SymPy_Multiple_Eqn_solution.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Solving a variant of problem 6.3, Gray, Costanzo, and Plesha (Gray, G.L., Costanzo, F., and Plesha, M.E., *Engineering Mechanics: Dynamics*, McGraw Hill, New York, 2010.), where kinetics are included ($I_1, I_2, I_3$) and a moment $M$ is applied to the rightmost gear.\n",
    "\n",
    "The problem statement is ``A Moment $M$ is applied to the rightmost gear (gear 3). Find the acceleration of the gear as a resulting moment presuming inertias $I_1, I_2, I_3$.''\n",
    "\n",
    "For my class: this is Wright State  University, ME 3210, Quiz 2, Spring 2016\n",
    "\n",
    "This notebook uses Python 3.5 in iPython (now called Jupyter).  \n",
    "\n",
    "Image awaiting copyright approval."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "#<img src=\"Gears.png\">"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# import symbolic capability to Python\n",
    "from sympy import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# print things all pretty\n",
    "from sympy.abc import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# Need to define variables as symbolic for sympy to use them. \n",
    "r_A, r_B, r_C, r_D, F_AB, F_DC, M= symbols(\"r_A r_B r_C r_D F_AB F_DC M\", real = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "I_1, I_2, I_3 = symbols(\"I_1 I_2 I_3\", real = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "theta_1, theta_2, theta_3 = symbols(\"theta_1 theta_2 theta_3\", real = True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The Free Body Diagrams, and kinematic relationships. \n",
    "<img src=\"images/Quiz2.png\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Defining equations. In SymPy, and equation is an expression that is equal to zero. All terms must be brought to the same side in defining it. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALcAAAAvBAMAAABTdUaCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$F_{DC} r_{D} - I_{3} \\frac{d^{2}}{d t^{2}}  \\theta_{3}{\\left (t \\right )}$$"
      ],
      "text/plain": [
       "                2       \n",
       "               d        \n",
       "F_DC⋅r_D - I₃⋅───(θ₃(t))\n",
       "                2       \n",
       "              dt        "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "EA = F_DC *r_D  - I_3 * diff(theta_3(t),t,t)\n",
    "EA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQ8AAAAvBAMAAAAP2lKPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
      "text/latex": [
       "$$F_{AB} r_{B} - F_{DC} r_{C} - I_{2} \\frac{d^{2}}{d t^{2}}  \\theta_{2}{\\left (t \\right )}$$"
      ],
      "text/plain": [
       "                           2       \n",
       "                          d        \n",
       "F_AB⋅r_B - F_DC⋅r_C - I₂⋅───(θ₂(t))\n",
       "                           2       \n",
       "                         dt        "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "EB = r_B* F_AB - r_C* F_DC - I_2 * diff(theta_2(t),t,t)\n",
    "EB"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPQAAAAvBAMAAAAm+PohAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$- F_{AB} r_{A} - I_{1} \\frac{d^{2}}{d t^{2}}  \\theta_{1}{\\left (t \\right )} + M$$"
