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This paper investigates numerically the active tendon control of a cable-stayed bridge in a construction phase. A linear Finite Element model of small scale mock-up of the bridge is first presented. Active damping is added to the structure by using pairs of collocated force actuator-displacement sensors located on each active cable and decentralized first order positive position feedback (PPF) or direct velocity feedback (DVF). A comparison between these two compensators showed that each one has good performance for some modes and performs inadequately with the other modes. A decentralized parallel PPF-DVF is proposed to get the better of the two compensators. The proposed strategy is then compared to the one using decentralized integral force feedback (IFF) and showed better performance. The Finite Element model of the bridge is coupled with a nonlinear cable taking into 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.

In the past few decades, design and construction of civil structures showed a very deep evolution because of the technological progress in materials and devices. Cable-stayed bridges increased considerably their center span from 182.6 m (Stromsund Bridge in Sweden) to 1104 m (Russky Bridge in Russia). These structures are getting more slender, light, and flexible which makes them sensitive to vibrations induced by wind, traffic, waves, or even earthquakes. Consequently, vibration control has become a major issue in civil engineering.

Vibrations in cable-stayed bridges may be reduced using passive [

All the studies on active tendon control presented above used noncollocated pairs of actuator sensor which may destabilize the structure for certain gain values and may also cause spillover instability. Achkire and Preumont [

This paper investigates numerically the active tendon control of a small scale mock-up of a cable-stayed bridge in a construction phase. Active damping is added to the structure by using pairs of collocated force actuator-displacement sensors located on each active cable. This configuration is first examined with decentralized PPF and DVF. Then, a parallel PPF-DVF is proposed to get the better of the two compensators and compared to the one using decentralized IFF. A Finite Element model of the bridge is coupled with a nonlinear cable which takes into 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 excitations.

A model of a smart cable-stayed bridge was developed in Active Structure Laboratory at ULB [

Description of the main components of the bridge.

FE model of the bridge created by Matlab/SDT.

The global equation of motion of the linear cable-stayed bridge equipped with pairs of a force actuator and a displacement sensor in the chosen active cables (

The control forces of the decentralized DVF [

The control forces of the decentralized first order PPF [

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

The control forces of the proposed decentralized parallel PPF-DVF strategy are

Block diagram of the proposed control system.

Figure

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

Figure

The maximum damping ratio for decentralized DVF, PPF, and parallel PPF-DVF is determined for the first 17 modes using the root locus technique and is plotted in Figure

Maximum damping ratio as a function of mode number for different active control strategies.

FRF between the force of excitation (Fexcit) and the vertical displacement of the deck (

The decentralized IFF [_{z}) in point A is plotted in Figure

Maximum damping ratio as a function of mode number for IFF and parallel PPF-DVF concepts.

FRF between the force of excitation (Fexcit) and the vertical displacement of the deck (

The nonlinear model of the inclined cable takes into account general support movement, sag effect, and quadratic and cubic nonlinear couplings between in-plane and out-of-plane motions. The cable model is presented in Figure

3D model of an inclined cable with general support movements.

The transverse out-of-plane displacements of the cable are described by the following equation of motion governing the generalized coordinates_{n} of the

The expressions of

The in-plane displacements of the cable (perpendicular to its chord line) is described by the following equation of motion governing the generalized coordinates

As an alternative to a general nonlinear Finite Element approach which would be extremely time consuming, we had developed, using SDTools [

Principle of coupling between the FE model of the bridge and the nonlinear cables.

Taking into account the nonlinear dynamics of the_{c} cables and active damping, the global equation of motion of the cable-stayed bridge can be expressed in modal coordinates as follows:

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

In cable-stayed bridges, the presence of many low frequencies in the deck or tower and in the stay cables may give rise to parametric excitation. The coupling between a local cable and a global structure makes the bridge sensitive to very small motion of the deck or tower which may cause dynamic instabilities and very large oscillations of the stay cables (see Figure

Description of the numerical experience.

In order to produce a principal (first order) parametric excitation corresponding to a fundamental natural frequency of the in-plane mode (6.77 Hz) equal to the half of the frequency of the first symmetric flexural mode shape of the bridge (13.55 Hz), the tension of cable number 2 is tuned. Active damping is added through the four short active cables using decentralized parallel PPF-DVF strategy. Then, the global flexural mode had been harmonically excited by a frequency equal to 13.55 Hz and force amplitude of 2 N through the actuator of cable number 1. Finally, the in-plane midspan motion of cable number 2 and the deck vibration in the anchorage point A in the vertical direction had been recorded. Figure

(a) Evolution in time of the vertical deck vibration and the in-plane cable vibration at

(a) Evolution in time of the vertical deck vibration and the in-plane cable vibration at

The active tendon control of a cable-stayed bridge in a construction phase had been investigated numerically. Active damping is added to the structure by using pairs of collocated force actuator-displacement sensor located on each active cable and decentralized first order positive position feedback (PPF) or direct velocity feedback (DVF). A comparison between these two compensators showed that each one has good performance for some modes and performs inadequately with the other modes. A parallel PPF-DVF is proposed to get the better of the two compensators. The proposed strategy is then compared to the one using decentralized integral force feedback and showed better performance. Finally, the proposed strategy is applied to a nonlinear model of a cable-stayed bridge in order to control both deck and cable vibrations induced by parametric excitation. Both cable and deck vibrations are attractively damped. As a future work, a modal analysis of the cable-stayed bridge will be carried out during all the construction phases. The proposed control strategy will be improved to be adaptive to different phases of construction and semiactive tendon control of the cable-stayed bridge using MR dampers will also be investigated.

The Irvine parameter is

The effective modulus of elasticity is

The tension increment induced by the support movement is

The tension increment induced by the dynamic motion of the cable is

The reaction forces on the cable anchorage points

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

The authors would like to acknowledge the support and advice of Professor André Preumont and Professor Arnaud Deraemaeker from Free University of Brussels (ULB), Belgium. The smart bridge demonstrator has been developed in the framework of the S3T Eurocores S3HM project, funded by the FNRS and the FP6-RTN-Smart Structures project funded by the European Commission.