As observed from Equation 6–1, the drag force component in the Morison equation consists of the factor of . It is a nonlinear term in which can be replaced by a factor multiplied by the root mean square of relative velocity in order to create an equivalent linear term. In the literature (for example, [6]) this factor is given by . By choosing a proper factor , the linearized drag force at a cross section of a tube is expressed as

(6–5)

where is the root mean square of transverse directional relative velocity at that location.

The first item in the right hand side of Equation 6–5 is a linear function of fluid particle velocity and could be considered as the external fluid force acting on the structure. The corresponding total force and moment on the whole submerged tube element are

(6–6)

The second item in the right hand side of Equation 6–5 linearly relates to the structure velocity, so that its coefficient could be considered as the damping factor due to the linearized drag force.

Denoting the structural velocity vector of a tube in the tube local axis frame as

(6–7)

where is the translational velocity at node 1 in the local x-, y-, and z-directions and is the rotational velocity of tube about the local x-, y-, and z-axes.

The damping force from the linearized drag in the local y-direction and the moment about the local z-axis at the location of (x, 0, 0 ) are

(6–8)

the total damping force and moment on the whole submerged tube are

(6–9)

Similarly the total z-direction damping force and moment about local y axis on the whole submerged tube are

(6–10)

Introducing a linearized drag damping coefficient matrix

(6–11)

and setting

(6–12)

from Equation 6–6, Equation 6–9, and Equation 6–10 we will have

(6–13)

where is the tube wetted surface, is the axial drag coefficient, is the linearized factor in the axial direction, and is the root mean square of axial motion velocity. Aqwa sets as an unamendable default value for both circular tube and slender tube elements.

In the above discussion of the tube drag linearization method, the correlation effects between two transverse relative velocity components are not included, hence a one-dimensional linearization formula is employed. This method makes the total dissipating energy due to the linearized drag force approximately equal to the associated free energy dissipated in the exact time domain analysis. However, each dissipating energy component in either the local tube y-direction or z-direction may not be inherently compatible between the exact time domain analysis and the linearized frequency domain analysis.

Because of these limitations, Aqwa uses a two-dimensional method, which includes the correlation between two transverse relative velocities, which ensures equal dissipating energy components in both the local tube y- and z-directions between the time domain and the linearized frequency domain analyses.

In this two-dimensional approach, the linearized drag force at a cross-section position x in the local tube axes is given by

(6–14)

where the damping factors , , , and are the functions of the root of the mean square of structural motion at the position x, the tube cross-sectional geometric properties, and the drag coefficients of the tube in both local tube y- and z-directions. and are the fluid particle velocity components in the tube local y- and z-directions, and , and are the structural velocity components in the local tube y- and z-directions.

From Equation 6–14, the linearized drag force/moment due to fluid particle velocity can be written as

(6–15)

In Equation 6–14, the term linearly relates to the structural velocity: its coefficient can be considered the damping factor due to linearized drag force. Employing Equation 6–7 as the definition of the structural velocity vector of a tube in the tube local axis frame, the linearized damping force and moment in/about the local y- and z-directions is written as

(6–16)

Using the linearized drag damping coefficient matrix definition given in Equation 6–11 and Equation 6–12, the linearized tube damping coefficient matrix in the tube local axis frame can be derived from Equation 6–16:

(6–17)

Additionally, the axial translational motion linearized drag damping coefficient defined by Equation 6–13 should be included.

Note that the drag damping coefficient matrix and the force and moment in Equation 6–11 through Equation 6–17 are defined in the tube local axis frame. They should be transferred to the fixed reference axes and with respect to the center of gravity of the attached floating structure when the motion response is solved. To do so, denote as the coordinates of the tube element's first node and the center of gravity of the structure respectively, the relative location vector

(6–18)

and the unit vectors of the tube local axes reference to the global axes as

(6–19)

From Equation 6–12, the linearized tube drag force and moment with respect to the center of gravity in the global axes is above equations can be expressed as

(6–20)

where