      ],
      "text/plain": [
       "                 2           \n",
       "                d            \n",
       "-F_AB⋅r_A - I₁⋅───(θ₁(t)) + M\n",
       "                 2           \n",
       "               dt            "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "EC = r_A *(-F_AB) + M - I_1 * diff(theta_1(t),t,t)\n",
    "EC"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOwAAAAvBAMAAAASgTs9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ncGFAUmrscGqtDEhKzc3g/wEHB2rlTio5hQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- r_{C} \\frac{d^{2}}{d t^{2}}  \\theta_{2}{\\left (t \\right )} + r_{D} \\frac{d^{2}}{d t^{2}}  \\theta_{3}{\\left (t \\right )}$$"
      ],
      "text/plain": [
       "        2                2       \n",
       "       d                d        \n",
       "- r_C⋅───(θ₂(t)) + r_D⋅───(θ₃(t))\n",
       "        2                2       \n",
       "      dt               dt        "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E1 = diff(theta_3(t),t,t)* r_D - diff(theta_2(t),t,t) *r_C\n",
    "E1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOsAAAAvBAMAAADwXSBEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nuYbhq0YQFmKaaxi+aiRhW6D/A8F3arDoJc6hAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- r_{A} \\frac{d^{2}}{d t^{2}}  \\theta_{1}{\\left (t \\right )} + r_{B} \\frac{d^{2}}{d t^{2}}  \\theta_{2}{\\left (t \\right )}$$"
      ],
      "text/plain": [
       "        2                2       \n",
       "       d                d        \n",
       "- r_A⋅───(θ₁(t)) + r_B⋅───(θ₂(t))\n",
       "        2                2       \n",
       "      dt               dt        "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E2 = r_B * diff(theta_2(t),t,t) - diff(theta_1(t),t,t) *r_A\n",
    "E2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAyBAMAAADSNPrMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nX+NUEFOtx3jJq/91+AMIcphHJPRSzQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ \\frac{F_{DC} r_{D}}{I_{3}}\\right ]$$"
      ],
      "text/plain": [
       "⎡F_DC⋅r_D⎤\n",
       "⎢────────⎥\n",
       "⎣   I₃   ⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(EA,diff(theta_3(t),t,t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPAAAAA1BAMAAACNXnumAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nz+oT0Z5kjv8B03SEMqmS1QoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{1}{r_{B}} \\left(F_{DC} r_{C} + I_{2} \\frac{d^{2}}{d t^{2}}  \\theta_{2}{\\left (t \\right )}\\right)$$"
      ],
      "text/plain": [
       "                2       \n",
       "               d        \n",
       "F_DC⋅r_C + I₂⋅───(θ₂(t))\n",
       "                2       \n",
       "              dt        \n",
       "────────────────────────\n",
       "          r_B           "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(EB,F_AB)[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "I'm going to do substitutions in stages, eliminating one variable at a time for clarity."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAagAAAA1BAMAAADmE2LaAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
      "text/latex": [
