aeromecanica_rotor.ipynb

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  "name": ""
 },
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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Mec\u00e1nica con SymPy - Aeromec\u00e1nica de un rotor"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "* **Helic\u00f3pteros y Aeronaves Diversas I**\n",
      "* **\u00daltima revisi\u00f3n**: 2014-03-03\n",
      "* **Autor**: Juan Luis Cano <jl.cano@alumnos.upm.es>"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Introducci\u00f3n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Cinem\u00e1tica de la pala de un rotor:\n",
      "\n",
      "* **Azimut**: $\\psi$ alrededor de $z_{A}$ en el acoplamiento $A$.\n",
      "* **Batimiento**: $\\beta$ alrededor de $-y_{A1}$ en el acoplamiento $E_b$, situado a distancia $e_b$ de $A$.\n",
      "* **Arrastre**: $\\zeta$ alrededor de $z_{A2}$ en el acoplamiento $E_a$, situado a distancia $e_a - e_b$ de $E_b$.\n",
      "* **Paso**: $\\theta$ alrededor de $x_{A3}$, que coincide con la l\u00ednea $c / 4$ de la pala.\n",
      "\n",
      "Vamos a considerar ese orden para los movimientos aunque no tiene porqu\u00e9 ser el caso siempre."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import init_session\n",
      "init_session(use_latex=True, quiet=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "IPython console for SymPy 0.7.5 (Python 3.4.0-64-bit) (ground types: python)\n"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stderr",
       "text": [
        "WARNING: Hook shutdown_hook is deprecated. Use the atexit module instead.\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Datos iniciales"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Algunas variables:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "M_b = Symbol('M_b')  # Masa de la pala\n",
      "g = Symbol('g')  # Gravedad\n",
      "e = Symbol('e')  # Excentricidad de batimiento"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Sistemas de referencia"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy.physics.mechanics import ReferenceFrame, Point, dynamicsymbols"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Definimos nuestra propia clase para que los versores sean IJK\n",
      "class IJKReferenceFrame(ReferenceFrame):\n",
      "    def __init__(self, name):\n",
      "        super().__init__(name, latexs=['\\mathbf{%s}_{%s}' % (idx, name) for idx in (\"i\", \"j\", \"k\")])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Sistema de referencia \u00e1rbol:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = IJKReferenceFrame(\"A\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Definimos la velocidad de giro del rotor principal $\\Omega$:\n",
      "Omega = Symbol('Omega')\n",
      "Omega"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\Omega$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAA8AAAAPBAMAAADJ+Ih5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIlR2RM2ZMmarid27\nEO8D7p43AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAe0lEQVQIHWNgYGBUVmQAAZYwAbFwEKO/gYHB\nXoGBgf0rkMP6kYGB8xeQwfWNgYH7E5DB8YeBgecHkMH8E4iBogycQB7vayCD+wGQ6BdgYKhXADJY\nDRgY1gBpRsYABoaNjEAjGOYyMCZwATU0LWFgWmQGFPn/g4Hj/2cGALOyGJfGxDbaAAAAAElFTkSu\nQmCC\n",
