Importance of the Inertial Components in Modal State Covariances

The modal state responses of very large floating structures subjected to sea waves are addressed in this paper. Semi-analytical approximations for their second order statistics are first provided. They are derived by using the multiple timescale spectral analysis and they aim at calculating the variances of the nodal state responses much more rapidly than with the traditional time and frequency domain methods. Based on these approximate formulas, such expressions are developed for the correlation coefficients in this paper. They allow to understand in which cases the covariances between two modal state responses are significant and cannot legitimately be neglected. They are for instance important to consider when the natural frequencies of the corresponding modes are close to one another or when their shapes are similar. The accuracy of the proposed expressions is then verified on a realistic example inspired by an actual floating pontoon bridge. The results are shown to be less precise, although still acceptable, when the peak frequency of the loading and the natural frequencies of the structure are of the same order of magnitude. This is to be expected since the validity of the proposed approximations is conditioned upon the separation of these timescales.