Keywords: catenary riser, linearized dynamics, bending moment, complex singular value decomposition, periodic extension, finite impulse response filter.

This paper presents a solution for the problem of dynamic equilibrium for a catenary riser in the frequency domain. The problem is set in 2D space and is described by a system of six partial differential equations in terms of the variables of interest, i.e., the tension, the shear force, the curvature, the angle and the motions which the riser performs on the reference plane. As the structure is considered to operate within a wet environment, the hydrodynamic drag effect is incorporated in the mathematical formulation using Morison's formula for slender structures. The static configuration of the structure is derived by solving the corresponding problem of static equilibrium using the fourth-order Runge-Kutta method.

The set of the coupled PDEs, that depicts the system dynamics, is then ported into the frequency domain [1,2]. The system is simplified by omitting all higher-order terms that describe structural nonlinearities and the only nonlinear components which are retained are the proper linearized forms of the hydrodynamic drag forces. Thus, the initial set of spatiotemporal PDEs is converted to a set of spatial differential equations, where the unknowns are the linear (first-order) spectra of the spatial distribution along the riser's length of the involved variables. This set is treated by using a central difference numerical scheme. The outcome of the solution procedure is therefore the frequency response of the structure to specified motions at the riser's top.

The frequency response obtained in the manner above, can be interpreted as the spatially-distributed transfer function, i.e. an input-output frequency-domain relationship, connecting any specified motion at the riser's top to any spatiotemporal variable of interest as is e.g. the tension or the curvature. The accuracy of this relationship is evidently limited by the validity of the linearity assumption, employed for simplifying the formulation and solution procedure.

The next step is to appropriately decompose the frequency response with respect to the spatial coordinate. This can be achieved by applying Complex Singular Value Decomposition (cSVD) on the frequency-response matrix. The frequency-response matrix is a complex matrix containing the column vectors of the spatial response at pre-selected frequencies. It is obtained by the numerical solution of the derived spatial-only ODEs for the pre-selected frequency values. Employing cSVD [3] a set of modes of the riser's response to specified harmonic motions at its top, are obtained. A number of these modes are expected to be dominant, with respect to their effect on the accuracy of response reproduction when all other modes are neglected.

The main profit, by introducing cSVD, is the fact that for each one of the dominant modes obtained by the decomposition, the corresponding frequency characteristic and mode shape are determined as well. Therefore, one can approximate each frequency response of the dominant modes by a discrete-time, linear time-invariant (LTI), finite-impulse-response (FIR) system by using powerful techniques from digital signal processing [4].

On the other hand, the mode shapes can be further decomposed to Fourier components, by applying periodic extension outside the boundaries of the spatial coordinate. In effect, the periodically-extended, mode shapes can be expressed as Fourier sums of spatial harmonics with fundamental wave number the inverse of the unstretched length of the cable.

The derived LTI-FIR systems in combination with the mode shape Fourier expansion can be used for predicting the structure's response to either multi-tone or even transient excitations at its top with accuracy limited by the validity of the linearity assumption and the relative power of the dominant modes when compared to the neglected ones. It is worth-mentioning that the number of parameters and the complexity of calculations involved are significantly smaller than the ones required when direct numerical simulation is employed. Furthermore, an upper bound to the H-infinity norm of the associated approximation error may be obtained. Such an upper bound is necessary when examining the structure's response in closed-loop as in the cases when feedback control is employed for the stabilization or performance improvement of the system.

1

M.S. Triantafyllou, A. Bliek and H. Shin, "Dynamic analysis as a tool for open sea mooring system design", Transactions SNAME, Vol. 93, 1985.

2

S.A. Mavrakos, V.J. Papazoglou, M.S. Triantafyllou and I.K. Chatjigeorgiou, "Deep water mooring dynamics", Marine Structures, 9, 181-209, 1996. doi:10.1016/0951-8339(94)00019-O

3

A.C. Antoulas, "Approximation of Large-Scale Dynamical Systems", SIAM (Advances in Design and Control Series), USA, 2005.

4

L. Ljung, "System Identification: Theory for the User", Prentice-Hall (Information and System Sciences Series), NJ, USA, 1998.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £140 +P&P)