MCHE513 - Example 4p2.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h1 style=\"text-align:center\">Example 4.2</h1>\n",
    "<h3 style=\"text-align:center\">MCHE 513: Intermediate Dynamics &mdash; Fall 2015</h3> \n",
    "<p style=\"text-align:center\">Dr. Joshua Vaughan <br>\n",
    "<a href=\"mailto:joshua.vaughan@louisiana.edu\">joshua.vaughan@louisiana.edu</a><br>\n",
    "<a href=\"http://www.ucs.louisiana.edu/~jev9637/\">http://www.ucs.louisiana.edu/~jev9637/</a></p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy\n",
    "sympy.init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "beta, gamma, phi = sympy.symbols('beta gamma phi')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define the precession rotation matrix\n",
    "R_beta = sympy.Matrix([[sympy.cos(beta), 0, -sympy.sin(beta)],\n",
    "                       [0, 1, 0],\n",
    "                       [sympy.sin(beta), 0, sympy.cos(beta)]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define the nutation rotation matrix\n",
    "R_gamma = sympy.Matrix([[1, 0, 0],\n",
    "                        [0, sympy.cos(gamma), sympy.sin(gamma)],\n",
    "                        [0, -sympy.sin(gamma), sympy.cos(gamma)]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Define the sping rotation matrix\n",
    "R_phi = sympy.Matrix([[sympy.cos(phi), 0, -sympy.sin(phi)],\n",
    "                      [0, 1, 0],\n",
    "                      [sympy.sin(phi), 0, sympy.cos(phi)]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left[\\begin{matrix}\\sin{\\left (\\gamma \\right )} \\sin{\\left (\\phi \\right )} \\frac{d}{d t} \\beta\\\\\\cos{\\left (\\gamma \\right )} \\frac{d}{d t} \\beta\\\\- \\sin{\\left (\\gamma \\right )} \\cos{\\left (\\phi \\right )} \\frac{d}{d t} \\beta\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡              d     ⎤\n",
       "⎢sin(γ)⋅sin(φ)⋅──(β) ⎥\n",
       "⎢              dt    ⎥\n",
       "⎢                    ⎥\n",
       "⎢           d        ⎥\n",
       "⎢    cos(γ)⋅──(β)    ⎥\n",
       "⎢           dt       ⎥\n",
       "⎢                    ⎥\n",
       "⎢               d    ⎥\n",
       "⎢-sin(γ)⋅cos(φ)⋅──(β)⎥\n",
       "⎣               dt   ⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Write the precession term in XYZ, then use the rotation matrices to transform to xyz\n",
    "precession_vel = sympy.Matrix([[0], [sympy.Derivative(beta, sympy.Symbol('t'))], [0]])\n",
    "\n",
    "precession_term = R_phi * R_gamma * R_phi * precession_vel\n",
    "\n",
    "# Print the answer - not using the print() function causes it to default to \"pretty\" printing\n",
    "precession_term"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left[\\begin{matrix}\\cos{\\left (\\phi \\right )} \\frac{d}{d t} \\gamma\\\\0\\\\\\sin{\\left (\\phi \\right )} \\frac{d}{d t} \\gamma\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡       d    ⎤\n",
       "⎢cos(φ)⋅──(γ)⎥\n",
       "⎢       dt   ⎥\n",
       "⎢            ⎥\n",
       "⎢     0      ⎥\n",
       "⎢            ⎥\n",
       "⎢       d    ⎥\n",
       "⎢sin(φ)⋅──(γ)⎥\n",
       "⎣       dt   ⎦"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Write the precession term in x'y'z', then use the rotation matrices to transform to xyz\n",
    "nutation_vel = sympy.Matrix([[sympy.Derivative(gamma, sympy.Symbol('t'))], [0], [0]])\n",
    "\n",
    "nutation_term = R_phi * R_gamma * nutation_vel\n",
    "\n",
    "# Print the answer - not using the print() function causes it to default to \"pretty\" printing\n",
    "nutation_term"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# The spin component of the angular velocity is already written in xyz\n",
    "# No transformation is needed\n",
    "spin_vel = sympy.Matrix([[0], [sympy.Derivative(phi, sympy.Symbol('t'))], [0]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAS8AAABOCAMAAABG32ChAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nWMwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\sin{\\left (\\gamma \\right )} \\sin{\\left (\\phi \\right )} \\frac{d}{d t} \\beta + \\cos{\\left (\\phi \\right )} \\frac{d}{d t} \\gamma\\\\\\cos{\\left (\\gamma \\right )} \\frac{d}{d t} \\beta + \\frac{d}{d t} \\phi\\\\- \\sin{\\left (\\gamma \\right )} \\cos{\\left (\\phi \\right )} \\frac{d}{d t} \\beta + \\sin{\\left (\\phi \\right )} \\frac{d}{d t} \\gamma\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡               d              d     ⎤\n",
       "⎢ sin(γ)⋅sin(φ)⋅──(β) + cos(φ)⋅──(γ) ⎥\n",
