The current or wind drag forces on a structure in the relative current or wind direction can be expressed in the form of Equation 7–17.

Denoting the horizontal structure translational velocity in the local structure axes (LSA) as and ignoring the constant component of the hull drag force for the frequency domain analysis, the linearized hull drag force is given by

(12–10)

where is the 6×1 force/moment matrix, and is the 6×2 linearized damping matrix.

The fundamental requirement of the hull drag linearization is an equal dissipation of associated energy between the exact time domain analysis and the linearized hull drag frequency domain analysis. The energy dissipation ratio (EDR), which represents the ratio of energy in the time history to the linearized drag dissipated energy in the frequency domain, must have a unit value. By employing Equation 12–10 and Equation 7–2, this requirement can be expressed as

(12–11)

Where is the 6×1 matrix of structure translational and rotational motions.

By satisfying the above requirement the linearized damping matrix can be calculated. This matrix relates to the current and wind speed, the user-defined hull drag coefficient database, and the root mean square values of the structure horizontal translational (surge and sway) motions.

As discussed in Yaw Rate Drag Force, the current drag load components that depend on yaw rotational velocity are called yaw rate drag.

By satisfying the equivalence of energy dissipation requirement, the total damping matrix due to the yaw rate drag in the local structure axes is expressed as

(12–12)

where the damping matrix due to the constant term without integration is

(12–13)

in which and are functions of the current and wind speed, the user defined yaw rate drag coefficient, and the root mean square values of the surge and sway motions at the center of gravity of the structure. The integrand in Equation 12–12 at a point along the line between is given by

(12–14)

where and are functions of the current and wind speed, the user defined yaw rate drag coefficient, and the root mean square values of the structure horizontal translational motion components normal to the line between at that point.

The Morison hull drag force and moment components expressed by Equation 7–17 can also be linearized for a frequency domain analysis. The linearized damping matrix in the local structure axes is given by

(12–15)

where

in which the coefficients (j = 1,6) are a function of the current speed (for j = 1,2 only) and the root mean square value of the structure motion in the j-th degree of freedom.

For a given relative heading, the steady hull drag force and moment components can be expressed in the local structure axes by Equation 7–2. Similar to the wind drag force-induced stiffness in Equation 12–2, the additional stiffness matrix components due to the constant current drag forces and yaw motion are

(12–16)

where (j = 1,2) are the wind hull drag coefficients at the relative angle , and is the current velocity at a specified water depth.