diff --git a/.gitignore b/.gitignore index 0cab372ccdb5f884c1048789d724b00e01e95ca5..717299b8a591fb73099a88e8b33dcd15961094e1 100644 --- a/.gitignore +++ b/.gitignore @@ -23,3 +23,4 @@ *.synctex.gz *.bak *.mp4 +*.ps diff --git a/modeling.pdf b/modeling.pdf new file mode 100644 index 0000000000000000000000000000000000000000..977cf7a898644f81add749b2b4f5e1d68fdf13ce Binary files /dev/null and b/modeling.pdf differ diff --git a/modeling.tex b/modeling.tex new file mode 100644 index 0000000000000000000000000000000000000000..a5e2a2a3156b059d0a8fa475edaf7810597adb8e --- /dev/null +++ b/modeling.tex @@ -0,0 +1,72 @@ +\documentclass[12pt,a4paper]{report} +\usepackage[utf8]{inputenc} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{tikz} +\usetikzlibrary{patterns,angles,calc,quotes} + +\begin{document} + +\section*{Dynamics from Euler-Lagrange equations} + +\begin{figure} +\begin{center} +\begin{tikzpicture}[thick,>=latex] + \begin{scope} + %\draw[step=1cm,gray,very thin] (-1,-1) grid (4,5); + \draw[->,thick] (-0.1,0) -- (4,0) node[anchor=west]{x}; + \draw[->,thick] (0,-0.1) -- (0,4.5) node[anchor=south]{y}; + \coordinate (cart) at (2,0); + \coordinate (mass1) at ($(cart) + (104:4)$); + \coordinate (mass2) at ($(cart) + (127:2.5)$); + \coordinate (upp) at ($(cart) + (0,5)$); + + \draw[draw=black,fill=black] (cart) circle (.1cm) node (cart){}; + \draw[dashed] (cart.center) -- (upp) {}; + \node at (cart.south) [anchor=north] {$(x,0)$}; + + \draw (cart.center) -- (mass1) node[midway,xshift=2mm,yshift=5mm] (mid1) {}; + \draw[draw=black,fill=white] (mass1.center) circle (.15cm); + \node at (mass1.north) [anchor=south,xshift=1mm,yshift=1mm] { $(x_1,y_1)$}; + \pic["$\theta_1$", draw=black, ->, angle eccentricity=1.15, angle radius=2.2cm] + {angle = upp--cart--mass1}; + \draw (cart.center) -- (mass2) node[midway,xshift=1mm,yshift=3mm] (mid2) {}; + \draw[draw=black,fill=white] (mass2.center) circle (.15cm); + \node at (mass2.north) [anchor=south,xshift=1mm,yshift=1mm] { $(x_2,y_2)$}; + %\pic [draw, ->, angle eccentricity=1] {angle = upp--cart--mass2}; + \pic["$\theta_2\;$", draw=black, ->, angle eccentricity=1.25, angle radius=1.2cm] + {angle = upp--cart--mass2}; + \end{scope} +\end{tikzpicture} +\label{fig01} +\caption{Two pendulums on a moving cart} +\end{center} +\end{figure} +Two pendulums with lengths $l_1$ and $l_2$ and masses $m_1$ and $m_2$ are mounted on a moving cart with mass $M$, see Figure~\ref{fig01}. Introduce $L = T - V$ where kinetick and potential energies are given by +\begin{align} +T &= \frac{1}{2}M\dot x^2 + \frac{1}{2}m_1(\dot x_1^2 + \dot y_1^2) + \frac{1}{2}m_2(\dot x_2^2 + \dot y_2^2)\\ +V &= m_1gl_1 c_1 + m_2gl_2c_2 +\end{align} +where $c_i$ and $s_i$ and short for $\cos(\theta_i)$ and $\sin(\theta_i)$, $i=1,2$ respectively. +Note that $x_i = x-l_is_i$ and $y_i = l_ic_i$. +The Euler-Lagrange equations +\begin{align} +\begin{cases} +0=\dfrac{\partial L}{\partial \theta_i} - \dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot{\theta}_i} \right), & i=1,2 \\ +F=\dfrac{\partial L}{\partial x} - \dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot{x}} \right). +\end{cases} +\end{align} +give after some calculations that +\begin{align} +l_i \ddot \theta_i &= gs_i + c_i \ddot x, \quad i=1,2\\ +F&=M\ddot x + m_1(\ddot x - l_1c_1\ddot \theta_1 + l_1s_1\dot{ \theta}_1^2) + m_2(\ddot x - l_2c_2\ddot \theta_2+ l_2s_2\dot{ \theta}_2^2). +\end{align} +For $m_i\approx 0$ and setting $u = \frac{F}{M}$, this simplifies to +\begin{align} +l_1 \ddot \theta_1 &= g\sin(\theta_1) + \cos(\theta_1) u\\ +l_2 \ddot \theta_2 &= g\sin(\theta_2) + \cos(\theta_2) u\\ +\ddot x &= u +\end{align} + +\end{document} \ No newline at end of file