       "⎢               dt             dt    ⎥\n",
       "⎢                                    ⎥\n",
       "⎢               d       d            ⎥\n",
       "⎢        cos(γ)⋅──(β) + ──(φ)        ⎥\n",
       "⎢               dt      dt           ⎥\n",
       "⎢                                    ⎥\n",
       "⎢                d              d    ⎥\n",
       "⎢- sin(γ)⋅cos(φ)⋅──(β) + sin(φ)⋅──(γ)⎥\n",
       "⎣                dt             dt   ⎦"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The total angular velocity is just the sum of the precsesion, nutation, and spin terms\n",
    "ang_vel_xyz = precession_term + nutation_term + spin_vel\n",
    "\n",
    "# Print the answer - not using the print() function causes it to default to \"pretty\" printing\n",
    "ang_vel_xyz"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can subtitute the values for the angles using the ```.subs()``` method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMEAAABLCAMAAAD+txMQAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nPAWERAlRC40AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}-21.4891498386867\\\\900.194027703747\\\\7.71455780421365\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-21.4891498386867⎤\n",
       "⎢                 ⎥\n",
       "⎢900.194027703747 ⎥\n",
       "⎢                 ⎥\n",
       "⎣7.71455780421365 ⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ang_vel_xyz.subs([(sympy.Derivative(beta, sympy.Symbol('t')), 63.54),\n",
    "                  (sympy.Derivative(gamma, sympy.Symbol('t')), 19.416),\n",
    "                  (sympy.Derivative(phi, sympy.Symbol('t')), 837.8), \n",
    "                  (beta, 0.2939), \n",
    "                  (gamma,0.19021), \n",
    "                  (phi, 3.351)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is currently written in the $xyz$ frame. Using the rotation matrices developed, we can easily transform it into the $XYZ$ frame."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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+cTns0TB7tn3A0EPRGJuyABjtvHu9dqf4fz5eXl/5tg0zfIzeiZWBIThKwwVo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\nRK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\left(\\cos{\\left (\\gamma \\right )} \\frac{d}{d t} \\beta + \\frac{d}{d t} \\phi\\right) \\sin{\\left (\\beta \\right )} \\sin{\\left (\\gamma \\right )} + \\left(- \\sin{\\left (\\beta \\right )} \\sin{\\left (\\phi \\right )} \\cos{\\left (\\gamma \\right )} + \\cos{\\left (\\beta \\right )} \\cos{\\left (\\phi \\right )}\\right) \\left(\\sin{\\left (\\gamma \\right )} \\sin{\\left (\\phi \\right )} \\frac{d}{d t} \\beta + \\cos{\\left (\\phi \\right )} \\frac{d}{d t} \\gamma\\right) + \\left(\\sin{\\left (\\beta \\right )} \\cos{\\left (\\gamma \\right )} \\cos{\\left (\\phi \\right )} + \\sin{\\left (\\phi \\right )} \\cos{\\left (\\beta \\right )}\\right) \\left(- \\sin{\\left (\\gamma \\right )} \\cos{\\left (\\phi \\right )} \\frac{d}{d t} \\beta + \\sin{\\left (\\phi \\right )} \\frac{d}{d t} \\gamma\\right)\\\\\\left(\\cos{\\left (\\gamma \\right )} \\frac{d}{d t} \\beta + \\frac{d}{d t} \\phi\\right) \\cos{\\left (\\gamma \\right )} + \\left(\\sin{\\left (\\gamma \\right )} \\sin{\\left (\\phi \\right )} \\frac{d}{d t} \\beta + \\cos{\\left (\\phi \\right )} \\frac{d}{d t} \\gamma\\right) \\sin{\\left (\\gamma \\right )} \\sin{\\left (\\phi \\right )} - \\left(- \\sin{\\left (\\gamma \\right )} \\cos{\\left (\\phi \\right )} \\frac{d}{d t} \\beta + \\sin{\\left (\\phi \\right )} \\frac{d}{d t} \\gamma\\right) \\sin{\\left (\\gamma \\right )} \\cos{\\left (\\phi \\right )}\\\\\\left(- \\sin{\\left (\\beta \\right )} \\sin{\\left (\\phi \\right )} + \\cos{\\left (\\beta \\right )} \\cos{\\left (\\gamma \\right )} \\cos{\\left (\\phi \\right )}\\right) \\left(- \\sin{\\left (\\gamma \\right )} \\cos{\\left (\\phi \\right )} \\frac{d}{d t} \\beta + \\sin{\\left (\\phi \\right )} \\frac{d}{d t} \\gamma\\right) + \\left(- \\sin{\\left (\\beta \\right )} \\cos{\\left (\\phi \\right )} - \\sin{\\left (\\phi \\right )} \\cos{\\left (\\beta \\right )} \\cos{\\left (\\gamma \\right )}\\right) \\left(\\sin{\\left (\\gamma \\right )} \\sin{\\left (\\phi \\right )} \\frac{d}{d t} \\beta + \\cos{\\left (\\phi \\right )} \\frac{d}{d t} \\gamma\\right) + \\left(\\cos{\\left (\\gamma \\right )} \\frac{d}{d t} \\beta + \\frac{d}{d t} \\phi\\right) \\sin{\\left (\\gamma \\right )} \\cos{\\left (\\beta \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡⎛       d       d    ⎞                                                       \n",
