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\title{La ligne C du réseau métropolitain Lyonnais\\et sa crémaillère centrale
}


\subtitle[]{Analyse et modélisation d’une solution unique au monde}
\author{Louca Malerba - Candidat n°30501}
\titlegraphic{\includegraphics[scale=0.3]{Capture d'écran 2023-03-20 115400.png} }

\pgfdeclareimage[height=1cm]{logo}{dog1}%<- change dog1 for yor logo image file
\logo{\pgfuseimage{logo}}

\begin{document}



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\begin{frame}
\maketitle
\end{frame}

\section{Introduction}
\begin{frame}
I - Introduction

II - Réalisation d'une maquette

III - Expérimentations sur la maquette
	
$\;\;\;\;\;\;\;\;$	III - A. Coefficient de frottement : Étude statique du système
	
$\;\;\;\;\;\;\;\;$	III - B. Mise en mouvement du système : Étude dynamique

IV - Étude de résistance des matériaux : validation des performances du système réel

$\;\;\;\;\;\;\;\;$	IV - A. Modélisation du pignon de la crémaillère en poutre droite et dimensionnement

$\;\;\;\;\;\;\;\;$ IV - B. Modélisation informatique du pignon de la crémaillère et validation du dimensionnement réel


V - Conclusion

\end{frame}


\begin{frame}
\frametitle{Introduction}

\underline{A. De la ficelle de la Croix-Rousse au métro C actuel}

\footnotesize{Jusqu’en 1960 : la “ficelle” de Croix-Rousse :}

\begin{columns}
\column{0.5\textwidth}
	- Besoin de mobilité dès la fin du 19ème siècle.
	
	- Pente maximale de 17,3$\%$

	\begin{figure}
	\includegraphics[scale=0.30]{Capture d'écran 2023-03-20 112924.png} 
	
\tiny{\textbf{“Ficelle” de la Croix-Rousse} - \textit{crédit photo : Albert-Léon Lévy-Lambert - 1908}}
	\end{figure}
\column{0.5\textwidth}
\begin{figure}
\includegraphics[scale=0.35]{Capture d'écran 2023-03-20 113620.png}  
	
\tiny{\textbf{Funiculaire similaire présent sur la colline de Fourvière} - \textit{crédit photo : Poma.net}}
\end{figure}
\end{columns}

\end{frame}

\begin{frame}

\footnotesize{Aujourd’hui : le métro C}

\begin{columns}

\column{0.4\textwidth}
\begin{figure}
\includegraphics[scale=0.4]{Capture d'écran 2023-03-20 115400.png}   
	
\tiny{\textbf{Métro C de la Croix-Rousse} - \textit{crédit photo : ferro-lyon.net}}
\end{figure}


\column{0.5\textwidth}
\begin{figure}
\includegraphics[scale=0.5]{Capture d'écran 2023-03-20 115440.png} 
	
\tiny{\textbf{Principe de fonctionnement d’une crémaillère (vue
de côté)} - \textit{crédit image : Académie de Limoge}}
\end{figure}


\end{columns}
\end{frame}

\begin{frame}
\underline{B. Généralités sur les trains à crémaillère.}

\begin{enumerate}
\item Adhérence limitée des roues en acier $\Rightarrow$ pentes praticables en général en dessous de $4\%$.

\item Chemin de fer à crémaillère = deux rails parallèles en acier + un troisième rail denté, placé entre les rails lisses.

\item Cette solution permet de gravir des pentes allant jusqu'à 48\%. 

\end{enumerate}

\end{frame}

\begin{frame}
\uline{C. Problématique\\}

Dans quelles mesures le choix d’un métro à crémaillère a-t-il été un choix pertinent pour l’amélioration de la ligne C du réseau métropolitain ?