       "prompt_number": 6,
       "text": [
        "\u03a9"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Vamos introduciendo sistemas de referencia y sus or\u00edgenes:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A1 = IJKReferenceFrame(\"A1\")\n",
      "psi = dynamicsymbols('psi')\n",
      "A1.orient(A, 'Axis', [psi, A.z])\n",
      "A1.dcm(A)  # T_{A1A}"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\cos{\\left (\\psi{\\left (t \\right )} \\right )} & \\sin{\\left (\\psi{\\left (t \\right )} \\right )} & 0\\\\- \\sin{\\left (\\psi{\\left (t \\right )} \\right )} & \\cos{\\left (\\psi{\\left (t \\right )} \\right )} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQAAAABLCAMAAACoRKlPAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
       "prompt_number": 7,
       "text": [
        "\u23a1cos(\u03c8(t))   sin(\u03c8(t))  0\u23a4\n",
        "\u23a2                        \u23a5\n",
        "\u23a2-sin(\u03c8(t))  cos(\u03c8(t))  0\u23a5\n",
        "\u23a2                        \u23a5\n",
        "\u23a3    0           0      1\u23a6"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "T_a, T_k, T_h = dynamicsymbols('T_a, T_k, T_h')\n",
      "T_a"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\operatorname{T_{a}}{\\left (t \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADEAAAAVBAMAAAAdjxsPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA782r3SJ2ZjIQmUS7\niVSMai2aAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABHklEQVQoFWNgIBpwFICUVgAx4ydjeWejL3Cd\nMWAWcwMDA38AQ34BQyNcRpOBgdOVgUGBgWEWA0iGGSbDdoGBge8DA0MuA0M4WIZtAlSKMYCBgQuo\nYSIDQwRYhgMmswWoguUCAwPTA5BSoGlwcIuBYfP5wwkMvAtAQsgyx4H890DMrgAkUGT8gPxFQMzn\nACRQZKSBfF8g5viARYbzN6rM3iqwO4CmcX9gmMDA7QDXk8agDWIDXcCrwNeA7AKONDeQzGWgRw2m\nAYkFQM1r5GVWMDDcmWAGkgH6lPVYAtCnG0A8MPjKcOwBkMESAOEGQiggyfmVwS0BSLMvABJAcA1C\ngciatKYEEH0URDBwKoApFAISczxQQ5GlWMHeKmVgAAArlzzgj5ypEgAAAABJRU5ErkJggg==\n",
       "prompt_number": 8,
       "text": [
        "T\u2090(t)"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "t = symbols('t')\n",
      "omega1, omega2, omega3 = dynamicsymbols('omega1, omega2, omega3')\n",
      "theta1, theta2, theta3 = dynamicsymbols('theta1, theta2, theta3')\n",
      "eqs = [omega1 - theta1.diff(t), omega2 - theta2.diff(t), omega3 - theta3.diff(t)]\n",
      "eqs"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}\\omega_{1}{\\left (t \\right )} - \\frac{d}{d t} \\theta_{1}{\\left (t \\right )}, & \\omega_{2}{\\left (t \\right )} - \\frac{d}{d t} \\theta_{2}{\\left (t \\right )}, & \\omega_{3}{\\left (t \\right )} - \\frac{d}{d t} \\theta_{3}{\\left (t \\right )}\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAdAAAAAaBAMAAAD1dxBVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nc+E0jKIAAAAASUVORK5CYII=\n",
       "prompt_number": 9,
       "text": [
        "\u23a1        d                  d                  d        \u23a4\n",
        "\u23a2\u03c9\u2081(t) - \u2500\u2500(\u03b8\u2081(t)), \u03c9\u2082(t) - \u2500\u2500(\u03b8\u2082(t)), \u03c9\u2083(t) - \u2500\u2500(\u03b8\u2083(t))\u23a5\n",
