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Research Papers

Aeroelastic Control of Long-Span Suspension Bridges

[+] Author and Article Information
J. Michael R. Graham

Department of Aeronautical Engineering, Imperial College London, Exhibition Road, London SW7 2BT, UKm.graham@imperial.ac.uk

David J. N. Limebeer

Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UKdavid.limebeer@eng.ox.ac.uk

Xiaowei Zhao

Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UKxiaowei.zhao@eng.ox.ac.uk

J. Appl. Mech 78(4), 041018 (Apr 15, 2011) (12 pages) doi:10.1115/1.4003723 History: Received November 22, 2010; Revised January 14, 2011; Posted February 28, 2011; Published April 15, 2011; Online April 15, 2011

The modeling, control, and dynamic stabilization of long-span suspension bridges are considered. By employing leading- and trailing-edge flaps in combination, we show that the critical wind speeds for flutter and torsional divergence can be increased significantly. The relatively less well known aerodynamic properties of leading-edge flaps will be studied in detail prior to their utilization in aeroelastic stability and control system design studies. The optimal approximation of the classical Theodorsen circulation function will be studied as part of the bridge section model building exercise. 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, gearboxes, and levers. A single-loop control system that controls the leading- and trailing-edge flaps by sensing the main deck pitch angle is investigated. The key finding is that the critical wind speeds for flutter and torsional divergence of the sectional model of the bridge can be greatly increased, with good robustness characteristics, through passive feedback control. Static winglets are shown to be relatively ineffective.

Copyright © 2011 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Cross section of a long-span suspension bridge with flutter suppression winglets and controllable flaps. The wind speed is denoted as U, while the leading- and trailing-edge flap angles are denoted as βl and βt, respectively.

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Figure 2

Kinematic model of the bridge deck. The origin of the inertial axis system is O. The wind velocity U is assumed positive to the right (positive x-direction), the heave h and lift force L are assumed to be positive downward, and moments M are positive clockwise, as are the pitch and trailing-edge flap angles α and βt, respectively. The leading-edge flap angle βl is positive counterclockwise. The deck chord (including the flaps) is 2b. The leading- and trailing-edge flap chords are (1+cl)b and (1−ct)b, respectively; note that cl is a negative quantity.

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Figure 3

Aerodynamic derivatives for the trailing-edge flap. The blue dashed curves (red solid) correspond to the real (imaginary) parts of the aerodynamic derivatives computed using thin aerofoil theory. The (red) stars and (blue) hexagons were computed using a discrete vortex panel code.

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Figure 4

Transformation of the Theodorsen–Garrick wing-aileron-tab configuration (18) into a controlled bridge deck. (a) The wing pitch angle is α, the aileron angle is βl, and the tab angle is βt. The wind speed is U (from the left); the heave h and lift L are positive downward, while moments and angles are positive clockwise. The wing chord is 2b; the widths of the tab and aileron are described in terms of ct and cl, respectively. (b) The wing-aileron-tab configuration is transformed into the controlled bridge deck by making cl negative, thereby forcing the flap hinge to the left of the origin. In this new configuration, the aileron becomes the bridge deck, the wing becomes the leading-edge flap, and the tab becomes the trailing-edge flap. In order to relevel the bridge and return its mass center to the correct position, pitch and heave corrections must be applied.

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Figure 5

Aerodynamic derivatives for the leading-edge flap. The blue dashed curves (red solid) correspond to the real (imaginary) parts of the aerodynamic derivatives computed using Theodorsen–Garrick potential theory. The (red) stars and (blue) hexagons were computed using a linear discrete vortex code.

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Figure 6

The Theodorsen function and its rational approximations. In the left- hand diagram, the Theodorsen function (15) is the (blue) dot-dash curve, the quartic approximation is the (red) dashed curve, and the Jones (20) function is the (black) dotted curve. The right-hand diagram is the step response of the quartic approximation—this is the Wagner step response curve (14).

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Figure 14

Root-loci of the Akashi Kaikyo suspension bridge section with a trailing-edge flap of 2.5 m under feedback control. The feedback gains are Kt=0, 5, 10, 15, and 20, with the wind speed swept from 30 m/s to 80 m/s. The left-hand diagram shows the heave and pitch modes with the low-speed ends of the root loci marked with diamonds and the high-speed ends marked with hexagons. The critical wind speeds for flutter, in increasing order of feedback gain, are 51 m/s, 55 m/s, 60 m/s, 66 m/s, and 74 m/s, respectively. The right-hand plot shows the location of the torsional divergence mode eigenvalue as a function of Kt and wind speed.

