Memory Effect Function

Frequency-domain equation of motion

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by McIver 2005, i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is [math]\displaystyle{ v_{\mu}(\omega) }[/math] then the Fourier transform of the acceleration is given by

[math]\displaystyle{ \int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0) }[/math]

the time-derivative of the potential obeys a similar relation

[math]\displaystyle{ \int_{0}^{\infty}\frac{\partial\Phi}{\partial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mathbf{x},0). }[/math]

Therefore, the ini\partial conditions of the structure [math]\displaystyle{ (X_{\mu}(0),V_{\mu}(0)) }[/math] and of the potential [math]\displaystyle{ \Phi(\mathbf{x},0) }[/math] will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before then the Fourier transform of the equation of motion for the structure is

[math]\displaystyle{ M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right] =-\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+ i\omega \sum_{\nu} (f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)- \sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega) }[/math]

for [math]\displaystyle{ \mu=1,\ldots,6 }[/math], where [math]\displaystyle{ f^{A}_{\mu}(\omega) }[/math] is the Fourier transform of the applied force [math]\displaystyle{ F_{\mu}(t) }[/math] in equation~(\ref{linearisedmotion}). Although it is assumed that [math]\displaystyle{ \Phi(\mathbf{x},t)=0 }[/math] for [math]\displaystyle{ t\lt 0 }[/math], for a non-zero initial velocity [math]\displaystyle{ \lim_{t^{+}\rightarrow 0}\Phi(\mathbf{x},t)\neq 0 }[/math] because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by

[math]\displaystyle{ \rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS=a(\infty)V(0) }[/math]

Mei 1983 where [math]\displaystyle{ a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega) }[/math] is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form

[math]\displaystyle{ \sum_{\nu} \{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega) = -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0) }[/math]

for [math]\displaystyle{ \mu=1,\ldots,6 }[/math]. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by

[math]\displaystyle{ v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}. }[/math]

As described by McIver 2006, the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient [math]\displaystyle{ f_{\mu\nu}(\omega) }[/math] will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see McIver 2006) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving for [math]\displaystyle{ v(\omega) }[/math] then the time-domain velocity can be recovered using the inverse transform

[math]\displaystyle{ V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt. }[/math]