Consider the model of an arbitrary mass whose center of gravity moves with the velocity array in the fixed reference axes (FRA). This velocity array at time can be written as

(13–15)

which consists of three translational and three rotational velocity components.

The momentum at time is expressed as

(13–16)

where is the mass matrix at time in the fixed reference axes.

At the next time step , the momentum can be found from

(13–17)

where the matrix is the direction cosine matrix of the structure from its location at time to its position at the next time step . The first order direction cosine matrix with respect to Δt can be simplified as

(13–18)

where .

As is skew symmetric, i.e.

(13–19)

and by substituting Equation 13–18 and Equation 13–19 into Equation 13–17

(13–20)

Further splitting this expression into terms up to the first order with respect to , we have

(13–21)

Between the times and , the variation of the structure position is . Due to this vector change in position, there is an additional rotational momentum term that may be written (up to the first order with respect to ):

(13–22)

where

As , the force at the center of gravity can be written up to the first order with respect to as

(13–23)

If the added mass is omitted, the mass matrix in Equation 13–23 only includes the structure mass and moment of inertia matrix at the center of gravity:

(13–24)

where is the structure mass.

Substituting Equation 13–24 into Equation 13–23, we have

(13–25)

It is observed that there is no translational force component in the first term on the right hand side of Equation 13–25. The moment component in this term is named as the structural gyroscopic moment.

For each Morison element, the fluid forces and moments are calculated individually in the Morison element local axes, and then moved to the center of gravity. In such a case, the mass matrix in Equation 13–23 is replaced by representing the added mass matrix of the Morison element and the internal fluid mass matrix for flooded tubes, while is the tube velocity in the Morison element local axes.

The fluid gyroscopic force and moment with respect to the origin of the Morison element local axis system are defined as

(13–26)

The fluid momentum force and wave inertia force are denoted as

(13–27)

The corresponding forces and moments with respect to the center of gravity of the structure in the fixed reference axes can be determined by transformations similar to those applied in Equation 6–20.