        "\u23a3        dt                 dt                 dt       \u23a6"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "psi.diff() * A.z"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\dot{\\psi}\\mathbf{k}_{A}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACcAAAAYBAMAAACLofl+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA3asQzTKJImaZdlTv\nRLsxlI82AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+klEQVQYGa2QsWoCQRRFLww7rrNhWVKlzCeM\nXzK9TTal3ZaCEoQUKd0uBEG3SxUyPxAI/oB+QCCWdmo0JN3kzupabCy9xZ33zsx9PAagLr3VJHY1\ncI42AG5cUZskgWB2AsLUYZfJf/DhFExLGLkvnqVEp0CqEsbFVFdQRuuGVVvCflqyOAN6zSTUeId5\n+d2/G84BfZFyzRzG7CeKyYCXn5mEKGCunO8gWzn9Fj1IDWNffzxsxwv6GPd44/I23nh43UzoHyhC\nzja24RgMrVoTRqPls/+lnXQuQ+tpvPWBwHo/6BHBty+VrghjA6Dcl2setVrgbsMLqCOqij9kwEJK\nCjB9QQAAAABJRU5ErkJggg==\n",
       "prompt_number": 10,
       "text": [
        "\u03c8\u0307 \u001b[94m\u001b[1ma_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A1.ang_vel_in(A)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\dot{\\psi}\\mathbf{k}_{A}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACcAAAAYBAMAAACLofl+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA3asQzTKJImaZdlTv\nRLsxlI82AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+klEQVQYGa2QsWoCQRRFLww7rrNhWVKlzCeM\nXzK9TTal3ZaCEoQUKd0uBEG3SxUyPxAI/oB+QCCWdmo0JN3kzupabCy9xZ33zsx9PAagLr3VJHY1\ncI42AG5cUZskgWB2AsLUYZfJf/DhFExLGLkvnqVEp0CqEsbFVFdQRuuGVVvCflqyOAN6zSTUeId5\n+d2/G84BfZFyzRzG7CeKyYCXn5mEKGCunO8gWzn9Fj1IDWNffzxsxwv6GPd44/I23nh43UzoHyhC\nzja24RgMrVoTRqPls/+lnXQuQ+tpvPWBwHo/6BHBty+VrghjA6Dcl2setVrgbsMLqCOqij9kwEJK\nCjB9QQAAAABJRU5ErkJggg==\n",
       "prompt_number": 11,
       "text": [
        "\u03c8\u0307 \u001b[94m\u001b[1ma_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A2 = IJKReferenceFrame(\"A2\")\n",
      "beta = dynamicsymbols('beta')\n",
      "A2.orient(A1, 'Axis', [beta, -A1.y])\n",
      "A2.dcm(A1)  # T_{A2A1}"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\cos{\\left (\\beta{\\left (t \\right )} \\right )} & 0 & \\sin{\\left (\\beta{\\left (t \\right )} \\right )}\\\\0 & 1 & 0\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} & 0 & \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAP0AAABLCAMAAACMeHGmAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 12,
       "text": [
        "\u23a1cos(\u03b2(t))   0  sin(\u03b2(t))\u23a4\n",
        "\u23a2                        \u23a5\n",
        "\u23a2    0       1      0    \u23a5\n",
        "\u23a2                        \u23a5\n",
        "\u23a3-sin(\u03b2(t))  0  cos(\u03b2(t))\u23a6"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A2.ang_vel_in(A1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-  \\dot{\\beta}\\mathbf{j}_{A1}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADYAAAAYBAMAAABDzROoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMqsiZnZUu0Tv\nmYm/b9svAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABB0lEQVQoFWNgQABTBBOdxfgJXYTG/PQyBgYf\nQ6yWcEyQcmCI/4FVroSBewFD/29scmwFDJwGDJwbsMlxMDCwBmCTAIoB5fYkcP//h016S26TCQND\nMla5Brm3RxgYuMByXAkMDF5IBjxgYGiCydVvYPANZmAQMgYBFQaGBgaG/VA5xvsFDAyTEfq4HRgY\naqByfMIXUOT4gMpCoXIeXAtQ5JgYGNiOQeUCeA1Q5LgKGJw3QOTYJ3B8QJHbUtf3EGgsyA/Cd299\nRJF7AJRgYHQAyV1jYPmCKcf/hecnA2cBA8N3BoY5IMVgwNgAovh/+xswrF/AkPW5INc+AiwBJEDe\nA5LKtgIgGgIAiZBCdogCYUAAAAAASUVORK5CYII=\n",
