The forces involved in the low frequency ship manoeuvring motion estimation normally consist of the structure inertia force, and low frequency hydrodynamic forces on the hull, propeller and rudder. In this section only the hydrodynamic force and moment on the hull due to low frequency motion are covered. The gyroscopic moment is included in Equation 13–25.

In the local structure axes (LSA), the hydrodynamic force and moment on the hull due to the low frequency ship motion in an unbounded ideal fluid are given by Newman (2017) [31],

(13–91)

where the indices j, k, l have the values 1,2,3 and i=1,2, …,6, is the alternative tensor, U_i is the translational term (i=1,3) related to the current speed or rotational velocity (i=4,6) in the LSA, and m_li is the low (or zero) frequency added mass in the LSA.

The term in the hydrodynamic moment M_j is named as the Munk moment. Note that the off-diagonal 6x6 submatrices of the added mass due to the hydrodynamic interaction of structures are not accounted for in Equation 13–91.

The inertia force and moment terms in Equation 13–91 could be expressed as the derivatives of the linear and angular momentums,

(13–92)

Denoting as the yaw angle, the yaw motion Euler transformation matrix is given by

(13–93)

The matrix form of the translational and rotational velocity in the LSA is

(13–94)

where V _g(t) is the translational and rotational velocity related to the current speed in the fixed reference axes (FRA) at the time t.

Defining the rotational velocity and translational velocity matrices in the LSA as

(13–95)

the maneuvering force and moment in the FRA are written in the matrix form as

(13–96)

in which is the low (or zero) frequency added mass matrix in the LSA.

Denoting

(13–97)

Equation 13–96 is further expressed as

(13–98)