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A new decoupling strategy for structures with frequency-dependent properties

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Abstract

In this paper, a new procedure is developed to decouple the governing equations of non-classically damped structures with frequency dependent properties. To start, a modal state formulation is used to eliminate the off-diagonal elements that damping might otherwise create in the transfer matrix. From there on, the transfer matrix is expanded in series and decoupling is achieved through an iterative scheme, which relies on the successive inversions of a diagonal matrix only. This approach is finally shown to converge fast and to perform well on the hydroelastic responses of a floating bridge.

Section snippets

Nomenclature

Lowercase and capital bold letters are respectively used to denote vectors and matrices while italic letters with indices designate their elements. The superscripts ., . and . stand for the conjugate, the transpose and the hermitian operators. The imaginary unit is noted ι̇ and ω stands for the circular frequency.

State space eigenproblem

First, the state variables yω=Iι̇ωITxω and the state forces gω=I0Tfω, with I and 0 being respectively the N×N identity and zero matrices, are introduced. Doing so allows to recast the initial set of N second-order equations into 2N first-order equations. Indeed, it yields yω=Aω+ι̇ωBω1gωwhere Aω=Kω00MωandBω=CωMωMω0are referred to as the state matrices. These definitions of the state properties are chosen among others to conserve the symmetry of the stiffness, mass and damping matrices when

Models

The methodology proposed in this paper is now used to perform the hydroelastic analysis of an end-anchored floating pontoon bridge subjected to first order wave loads [3]. This example is based on a two-dimensional finite element model of the Bergsøysund Bridge, which crosses a 300-m deep fjord in the Northwestern coast of Norway and is currently the longest of its kind.

As illustrated in Fig. 2-(a) and 2-(b), this bridge is composed of seven pontoons which are connected to each others and to

Conclusions

The present paper proposes to combine a modal state formulation with a series expansion of the frequency response matrix to decouple the governing equations of a structure with frequency-dependent properties. This procedure indeed allows to compute the power spectral densities and the second order statistics of both the displacements and the velocities based on the successive inversions of a diagonal matrix only.

This approach is also iterative. It can be stopped at any order according to the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The first and the second authors received financial support from the F.R.S.-FNRS (Belgian Fund for Scientific Research). The work of the first author at NTNU Trondheim was also funded by two research stay grants of the FWB (Fédération Wallonie-Bruxelles), Belgium and the Rotary, USA . The contribution of the Norwegian Public Road Administration is acknowledged as well by the third author.

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