Problems in Linear Water-Wave theory in which there is an elastic body.
Expansion in Modes
The basic idea is to use the same solution method as for a rigid body)
except to include elastic modes.
While there will be an infinite number of these modes in general, in practice only a few of the lowest
modes will be important unless the body is very flexible.
Finite Element Method
The finite element method is ideally suited to analyse flexible bodies. In the standard FEM notation
the dynamic equation of motion in matrix form can be expressed as:
[math]\displaystyle{
\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}D\end{bmatrix}+
\begin{bmatrix}S\end{bmatrix}\begin{bmatrix}\dot D\end{bmatrix}+
\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}\ddot D\end{bmatrix}=
\begin{bmatrix}F(t)\end{bmatrix} }[/math]
where
[math]\displaystyle{ \begin{bmatrix}K\end{bmatrix} }[/math] is structural stiffness matrix,
[math]\displaystyle{ \begin{bmatrix}S\end{bmatrix} }[/math] is structural damping matrix,
[math]\displaystyle{ \begin{bmatrix}M\end{bmatrix} }[/math] is structural mass matrix,
[math]\displaystyle{ \begin{bmatrix}D\end{bmatrix} }[/math] is generalized nodal displacements vector,
[math]\displaystyle{ \begin{bmatrix}F\end{bmatrix} }[/math] is generalized force vector (fluid forces, gravity forces,...).
Left-hand side of the global FEM matrix equation represents "dry" (in vacuuo) structure, while the right-hand side includes fluid forces (and coupling between the surrounding fluid and the structure).
Frequency Domain Problem
We consider the problem in the Frequency Domain so that
[math]\displaystyle{ \begin{bmatrix}D\end{bmatrix}=\begin{bmatrix}w\end{bmatrix}\,e^{-i\omega t} }[/math].
The eigenvalue problem for the "dry" natural vibrations yields:
[math]\displaystyle{
\left( \begin{bmatrix}K\end{bmatrix}-\omega^2 \begin{bmatrix}M\end{bmatrix} \right )\begin{bmatrix}w\end{bmatrix}=\begin{bmatrix}0\end{bmatrix}
}[/math]
As a solution of the eigenvalue problem for each natural mode one obtains [math]\displaystyle{ \omega_n }[/math], the n-th dry natural frequency and [math]\displaystyle{ \begin{bmatrix}w_n\end{bmatrix} }[/math], the corresponding dry natural mode. Note that only the first modes
are accurate approximations.
We expand the displacement in [math]\displaystyle{ N }[/math] modes,
[math]\displaystyle{ w = \sum_{n} \xi_n
}[/math]
[math]\displaystyle{ \begin{bmatrix}W\end{bmatrix} }[/math]=[math]\displaystyle{ \begin{bmatrix}\mathbf{w_1}\,\mathbf{w_2}\,\ldots \end{bmatrix} }[/math] is matrix of dry natural modes, with modes being sorted column-wise,
[math]\displaystyle{ \begin{bmatrix}\xi\end{bmatrix} }[/math]is natural modes coefficients vector (modal amplitudes).
[math]\displaystyle{ \begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}K\end{bmatrix} \begin{bmatrix}W\end{bmatrix}
\begin{bmatrix}\xi\end{bmatrix}+\begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}S\end{bmatrix} \begin{bmatrix}W\end{bmatrix}
\begin{bmatrix}\dot\xi\end{bmatrix}
+\begin{bmatrix}W\end{bmatrix}^T \begin{bmatrix}M\end{bmatrix} \begin{bmatrix}W\end{bmatrix}
\begin{bmatrix}\ddot\xi\end{bmatrix} }[/math]=[math]\displaystyle{ \begin{bmatrix}W\end{bmatrix}^T\begin{bmatrix}F(t)\end{bmatrix} }[/math]
[math]\displaystyle{ \begin{bmatrix}k\end{bmatrix} \begin{bmatrix}\xi\end{bmatrix}+\begin{bmatrix}s\end{bmatrix} \begin{bmatrix}\dot\xi\end{bmatrix}+\begin{bmatrix}m\end{bmatrix} \begin{bmatrix}\ddot\xi\end{bmatrix} }[/math]=[math]\displaystyle{ \begin{bmatrix}f(t)\end{bmatrix} }[/math]
[math]\displaystyle{ \begin{bmatrix}k\end{bmatrix}=\begin{bmatrix}W^T\end{bmatrix}\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}W\end{bmatrix} }[/math]is the modal stiffness matrix,
[math]\displaystyle{ \begin{bmatrix}m\end{bmatrix}=\begin{bmatrix}W^T\end{bmatrix}\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}W\end{bmatrix} }[/math]is the modal mass matrix.
Hydroelastic analysis of the general 3D structure is thus preformed using the modal superposition method.
Let us assume time-harmonic motion. Then the following is valid:
[math]\displaystyle{ \begin{bmatrix}\xi(t)\end{bmatrix}=\begin{bmatrix}\tilde{\xi}(\omega)\end{bmatrix}\cdot e^{i \omega t},
\; \begin{bmatrix}f(t)\end{bmatrix}=\begin{bmatrix}\tilde{f}(\omega)\end{bmatrix}\cdot e^{i \omega t} }[/math]
[math]\displaystyle{ \left ( \begin{bmatrix}k\end{bmatrix}+i\omega\begin{bmatrix}s\end{bmatrix}-\omega^2\begin{bmatrix}m\end{bmatrix} \right ) \begin{bmatrix}\tilde{\xi}\end{bmatrix}=\begin{bmatrix}\tilde{f}\end{bmatrix} }[/math]
Modal hydrodynamic forces are calculated by pressure work integration over the wetted surface:
[math]\displaystyle{ \tilde{f}^{hd}_i(t)=-i\omega\rho\iint_{S}\tilde{\phi}\,\mathbf{h_i}\mathbf{n}\,\mbox{d}S }[/math]
Total velocity potential can be decomposed as:
[math]\displaystyle{ \tilde{\phi}=\tilde{\phi}^I+\tilde{\phi}^D-i\omega\sum_{j=1}^N\tilde{\xi}_j\,\tilde{\phi}_j^R }[/math]
to be continued............
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