temp_explication_pelamis.lyx~

#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\begin_body

\begin_layout Standard
Expression de la vitesse des vagues:
\end_layout

\begin_layout Standard
\begin_inset Formula $\overrightarrow{v}=\eta_{0}\omega e^{kz}cos(\omega t-kx)\vec{e_{x}}-\eta_{0}\omega e^{kz}sin(\omega t-kx)\vec{e_{z}}$
\end_inset

 avec 
\begin_inset Formula $k\eta_{0}\ll1$
\end_inset


\end_layout

\begin_layout Standard
Expression de la relation de dispersion
\begin_inset Formula 
\[
w^{2}=gk
\]

\end_inset


\end_layout

\begin_layout Standard
Expression de la hauteur des vagues:
\begin_inset Formula 
\[
\eta(x,t)=\eta_{0}\cos\left(\omega t-kx\right)
\]

\end_inset

 
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x)  r)});
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vec{e_z}$} |- (9,0) node[below]{$
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draw[color=blue] (8,0) node{$
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times$};
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vec{e_y}$};
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vec{e_{zc}}$} |- (1,0) node[right]{$
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vec{e_{xc}}$};
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draw[color=red,- latex] (0,0) --  (4,0);
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draw[red] (0,0) node{$
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times$};
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vec{e_{yc}}$};
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end{scope}
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draw[->] (-3.8,0) -- (-3.8,-0.4) node[midway,left]{$z_{0}$};
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draw[-latex] (2.5,-0.4) arc(0:12:2.5) node[midway, right]{$
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alpha$};
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end{center}
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\end_inset


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\begin_layout Standard
L'équation du mouvement est:
\begin_inset Formula 
\begin{eqnarray*}
PFD & (Oz): & m\ddot{z}=-mg+\Pi_{a,z}\\
TMC & (Oy): & J\ddot{\alpha}=\mathcal{M}\left(\Pi_{a}\right)_{y}
\end{eqnarray*}

\end_inset


\begin_inset Formula 
\[
-\iintop_{M\in S}P(M)\vec{dS_{M}}
\]

\end_inset


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vec{e_{xc}}$};
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node[below right] (0,0) {$
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vec{u_{z}}$};
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vec{u_{r}}$};
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vec{u_{
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theta}}$};
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theta$};
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end{tikzpicture}
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end{center}
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\begin_layout Standard
\begin_inset Formula 
\[
\vec{dF}(M)=-P(M)\vec{dS}
\]

\end_inset

 La composante suivant 
\begin_inset Formula $\vec{e_{z}}$
\end_inset

 est alors: 
\begin_inset Formula 
\[
dF(M)_{z}=-P(M)dS\left(\vec{e_{z}}\cdot\vec{u_{r}}\right)=-P(M)dS\sin\theta\cos\alpha
\]

\end_inset

De même pour le moment des forces de pression:
\begin_inset Formula 
\begin{eqnarray*}
\vec{\mathcal{M}}(\vec{dF})(M) & = & \vec{OM}\land\vec{dF}\\
 & = & \left(R\vec{u_{r}}+x_{c}\vec{u_{z}}\right)\land\left(-P(M)dS\vec{u_{r}}\right)\\
 & = & -P(M)x_{c}dS\vec{u_{\theta}}
\end{eqnarray*}

\end_inset

La composante suivant 
\begin_inset Formula $\vec{e_{y}}$
\end_inset

 est alors:
\begin_inset Formula 
\[
\mathcal{M}_{y}(\vec{dF})(M)=-P(M)x_{c}dS\left(\vec{e_{y}}\cdot\vec{u_{\theta}}\right)=P(M)x_{c}\sin\theta dS
\]

\end_inset


\end_layout

\begin_layout Standard
On a trois repère différents.
 Montrons comment on peut passer de l'un à l'autre:
\end_layout

\begin_layout Standard
On a:
\begin_inset Formula 
\begin{eqnarray*}
\vec{e_{xc}} & = & \cos\alpha\vec{e_{x}}+\sin\alpha\vec{e_{z}}\\
\vec{e_{yc}} & = & \vec{e_{y}}\\
\vec{e_{zc}} & = & -\sin\alpha\vec{e_{x}}+\cos\alpha\vec{e_{z}}
\end{eqnarray*}

\end_inset

soit la matrice de passage de 
\begin_inset Formula $\left(\vec{e_{xc}};\vec{e_{yc}};\vec{e_{zc}}\right)$
\end_inset

 à 
\begin_inset Formula $\left(\vec{e_{x}};\vec{e_{y}};\vec{e_{z}}\right)$
\end_inset

:
\begin_inset Formula 
\[
A_{cyl\rightarrow mer}=\left(\begin{array}{ccc}
\cos\alpha & 0 & -\sin\alpha\\
0 & 1 & 0\\
\sin\alpha & 0 & \cos\alpha
\end{array}\right)
\]

\end_inset


\end_layout

\begin_layout Standard
De même:
\begin_inset Formula 
\begin{eqnarray*}
\vec{u_{r}} & = & \cos\theta\vec{e_{yc}}+\sin\theta\vec{e_{zc}}\\
\vec{u_{\theta}} & = & -\sin\theta\vec{e_{yc}}+\cos\theta\vec{e_{zc}}\\
\vec{u_{z}} & = & \vec{e_{xc}}
\end{eqnarray*}

\end_inset

soit la matrice de passage de 
\begin_inset Formula $\left(\vec{u_{r}};\vec{u_{\theta}};\vec{u_{z}}\right)$
\end_inset

 à 
\begin_inset Formula $\left(\vec{e_{xc}};\vec{e_{yc}};\vec{e_{zc}}\right)$
\end_inset

:
\begin_inset Formula 
\[
A_{pol\rightarrow cyl}=\left(\begin{array}{ccc}
0 & 0 & 1\\
\cos\theta & -\sin\theta & 0\\
\sin\theta & \cos\theta & 0
\end{array}\right)
\]

\end_inset


\end_layout

\begin_layout Standard
Matrice de passage de la base polaire liée au cylindre à la base cartésienne
 de référence (la mer):
\begin_inset Formula 
\[
\left(\begin{array}{ccc}
-\sin\theta\sin\alpha & -\cos\theta\sin\alpha & \cos\alpha\\
\cos\theta & -\sin\theta & 0\\
\sin\theta\cos\alpha & \cos\theta\cos\alpha & \sin\alpha
\end{array}\right)
\]

\end_inset


\end_layout

\begin_layout Standard
Cette matrice s'obtient en faisant: 
\begin_inset Formula $A_{cyl\rightarrow mer}\times A_{pol\rightarrow cyl}$
\end_inset


\end_layout

\begin_layout Standard
Méthode d'Euler:
\end_layout

\begin_layout Standard
Les équations d'Euler sont:
\begin_inset Formula 
\begin{eqnarray*}
v_{z}[i+1] & = & v_{z}[i]+\left(\frac{1}{m}\Pi_{a,z}-g\right)dt\\
\dot{\alpha}[i+1] & = & \dot{\alpha}[i]+\frac{1}{J}\mathcal{M}\left(\vec{\Pi_{a}}\right)_{y}dt\\
z[i+1] & = & Z[i]+v_{z}[i]dt\\
\alpha[i+1] & = & \alpha[i]+\dot{\alpha}[i]dt
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
Détermination de la position en statique du cylindre:
\end_layout

\begin_layout Standard
\begin_inset ERT
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\end_body
\end_document