The free trim large angle stability is similar to the concept discussed in Large Angle Stability. It is assumed that a free-floating body rotates at a specified angle about a prescribed horizontal hinge axis, and the body's righting moments about this hinge axis are calculated under a free trim large angle position when:

The buoyance and gravity centers stay on the rotation plane, which is perpendicular to the hinge axis. This renders the least work to be performed in order to heel the body up to the prescribed rotation angle, and the direction of the righting moment is always parallel to the hinge axis [36].

The free trim large angle position is defined in the hinge axis local coordinate system, which has the hinge axis as the X-axis, the FRA’s Z axis as the Z-axis, and the Y-axis following the right-hand rule. The trim, heel, and heave discussed in this section all refer to the hinge axis local coordinate system.

The iterative approach discussed in Iteration Towards Equilibrium calculates the free trim large angle heave and trim positions. The righting moment is formed by the hydrostatic and gravity forces in the rotation plane. The hydrostatic force is calculated by Equation 3–4 and Equation 3–29 through Equation 3–30 under the free trim large angle position (including internal tanks, moonpools, and tubes). Furthermore, the additional hydrostatic stiffness-related forces and point buoyancy force are included. The righting moment, , is positive when its direction is opposite to the heeling moment, which rotates the body about the hinge axis to the prescribed angle.

(3–46)

where is the hydrostatic moment about the hinge axis calculated by Equation 3–4 and Equation 3–29 through Equation 3–30;

, , and are with respect to the hinge axis local coordinate system;

is the heeling term of the additional hydrostatic stiffness matrix;

is the heel and heave coupled term of the additional hydrostatic stiffness matrix;

is the heel and trim coupled term of the additional hydrostatic stiffness matrix;

is the displacement of the combined gravity center in the Z-axis direction of the hinge axis local coordinate system;

is the heeling angle in the hinge axis local coordinate system;

is the trim angle in the hinge axis local coordinate system;

is the i-th point buoyancy with the Y-axis coordinate value of in the hinge axis local coordinate system;

is the j-th point mass with the Y-axis coordinate value of in the hinge axis local coordinate system;

and is the combined gravity center's Y-axis coordinate value in the hinge axis local coordinate system.

After the right moment is obtained, the conventional GZ value is calculated by

(3–47)

where is the total gravity force of the body including the internal tank fluid, i.e., the total buoyancy force.

The area under the GZ curve at the rotation angle is calculated by

(3–48)

where is the smallest prescribed heeling angle where the GZ curve starts.