       "$$- I_{1} \\frac{d^{2}}{d t^{2}}  \\theta_{1}{\\left (t \\right )} + M - \\frac{r_{A}}{r_{B}} \\left(F_{DC} r_{C} + I_{2} \\frac{d^{2}}{d t^{2}}  \\theta_{2}{\\left (t \\right )}\\right)$$"
      ],
      "text/plain": [
       "                          ⎛                2       ⎞\n",
       "                          ⎜               d        ⎟\n",
       "                      r_A⋅⎜F_DC⋅r_C + I₂⋅───(θ₂(t))⎟\n",
       "       2                  ⎜                2       ⎟\n",
       "      d                   ⎝              dt        ⎠\n",
       "- I₁⋅───(θ₁(t)) + M - ──────────────────────────────\n",
       "       2                           r_B              \n",
       "     dt                                             "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "step1 = EC.subs(F_AB, solve(EB,F_AB)[0])\n",
    "step1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAeIAAAA1BAMAAABvi/WzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\niNFmfL00+Bzuf03qCjuGNH9DAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- I_{1} \\frac{d^{2}}{d t^{2}}  \\theta_{1}{\\left (t \\right )} + M - \\frac{r_{A}}{r_{B}} \\left(I_{2} \\frac{d^{2}}{d t^{2}}  \\theta_{2}{\\left (t \\right )} + \\frac{I_{3} r_{C}}{r_{D}} \\frac{d^{2}}{d t^{2}}  \\theta_{3}{\\left (t \\right )}\\right)$$"
      ],
      "text/plain": [
       "                          ⎛                         2       ⎞\n",
       "                          ⎜                        d        ⎟\n",
       "                          ⎜                I₃⋅r_C⋅───(θ₃(t))⎟\n",
       "                          ⎜     2                   2       ⎟\n",
       "                          ⎜    d                  dt        ⎟\n",
       "                      r_A⋅⎜I₂⋅───(θ₂(t)) + ─────────────────⎟\n",
       "       2                  ⎜     2                 r_D       ⎟\n",
       "      d                   ⎝   dt                            ⎠\n",
       "- I₁⋅───(θ₁(t)) + M - ───────────────────────────────────────\n",
       "       2                                r_B                  \n",
       "     dt                                                      "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "step2 = step1.subs(F_DC,solve(EA,F_DC)[0])\n",
    "step2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAf4AAAA1BAMAAABSGZTVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMnarmSJEibtU\n72aoTn1kAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAKSUlEQVRoBcVafWxcRxGf57vzffjufCKCig/j\nq0sV/giKadUmUGhPlRCID/loKYEilEtREqSo8rWoNiSCPIHU/hE1tgpCKBXytWkSQEK4QWqgKPUh\nkKhEwU74ShuJHOKPRoLWlyZN48TCzMx+vH1fd8/vru3K93ZmdvY3s+/t7tt5Y4CIJbfvgYiavasd\nO1HqHYQQ+uj0T+C+/vjUHSVnD5a7a0XR6KPTW2HCjmKyDzoJO3mlDzAI0UenPwIzs/1xyo3yUzdL\nXGI2s+KXxpF0ctqqrRNxsU+L0mU2V3exkslfCJLGkXVw+jEv3hfXNtpemeCTJ9HNW4LbepMe9XX/\nzD9wBlR94jBBfKeLZS/ma16B5oebEPyktEY8In/K328V4JhfGiqJ7/SzHszsVY/AYedK8LLD9Y8a\nmfVhZS9AMXBR+DRZ0IPTI003ZC581zkM2Ua24lbvB3fCD5Ibh0PwA788RNKD0/klN+ZA280b3Ea4\n84U/lgxBf8igfW6gntz0wtOR8Xtx+qNuK8Nl5rNH3/2zqtmS2fuFVTi3tmbK4tA+YEjUvTiPHJlo\nFtbWwp+Et0MvTi/aLrQ5HPWdeGgc2jSwBI9fnt4jm39sD/XlfSSAAYa02WVNSUeyZ2ChJOloFTqd\n27ZDOg3/2f3El0S/KE57bv8vKvDJrwF8NlHDpV5cgtwSQ1n/hHQ5mjOdtQQwwPKs0rtNEaoeacGD\nio5Wo9PfgX3S6e9W4b3rcLrofqyb0eLvAEqj5F+6BnJ1pi9AohHNmc5aAhis/S2pl/RtuPtL4FmU\nnSEB0Om9sCycpmNjcZ57RHLaet2Ffg05HD8skhSPIPKpF+aB7whJeysMDLkNdQmTFs/KAD0Llu+e\nGM0BJDmNz5+dpseUrLJSNKfHWFdesnQ3aPw8A0crcFeJW0Yb4o5ItR4qMbU/MTQvMQo1D5h1FfJt\nj6wzy05bfxVOb0eHrfU4vd020HOXkKHxv0TCuXu+UqMaYLQJx/M2kz1eGBhq6XGJM9HyAFqrMFR+\n3iPsyLLTd1WF07c5qtGcnmk5PSDRRgbHL4KvJwHuFrdyoFq8NWfoxSYFcLGZIkNUXLefJTfCv2sN\npiJe2Gmc/4z9htMpmtPmBHx482W0jOPPjhMMnsxmmkSAtfOph7/MlLi8vyWZDBPnJde9EsAbpvep\nbfekr8+77v3UzllH2tWUcBo9ZWzaOg+K3h6nrVcVqGiXTg/xUFUT1TT/uWAU4jkeyIbs03+Q1AGu\nB23J6qo4iZ3x6Z5tapFBfBOStNConBVVaAjX3RQB4OqX83gfcg0S+crj76sLWWaea+l0ru3V/JMU\nJC9ARixXr8bnSsMlIfsg2t4KVllwxnXwKjLJyVlDpMl8C+B/grPUfA0L4SKYIqSt8FXhUaIF77EF\ntvuaPAVyzg1gu+N0ccWtB4/csVNIDr8xvSfQf7gdMBimQlEaHRGOMId0VRJw8O/UPKlYV70wDwcu\ntliUlK/f0BAugikC+vS9X2c8gKN7Py8pd4U3hqckcBTrOK08cGt35IpLMNxijYSNp6QywAhz5vjP\nH0fRv+Tslq0BVV76FBbCRTEVABsgekXPua3Y6jitZ2BAnxARnpknmtz2PbzSiWNAzhPn+de/jW+R\nH7kPV9zFfSnKfSAshItiyo0Yxm2BpLjXmXFUMZy+HNYjVD43C9vFgLcB/HDhGQw/6kLZGP92gJQa\nXSgSpK6ItuFysE4UU8E9PdLsCqTESk+hr6bT1zya3dnJXbvOibzCc6hMJ9piWfTS48+0Fivw28G2\nEIdfc3L8FMIFZVcimAoHN1sGL++6vw1Hpkow1EK54fRpUy0SfRPAa8B5hY+h/oP4y4+Ljnr8RXum\naTUL893wMEbhgiFcYKIigqluJkR7ogzD5XQrPQ+JJkoMp/8C77iZygcA1oJKG4Zd4jZcpaXEeYUX\nEYpOHEl60LmxsRtOj43VkIQ0DFdTMFMl2rqB4G+qIGkiURdQM2SzTFSYCvQJJIIpl3OK8Y4Fhhu4\nZ6UbxTYUZtGudhrg4+THegqe0NNLwHkFHD8Hazx+xNDPPw2F+vOwXOmGq8aPizAoURHFVDcTon20\nCbgjAz5/Gr/p9PrHv4IPF7Ewr4DzP9uGkjwvG+M/CAPXt2Cyq29y/nMIR4Cegn52NeXpEsJi+E4v\no5EKz3/T6ZtDegSJxR71IhymxlsAcP/LlfMV//53HtJ/04fbICQpk/sfh3Du7EpkUx3QRROnkguz\nuXF8I92In95abqdPdwVwFMQe9VDmDIronXcMZ1TtnXhpCB09/8chdQkiHK1QiwqHcATolMimnC7B\nlEglp6uP2tiO85/MmE53PaMZsCKZet22Espext/38ZDzXAvPPxXgosZ//8Vqvpo/eXGHEIdfxelb\nhHAE6JSoppweIZRIJVsPPUDtmVVeq6bTKgIJ6S7FlJJz7VGcDKHzLxU6B1JR4xdchGt+RSvp7Erf\nTHHK0kwlj1QyGHfR+ZeKcBq/N0UqqxQz8aYn1DkZUpRTVsU/lh0Jy1EyloiTXembKYzSNqBHOsWS\nsFOnxMQlD4TThgeOW34KU3IMg5ueKOc4GfIrZqyykMW4OrNPACJE/0zNlfidkqhKx5JHjto692CV\nWaoiMGbCL7h7Uszk3qMAxPePITu8Y5cWPOB5S/9M4WuKDnq4S5slM8+cdFqdwE2NAHqgzjGTe4/C\n/aRFuucDOkQU7ffr9c/URj7oqUWqLbm+f6kTmG49tOOQWs5aRik5ipn0HqVb1kn4wJfpVWKWWKb8\naUV8NJiyzO65trNLKtkXozSONz9EDomlza6JlNwipgZ7zQAzeHZh95RClx8S9B2IZ8qXr0Q8zv7l\n29AllTyjNgfpQrJ5K6feze9iIy1aR/in9yjt7/oICf4KwM9lx0LNjRDPlM5XJs8oOJGyxD2rSyp5\nu616iNqS2afr8E2pikjJUczUY5HgU857NN12Q8YzJdKKgzWAXQpOZP8SDcWH1b/0NuCUofLEFqeB\nU3LyvjjSOJQAx5co7ctU8LuMq8Q0hWsTT9E4le9WaLiuMebBv87Fk/9EZXrNUQ7xN1RxESk5jpmU\nKG7N4NYlSF2vEIzbjKK4pvh2jlacdQWjDfrIMweWMhRcp+QHGKe10CB6EL6hRSIlx4GelsUkGDz7\n+j3P6P7uF0BcUy8R3hz+7lPAGPBjyvJJ+L0SBNfe/QdTXjxl/gv4AVMVTslxzKQkcWsGz5Uh9WGF\nUKgriut4pnS+UiRuCUlk/6YsXBOdyqLtbcWYDufhlumFkm7hlBzHTFoUk2DwoTrFYbLow7ng45nK\njlPvZ/GovyRhZcrywKuKD6nxiBRU0jZMVIIa+iHDsCR/RQOFeKDboxN4TwP+mbBj/6B/vqQOf8Zv\nla2OPXtonCjBow3df6aiyd4IylduWidEyP0augaZhdvXiRVVPXfH9NSvHeU8z12Hj01hvnK3s2aj\nwZyIpvamauFp6O0qxfm3y7Jhd7BqMG8t+dhbay7E2rdC5G+62OLvk/8Hunhf8Nevim0AAAAASUVO\nRK5CYII=\n",
      "text/latex": [
       "$$- \\frac{I_{1} r_{B}}{r_{A}} \\frac{d^{2}}{d t^{2}}  \\theta_{2}{\\left (t \\right )} + M - \\frac{r_{A}}{r_{B}} \\left(I_{2} \\frac{d^{2}}{d t^{2}}  \\theta_{2}{\\left (t \\right )} + \\frac{I_{3} r_{C}}{r_{D}} \\frac{d^{2}}{d t^{2}}  \\theta_{3}{\\left (t \\right )}\\right)$$"
