Difference between revisions of "Category:Linear Hydroelasticity"

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Finite Element Method (FEM) can be used to analyze any general 3D elastic structure using linear hydroelastic theory.
 
Finite Element Method (FEM) can be used to analyze any general 3D elastic structure using linear hydroelastic theory.
 +
 
<center>  
 
<center>  
 
local FE <math>\Rightarrow </math> global FE model
 
local FE <math>\Rightarrow </math> global FE model
 
</center>
 
</center>
 +
 +
Dynamic equation of motion in matrix form can be expressed as:
 +
<center>
 
<math>
 
<math>
 
\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}D\end{bmatrix}+
 
\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}D\end{bmatrix}+
 
\begin{bmatrix}S\end{bmatrix}\begin{bmatrix}\dot D\end{bmatrix}+
 
\begin{bmatrix}S\end{bmatrix}\begin{bmatrix}\dot D\end{bmatrix}+
−
\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}\dot\dot D\end{bmatrix}=
+
\begin{bmatrix}M\end{bmatrix}\begin{bmatrix}\ddot D\end{bmatrix}=
 +
\begin{bmatrix}F(t)\end{bmatrix}</math>, where:</center>
 +
 
 +
<center><math>\begin{bmatrix}K\end{bmatrix}</math> is structural stiffness matrix,</center>
 +
<center><math>\begin{bmatrix}S\end{bmatrix}</math> is structural damping matrix,</center>
 +
<center><math>\begin{bmatrix}M\end{bmatrix}</math> is structural mass matrix,</center>
 +
<center><math>\begin{bmatrix}D\end{bmatrix}</math> is generalized nodal displacements vector,</center>
 +
<center><math>\begin{bmatrix}F\end{bmatrix}</math> is loading vector.</center>
 +
 
 +
 
 +
The eigenvalue problem for the "dry" natural vibrations yields:
 +
 
 +
<center><math>\begin{bmatrix}K\end{bmatrix}</math> is loading vector.</center>
 +
 
 +
 
 +
Generalized nodal displacements vector can be expressed using "dry" structure natural modes:
 +
 
 +
 
 +
<center><math>\begin{bmatrix}D\end{bmatrix}</math>=<math>\begin{bmatrix}W\end{bmatrix}\cdot\begin{bmatrix}\xi\end{bmatrix}</math></center>
 +
 
 +
 
 +
<center><math>\begin{bmatrix}W\end{bmatrix}</math>=<math>\begin{bmatrix}\mathbf{w_1}\,\mathbf{w_2}\,\ldots \end{bmatrix}</math> is matrix of dry natural modes, with modes being sorted column-wise,</center>
 +
 
 +
<center><math>\begin{bmatrix}\xi\end{bmatrix}</math>is natural modes coefficients vector.</center>
 +
 
 +
 
 +
<center><math>\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>\begin{bmatrix}W\end{bmatrix}^T\begin{bmatrix}F(t)\end{bmatrix}</math></center>
 +
 
 +
 
 +
<center><math>\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>\begin{bmatrix}f(t)\end{bmatrix}</math></center>
 +
 
 +
to be continued...

Revision as of 07:02, 13 November 2008

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} }[/math] is loading vector.


Generalized nodal displacements vector can be expressed using "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.

F

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