Quelle limite de pente peut atteindre par un train à adhérence simple ?
\end{frame}

\section{Réalisation d'une maquette}
\begin{frame}
\frametitle{II - Réalisation d'une maquette}
\underline{Modélisation d'un bogie moteur d'une rame}
\begin{columns}
\column{0.55\textwidth}
\begin{figure}
\includegraphics[scale=0.3]{Capture d'écran 2023-03-25 214202.png} 	
\tiny{\textbf{Proposition de modèle 3D, vue de côté}}
\end{figure}

\column{0.55\textwidth}
\begin{figure}
\includegraphics[scale=0.3]{Capture d'écran 2023-03-25 214456.png} 
	
\tiny{\textbf{Proposition de modèle 3D, vue de dessus}}
\end{figure}
\end{columns}

\end{frame}

\begin{frame}

\footnotesize{Images de la maquette} :

\begin{columns}

\column{0.4\textwidth}
\begin{figure}
\includegraphics[scale=0.13]{Capture d'écran 2023-05-16 201745.png}    
	
\tiny{\textbf{Maquette} - vue de dessus}
\end{figure}

\column{0.4\textwidth}
\begin{figure}
\includegraphics[scale=0.075]{Capture d'écran 2023-05-16 200508.png}  
	
\tiny{\textbf{Vue d'ensemble de la maquette}}
\end{figure}

\column{0.4\textwidth}
\begin{figure}
\includegraphics[scale=0.22]{Capture d'écran 2023-05-16 200935.png} 
	
\tiny{\textbf{Supports de rail sans crémaillère centrale} - (montre le système de réglage et la conception démontable des rails)}
\end{figure}


\end{columns}
\end{frame}

\begin{frame}
\begin{columns}
\column{0.55\textwidth}
\begin{figure}
\includegraphics[scale=0.3]{Capture d'écran 2023-05-16 195741.png} 
	
\tiny{\textbf{Maquette, crémaillère et transmission de la puissance du moteur par système poulie courroie}}
\end{figure}

\column{0.55\textwidth}
\begin{figure}
\includegraphics[scale=0.3]{Capture d'écran 2023-05-16 195741.png} 
	
\tiny{\textbf{Maquette, crémaillère centrale}}
\end{figure}
\end{columns}
\end{frame}

\begin{frame}
\begin{columns}
\column{0.25\textwidth}
\begin{figure}
\includegraphics[scale=0.18]{Capture d'écran 2023-05-16 195741.png} 
	
\tiny{\textbf{Maquette, mise en évidence de l'hyperstatisme au niveau de la crémaillère}}
\end{figure}

\column{0.25\textwidth}
\begin{figure}
\includegraphics[scale=0.18]{Capture d'écran 2023-05-16 195741.png} 
	
\tiny{\textbf{Maquette, mise en évidence de l'hyperstatisme au niveau d'un contact roue/rail}}
\end{figure}

\column{0.25\textwidth}
\begin{figure}
\includegraphics[scale=0.18]{Capture d'écran 2023-05-16 195741.png} 
	
\tiny{\textbf{Maquette, rotule comme solution à l'hyperstatisme}}
\end{figure}

\column{0.25\textwidth}
\begin{figure}
\includegraphics[scale=0.18]{Capture d'écran 2023-05-16 195741.png} 
	
\tiny{\textbf{Maquette, supports des rails} - cales de réglage visibles}
\end{figure}
\end{columns}
\end{frame}

\section{Expérimentations sur la maquette}
\begin{frame}
\frametitle{III - Expérimentations sur la maquette}

Le système doit être capable de gravir une pente élevée.\newline

$\hookrightarrow$ \uline{Exigence associée} : pente de 17\% doit pouvoir être empruntée. (Choix délibéré d'une pente proche du système réel malgré la différence de matériaux)\newline

$\hookrightarrow$ \uline{Objectif de cette partie} : valider l'utilité d'une crémaillère et obtention des limites de cette solution.

\end{frame}

\begin{frame}
\underline{III - A. Coefficient de frottement : Étude statique du système.}\\

\underline{Protocole expérimental :}

- Enlever la crémaillère centrale.

- Mesurer les hauteurs aux extrémités des rails.

- Poser le bogie dessus et le lâcher le plus lentement possible. Veiller à ce que les boudins des roues ne soient pas en contact avec les rails (ajouterait un contact ponctuel supplémentaire).

- Répéter l’opération, en changeant la hauteur mm par mm, jusqu’à qu’il soit impossible de garder le bogie fixe sur les rails.