       "prompt_number": 13,
       "text": [
        "- \u03b2\u0307 \u001b[94m\u001b[1ma1_y\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A2.dcm(A)  # T_{A2A}"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\psi{\\left (t \\right )} \\right )} & \\sin{\\left (\\psi{\\left (t \\right )} \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} & \\sin{\\left (\\beta{\\left (t \\right )} \\right )}\\\\- \\sin{\\left (\\psi{\\left (t \\right )} \\right )} & \\cos{\\left (\\psi{\\left (t \\right )} \\right )} & 0\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\psi{\\left (t \\right )} \\right )} & - \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\psi{\\left (t \\right )} \\right )} & \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAgcAAABLCAMAAADTetQ3AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n9T3n943A3XS/+A/1RtgKP/nZ1wAAAABJRU5ErkJggg==\n",
       "prompt_number": 14,
       "text": [
        "\u23a1cos(\u03b2(t))\u22c5cos(\u03c8(t))   sin(\u03c8(t))\u22c5cos(\u03b2(t))   sin(\u03b2(t))\u23a4\n",
        "\u23a2                                                     \u23a5\n",
        "\u23a2     -sin(\u03c8(t))            cos(\u03c8(t))            0    \u23a5\n",
        "\u23a2                                                     \u23a5\n",
        "\u23a3-sin(\u03b2(t))\u22c5cos(\u03c8(t))  -sin(\u03b2(t))\u22c5sin(\u03c8(t))  cos(\u03b2(t))\u23a6"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A2.ang_vel_in(A)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-  \\dot{\\beta}\\mathbf{j}_{A1} + \\dot{\\psi}\\mathbf{k}_{A}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHkAAAAYBAMAAAAsfE+FAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nwmORykLyxOMGqpdHj+0vnqx0C0/UX83Ii7J7OYjGAAAAAElFTkSuQmCC\n",
       "prompt_number": 15,
       "text": [
        "- \u03b2\u0307 \u001b[94m\u001b[1ma1_y\u001b[0;0m\u001b[0;0m + \u03c8\u0307 \u001b[94m\u001b[1ma_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A3 = IJKReferenceFrame(\"A3\")\n",
      "zeta = dynamicsymbols('zeta')\n",
      "A3.orient(A2, 'Axis', [zeta, A2.z])\n",
      "A3.dcm(A2)  # T_{A3A2}"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\zeta\\left(t\\right) \\right )} & 0\\\\- \\sin{\\left (\\zeta\\left(t\\right) \\right )} & \\cos{\\left (\\zeta\\left(t\\right) \\right )} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAP8AAABLCAMAAACIjaGbAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 16,
       "text": [
        "\u23a1cos(\u03b6(t))   sin(\u03b6(t))  0\u23a4\n",
        "\u23a2                        \u23a5\n",
        "\u23a2-sin(\u03b6(t))  cos(\u03b6(t))  0\u23a5\n",
        "\u23a2                        \u23a5\n",
