If the external force in a time domain analysis is not periodic with constant amplitude, the equation of motion in the frequency domain (i.e. Equation 12–19) cannot be directly converted into the following form in the time domain:

(13–1)

as the added mass in the mass matrix and the hydrodynamic damping in the damping matrix are frequency dependent.

Instead, the equation of motion of the floating structure system is expressed in a convolution integral form [10]

(13–2)

where is the structural mass matrix, is the fluid added mass matrix at infinite frequency, is the damping matrix except the linear radiation damping effects due to diffraction panels, is the total stiffness matrix, and is the velocity impulse function matrix.

Alternatively, the acceleration impulse function matrix can be employed in the equation of motion, such as

(13–3)

in which the acceleration impulse function matrix is defined by

(13–4)

where and are the added mass and hydrodynamic damping matrices, respectively, defined in Equation 4–48.

Because the complex function is analytic in the upper half plane, the real and imaginary parts of this function are the Cauchy principle values of the Hilbert transforms of each other:

(13–5)

where denotes the Cauchy principle value. Generally, the values of the added damping coefficients at zero frequency and infinite frequency are null. It is therefore more practical to use the added damping coefficient matrix in Equation 13–4 to obtain the required impulse response function matrix.

The integration of Equation 13–4 is numerically truncated at a finite upper frequency limit. In Aqwa, this upper limit is set to be approximately 10 radians/s, or a 0.5 second period, for a relatively large structure. For a small diffracting structure with , where is the characteristic structure size (in meters), this upper limit is increased by a factor of .

The selected frequency region applied for the calculation of the diffraction and radiation potential in a conventional three-dimensional frequency domain hydrodynamic analysis may not correspond exactly to the total required range . An automatic extrapolation is therefore carried out to estimate the added damping values in the extended frequency ranges of and .

The extended set of added damping coefficients in the frequency region of is also used in the Hilbert transform shown in Equation 13–5 to obtain the fitted frequency-dependent added mass matrix . A best fit is then obtained between the fitted added mass transformed from the damping coefficients and the directly-calculated added mass from an Aqwa hydrodynamic analysis. This gives the effective asymptotic added mass matrix at infinite frequency.

The average 'fitting quality' of the added mass coefficient is defined as

(13–6)

where is the number of frequency points in the hydrodynamic database and is the normalized mass factor, which may be related to the structure and added mass matrices as

(13–7)

where and are the diagonal elements of the structure mass matrix in the j-th and k-th degrees of freedom, and are the diagonal elements of the added mass matrix in the j-th and k-th degrees of freedom, and and are the first and last frequency points in the hydrodynamic database used for the impulse function calculation.

The generalized damping is also checked when the convolution method is used. It should generally be positive.

The impulse function convolution described above provides a more rigorous approach to the radiation force calculation in the time domain, and enhances the capability of the program in handling the nonlinear response of structures.

Alternatively, the RAO-based radiation forces at each time step can be estimated by the following approach. Denoting as the position of the structure center of gravity and as the yaw angle at time (both in the fixed reference axes), when the incident wave elevation is numerically expressed by Equation 2–28 and Equation 2–29, the free-floating structure RAO-based radiation force is given by

(13–8)

where represents the motion RAOs at frequency and relative wave direction , as calculated by a frequency domain hydrodynamic diffraction analysis.

The motion RAOs appearing in Equation 13–8 do not include the nonlinear effects of mooring lines, articulations and other non-harmonic external forces. It is also assumed that at time , the structure steadily oscillates with the incident wave component frequencies. Hence the RAO-based radiation force calculation of Equation 13–8 is considered to be a simplified, but approximate, method.