\end{frame}

\begin{frame}
\begin{columns}

\column{0.6\textwidth}
\begin{figure}
\includegraphics[scale=0.12]{Capture d'écran 2023-05-16 200508.png} 
	
\tiny{\textbf{Vue d'ensemble}}
\end{figure}

\begin{figure}
\includegraphics[scale=0.2]{Capture d'écran 2023-05-18 184242.png} 
	
\tiny{\textbf{Boudin d'une roue}}
\end{figure}

\column{0.4\textwidth}
\begin{figure}
\includegraphics[scale=0.2]{Capture d'écran 2023-05-16 202701.png} 
	
\tiny{\textbf{Lecture d'une valeur de $h$}}
\end{figure}
\end{columns}
\end{frame}



\begin{frame}
\footnotesize{
\underline{Principe fondamental de la statique :}
\begin{align}
[...]
\iff
    f > \frac{sin(\alpha)}{cos(\alpha)} = tan(\alpha)
\end{align}

Expérimentalement : \\
- \uline{Sur rails secs} : $\alpha \approx$ 12,8° (22,7\%)\\
- \uline{Sur rails mouillés} : $\alpha \approx$ 10,3° (18,2\%) \\

Donc, sachant qu'on se trouve à la limite de l'angle avant qu'il y ait glissement du boggie :

\begin{align}
\left\lbrace
    \begin{array}{ll}
        f = 0.454 \pm 0.003\:sur\:rails\:secs \\
        f = 0.362 \pm 0.003\:sur\:rails\:mouilles
    \end{array}
\right.
\end{align}}

De plus, avec la crémaillère, la pente peut atteindre $45\% \; (i.e.) \approx 26$° 
\end{frame}

\begin{frame}
\uline{III - C. Mise en mouvement de la maquette : Étude dynamique. \\}

Les pentes obtenues en statiques, aussi bien sur pente humide que sur pente sèche respectent le cahier des charges (pour rappel, la pente maximale à gravir est de $17\%$).

Toutefois, ces calculs sont optimistes car ils négligent les effets d'inertie.

L'étude dynamique permet de les prendre en compte.

\end{frame}

\begin{frame}
Il est difficile de modéliser une correspondance entre les performances du système réel et celle du modèle.

On s'intéresse donc à l'évolution de l'accélération maximale sans glissement en fonction de la pente à gravir.
\end{frame}

\begin{frame}
\underline{Protocole expérimental} :

- Poser le bogie sur les rails \textbf{sans crémaillère centrale}.

- Placer des repères pour repérer un éventuel glissement plus facilement.

- Lancer un enregistrement vidéo (à 240 images par secondes).

- Lancer le programme d'asservissement avec une accélération la plus basse possible.

- Regarder la vidéo, s'il n'y a pas de glissement, recommencer avec une accélération plus grande. Sinon, noter les deux dernières valeurs d'accélération et augmenter la pente puis recommencer.

\end{frame}

\begin{frame}
\begin{figure}
\begin{center}
\includegraphics[scale=0.3]{Capture d'écran 2023-05-16 202008.png} 
\end{center}

\tiny{\textbf{Extrait d'une vidéo utilisée dans le protocole} - (montre les marquages et la vue utilisée)}
\end{figure}
\end{frame}

\begin{frame}
\begin{columns}
\column{0.45\textwidth}
En parallèle, application du principe fondamental de la dynamique

$\Rightarrow$ Valeurs maximales d'accélération pour qu'il y ait roulement sans glissement.

$\Rightarrow$ Condition mathématique : 


\fbox{$\Gamma(G, \Sigma / 0) < \frac{b.g.cos(\theta)f}{a + b} - g sin(\theta)$}

\column{0.5\textwidth}
\begin{figure}
\includegraphics[scale=0.35]{Capture d'écran 2023-05-17 103710.png} 

\tiny{\textbf{Accélération linéaire maximale en fonction de l'angle d'inclinaison des rails}, sans crémaillère centrale}
\end{figure}
\end{columns}
\end{frame}

\begin{frame}
\begin{figure}
\includegraphics[scale=0.3]{Capture d'écran 2023-05-17 134305.png} 


\tiny{\textbf{Accélération linéaire maximale en fonction de l'angle d'inclinaison des rails}, sans crémaillère centrale} 
\end{figure}



\end{frame}

\begin{frame}
Les résultats de cette expérience ne sont pas satisfaisants.\newline

\uline{Causes possibles} :
\begin{itemize}
\item La liaison pivot arrière non idéale.

\item Résolution du capteur de position trop faible. (Fonctionne pour mesurer la position, permet une approximation intéressante de la vitesse, mais incertitude liée à l'accélération trop grande).

\item Transmission de puissance imparfaite. (à-coups à faible tension en sortie du hacheur)
\end{itemize}

\begin{frame}
\underline{Réalisation d'une expérience moins complète} :

- Programme modifié pour augmenter la tension très lentement.