      ],
      "text/plain": [
       "                              ⎛                         2       ⎞\n",
       "                              ⎜                        d        ⎟\n",
       "                              ⎜                I₃⋅r_C⋅───(θ₃(t))⎟\n",
       "           2                  ⎜     2                   2       ⎟\n",
       "          d                   ⎜    d                  dt        ⎟\n",
       "  I₁⋅r_B⋅───(θ₂(t))       r_A⋅⎜I₂⋅───(θ₂(t)) + ─────────────────⎟\n",
       "           2                  ⎜     2                 r_D       ⎟\n",
       "         dt                   ⎝   dt                            ⎠\n",
       "- ───────────────── + M - ───────────────────────────────────────\n",
       "         r_A                                r_B                  "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "step3 = step2.subs(diff(theta_1(t),t,t),solve(E2,diff(theta_1(t),t,t))[0])\n",
    "step3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$- \\frac{I_{1} r_{B} r_{D}}{r_{A} r_{C}} \\frac{d^{2}}{d t^{2}}  \\theta_{3}{\\left (t \\right )} + M - \\frac{r_{A}}{r_{B}} \\left(\\frac{I_{2} r_{D}}{r_{C}} \\frac{d^{2}}{d t^{2}}  \\theta_{3}{\\left (t \\right )} + \\frac{I_{3} r_{C}}{r_{D}} \\frac{d^{2}}{d t^{2}}  \\theta_{3}{\\left (t \\right )}\\right)$$"
      ],
      "text/plain": [
       "                                  ⎛         2                   2       ⎞\n",
       "                                  ⎜        d                   d        ⎟\n",
       "               2                  ⎜I₂⋅r_D⋅───(θ₃(t))   I₃⋅r_C⋅───(θ₃(t))⎟\n",
       "              d                   ⎜         2                   2       ⎟\n",
       "  I₁⋅r_B⋅r_D⋅───(θ₃(t))           ⎜       dt                  dt        ⎟\n",
       "               2              r_A⋅⎜───────────────── + ─────────────────⎟\n",
       "             dt                   ⎝       r_C                 r_D       ⎠\n",
       "- ───────────────────── + M - ───────────────────────────────────────────\n",
       "         r_A⋅r_C                                  r_B                    "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "step4 = step3.subs(diff(theta_2(t),t,t),solve(E1,diff(theta_2(t),t,t))[0])\n",
    "step4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Finally, the acceleration of the right most gear as a function of the input moment"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQkAAAAyBAMAAAC9jDONAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nu2X9ZXxrMzSII/+VtFfKDPwvzf88lWHXgPia0AAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ \\frac{M r_{A} r_{B} r_{C} r_{D}}{I_{1} r_{B}^{2} r_{D}^{2} + I_{2} r_{A}^{2} r_{D}^{2} + I_{3} r_{A}^{2} r_{C}^{2}}\\right ]$$"
      ],
      "text/plain": [
       "⎡            M⋅r_A⋅r_B⋅r_C⋅r_D             ⎤\n",
       "⎢──────────────────────────────────────────⎥\n",
       "⎢      2    2         2    2         2    2⎥\n",
       "⎣I₁⋅r_B ⋅r_D  + I₂⋅r_A ⋅r_D  + I₃⋅r_A ⋅r_C ⎦"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(step4,diff(theta_3(t),t,t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": []
  }
 ],
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