10−1

Control off20 to 30 s)Control on (30 to 40 s)Control on (

(b)

Figure 13: (a) Evolution in time of the vertical deck vibration and the in-plane cable vibration at L/2 (control on at 20 s); (b) cable trajectoryat L/2 before and after control (case of fundamental parametric excitation).


where 𝑢𝑎, 𝑢𝑏, V𝑎, V𝑏, 𝑤𝑎, and 𝑤

𝑏are the movements imposed

to anchorage points.The tension increment induced by the dynamicmotion of

the cable is

𝑇𝑑= 𝑇(1)

𝑑+ 𝑇(2)

𝑑, (A.5)

where

𝑇(1)

𝑑=

𝐸𝐴2

𝛾

𝑇0

∑

𝑛

[𝑧𝑛

𝑛𝜋(1 + (−1)

𝑛+1

)] ,

𝑇(2)

𝑑=

𝐸𝐴

2∑

𝑛

(𝑦2

𝑛

𝑛2

𝜋2

2𝑙2) +

𝐸𝐴

2∑

𝑛

(𝑧2

𝑛

𝑛2

𝜋2

2𝑙2)

−

𝐸𝑞𝐸𝐴𝛾

(𝜎𝑠)2

𝑢𝑏− 𝑢𝑎

𝑙∑

𝑛

[𝑧𝑛

𝑛𝜋(1 + (−1)

𝑛+1

)] .

(A.6)

The reaction forces on the cable anchorage points a and bare expressed as follow:

𝐹𝑢𝑎

=1

2𝑚𝑙{𝑐1(��𝑏− ��𝑎) + ��𝑏+ 𝑐2(��𝑏− ��𝑎)

+𝑐3��𝑎+ 𝑐4∑

𝑛

1 + (−1)𝑛+1

(𝑛𝜋)3

��𝑛}

+ (𝑇0+ 𝑇𝑞+ 𝑇𝑑)

× {𝑐5+ 𝑐6∑

𝑛

1 + (−1)𝑛+1

(𝑛𝜋)𝑧𝑛+ 𝑐7(𝑢𝑏− 𝑢𝑎)} ,

𝐹𝑢𝑏

=1

2𝑚𝑙{ − 𝑐

1(��𝑏− ��𝑎) + ��𝑏+ 𝑐8(��𝑏− ��𝑎)

−𝑐3��𝑎− 𝑐4∑

𝑛

1 + (−1)𝑛+1

(𝑛𝜋)3

��𝑛}

− (𝑇0+ 𝑇𝑞+ 𝑇𝑑)

× {𝑐5+ 𝑐6∑

𝑛

1 + (−1)𝑛+1

(𝑛𝜋)𝑧𝑛+ 𝑐7(𝑢𝑏− 𝑢𝑎)} ,

𝐹V𝑎

=1

2𝑚𝑙{−

2

3(V𝑏− V𝑎) + V𝑏+ 2∑

𝑛

1

𝑛𝜋𝑦𝑛}

− (𝑇0+ 𝑇𝑞+ 𝑇𝑑)V𝑏− V𝑎

𝑙,

𝐹V𝑏

=1

2𝑚𝑙{

2

3(V𝑏− V𝑎) + V𝑎+ 2∑

𝑛

(−1)𝑛+1

𝑛𝜋𝑦𝑛}

+ (𝑇0+ 𝑇𝑞+ 𝑇𝑑)V𝑏− V𝑎

𝑙,

𝐹𝑤𝑎

=1

2𝑚𝑙{𝑐9(��𝑏− ��𝑎) + ��𝑏− 𝑐10(��𝑏− ��𝑎)

+𝑐11��𝑎+ 2∑

𝑛

1

𝑛𝜋��𝑛}

+ 𝑐12(𝑇0+ 𝑇𝑞+ 𝑇𝑑) (𝑤𝑏− 𝑤𝑎) ,

𝐹𝑤𝑏

=1

2𝑚𝑙{ − 𝑐

9(��𝑏− ��𝑎) + ��𝑎+ 𝑐8(��𝑏− ��𝑎)

−𝑐11��𝑎+ 2∑

𝑛

(−1)𝑛+1

𝑛𝜋��𝑛}

− 𝑐12(𝑇0+ 𝑇𝑞+ 𝑇𝑑) (𝑤𝑏− 𝑤𝑎) ,

(A.7)

where

𝑐1= − 2(

1

1 + (𝜆2/12))

2

(1

3+𝜆2

18+

𝜆4

432+

𝜆2

120

𝐸

𝜎𝑠) ,

𝑐2=

𝜆

12 + 𝜆2√𝜎𝑠

𝐸(−

𝜆2

12− 1 +

𝐸

𝜎𝑠) ,

𝑐3=

2𝜆

12 + 𝜆2√

𝐸

𝜎𝑠,

𝑐4=

2𝛾𝑙𝐸𝑞

(𝜎𝑠)2,

𝑐5= −

1

1 + (𝜆2/12),

𝑐6=

𝛾𝐴2

𝐸𝑞

𝑇0

2,

𝑐7= −

1

𝑙(

𝜆2

12 + 𝜆2

𝐸

𝜎𝑠+ 1) ,

𝑐8= −

𝜆

12 + 𝜆2√𝜎𝑠

𝐸(𝐸

𝜎𝑠+𝜆2

12+ 1) ,

𝑐9= −

2

3−𝜆2

𝜎𝑠

60𝐸,

𝑐10

=𝜆

12 + 𝜆2√𝜎𝑠

𝐸(−

𝐸

𝜎𝑠+𝜆2

12+ 1) ,

𝑐11

= −𝜆

6

√𝜎𝑠

𝐸,

𝑐12

= −1

𝑙(1 +

𝜆2

𝜎𝑠

12𝐸) .

(A.8)


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors would like to acknowledge the support andadvice of Professor Andre Preumont and Professor ArnaudDeraemaeker from Free University of Brussels (ULB), Bel-gium. The smart bridge demonstrator has been developed inthe framework of the S3TEurocores S3HMproject, funded bythe FNRS and the FP6-RTN-Smart Structures project fundedby the European Commission.

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[29] http://www.sdtools.com.

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