- Toujours le même protocole pour vérifier le non-glissement.

$\Rightarrow$ Résultat : angle limite de démarrage sans glissement :

$\theta_{limite} = 3,3 \pm 0,1°$
 
\end{frame}


\begin{frame}
\uline{En conclusion} : 

Expérimentalement (liaisons non idéales, contrôle de l'accélération imparfait, ...), le système est incapable de gravir une pente de plus de $5,8\% \pm 2$ ($3,3\pm 0.1$°) sans glissement.

De plus, le système réel a besoin d'être conçu avec un coefficient de sécurité, et doit être capable de gravir une partie mouillée. Le système ne répond donc pas aux exigences.

Il y a donc besoin d'utiliser une crémaillère :  l'expérience démontre le fonctionnement du système avec la crémaillère avec des pentes allant jusqu'à 45$\%$.
\end{frame}

\section{Etude de résistance des matériaux}
\begin{frame}
\frametitle{III - Étude de résistance des matériaux}

Le système doit être sécurisé.

La crémaillère et le pignon associé sont très sollicités.

$\hookrightarrow$ \uline{Objectif de cette partie} : valider le dimensionnement du système réel.

\end{frame}

\begin{frame}

\underline{A - Recherche d'informations.}

- Peu d'informations disponibles sur le système réel.

\begin{columns}
\column{0.5\textwidth}
\begin{figure}
\includegraphics[scale=0.20]{UTMC-09-08-11.jpg}  
	
\tiny{\textbf{Image du pignon de la crémaillère} - \textit{crédit photo : www.ferro-lyon.net - 09/08/2011}}
\end{figure}

\column{0.5\textwidth}
- Diamètre des roues neuves connu : $690mm$

- Photo $\Rightarrow$ $14\pm1$ dents sur le pignon.

- $d = m\times Z \iff m = \frac{d}{Z} = \frac{573}{14} = 40.9 mm$

- $\Rightarrow h = m\times 2.25 = 92$mm

\end{columns}

\end{frame}

\begin{frame}

\underline{B - Modélisation en poutre droite d'une dent du pignon de la crémaillère.}

Hypothèses :

\begin{enumerate}
\item On considère une dent comme une poutre de section rectangulaire, "proche" des dimensions d'une dent du système réel.

\item L'effort est orienté de 20° par rapport à l'axe $\vec{y}$.

\item L'ensemble de l'effort du la rame s'applique sur \uline{une seule dent}.
\end{enumerate}

\begin{columns}
\column{0.5\textwidth}
\begin{figure}

\begin{center}
\includegraphics[scale=0.3]{Capture d'écran 2023-03-23 095158.png} 
\end{center}
\tiny{\textbf{Modélisation en poutre droite}}
\end{figure}

\column{0.5\textwidth}
\begin{figure}
\begin{center}
\includegraphics[scale=0.25]{RDM TIPE.png} 
\end{center}
\tiny{\textbf{Principe de superposition}}
\end{figure}
\end{columns}
\end{frame}

\begin{frame}
\begin{columns}

\column{0.5\textwidth}
\begin{figure}
\includegraphics[scale=0.3]{Capture d'écran 2023-03-25 204906.png} 
	
\tiny{\textbf{Courbe de $\sigma_{xx,max} = f(x)$}}
\end{figure}


\column{0.5\textwidth}
\footnotesize{On en déduit la valeur maximale de la contrainte normale :}\\

\begin{center}
\fbox{$\sigma_{xx,max} = 1,62.10^{8}\; Pa$}
\end{center}
\end{columns}

\end{frame}


\begin{frame}
\underline{C - Résolution numérique sous SolidWorks}

\begin{columns}

\column{0.5\textwidth}
\begin{figure}
\includegraphics[scale=0.3]{Capture d'écran 2023-03-23 095648.png}  
	
\tiny{\textbf{Image du pignon de la crémaillère sous Solidworks}}
\end{figure}


\column{0.5\textwidth}
\footnotesize{- Matériau appliqué : acier peu allié.}

\footnotesize{- Effort à 20° comme pour le modèle en poutre droite.}\\

\footnotesize{- Conditions les plus défavorables possible :}

$\;\;\;\;$ $\rightarrow$ \footnotesize{Effort entier sur une seule dent (irréel mais simplifie les calculs)}

