Memory Effect Function

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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}\mathrm{d}t=-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}\mathrm{d}t=-i\omega \phi_{\mu}(\omega) - \Phi(\mathbf{x},0). }[/math]

Therefore, the initial 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}\mathrm{d}S+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}\mathrm{d}S=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} \mathrm{d}t. }[/math]

Integro-differential equation method

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by [math]\displaystyle{ \delta(t-\tau)V_{\alpha}(\tau) }[/math] where [math]\displaystyle{ \dot{X_{\alpha}}=V_{\alpha} }[/math]. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function [math]\displaystyle{ V_{\alpha}(t) }[/math] an integral of the impulse response over the range [math]\displaystyle{ -\infty\lt \tau\lt \infty }[/math] must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

[math]\displaystyle{ (11) \mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,\mathrm{d}\tau }[/math]

where [math]\displaystyle{ m_{\beta\alpha}(\infty) }[/math] is an infinite frequency added mass coefficient and [math]\displaystyle{ L_{\beta\alpha}(t-\tau) }[/math] is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. [math]\displaystyle{ V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \} }[/math], the impulse response function [math]\displaystyle{ L_{\beta\alpha} }[/math] can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

[math]\displaystyle{ \begin{align} a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, \mathrm{d}\tau \\ b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, \mathrm{d}\tau \end{align} }[/math]

and so if the added mass or damping coefficients are known for all frequencies [math]\displaystyle{ 0\leq\omega\leq\infty }[/math], the impulse response or memory function [math]\displaystyle{ L_{\alpha\beta}(t) }[/math] can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

[math]\displaystyle{ \left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,\mathrm{d}\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t), }[/math]

for [math]\displaystyle{ \alpha=1,\ldots,6 }[/math], where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency [math]\displaystyle{ \omega }[/math] on the interval [math]\displaystyle{ [0,\infty) }[/math]. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.