        "\u23a3    0           0      1\u23a6"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A3.dcm(A1)  # T_{A3A1}"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )}\\\\- \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} & \\cos{\\left (\\zeta\\left(t\\right) \\right )} & - \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )}\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} & 0 & \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAggAAABLCAMAAAAicY+6AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRIki781m3btsR7x/vAAAAAlwSFlzAAAOxAAADsQBlSsOGwAADAVJREFUeAHtXeu6\nqyoSxMQ4Z3LP+P7vOghdDSJIQzDHfCv5sSEtVtFNB9FFbVU3ms9B/T5/MgJPO/5KdeOx15/Tn4zC\nz2n1mEb/ME6J0JWHo3vgnOvjcaW6M6oHYaJE60zJCIA07dmqGNdVMoiZwww9I1RsdoyFniR5HfSe\nIveoTITrC47qmWR42VF3Rn3wQJmAEu1XS4dw8do5q4fLBF7DdLW79dGDDtonVM7sMRZ5ovKMu4pc\nbSLcB4qr+Z2cbVqQcXiaX9edWqCMDkRgZFh16Huec+K4qgS4G2cDzbRxQhVnLCHU82yGcV+Rq0yE\nHk5ezWidnlNkYTzpWcb7CrMxrv/DTYeDXrZg0QJrgMt865irRwGtZoSMHDBy61XM9YOMsbPIVSbC\nDVdUO+EezB0HjEf7yxluNiQo1wNkjgJhuJ+91rAGuKoA2EObVQE9J1QwB4wNCBla7SxydYlwNjPA\nFFLjztkEjo1Pmi7u9JNGORuC2BdGuPNVQTdja4irxMBKXbvYgpihZ4RpxgLCHOPeIhdPhOF4uVym\nYRwufX+Zfp1df+p7vkZejjSO5+567O9PE2RrPB9e4+tg8gOtUGq4dVw0RGlJ7LcIrpq3oy7NC+r3\n9TXqSau7346dvlOyP0bdEAAo6dwkY9BuTkXfZIxqb5GLJsL1ppOg0z/zziz7Xr2+L5jc5HnghV+s\nPnK5HG8mEWCkC6u+76LEQalyuIQwjFiJmtgmcZmAhiBSuH6/zNXrrn1R1xEXtjihSjKyJxEqMskY\np6l0X5GLJsJ9GsGTfrpgJ2P9nOF8m8YGK0T1xDrOWuxUASNdWDUCJQ5KlcMlhOB3l8RlgvSwuH7b\nRHiZdcuIPI4TsntpT95lRCz3E7lYIgwjzfzdaAdc3wg9x1eP35FSN6zl7CR7NrcJMOJSrm+gbLxQ\nZnEJASNP0U7iMkF6WFy/KRHMJDXi2hAnZPeSnqwQyhhpcaX2E7lYIpwRqNNoB3x86XnsPiI/XCKc\n7QrsYX5nFNWBTpqeWM4TIYtLCMGDziQuE6yMC/ebEsFc4uAfRjwghDntyQqhi9Qao14iGIz9RC6W\nCMtf7vE6zQYPHmL8ZulaYe8eyWiS3FzksSZHmcUlBL6E23gncd3qPjkwrt/RYYkT4tKQ9iTJpxcg\nHKk1Rlxl9xO5WCLYa7m+xqubuUacx/PJDPkB11aspuxq8Gpvu8l40caTyXcsrVBmcQkB6BTuJK5g\nsej6HR2WOCEWiyuepDNBxkhPRXcUuWgiXG/TiF/0XYO5H7hf1MksFl9YJVzoT9YmkN3TXkDIeDyo\nwV6Dj7TUQKnvGtZxCcE241gncRUDc9uw4vr9MhcF+y9fGuKEKsmYJ/QitcZoc21PkYsmgnngqvNA\nT3T6xv+gx/l8efQP80DBhJom+3P3uPSXC93skfH66mkthseDKPVzBP3keAWXryH9fSKlTxJXOWC0\nDUv0uzuM42Ew/3an13ijRI4T4pKz5klIxN9ljPuLXDwR2K1Uxf5ZiW8nbTNr5FOu9IgZJR9IVxxC\n98QzK/34AhORPZHxuJIGzB1x0D5hirEBIUHvL3KVifAwwxS4Y40c+wtNDCj5QLriIXSURrqxZ51O\nZTyupAFzRzxojzDF2ICQoPcXucpEUNNygZYCLti8hphMg11JcumardQIYRgU/WXbtI3ilgGnOOOE\nxj0+pcoTPjusTIw7jFxtIkw7N+he2Hk6286B0UPpmq3UgND3R+96AKs5kfG4soKXPQToOeF8YwqI\nUGZBVxtMjDuMXG0iKPvn9MBlz4jtayiDlqmvHoLXxLMyHle8dhVVD9o/2zODCKXfrKbuQbvTPSN4\nULpGqzUPwWvnWRmPK147/ZBoegJoH3P59l/9j0Xglwh/bMBT7v4SIRWZP2b/JcIfG/CUu79ESEXm\nj9nfTgSWa3i3ezqGbK4WiDiEpjqQ5fgy0cwDtlY7sGRaWBxJQx8ZNOfPTCMkTYS8XGP2sAy358Z1\nCENQLuIRMziEpjqQJZUj8j1wVn0GOo5yCbJuyQevwsc8aNYfdkx3X5wIObnGTJHSQCDCypO2OpDl\niDHRzIMGChdHtYnWJQsq8MfTCEkTwXk1q7Fco7VAhIHtE5FWCppZ580XJpp5AOsGChfXBZDQw7k2\nPjKoxB9W8shnBNf9WQ1KkOYCEQA31oHMOm++gGjuAawbKFxcF0DS1EeAivzxNELSGeHDAhFsFGis\noHGDgBoTzSQubMX2VShbUOJ0WZkJXp3WJQMq88dphHKJIJNrBBvQBQIRodSlVAeyGJgMT0LiInBg\nwRQxyIJX6KMMdD4iSX/Y/9xiUSbXCBQp2POnpREUHexaRCmVupTqQBaDkZPUoKuBB9g1mXZgwRQx\nyIJX6KMMVOqP2/aZmRG2EogIpS6lOpDFaOR4sGF5/gOCVdUoXFwfZMEr9FEGKvXHaYRmiTDcn/y5\n279IsrCFdgG3EYgsN7YHEhrSMhTqQNwYUC3LAwkD7WsHAOixRGAFBaQaaOjKSOw20brIQKX+sGPm\n0vCff/7rHApq2whEpFIX7OAQ6kCCvusNINivnJPqBH+IT2pq0omw4NYGUfBKfRSBSv1xifC/f1b3\nI8jkGoEiBTNrWiCy/KUGEhpkND0cE+pAFoOR5UFXAw+IPu3AgilmkAWvUOsiA5X6gw3b2ecIMrkG\n1laIBn1fEYiYa/eKhAaIhQoadIDLHA8Wi+CjE/MOMMNKRRa8Qq2LDFTqj3ixuJVARCh1KdWBLEYl\nxwMti1RTI1C4uD7IglfoowxU6o/TCM0Wi84H1GRyDb0r11ekYL5ZE4jIpC7FOhD0m8sMD7oaeEAP\nlNYcYIZ0RRa8Qh9loFJ/nEYokwhpL+2RzQQiFtj/+5kmdGyGHXoTlLnORo87TF/i4qztmBb0m/jo\nei7wRz/PQa/eTITNBCIWOEgEj23qP/QmKOFTUelhehIXz9qMadGtTXz0ep73h0OYXSwuOh8aaK9/\noEhpIBCZgDfQgYT9V+hq4MFMw9BW4eK6sImP8RGJ+uNrhN6cEVgJ0lwgMu0M2UAH4kYBNWxBmXsA\nq2mFKKLEqe+Wm/iIngv8wY9g8uPdRIgLXbSMGv+VgqagLVMoheHzENwZnhFwKF2jwpqH6Z3pWcGA\n0mv1ZtUjcUieEYwoXaOVmne+18qzMhpXdLu3E8Hj+lW/OAK/RPjiwWvZ9V8itIzmF2P9EuGLB69l\n13+J0DKaX4z12URg7YW+q+AlqzOy4qLVy1K+eGCo6y446xHj0NW6vEki5LUXFYKOWg8l5+mdjf1x\nj/83AB4K6B06mRfl+GIVicthm20SIaeG2Vq0EnqZ+z5tTqb3zuSafvY4q2/yEfPEKjV93CQREh1h\n7YV9uNFG0JHgKjLbDVDT9omdfUoi5olVarz4ZCJAe9FU0FHjdHiO3Y95mr8dIGz0b3wviZgnVqnp\n6iaJkNFe1Ak6arwTnjO91GPa4IiXDwhP27wZ62xEEXNilZqONU0EmfaiUNBR41XZObSzkXe6lp39\nTuuM/IY3pYte9+LEKjVdapkIMu1FoaCjxqnCc652bZuSFxeiFTTPyW9456HodS9u/2FBF7hpy0SQ\naS8KBR3c0+0q/1oi5OQ32MtNG50zr3txYpWaUFUnwscEHTVelZ0z2Bnh45eG5Wb7t2Q+TqNQ5r5t\nXZ0IMTKR9gLbTWpFKzHiN2321TSnTy8WOfNS8hsIrmSve9lPIsi0F4WCjjfHWHT607zL4fHp28fl\njPCWzId3ZIt8Dhu1nBFk2otCQUfY4S2+92YvLytet6CIYubkN1gsymQ++1ksyrQXhYKOaARbG6dV\n2WDfXtkaeg0vJ7+hN8kII+bEKmucqWMtZwSZ9qJQ0JHqeFP79MLb2v8y7Z2OZOQ39EBJGDEnVqnp\nUstEyPFvIujIkX718ZKIeWKVGp8/mQibCDpqnP6ac0oi5olVavz7ZCKYbfQfEa3URGKX50xKClnE\nfLFKjSsfTYRNBB01Xn/NOfKI+WKVGvc+mgi+7MV1NiK94G1srtUfrXnBcRHwjBwprrhmRbXPJkJR\n136NPxkBmwjj9KGXYn6S/ce1hwg8zfDr/wpx6M3HvXt1D7379eFjEXjY8Vf/B7pa03jCuJK4AAAA\nAElFTkSuQmCC\n",
       "prompt_number": 17,
       "text": [
        "\u23a1cos(\u03b2(t))\u22c5cos(\u03b6(t))   sin(\u03b6(t))  sin(\u03b2(t))\u22c5cos(\u03b6(t)) \u23a4\n",
        "\u23a2                                                     \u23a5\n",
        "\u23a2-sin(\u03b6(t))\u22c5cos(\u03b2(t))  cos(\u03b6(t))  -sin(\u03b2(t))\u22c5sin(\u03b6(t))\u23a5\n",
        "\u23a2                                                     \u23a5\n",
        "\u23a3     -sin(\u03b2(t))           0           cos(\u03b2(t))      \u23a6"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B = IJKReferenceFrame(\"B\")\n",
      "theta = dynamicsymbols('theta')\n",
      "B.orient(A3, 'Axis', [theta, A3.x])\n",
      "B.dcm(A3)  # T_{BA3}"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & \\sin{\\left (\\theta{\\left (t \\right )} \\right )}\\\\0 & - \\sin{\\left (\\theta{\\left (t \\right )} \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPgAAABLCAMAAABqUbriAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
       "prompt_number": 18,
       "text": [
        "\u23a11      0           0    \u23a4\n",
        "\u23a2                        \u23a5\n",
        "\u23a20  cos(\u03b8(t))   sin(\u03b8(t))\u23a5\n",
        "\u23a2                        \u23a5\n",
        "\u23a30  -sin(\u03b8(t))  cos(\u03b8(t))\u23a6"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B.dcm(A1)  # T_{BA1}"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )}\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} - \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & - \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} & - \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} + \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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       "prompt_number": 19,
       "text": [
