Elsevier

Engineering Structures

Volume 33, Issue 8, August 2011, Pages 2237-2247
Engineering Structures

Model-based design and experimental validation of active vibration control for a stress ribbon bridge using pneumatic muscle actuators

https://doi.org/10.1016/j.engstruct.2011.02.035Get rights and content

Abstract

This paper describes the development of an active vibration control system for a light and flexible stress ribbon footbridge. The 13 m span carbon fiber reinforced plastic (CFRP) stress ribbon bridge was built in the laboratory of the Department of Civil and Structural Engineering, Berlin Institute of Technology. Its lightness and flexibility result in high vibration sensitivity. To reduce pedestrian-induced vibrations, very light pneumatic muscle actuators are placed at handrail level, introducing control forces. First, a reduced discretized analytical model is derived for the stress ribbon bridge. To verify the analytical prediction, experiments without feedback control are conducted. Based on this model, a delayed velocity feedback control strategy is designed. To handle the nonlinearities of the muscle actuator, a subsidiary force control is implemented. Then the control performance from numerical simulation is verified by experiments under free vibration. As a result, analytical analyses agree well with experimental results. It is demonstrated that handrail-introduced forces can efficiently control the first mode response.

Introduction

Stress ribbon bridges are among the most elegant and lightest bridges. Due to their static and dynamic characteristics, they have been mainly designed for pedestrian traffic rather than for road or rail traffic [1], [2]. The suspension cable and the bridge deck are combined into one stiffening element, which is anchored in the abutments. Usually, the cables are made of steel cables or steel plates. To show the potential of high-strength carbon fiber reinforced plastics (CFRPs), a 13 m span stress ribbon bridge with CFRP ribbons was built in the laboratory of the Department of Civil and Structural Engineering, Berlin Institute of Technology (TU Berlin); see Fig. 1. This composite material allows considerably smaller cross sections compared to steel. The bridge’s tensile force under dead and live load is carried by only 6 ribbons with a cross section of ≈1.1 mm×50 mm each. The combination of low extensional stiffness using CFRP for the ribbons and a lightweight bridge deck leads to considerable dynamic responses caused by pedestrian loads [3], [4].

Generally, to keep vibrations within acceptable limits, several conceptual design approaches such as increasing the stiffness or increasing the dead load/traffic load ratio have been applied in practice. Sometimes, additional passive dampers have been installed to reduce high vibrations [5], [6], [7], [8].

An alternative potential approach to ensure the structural serviceability respectively comfort criteria of footbridges is to use active vibration control (AVC). In particular, this is necessary for extremely light structures, where system properties such as the mass and stiffness become time-variant with changing pedestrian traffic. Then the natural frequencies start to depend on live loads and some passive damping techniques can no longer operate optimally.

In civil engineering structures, active vibration control has been achieved and implemented by active mass dampers (AMDs) or hybrid mass dampers (HMDs) in towers, tall buildings, and pylons of cable-stayed bridges. Controllable fluids such as electro-rheological (ER) and magneto-rheological (MR) ones have been used as semi-active dampers to control wind-induced cable vibrations and buildings under earthquake excitations [9], [10], [11]. Studies on the active control of cable vibrations have been conducted in [12], [13], [14]. By axial support movements, sag-induced forces can be applied which change the cable tension and control the in-plane vibrations. Experimental studies have confirmed this control strategy, limited to the symmetric modes and efficient only for the first in-plane mode.

The applied concept of active vibration control for the stress ribbon bridge is to control the symmetric modes as well as the asymmetric modes. The natural frequencies of the CFRP stress ribbon bridge that coincide with the dominant frequencies of pedestrian-induced loads range from 1.34 to 3.75 Hz due to walking, running, and jumping [4]. To reduce the dynamic responses in this range, control forces are introduced at handrail level by very light pneumatic muscle actuators (PMAs) placed at the midspan and quarter points (Fig. 2a, set-up B). The handrail posts are rigidly coupled with the concrete slabs and pin-jointed with each other, except for the actuator placement. In this way,control forces can be applied along the bridge deck to produce a countermovement.

In this study, one pair of actuators mainly controlled by one sensor is considered. The actuators are placed between two handrail posts on each side at the midspan, and the sensor is placed at the midspan in the centre line (Fig. 2b, Fig. 3a, Fig. 3b). This actuator/sensor configuration (set-up A) allows control of the symmetrical modes. The control system presented here focuses only on the first symmetric mode. To design a model-based controller, an analytical control-orientated model is derived for the bridge in Section 2. The nonlinear force contraction behavior of the extremely light pneumatic muscle actuator is described in Section 3. In Section 4, modal state control strategies are investigated, and a subsidiary force control is proposed to handle the nonlinearities of the pneumatic actuator. The efficiency of the obtained control designs is tested by numerical simulations and verified by full-scale experiments focusing on free vibrations in Section 5.

Section snippets

Parameters of the bridge and the derived eight-plate model

From the distributed system of the stress ribbon bridge, an analytical plane rigid body model for a multi-variable control system is developed. In order to get good agreement with experimental data for the modes to be controlled, a plane model with seven degrees of freedom qi is developed (Fig. 4). The stiffening effect of the railing is not included, as its influence on the modes to be controlled is negligible (Table 3).

The deck of the bridge consists of six CFRP ribbons, which are covered by

Physical model of a pneumatic muscle actuator (PMA)

Pneumatic muscle actuators (PMAs) are rather new in the field of active vibration control. They are increasingly used in the area of automation and robotics [17]. One of the major advantages is their extremely low self-weight because their core element consists of a membrane tube (Fig. 6). Thus the force/self-weight ratio is considerably higher in comparison to pneumatic cylinders and other actuators. For this reason, PMAs are used in this study.

The functionality and the force contraction

Modal state-space representation and state estimator

To design a model-based controller for specific modes, it is convenient to use a modal state-space representation [20]. This representation is obtained from the nodal state-space representation by transformation of Eq. (26) using a transformation matrix, here the modal matrix, R. The modal state variables xm are introduced such that x=Rxm. Hence, the modal state-space representation can be obtained, which is characterized by the block-diagonal state matrix Am, the input matrix Bm, and the

Details for simulation and implementation of the control design

First, the simulations are carried out using the open-source software package Scilab/Scicos for numerical computation (http://www.scilab.org/, http://www.scicos.org/). The controller gain for delayed velocity feedback control was determined and tested by using the eight-plate model in the simulation environment, as shown in Fig. 10. The excitation signal for the simulations is a sinusoidal pressure signal in the first mode for an initial 4 s. The pneumatic muscle actuators work first as an

Conclusion

Active vibration control for a stress ribbon bridge with an extremely light pneumatic actuator was investigated. Therefore, an analytical nonlinear bridge model was developed. By experiments, it was confirmed that the linearized model represents the nonlinear behavior of the stress ribbon well for multi-modal motions. By using sensors to measure the nodal displacement and pressure only, a subsidiary force control was presented which compensates the nonlinearities of the pneumatic actuator.

Acknowledgements

The authors would like to thank Festo AG & Co. KG, Esslingen, Germany for the generous material support. The first author would like to thank the JSPS (Japan Society for the Promotion of Science), who funded a grant for a research period at the Bridge and Structure Laboratory, Department of Civil Engineering, Tokyo University.

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