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Figure 15

Root-loci of the Akashi Kaikyo suspension bridge section with a leading-edge flap of 2.5 m under feedback control. The feedback gains are Kl=0, 5, and 10, with the wind speed swept from 30 m/s to 80 m/s. The left-hand diagram shows the heave and pitch modes with the low-speed ends of the root loci marked with diamonds and the high-speed ends marked with hexagons. The critical wind speeds for flutter, in increasing order of feedback gain, are 51 m/s, 56 m/s, and 62 m/s, respectively. The right-hand plot shows the eigenvalue location of the torsional divergence mode as a function of Kl and the wind speed.

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Figure 16

Nyquist diagram of the leading-edge flap control loop on the Akashi Kaikyo suspension bridge section at 75 m/s. To achieve the required winding number of N=3, the leading-edge flap loop gain must be at least 16.34.

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Figure 17

Compensation of the Akashi Kaikyo suspension bridge section with leading- and trailing-edge flaps with chord 2.5 m. The left-hand diagram shows the compensated solid (black) and the uncompensated dotted (red) Nyquist diagrams at a wind speed of 80 m/s. The leading-edge flap has feedback compensator Kl=10, while in the compensated case the trailing-edge flap has compensator (Eq. 21) with optimized parameters in Eq. 22. The stable regions correspond to N=3. The right-hand diagram shows the root loci of the compensated closed-loop system. The wind speed is swept from 30 m/s to 80 m/s, with the low-speed ends of the root loci marked with (blue) diamonds and the high-speed ends marked with (red) hexagons.

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Figure 18

Compensation of the Akashi Kaikyo suspension bridge section with leading- and trailing-edge flaps with chord 2.5 m. The left-hand diagram shows the compensated solid (black) and the uncompensated dotted (red) Nyquist diagrams at a wind speed of 80 m/s. The leading-edge flap has feedback compensator Kl=10, while in the compensated case the trailing-edge flap has compensator (Eq. 23) with optimized parameters in Eq. 24. The right-hand diagram shows the root loci of the compensated closed-loop system. The wind speed is swept from 30 m/s to 80 m/s, with the low-speed ends of the root loci marked with (blue) diamonds and the high-speed ends marked with (red) hexagons.

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Figure 12

Winglet effectiveness as a moment generator. “Moment effectiveness” is the ratio of actual moment generated by the deck and winglet to the moment generated by the deck alone.

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Figure 11

Lift-pitch aerodynamic derivatives for the winglet and winglet-deck assembly. The blue pentagrams show the steady-state lift-pitch derivative for the whole deck-winglet assembly as computed using a vortex code. The red hexagrams show the steady-state lift-pitch derivative for the winglet alone (×10); all coefficients are nondimensionalized by the bridge-deck chord. The separation between the winglet and the main deck is given in terms of main deck chords.

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Figure 10

Winglet generating a stabilizing moment

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Figure 9

Root-loci of the bridge section with only the pitch dynamics present. The wind speed is swept from 30 m/s to 80 m/s, with the low-speed ends of the root loci marked with (blue) diamonds and the high-speed ends marked with (red) hexagons. The torsional divergence mode goes unstable at approximately 70 m/s, as predicted by Eq. 18.

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Figure 8

Root-loci of the bridge section. The wind speed is swept from 30 m/s to 80 m/s, with the low-speed ends of the root loci marked with (blue) diamonds and the high-speed ends marked with (red) hexagons. The pitch mode goes unstable at approximately 52 m/s, while the torsional divergence mode goes unstable at approximately 70 m/s.

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Figure 7

Block diagram of the aeroelastic control system. The dynamics of the bridge are represented by the plant P(s) and the Theodorsen function C(s); “s” represents the Laplace variable. The leading and trailing flap control systems are given by Kl(s) and Kt(s), respectively. The leading- and trailing-edge flap angles are given by βl(s) and βt(s), respectively, and the deck pitch angle and heave are given by α(s) and h(s), respectively.

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Figure 13

Nyquist diagram of the Akashi Kaikyo suspension bridge section at two different wind speeds. The left-hand plot corresponds to a wind speed of 60 m/s, while the right-hand plot corresponds to a wind speed of 75 m/s. The winding number around each point on the real axis is given by N, with counterclockwise encirclements counted positive. For the left-hand diagram, point A corresponds to ω=0.792 r/s and Kt=9.45, point B corresponds to ω=0.0 r/s and Kt=13.8, point C corresponds to ω=0.384 r/s and Kt=−3.70, and point D corresponds to ω=0.562 r/s and Kt=−2.77. In the right-hand diagram, point A corresponds to ω=0.933 r/s and Kt=18.8, point B corresponds to ω=0.0 r/s and Kt=−5.37, and point C corresponds to ω=0.455 r/s and Kt=−2.17.

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