        "\u23a1                cos(\u03b2(t))\u22c5cos(\u03b6(t))                        sin(\u03b6(t))         \n",
        "\u23a2                                                                             \n",
        "\u23a2-sin(\u03b2(t))\u22c5sin(\u03b8(t)) - sin(\u03b6(t))\u22c5cos(\u03b2(t))\u22c5cos(\u03b8(t))  cos(\u03b8(t))\u22c5cos(\u03b6(t))   -\n",
        "\u23a2                                                                             \n",
        "\u23a3-sin(\u03b2(t))\u22c5cos(\u03b8(t)) + sin(\u03b8(t))\u22c5sin(\u03b6(t))\u22c5cos(\u03b2(t))  -sin(\u03b8(t))\u22c5cos(\u03b6(t))  s\n",
        "\n",
        "               sin(\u03b2(t))\u22c5cos(\u03b6(t))                 \u23a4\n",
        "                                                   \u23a5\n",
        "sin(\u03b2(t))\u22c5sin(\u03b6(t))\u22c5cos(\u03b8(t)) + sin(\u03b8(t))\u22c5cos(\u03b2(t))\u23a5\n",
        "                                                   \u23a5\n",
        "in(\u03b2(t))\u22c5sin(\u03b8(t))\u22c5sin(\u03b6(t)) + cos(\u03b2(t))\u22c5cos(\u03b8(t)) \u23a6"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B.dcm(A)  # T_{BA}"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}- \\sin{\\left (\\psi{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} + \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\psi{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\psi{\\left (t \\right )} \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} + \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\psi{\\left (t \\right )} \\right )} & \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )}\\\\\\left(- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} - \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\right) \\cos{\\left (\\psi{\\left (t \\right )} \\right )} - \\sin{\\left (\\psi{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\left(- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} - \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\right) \\sin{\\left (\\psi{\\left (t \\right )} \\right )} + \\cos{\\left (\\psi{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & - \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\\\\\left(- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\right) \\cos{\\left (\\psi{\\left (t \\right )} \\right )} + \\sin{\\left (\\psi{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\left(- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\right) \\sin{\\left (\\psi{\\left (t \\right )} \\right )} - \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\psi{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} + \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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3z1Qg3QJDOvVweX5o/YzBzBhQON76ag77ivtjtYVCx9aiSbss5JTbr1dCOtwP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\naMlCaQAAAABJRU5ErkJggg==\n",
       "prompt_number": 20,
       "text": [
        "\u23a1                      -sin(\u03c8(t))\u22c5sin(\u03b6(t)) + cos(\u03b2(t))\u22c5cos(\u03c8(t))\u22c5cos(\u03b6(t))   \n",
        "\u23a2                                                                             \n",
