Flutter Control of Long-Span Suspension Bridges Xiaowei Zhao, David J. N. Limebeer and J. Michael R. Graham Abstract— The dynamic stabilization of a sectional model of a long-span suspension bridge is considered. Feedback control is achieved using leading- and trailing-edge flaps as actuators. While a wide variety of control systems is possible, we focus on compensation schemes that can be implemented using passive mechanical components such as springs, dampers, and a rack and pinion mechanism. A single-loop control system is investigated that controls the flaps by sensing the main deck heave velocity. A symmetrical control scheme is used on both flaps to make the feedback system insensitive to the wind direction. The key finding is that the critical wind speed for the flutter instability of the sectional model of the bridge can be greatly increased, with good robustness characteristics, through passive feedback control. I. I NTRODUCTION The now iconic Tacoma Narrows bridge disaster (1940) was caused by the gradual growth, over a period of ap- proximately 45 minutes, of a torsional flutter oscillation. It was subsequently established that the Tacoma Narrows bridge failure resulted from the use of a structurally and aerodynamically inappropriate squat H-section structure for the main bridge deck [1]. One possible modification to the aeroelastic properties of the bridge structures is to introduce stationary, or actively controlled aerodynamic flutter suppres- sion surfaces. In [2] we present a 2D aerodynamic model of a long-span suspension bridge with controllable leading- and trailing-edge flaps; see Fig. 1. In this work we make use of the thin aerofoil theory first developed by Theodorsen [3] to study flutter - this theory also exposes a non-oscillatory instability known as torsional divergence. The torsional di- vergence mode is a bona fide aeroelastic mode that goes unstable when there is a loss of torsional rigidity due to the cancelation of the (positive) torsional stiffness of the structure by the negative pitch-related aerodynamic moment. This paper investigates the utility of a symmetric (with respect to wind direction) aerodynamic control system that will be evaluated by numerical simulation. There are sup- plementary issues to consider that relate to control surface flow separation and compromised actuator effectiveness due to their immersion in the main deck’s wake. A lot of good work has been done on the aerodynamic con- trol of cable-stayed suspension bridges. This literature falls into four broad categories, and while much progress has been This work was funded by the UK EPSRC. X. Zhao and D. J. N. Limebeer are with the Department of En- gineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, United Kingdom, e-mail: xiaowei.zhao@eng.ox.ac.uk, david.limebeer@eng.ox.ac.uk J. M. R. Graham is with the Department of Aeronautical Engineering, Imperial College London, London SW7 2AZ, United Kingdom, e-mail: m.graham@imperial.ac.uk U β l β t Fig. 1. Cross section of a long-span suspension bridge with controllable flaps. The wind speed is denoted U , while the leading- and trailing-edge flap angles are denoted β l and βt respectively. made, each approach has its short-comings. Passive pure-gain controllers [4], [5] are in principle easy to implement, but these systems forego the advantages that might accrue from some form of phase compensation. Fixed-phase controllers, such as those described in [6], are not physically realizable. Realizable systems that introduce frequency-dependent phase compensation may operate satisfactorily, but this has not thus far been established. Active controllers, such as those based on linear optimal control and H ∞ [7], face severe reliability questions, because they are relatively complicated and will require a power supply and probably also a computer system. Bad weather situations may well result simultaneously in high winds and power supply failures. Adaptive controllers, such as variable gain output feedback controllers [8], face the same difficulties. Our purpose is to address some of these issues and contribute to the better understanding of these systems. It is a truism that computer models can be useful, but they are never ‘right’. As a result, one must be mindful of robust stability and robust performance issues [9], because ultimately it is the bridge and not the bridge model that must be stabilized. Robustness is an important issue that appears not to have been considered in the long-span bridge design context. II. DYNAMIC MODEL We begin by describing the structural and aerodynamic models of the bridge section with leading- and trailing-edge controllable flaps developed in [10], [2]. Referring to the diagram of the system kinematics in Fig. 2, we see that the generalized coordinates are the deck’s heave h and pitch angle α, and the flap angles β t and β l . The heave and pitch dynamics are described by: M h ¨ h +2ω h ζ h ˙ h + K h h = L, (1) J p ¨ α +2ω p ζ p ˙ α + K p α = M, (2) where L is the aerodynamic lift force; M is the aerodynamic moment around O as shown in Fig. 2; M h and J p are the mass and the torsional mass moment of inertia, per unit length, respectively; ω h and ω p are the undamped natural frequencies of the heave mode and pitch mode; ζ h and ζ p 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) Orlando, FL, USA, December 12-15, 2011 978-1-61284-799-3/11/$26.00 ©2011 IEEE 4195