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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# O P\u00eandulo Simples\n",
      "Consideremos o p\u00eandulo conforme a figura abaixo\n",
      "\n",
      "![Um p\u00eandulo simples](http://www.ime.usp.br/~tonelli/imagens/pendulo.png)\n",
      "\n",
      "Para escrever as equa\u00e7\u00f5es do movimento deste P\u00eandulo escrevemos que:\n",
      "$$ \\begin{gather}\n",
      "\\mathbf{r}(t) = L(\\sin(\\theta) \\mathbf{i} - \\cos(\\theta)\\mathbf{j}) = L\\mathbf{e}_1(t) \\\\\n",
      "\\dot{\\mathbf{r}}(t) =L \\dot{\\mathbf{e}}_1(t) = L\\dot{\\theta}(\\cos{\\theta}\\mathbf{i} + \\sin{\\theta}\\mathbf{j})=L\\dot{\\theta}\\mathbf{e_2}(t) \\\\\n",
      "\\ddot{\\mathbf{r}}(t) = L\\ddot{\\theta}\\mathbf{e}_2(t) + L\\dot{\\theta}\\dot{\\mathbf{e}}_2(t)=-L\\dot{\\theta}^2\\mathbf{e}_1(t) + L\\ddot{\\theta}\\mathbf{e_2}(t)\n",
      "\\end{gather}$$\n",
      "As for\u00e7as que atuam na part\u00edcula s\u00e3o a for\u00e7a peso $-Mg\\mathbf{j}$ e for\u00e7a de rea\u00e7\u00e3o vincular da haste que segura a part\u00edcula $-\\alpha(t)\\mathbf{e}_1(t).$\n",
      "Como $\\mathbf{j}= -\\cos{\\theta}\\mathbf{e}_1 + \\sin{\\theta}\\mathbf{e}_2$, podemos reescrever a for\u00e7a   peso como\n",
      "$$ \\mathbf{P} = Mg\\cos{\\theta}\\mathbf{e}_1 - Mg\\sin{\\theta}\\mathbf{e}_2 $$ e aplicando a *lei de Newton* temos a equa\u00e7\u00e3o do movimento neste caso:\n",
      "$$ \\ddot{\\theta}(t) = -\\frac{g}{L}\\sin(\\theta) $$\n",
      "Esta equa\u00e7\u00e3o pode-se escrever como um sistema de EDO de primeira ordem:\n",
      "$$ \\begin{eqnarray}\n",
      "\\dot{\\theta} &= & \\omega\\\\\n",
      "\\dot{\\omega} & = & -\\frac{g}{L}\\sin(\\theta)\n",
      "\\end{eqnarray}$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## O p\u00eandulo e o problema de controle\n",
      "Os pontos da forma $({n\\pi},0)$ s\u00e3o pontos de equil\u00edbrio do sistema acima. Quando $n$ \u00e9 \u00edmpar os pontos de equil\u00edbrios s\u00e3o int\u00e1veis. Podemos acrescentar um torque ajust\u00e1vel no p\u00eandulo de forma que a equa\u00e7\u00e3o do sistema depende agora desta nova vari\u00e1vel:\n",
      "$$ \\begin{eqnarray}\n",
      "\\dot{\\theta} &= & \\omega\\\\\n",
      "\\dot{\\omega} & = & -\\frac{g}{L}\\sin(\\theta) + u(t)\n",
      "\\end{eqnarray}$$\n",
      "Podemos escolher este torque de modo que os pontos de equil\u00edbrios inst\u00e1veis fiquem est\u00e1veis?\n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Resolvendo a equa\u00e7\u00e3o com scipy"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "from matplotlib import pyplot as plt\n",
      "import numpy as np\n",
      "from scipy.integrate import odeint\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# \\theta = y[0] \\omega =y[1]\n",
      "g=9.8\n",
      "L=10\n",
      "def rhs(y,t):\n",
      "    return [y[1], -g/L* np.sin(y[0])]\n",
      "y0=[0.1, 1.0]\n",
      "t_out = np.arange(0,10,0.1)\n",
      "y_out = odeint(rhs,y0, t_out)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "theta=y_out[:,0]\n",
      "omega=y_out[:,1]\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(t_out,theta)\n",
      "plt.grid()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
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u7wqcU2N7JW8fz2DXh8Pj9ioc3ALsMc5rQ8CDVCdyZ6ITuZ0K3gEyEppp1l4F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\nERERERGRIv1/UxhWUIi4c7cAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f84c83f1e48>"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(t_out,omega)\n",
      "plt.grid()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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FdEueio/XgdWBNYF1gakUn+4uAz4Ijd9ZJHeuNzVeA17rpoXRin63U/heoigE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\nzjnnnHPOZen/AAWNMxmIR8fDAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f84c8384ba8>"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(theta,omega)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 15,
       "text": [
        "[<matplotlib.lines.Line2D at 0x7f84c8395c88>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x7f84c83609b0>"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}