       "⎢⎜cos(γ)⋅──(β) + ──(φ)⎟⋅sin(β)⋅sin(γ) + (-sin(β)⋅sin(φ)⋅cos(γ) + cos(β)⋅cos(φ)\n",
       "⎢⎝       dt      dt   ⎠                                                       \n",
       "⎢                                                                             \n",
       "⎢                             ⎛       d       d    ⎞          ⎛              d\n",
       "⎢                             ⎜cos(γ)⋅──(β) + ──(φ)⎟⋅cos(γ) + ⎜sin(γ)⋅sin(φ)⋅─\n",
       "⎢                             ⎝       dt      dt   ⎠          ⎝              d\n",
       "⎢                                                                             \n",
       "⎢                                        ⎛                d              d    \n",
       "⎢(-sin(β)⋅sin(φ) + cos(β)⋅cos(γ)⋅cos(φ))⋅⎜- sin(γ)⋅cos(φ)⋅──(β) + sin(φ)⋅──(γ)\n",
       "⎣                                        ⎝                dt             dt   \n",
       "\n",
       "  ⎛              d              d    ⎞                                        \n",
       ")⋅⎜sin(γ)⋅sin(φ)⋅──(β) + cos(φ)⋅──(γ)⎟ + (sin(β)⋅cos(γ)⋅cos(φ) + sin(φ)⋅cos(β)\n",
       "  ⎝              dt             dt   ⎠                                        \n",
       "                                                                              \n",
       "              d    ⎞                 ⎛                d              d    ⎞   \n",
       "─(β) + cos(φ)⋅──(γ)⎟⋅sin(γ)⋅sin(φ) - ⎜- sin(γ)⋅cos(φ)⋅──(β) + sin(φ)⋅──(γ)⎟⋅si\n",
       "t             dt   ⎠                 ⎝                dt             dt   ⎠   \n",
       "                                                                              \n",
       "⎞                                           ⎛              d              d   \n",
       "⎟ + (-sin(β)⋅cos(φ) - sin(φ)⋅cos(β)⋅cos(γ))⋅⎜sin(γ)⋅sin(φ)⋅──(β) + cos(φ)⋅──(γ\n",
       "⎠                                           ⎝              dt             dt  \n",
       "\n",
       "  ⎛                d              d    ⎞ ⎤\n",
       ")⋅⎜- sin(γ)⋅cos(φ)⋅──(β) + sin(φ)⋅──(γ)⎟ ⎥\n",
       "  ⎝                dt             dt   ⎠ ⎥\n",
       "                                         ⎥\n",
       "                                         ⎥\n",
       "n(γ)⋅cos(φ)                              ⎥\n",
       "                                         ⎥\n",
       "                                         ⎥\n",
       " ⎞   ⎛       d       d    ⎞              ⎥\n",
       ")⎟ + ⎜cos(γ)⋅──(β) + ──(φ)⎟⋅sin(γ)⋅cos(β)⎥\n",
       " ⎠   ⎝       dt      dt   ⎠              ⎦"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R = R_phi * R_gamma * R_beta\n",
    "\n",
    "ang_vel_XYZ = R.T * ang_vel_xyz\n",
    "\n",
    "# Print the answer - not using the print() function causes it to default to \"pretty\" printing\n",
    "ang_vel_XYZ"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The ```.subs()``` method can also be used here to get a numerical answer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left[\\begin{matrix}64.4695579456653\\\\886.229902584183\\\\145.982263786794\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡64.4695579456653⎤\n",
       "⎢                ⎥\n",
       "⎢886.229902584183⎥\n",
       "⎢                ⎥\n",
       "⎣145.982263786794⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ang_vel_XYZ.subs([(sympy.Derivative(beta, sympy.Symbol('t')), 63.54),\n",
    "                  (sympy.Derivative(gamma, sympy.Symbol('t')), 19.416),\n",
    "                  (sympy.Derivative(phi, sympy.Symbol('t')), 837.8), \n",
    "                  (beta, 0.2939), \n",
    "                  (gamma,0.19021), \n",
    "                  (phi, 3.351)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A similar procedure can be used to find the angular acceleration."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr style=\"border: 0px;\n",
    "        height: 1px;\n",
    "        text-align: center;\n",
    "        background: #333;\n",
    "        background-image: -webkit-linear-gradient(left, #ccc, #333, #ccc); \n",
    "        background-image:    -moz-linear-gradient(left, #ccc, #333, #ccc); \n",
    "        background-image:     -ms-linear-gradient(left, #ccc, #333, #ccc); \n",
    "        background-image:      -o-linear-gradient(left, #ccc, #333, #ccc);\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Licenses\n",
    "Code is licensed under a 3-clause BSD style license. See the licenses/LICENSE.md file.\n",
