# Time-domain representation of frequency dependent inertial forces on offshore structures

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## Abstract

The inertial wave force on a vertical cylinder decreases with decreasing wave length, when the wave length is less than about six times the diameter of the diameter of the cylinder. In structures with a largediameter component like mono-towers the resonance frequency of the structure is typically located above the peak frequency of the wave spectrum, and the frequency dependence of the inertial force coefficient can then result in a substantial reduction of the resonant part of the response. It is of interest to represent this effect in the time domain for response analysis including finite height waves and drag forces. The inertia coefficient has been determined within linear wave theory in terms of the wave-number by MacCamy and Fuchs. For diameters less than about half the water depth this solution can be transformed to frequency form by use of the deep-water dispersion relation. The frequency dependence is then approximated by a rational function, corresponding to a set of ordinary differential equations in the time domain. The MacCamy-Fuchs solution leads to a representation of the inertial force coefficient as a complex function with argument mainly corresponding to a 'phase lead', in contrast to the 'phase lag' obtained for the response of discrete mechanical systems. Two options are explored: introducing a corresponding phase lag in the components of the wave kinematics, or compensating the phase lag by combining a stable complex frequency with its complex conjugate. In the latter case the time history of the inertial force is determined by processing the stable part of the transformation by a forward time integration, followed by an integration in the negative time-direction to obtain the final inertial force time history. The differential equations of the local inertial force at a cross section are uncoupled, and they are easily integrated with e.g. a central difference scheme for the state-space variables. © 2013 Taylor & Francis Group, London.