Template:Two dimensional floating body time domain

The body boundary condition for a fixed body is given in terms of the 3 rigid body motions, namely surge, heave and pitch which are indexed as [math]\displaystyle{ \nu=1,3,5 }[/math] in order to be consistent with the three-dimensional problem. We have a kinematic condition

[math]\displaystyle{ \partial_{n}\Phi=\sum_{\nu}\partial_t \xi_{\nu}\mathbf{n}_{\nu},\ \mathbf{x}\in\partial\Omega_{B}, }[/math]

where [math]\displaystyle{ \xi_{\nu} }[/math] is the motion of the [math]\displaystyle{ \mu }[/math]th mode and [math]\displaystyle{ \mathbf{n}_{\nu} }[/math] is the normal associated with this mode. The dynamic condition is the equation of motion for the structure:

[math]\displaystyle{ \sum_{\nu} M_{\mu\nu}\partial_t^2 \xi_{\nu}=-\rho\iint_{\partial\Omega_{B}}\partial_t\Phi n_{\mu}\, dS - \sum_{\nu} C_{\mu\nu}\xi_{\nu},\quad \textrm{for} \qquad \mu=1,3,5, }[/math]

In this equation, [math]\displaystyle{ M_{\mu\nu} }[/math] are the elements of the mass matrix

[math]\displaystyle{ \mathbf{M}=\left[ \begin{matrix} M & 0 & M(z^c-Z^R) \\ 0 & M & -M(x^c-X^R) \\ M(z^c-Z^R)& -M(x^c-X^R) & I^b_{11}+I^b_{33} \end{matrix} \right] , }[/math]

for the structure and [math]\displaystyle{ c_{\mu\nu} }[/math] are the elements of the buoyancy matrix

[math]\displaystyle{ \mathbf{C}=\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & \rho g W & -\rho g I^A_{1} \\ 0 & -\rho g I^A_{1} & \begin{matrix}\rho g (I^A_{11}+I^V_3)-\\ Mg(z^c-Z^R) \end{matrix} \end{matrix} \right]. }[/math]

The terms [math]\displaystyle{ I^b_{11} }[/math], [math]\displaystyle{ I^b_{33} }[/math] are the moments of inertia of the body about the [math]\displaystyle{ x }[/math] and [math]\displaystyle{ z }[/math] axes and the terms [math]\displaystyle{ I_1^{A} }[/math], [math]\displaystyle{ I^{A}_{11} }[/math] are the first and second moments of the waterplane (the waterplane area is denoted [math]\displaystyle{ W }[/math]) about the [math]\displaystyle{ x }[/math]-axis (see Chapter 7, Mei 1983). In addition, [math]\displaystyle{ (x^c,z^c) }[/math] and [math]\displaystyle{ (X^R,Z^R) }[/math] are the positions of the centre of mass and centre of rotation of the body and [math]\displaystyle{ I^{V}_{3} }[/math] is [math]\displaystyle{ z }[/math]-component centre of buoyancy of the structure. Thus, the coupled equations of motion for a floating structure have been derived. (N.B. if is assumed that the centre of rotation and the centre of mass of the structure coincide, i.e. if it is assumed that the body is semi-submerged, the mass and buoyancy matrices become diagonal). Any wave incidence is assumed to be propagating in the positive [math]\displaystyle{ x }[/math] direction.) The scattering and radiation problems are simpler than the coupled problem because the motion of the the structure is then prescribed.