Given the static equilibrium position of the floating system, , an equation of small motions of the system about its equilibrium position can be written as

(11–13)

where is the acceleration vector, is the mass matrix, and is the total external force vector evaluated at the position .

Neglecting terms of second order or higher, the linearized equation of motion of the system can be expressed in terms of a general force, as

(11–14)

where is the total mass matrix, including the structural mass and the hydrodynamic added mass.

This equation of motion may also be rewritten in the Hamiltonian form

(11–15)

where is the velocity vector.

Denoting

(11–16)

the eigenvalues of Equation 11–15 can be determined from

(11–17)

Eigenvalues of the system given by Equation 11–16 and Equation 11–17 will indicate the modes of motion of the system, and may be interpreted as follows:

The natural period of a mode of motion is given by:

(11–18)

The critical damping percentage of a mode of motion is defined as

(11–19)

The normalized eigenvector output in Aqwa is defined as

(11–20)

For a single degree of freedom system, the equation of motion is written as

(11–21)

which may alternatively be expressed as

(11–22)

where and .

The percentage of critical damping can then be simplified as

(11–23)

The total mass matrix and linearized damping matrix may be frequency dependent. As an approximation, constant added mass and damping matrices at 'drift frequencies' with a wave period of 200 seconds are used in Equation 11–17. In order to evaluate a more accurate natural frequency and its corresponding mode of motion, an iterative approach may be employed. This picks up the hydrodynamic added mass and damping at a frequency close to that natural frequency, and is implemented in the Aqwa Graphical Supervisor (AGS) online dynamic stability calculation.