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Discovering time-varying aerodynamics of a prototype bridge by sparse identification of nonlinear dynamical systems

Shanwu Li, Eurika Kaiser, Shujin Laima, Hui Li, Steven L. Brunton, and J. Nathan Kutz
Phys. Rev. E 100, 022220 – Published 21 August 2019
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  • INTRODUCTION
  • VORTEX-INDUCED VIBRATION OF LONG-SPAN…
  • FIELD MEASUREMENTS AND DATA…
  • DATA-DRIVEN MODEL DISCOVERY: SPARSE…
  • DISTINGUISHED DYNAMICAL REGIMES…
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
    • References

    Abstract

    Vortex-induced vibrations (VIVs) have been observed on a long-span suspension bridge. The nonstationary wind in the field characterized by the time-varying mean wind speed is likely to lead to time-varying aerodynamics of the wind-bridge system during VIVs, which is different from VIVs induced by stationary or even steady wind in wind tunnels. In this paper, data-driven methods are proposed to reveal the time-varying aerodynamics of the wind-bridge system during VIV events based on field measurements on a long-span suspension bridge. First, a variant of the sparse identification of nonlinear dynamics algorithm is proposed to identify parsimonious, time-varying aerodynamical systems that capture VIV events of the bridge. Thus we are able to posit new, data-driven, and interpretable models highlighting the aeroelastic interactions between the wind and bridge. Second, a density-based clustering algorithm is applied to discovering the potential modes of dynamics during VIV events. As a result, the time-dependent model is obtained to reveal the evolution of the aerodynamics of the wind-bridge system over time during an entire VIV event. It is found that the level of self-excited effects of the wind-bridge system is significantly time varying with the real-time wind speed and bridge motion state. The simulations of VIVs by the obtained time-dependent models show high accuracies of the models with an averaged normalized mean-square error of 0.0023. The clustering of obtained models shows underlying distinct dynamical regimes of the wind-bridge system, which are distinguished by the level of self-excited effects.

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    • Received 29 May 2019

    DOI:https://doi.org/10.1103/PhysRevE.100.022220

    ©2019 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Continuum mechanics
    1. Physical Systems
    Dynamical systems
    Nonlinear Dynamics

    Authors & Affiliations

    Shanwu Li1, Eurika Kaiser2, Shujin Laima1, Hui Li1, Steven L. Brunton2, and J. Nathan Kutz3,*

    • 1School of Civil Engineering, Harbin Institute of Technology, Harbin, China
    • 2Department of Mechanical Engineering, University of Washington, Seattle, Washington 98195-2420, USA
    • 3Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-2420, USA

    • *kutz@uw.edu

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    Issue

    Vol. 100, Iss. 2 — August 2019

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    Images

    • Figure 1
      Figure 1

      Field monitoring on the bridge. Anemometers and accelerometers are installed at 1/4, 1/2, and 3/4 center span. Anemometers are installed on both sides of the bridge section.

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    • Figure 2
      Figure 2

      Preprocessing of wind data. S1, S2, and S3 indicate 1/4 span, midspan, and 3/4 span, respectively, as shown in Fig. 1. (a) Horizontal instantaneous wind speed V and wind direction θ are obtained from original measurements of wind speed; 90∘ and 270∘ indicate the perpendicular direction to the spanwise direction. (b) The wind speed component perpendicular to the spanwise direction is determined. (c) The time-varying mean wind speed is estimated by applying a low-pass filter.

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    • Figure 3
      Figure 3

      Time-frequency analysis of measured acceleration for a VIV event. (a) The PSD of the vibration displacement history. (b) Displacement history of a VIV event. (c) The mode shape and the natural frequency of the bridge obtained by an accompanying numerical simulation using FEM.

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    • Figure 4
      Figure 4

      Comparison of vehicle-induced vibrations (mean wind speed is close to zero) around (a) 00:00, (b) 06:00, (c) 12:00, (d) 18:00 and vehicle-wind-induced vibration under strong wind (mean wind speed is 10 m/s).

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    • Figure 5
      Figure 5

      Preprocessing of vibration data. (a) The time-varying displacement amplitude A is obtained by extracting the envelop from the displacement history y which is obtained by integration of acceleration ÿ in the frequency domain. (b) Vibration amplitudes are obtained for all the three sensor locations. (c) Time derivatives of the amplitudes are obtained.

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    • Figure 6
      Figure 6

      Schematic of the time-varying SINDy framework, demonstrated on the aerodynamics of a VIV event on a bridge. (a) Data are collected from the measurement system, including a history of time-varying mean wind speed U, amplitudes A, and time derivatives Ȧ. (b) A typical SINDy is conducted in a moving time window at each time instant. The time window is swept across the entire VIV event with a size of 50 s and a moving step size of 25 s. Each component of the obtained model ξ is reshaped into a three-column matrix, where each column corresponds to sensor measurements at one bridge section, respectively, for a more interpretable representation of the obtained time-varying aerodynamics. (c) A time series of the model in terms of ξ is obtained that captures the time-varying aerodynamics of an entire VIV event.

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    • Figure 7
      Figure 7

      Time-varying dynamics of three exemplary VIV events discovered by time-varying SINDy.

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    • Figure 8
      Figure 8

      The effects of window size and moving step size on the time-varying SINDy results for a VIV event.

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    • Figure 9
      Figure 9

      Interpretation of time-varying aerodynamics for a VIV event and the VIV wind speed range obtained in Ref. [52]. (a) The time series of model sets ξ2. (b) The history of time-varying mean wind speeds U compared with the VIV wind speed. (c) The history of displacement y2 with the amplitude A2.

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    • Figure 10
      Figure 10

      Simulations for three entire VIV events as examples by solving the ODE with the corresponding obtained time-dependent parameter Ξ(t) [see Eq. (15)] with only the measured initial state A(t=0) and the measured wind history U(t) given. (a) VIV event No. 1 with NMSE of 0.0011. (b) VIV event No. 2 with NMSE of 0.0058. (3) VIV event No. 3 with NMSE of 0.0003.

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    • Figure 11
      Figure 11

      Decision graph: Seven cluster centers (colored) are determined by points for which the value of δ is anomalously large.

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    • Figure 12
      Figure 12

      The obtained seven clusters in the model sets. The model sets in the same cluster have common donimant terms except Cluster 7 (C7). These clusters are distinguished by the polynomial order of the vibration amplitude A in the dominant terms.

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