^{1}

^{2}

^{3}

^{1}

^{1}

^{2}

^{3}

Technique of feedforward and feedback optimal vibration control and simulation for long-span cable-bridge coupled systems is developed. Buffeting loading systems of long-span cable-bridge structure are constructed by weighted amplitude wave superposition method. Nonlinear model of cable-bridge coupled vibration control system is established and the corresponding system of state space form is described. In order to reduce buffeting loading influence of the wind-induced vibration for the structure and improve the robust performance of the vibration control, based on semiactive vibration control devices and optimal control approach, a feedforward and feedback optimal vibration controller is designed, and an algorithm is presented for the vibration controller. Numerical simulation results are presented to illustrate the effectiveness of the proposed technique.

Bridge plays an irreplaceable role in the transportation system and has an important value in politics, economy, culture, and military. Influenced by winds, vehicles, pedestrians, and seism loads, structural vibration of bridges is inevitable. Sustained and severe vibration of bridge structures not only shortens the service life of bridges but also reduces the security of traffic. Combined with extreme weather situations and geological disasters, accidents and dangerous cases happen frequently to bridge structures. Therefore, in order to decrease bridge vibration and to improve the reliability and safety of bridge structures, constructing vibration control systems has become an urgent problem to be solved.

Most of the studies on bridge structure focus on dynamic behavior [

The remainder of this paper is structured as follows. In Section

As the span of the bridge extends, the stiffness of the bridge structure reduces, and the bridge has much more characteristics of the flexible structure. Therefore, the influence of the wind loading on the bridge structure cannot be neglected in engineering, and many scholars turn their attention to the influence of the wind-induced vibration on the structure. Bridge buffeting is a sort of random vibration of the bridge structure influenced by the fluctuating wind. Fluctuating wind field induced by natural wind can be described by ergodic and stationary Gauss random process and can be regarded as a single variable four-dimensional random field in mathematics.

We employ weighted amplitude wave superposition method to describe buffeting loading forces on long-span cable-bridge structures. According to weighted amplitude wave superposition method, buffeting loading forces on long-span cable-bridge structure can be simulated by superposition of weighted amplitude waves. The buffeting force of the

So, the total buffeting loading force acting on the bridge structure can be generated by the exosystem:

Considering cable-bridge structures with semiactive tune mass damper devices, the deck is simplified as the lumped mass influenced by the cable end and is denoted by the parameter

The mechanical model of cable and deck.

In order to simplify the problem and reflect the essence of the vibration control for the cable-bridge structure, we make some fundamental assumptions as follows:

the flexural stiffness, torsional stiffness, and shear stiffness are disregarded;

the gravity sag curve is considered as a parabola;

the constitutive relation of the deformation for the cable satisfies Hooke’s law and is uniform for each point;

the effect of the tower vibration on the cable is disregarded.

Generally, the fundamental mode is in the dominant position; therefore we consider the first mode as the main object to study the cable-bridge coupled vibration control problem. The dynamical system of the cable-bridge coupled vibration induced by the buffeting loading is as follows:

Choose state variables and state vector for the cable-bridge coupled vibration control system (

In order to study optimal vibration control problem for long-span cable-bridge, we choose an average performance index for system (

The objective of this paper is to find a control law

Applying the maximum principle to the optimal control problem in (

Unfortunately, for the nonlinear TPBV problem in (

Consider the nonlinear system

In order to use the SAA, we introduce two lemmas [

Define the vector function sequence

Define the vector function sequence

Then, we design vibration controller for system (

Consider the optimal control problem described by system (

To use the SAA to solve nonlinear TPBV problem (

Substituting the first equation of (

Correspondingly, the control sequence is given in the following form:

From (

Control block diagram of the system.

In fact, it is impossible to calculate feedforward and feedback optimal control law in (

In order to implement the feedforward and feedback control law described in Theorem

Consider the following.

In this section, we apply the proposed feedforward and feedback optimal vibration controller to a road suspension cable-bridge. The cable-bridge with three towers located in Yangtze River, whose span arrangement is 360 m + 1080 m + 1080 m + 360 m

Employing Matlab software, numerical experiments are carried out for the proposed optimal vibration controller. Figure

Buffeting loading force curve.

The main purpose of vibration control of long-span cable-bridge is to reduce the deck displacement which indicates the limit of the deck motion and to reduce the deck velocity which ensures the road holding ability for vehicles and pedestrians. So, to evaluate effectiveness of the proposed control strategy, the deck displacement and velocity are considered. Then, the corresponding curves of open loop system, feedback optimal control system, and the system controlled by the proposed optimal vibration controller are compared and shown in Figures

Displacement curves.

Velocity curves.

Control force curves.

The curves of displacement are shown in Figure

The effect of the wind-induced vibration on the long-span cable-bridge structure cannot be neglected in construction phase or in operation stage. The influence of buffeting loading and nonlinear factor for long-span cable-bridge structures is considered in this paper. Based on the semiactive vibration control devices, optimal vibration controller and control algorithm are designed for the long-span cable-bridge. Numerical simulation results show that the proposed strategy is efficient, real-time, and robust in reducing the vibration induced by buffeting loading.

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

This research was supported by Natural Science Foundation of China (nos. 11301009, U1204402, and 61374003) and Natural Science Foundations of Henan Province Education Department (nos. 12A120001, 13A520018, 14B520018, and 13A110022).