$\;\;\;\;$ $\rightarrow$ \footnotesize{Effort appliqué à l'extrémité de la dent.}

\end{columns}


\end{frame}

\begin{frame}
\begin{columns}
\column{1\textwidth}
\footnotesize{\uline{Deux cas considérés} : }

$\;\;\;\;$ \footnotesize{- Effort maximal réel appliqué sur la crémaillère :} $F = 100600 \; N$

$\;\;\;\;$ \footnotesize{- Effort pour un freinage entier sur la crémaillère :} $F = 64400 \; N$

\begin{figure}
\includegraphics[scale=0.4]{Capture d'écran 2023-05-17 213231.png}

\tiny{\textbf{Tableau récapitulatif des efforts maximaux} - \textit{Documentation Alsthom - publié dans la revue Chemins de fer}} 
\end{figure}


\end{columns}
\end{frame}

\begin{frame}
On obtient les résultats suivants selon l'exploitation de la rame : 
\begin{columns}

\column{0.55\textwidth}
\begin{figure}
\includegraphics[scale=0.35]{Capture d'écran 2023-03-23 100428.png}   
	
\tiny{\textbf{Application des efforts pour l'application de l'effort maximal réel sur la crémaillère}}
\end{figure}

\column{0.55\textwidth}
\begin{figure}
\includegraphics[scale=0.35]{Capture d'écran 2023-03-23 100645.png} 
	
\tiny{\textbf{Application des efforts pour l'application de l'effort maximal dans les conditions les plus défavorables (tout le freinage se repose sur une seule dent)}}
\end{figure}
\end{columns}

\end{frame}

\begin{frame}
\underline{Tableau récapitulatif des résultats} :
\begin{center}
\begin{tabular}{|p{4cm}|l|}
	\hline Méthode & $\sigma_{xx,max}$ \\
	\hline Poutre droite, freinage réel & $1,03.10^{8}\; Pa$\\
	\hline Poutre droite, freinage entier sur le pignon & $1,62.10^{8}\; Pa$ \\
	\hline SolidWorks,  freinage réel & $9,70.10^7\; Pa$\\
	\hline SolidWorks,  freinage entier sur le pignon& $1,52.10^8\; Pa$\\
	\hline
\end{tabular}
\end{center}
\footnotesize{Or, SolidWorks nous donne comme donnée pour la limite d'élasticité d'un acier peu allié : $6,20.10^8 Pa$.}

$\Rightarrow$ Dans le pire des cas, (très exagéré par rapport à la réalité), un coefficient de sécurité \fbox{$c = \frac{6,20.10^8}{1,62.10^{8}} = 3.8$}

\end{frame}

\begin{frame}
\uline{À noter} : Il est très important d'avoir un coefficient de sécurité élevé sur une rame neuve, car il y a ensuite usure des dents\newline \newline

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\end{frame}



\begin{frame}
À noter qu'il faut encore:

- Que je prenne des photos du système au laboratoire pour un meilleur rendu.

- Que je réalise les calculs d'incertitude.

- Que je commente et que j'insère mes programmes en annexe.

\end{frame}
\section{Annexes}
\begin{frame}
\frametitle{Annexes}
\begin{center}
\includegraphics[scale=0.4]{Capture d'écran 2023-05-16 192857.png}  
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\includegraphics[scale=0.4]{Capture d'écran 2023-05-16 192952.png}  
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\includegraphics[scale=0.5]{Capture d'écran 2023-05-16 193118.png}   
\end{center}
\end{frame}

\begin{frame}
\uline{Paramétrage Principe fondamental de la dynamique}:


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\uline{Principe fondamental de la Dynamique} :

\begin{center}
\includegraphics[scale=0.5]{Capture d'écran 2023-05-17 142150.png} 
\end{center}
\end{frame}

\begin{frame}
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\includegraphics[scale=0.5]{Capture d'écran 2023-05-17 142354.png} 
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\end{frame}

\begin{frame}
\tiny{Image supplémentaire : vue de dessus.}

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\includegraphics[scale=0.3]{Capture d'écran 2023-05-16 201745.png} 
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\begin{frame}
\uline{Schéma électrique}

