Problems in Linear Water-Wave theory in which there is an elastic body.
Finite Element Method (FEM) can be used to analyze any general 3D elastic structure using linear hydroelastic theory.
local FE [math]\displaystyle{ \Rightarrow }[/math] global FE model
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 loading vector.
The eigenvalue problem for the "dry" natural vibrations yields:
[math]\displaystyle{ \begin{bmatrix}K\end{bmatrix}\begin{bmatrix}D\end{bmatrix}+\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}\ddot D\end{bmatrix}=\begin{bmatrix}0\end{bmatrix} }[/math]
If one assumes trial solution as [math]\displaystyle{ \begin{bmatrix}D\end{bmatrix}=\begin{bmatrix}w\end{bmatrix}\,e^{i\omega t} }[/math] then the eigenvalue problem reduces to [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], where [math]\displaystyle{ \omega }[/math] is the dry natural frequency and [math]\displaystyle{ \begin{bmatrix}w\end{bmatrix} }[/math] is the dry natural vector.
Generalized nodal displacements vector can be expressed using calculated "dry" structure natural modes:
[math]\displaystyle{ \begin{bmatrix}D\end{bmatrix} }[/math]=[math]\displaystyle{ \begin{bmatrix}W\end{bmatrix}\cdot\begin{bmatrix}\xi\end{bmatrix} }[/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.
[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]
to be continued...
Subcategories
This category has only the following subcategory.