    "\n",
    "Other content is provided under a [Creative Commons Attribution-NonCommercial 4.0 International License](http://creativecommons.org/licenses/by-nc/4.0/), CC-BY-NC 4.0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<script>\n",
       "    MathJax.Hub.Config({\n",
       "                        TeX: {\n",
       "                           extensions: [\"AMSmath.js\"]\n",
       "                           },\n",
       "                tex2jax: {\n",
       "                    inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
       "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
       "                },\n",
       "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
       "                \"HTML-CSS\": {\n",
       "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
       "                }\n",
       "        });\n",
       "</script>\n",
       "\n",
       "<style>\n",
       "    @font-face {\n",
       "        font-family: \"Computer Modern\";\n",
       "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf');\n",
       "    }\n",
       "    @font-face {\n",
       "        font-family: \"Computer Modern\";\n",
       "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunsx.otf');\n",
       "        font-weight: bold;\n",
       "    }\n",
       "    @font-face {\n",
       "        font-family: \"Computer Modern\";\n",
       "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunsi.otf');\n",
       "        font-style: oblique;\n",
       "    }\n",
       "    @font-face {\n",
       "        font-family: \"Computer Modern\";\n",
       "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunso.otf');\n",
       "        font-weight: bold;\n",
       "        font-style: oblique;\n",
       "    }\n",
       "\n",
       "    div.cell {\n",
       "        max-width: 1100px;\n",
       "    }\n",
       "    \n",
       "    h1 {\n",
       "        font-family: Computer Modern;\n",
       "    }\n",
       "    \n",
       "    h4 {\n",
       "        margin-top: 12px;\n",
       "        margin-bottom: 3px;\n",
       "    }\n",
       "\n",
       "    div.text_cell_render {\n",
       "        font-family: Computer Modern, \"Helvetica Neue\", Arial, Helvetica, Geneva, sans-serif;\n",
       "        line-height: 145%;\n",
       "        font-size: 130%;\n",
       "        width: 100%;\n",
       "        max-width: 1100px;\n",
       "    }\n",
       "    \n",
       "    .CodeMirror {\n",
       "        font-family: \"Source Code Pro\", source-code-pro, Consolas, monospace;\n",
       "    }\n",
       "    \n",
       "    .warning {\n",
       "        color: rgb( 240, 20, 20 )\n",
       "    }  \n",
       "  \n",
       "   \n",
       "    hr.style-end {\n",
       "        border: 0px !important;\n",
       "        height: 1px !important;\n",
       "        text-align: center !important;\n",
       "        background: #333 !important;\n",
       "        background-image: -webkit-linear-gradient(left, #ccc, #333, #ccc) !important; \n",
       "        background-image:    -moz-linear-gradient(left, #ccc, #333, #ccc) !important; \n",
       "        background-image:     -ms-linear-gradient(left, #ccc, #333, #ccc) !important; \n",
       "        background-image:      -o-linear-gradient(left, #ccc, #333, #ccc) !important; \n",
       "    }\n",
       "\n",
       "    hr.style-end:after {\n",
       "        content: &#x269C !important;\n",
       "        left: 50% !important;\n",
       "        position: absolute !important;\n",
       "        /* Controls the whitespace around the symbol */\n",
       "        padding: 0px !important;\n",
       "        background: #fff !important;\n",
       "    }\n",
       "    \n",
       "/*  Center figures, etc\n",
       "    .ui-wrapper {\n",
       "        margin-left: auto !important;\n",
       "        margin-right: auto !important;\n",
       "    }\n",
       "*/\n",
       "    \n",
       "</style>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# This cell will just improve the styling of the notebook\n",
    "# You can ignore it, if you are okay with the default sytling\n",
    "from IPython.core.display import HTML\n",
    "import urllib.request\n",
    "response = urllib.request.urlopen(\"https://cl.ly/1B1y452Z1d35\")\n",
    "HTML(response.read().decode(\"utf-8\"))"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.2"
  }
 },
 "nbformat": 4,
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}