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% Text Node
\draw (376.82,28.77) node [anchor=north west][inner sep=0.75pt]  [rotate=-15.72] [align=left] {{\footnotesize Capteur GND}};
% Text Node
\draw (405.19,13.96) node [anchor=north west][inner sep=0.75pt]  [rotate=-13.64] [align=left] {{\footnotesize Capteur 3.3V}};
% Text Node
\draw (164.08,24.69) node [anchor=north west][inner sep=0.75pt]  [rotate=-359.67] [align=left] {{\footnotesize Capteurs phases A et B}};
% Text Node
\draw (534,115) node [anchor=north west][inner sep=0.75pt]  [font=\scriptsize] [align=left] {Motoréducteur\\+ capteurs à effet Hall};
% Text Node
\draw (562,57) node [anchor=north west][inner sep=0.75pt]  [font=\LARGE] [align=left] {\textbf{M}};
% Text Node
\draw (27,112) node [anchor=north west][inner sep=0.75pt]  [font=\scriptsize] [align=left] {Microcontrolleur\\Raspberry Pi};
% Text Node
\draw (299,267) node [anchor=north west][inner sep=0.75pt]  [font=\scriptsize] [align=left] {Hacheur};
% Text Node
\draw (292,174) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {GND};
% Text Node
\draw (329,173) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {GND};
% Text Node
\draw (330,185) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {VIN};
% Text Node
\draw (322,196) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {BOUT2};
% Text Node
\draw (323,207) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {BOUT1};
% Text Node
\draw (320.5,217.2) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {AOUT2};
% Text Node
\draw (322.5,228.2) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {AOUT1};
% Text Node
\draw (324.5,241.2) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {VMM};
% Text Node
\draw (291.5,243.2) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {MODE};
% Text Node
\draw (291.5,233) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {AIN1};
% Text Node
\draw (291.5,221) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {AIN2};
% Text Node
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% Text Node
\draw (290.5,200) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {BIN2};
% Text Node
\draw (292.5,188) node [anchor=north west][inner sep=0.75pt]  [font=\tiny] [align=left] {VCC};
% Text Node
\draw (522,178) node [anchor=north west][inner sep=0.75pt]   [align=left] {+};
% Text Node
\draw (558,178) node [anchor=north west][inner sep=0.75pt]   [align=left] {+};
% Text Node
\draw (565,247) node [anchor=north west][inner sep=0.75pt]   [align=left] {\mbox{-}};
% Text Node
\draw (529,247) node [anchor=north west][inner sep=0.75pt]   [align=left] {\mbox{-}};
% Text Node
\draw (583,213) node [anchor=north west][inner sep=0.75pt]  [font=\scriptsize] [align=left] {Accumulateur\\3.7V (x2)};
% Text Node
\draw (29,267) node [anchor=north west][inner sep=0.75pt]  [font=\scriptsize] [align=left] {Batterie Lithium 5V};
% Text Node
\draw (0.63,154.93) node [anchor=north west][inner sep=0.75pt]  [font=\scriptsize,rotate=-270.81] [align=left] {Micro-USB};


\end{tikzpicture}
\end{frame}

\begin{frame}
\uline{Code d'asservissement en différents modes}:
\begin{center}
\includegraphics[scale=0.35]{Capture d'écran 2023-05-18 194910.png} 
\end{center}
\end{frame}


\begin{frame}
\begin{center}
\includegraphics[scale=0.35]{Capture d'écran 2023-05-18 194936.png} 
\end{center}
\end{frame}


\begin{frame}
\begin{center}
\includegraphics[scale=0.35]{Capture d'écran 2023-05-18 194958.png}  
\end{center}
\end{frame}


\begin{frame}
\begin{center}
\includegraphics[scale=0.35]{Capture d'écran 2023-05-18 195025.png} 
\end{center}
\end{frame}

\begin{frame}
Valeurs de masse (en $g$):

masse totale: $1265 \pm 0.5$

bloc batterie + breadboard: $300 \pm 10$

raspberry + ventilateur: $80 \pm 10$

accumulateur (x1): $67 \pm 0.5$

chassis arrière + rotule + roues arrières: $252 \pm 0.5$

chassis avant + roues avant + moteur + rangement des accumulateurs: $467 \pm 1$

\end{frame}

\begin{frame}

\begin{center}
\begin{figure}
\includegraphics[scale=0.50]{Capture d'écran 2023-03-20 113816.png} 
\end{figure}

\tiny{\textbf{Schéma de principe d’un funiculaire} - \textit{crédit photo : https://www.remontees-mecaniques.net/}}

\end{center}

\end{frame}
\end{document}