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Dynamic Solution

This section discusses the solution of Mathematical Background (Eq.8), the equation for the dynamic response about the mean structure position. In the development that follows it is convenient to represent sinusoidal components in terms of complex variables, so it is understood that it is the real part of all terms containing which is actually being used in any equation.

The first step in deriving the frequency domain dynamic equation is to express the hydrodynamic forces on the structure and the resulting structure motions in terms of sinusoidal components. Consider a structure subjected to a regular (cosine) wave or a harmonic component of a discretised spectrum, whose circular frequency is w radians/second. The frequency domain development assumes the vector of nodal time-varying displacements can be expressed as:

               (1)

where is a complex vector of nodal displacements (the use of the subscript denotes a complex quantity). The complex entries of contain information about both the amplitudes of the nodal displacements and their phasing relative to the incident wave harmonic. The vectors of nodal velocities and accelerations are found by differentiating the Equation (1) above once to get:

       (2)

and twice to give:

       (3)

The left-hand side of Mathematical Background (Eq.8) can now be rewritten as follows:

       (4)

It is further assumed that the vector can be expressed in a similar manner to the first Equation above as:

               (5)

where again is a vector of complex quantities combining amplitude and phase information. The final frequency domain form of Mathematical Background (Eq.8) can be written as:

               (6)

Once the term is cancelled from both sides, Equation (6) above can be solved for the complex vector , from which the time-varying displacements can be found using Equation (1) above. Finally the total displacement vector is found by combining and using Equation (4).

The major feature underpinning the success of any frequency domain method is the derivation of an accurate formulation for the complex force vector to justify the assumption of Equation (5). This topic of Drag Linearisation is now discussed.