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Mathematical Background

This section provides some of the mathematical background to the frequency domain analysis procedure. Matrix algebra is used extensively in the derivations that follow. A matrix quantity is denoted by bold typeface, for example . An uppercase matrix variable, for example , represents a rectangular array or tensor, that is an array comprised of m rows and n columns, where m and n are both greater than 1. A lowercase matrix quantity, for example , represents a vector quantity, that is a matrix comprising m rows and one column only.

The dynamic equations of motion for a multi-degree of freedom offshore structure subjected to regular wave and current can be expressed in standard finite element fashion as:

               (1)

where is the total structure stiffness matrix; is the total structure damping matrix; is the total structure mass matrix; , and are respectively the vectors of nodal accelerations, velocities and displacements; and is the vector of applied nodal loads. Note that the damping matrix is given by:

                               (2)

where and are user-specified stiffness and mass damping coefficients.

The development of the frequency domain equations is based on a decomposition of the load vector into two parts, time-dependent and time-independent loads. The former are the hydrodynamic loads experienced by the structure due to waves and vessel motions, while the latter include buoyancy and gravity loads and fluid loading due to current. User-specified mechanical loads can fall into either category. In the same way the nodal displacements are decomposed into time-dependent and time-independent components, where the constant component is the mean displacement and the time-dependent displacements represent the periodic motion of the structure about this mean position. This division of loads and displacements into constant and time-varying can be represented by the equations:

                               (3)

and

                       (4)

where the subscript denotes static (time-independent) and dynamic (time-dependent). Of course:

                       (5)

so that:

and                (6)

Based on this, the first Equation above can be written as:

       (7)

This equation is also decomposed into two parts, representing time-dependent and time-independent response as follows:

                       (8)

and:

                                               (9)

The solution procedure entails the assembly and solution of the two Equations (8) and (9) above to find and . The total solution is then found from Equation (4). The force vectors and both contain non-linear hydrodynamic drag components, and a linearisation procedure must be adopted if the equations are to be solved directly in closed form. The technique adopted here has the effect of coupling the two equations through the linearisation procedure, that is the two linearised equations are not independent, and the solutions and must be found iteratively. Further details are provided later in the section.

The development to this point presumed a regular wave analysis, with or without current. In the case of a random sea analysis, where the seastate is characterised by a wave energy spectrum, the analysis begins by decomposing the input wave spectrum into a series of harmonic components. A pair of linearised equations, similar to Equations (8) and (9), is then assembled and solved at each component frequency, with the iteration loop in this case extending over all frequencies in the seastate discretisation. The drag force linearisation has the effect in this case of coupling the linearised equations at all of the component harmonics. This point is elaborated later.