118 O. Øiseth et al. Fig. 13.1 The Hardanger Bridge performance [2]. A finite element model is combined with the aerodynamic derivatives of the bridge deck to predict the in-wind natural frequencies and damping ratios of the bridge in this paper. Similar analysis has been presented by the authors previously focusing on the flutter stability limit of the combined structure and flow system [3–5]. Even though the finite element method and increased computational resources is readily available, verification of the analysis procedures is still a crucial issue. This is particularly important when modelling structures subjected to environmental loading where a coupled analysis is necessary due to fluid structure interaction. Operational modal analysis has become a versatile tool for inverse modelling of structures when experimental modal analysis is hard or impossible. A comprehensive overview of relevant methods and how they can be implemented is given by Rainieri and Fabbrocino [6]. A comprehensive study of the performance of different operational modal analysis methods using the Humber Bridge as case study have been presented by Brownjohn et al. [7]. It is concluded that the stochastic subspace identification outperforms the other methods, in particular for the estimated damping ratios. The in-wind natural frequencies and damping ratios of the Hardanger Bridge have therefore been estimated using data-driven and covariance-driven stochastic subspace identification in this paper. The estimates are compared with results predicted using experimental data of aerodynamic derivatives and the finite element method. 13.2 Prediction of Frequencies and Damping Ratios Due to the fluid structure interaction the natural frequencies and damping ratios of a suspension bridge is in general dependent on the aerodynamic properties of the structure and the mean wind direction and velocity. The wind loading can be divided into four parts. (i) Static loading originating from the mean wind velocity, (ii) dynamic wind loading caused by vortex shedding, (iii) buffeting loading caused by the turbulence in the wind field and (iv) self-excited loading generated by the motion of the structure. Since the self-excited forces are dependent on the motion of the structure these needs to be taken into account when calculating natural frequencies and damping ratios when the mean wind velocity is different from zero. 13.2.1 Self-Excited Forces The self-excited forces acting on a bridge deck section are commonly represented by the aerodynamic derivatives introduced in bridge engineering by Scanlan and Tomoko [1]. For a two-dimensional bridge deck section (see Fig. 13.2), this can be expressed in matrix notation as follows:
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