        "\u23a2(-sin(\u03b2(t))\u22c5sin(\u03b8(t)) - sin(\u03b6(t))\u22c5cos(\u03b2(t))\u22c5cos(\u03b8(t)))\u22c5cos(\u03c8(t)) - sin(\u03c8(t))\u22c5\n",
        "\u23a2                                                                             \n",
        "\u23a3(-sin(\u03b2(t))\u22c5cos(\u03b8(t)) + sin(\u03b8(t))\u22c5sin(\u03b6(t))\u22c5cos(\u03b2(t)))\u22c5cos(\u03c8(t)) + sin(\u03c8(t))\u22c5\n",
        "\n",
        "                                           sin(\u03c8(t))\u22c5cos(\u03b2(t))\u22c5cos(\u03b6(t)) + sin\n",
        "                                                                              \n",
        "cos(\u03b8(t))\u22c5cos(\u03b6(t))  (-sin(\u03b2(t))\u22c5sin(\u03b8(t)) - sin(\u03b6(t))\u22c5cos(\u03b2(t))\u22c5cos(\u03b8(t)))\u22c5si\n",
        "                                                                              \n",
        "sin(\u03b8(t))\u22c5cos(\u03b6(t))  (-sin(\u03b2(t))\u22c5cos(\u03b8(t)) + sin(\u03b8(t))\u22c5sin(\u03b6(t))\u22c5cos(\u03b2(t)))\u22c5si\n",
        "\n",
        "(\u03b6(t))\u22c5cos(\u03c8(t))                                         sin(\u03b2(t))\u22c5cos(\u03b6(t))  \n",
        "                                                                              \n",
        "n(\u03c8(t)) + cos(\u03c8(t))\u22c5cos(\u03b8(t))\u22c5cos(\u03b6(t))  -sin(\u03b2(t))\u22c5sin(\u03b6(t))\u22c5cos(\u03b8(t)) + sin(\n",
        "                                                                              \n",
        "n(\u03c8(t)) - sin(\u03b8(t))\u22c5cos(\u03c8(t))\u22c5cos(\u03b6(t))  sin(\u03b2(t))\u22c5sin(\u03b8(t))\u22c5sin(\u03b6(t)) + cos(\u03b2\n",
        "\n",
        "               \u23a4\n",
        "               \u23a5\n",
        "\u03b8(t))\u22c5cos(\u03b2(t))\u23a5\n",
        "               \u23a5\n",
        "(t))\u22c5cos(\u03b8(t)) \u23a6"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Con SymPy podemos obtener matrices de giro complicadas de forma casi instant\u00e1nea. Definamos ahora el **punto de acoplamiento de la pala** $E$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "O = Point(\"O\")\n",
      "O.set_vel(A, 0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "E = O.locatenew('E', e * A1.x)\n",
      "E.pos_from(O)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$e\\mathbf{i}_{A1}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACQAAAASBAMAAADBDWEbAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAt0lEQVQYGWNgAIMXEyA0guT8fwHBgbBY/29A\nF2JYiSGCRYAfVSOjskkA630Us9IDWBUY/MFCfAEMDOsYGFg7GUQWQIXOJzAs2c7AwN1jfJABIsTo\nf4CBYToDAz9YC1gjk7gDREgB5AKw0GK+B2AhDqCQAURoA0sBWIj5AoNIAFiIZwLzB7AQg4nNAQb+\n//8ZGMRd/D5ChEBGgYErA9cvVCHOAwwMfxgYZkAVgKj3Dxjivh8Iqd8MAJGqMhLLllA0AAAAAElF\nTkSuQmCC\n",
       "prompt_number": 22,
       "text": [
        "e \u001b[94m\u001b[1ma1_x\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Y su centro de masas $GB$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x_GB = Symbol('x_GB')\n",
      "GB = E.locatenew('GB', x_GB * B.x)\n",
      "GB.pos_from(E)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x_{GB}\\mathbf{i}_{B}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADsAAAASBAMAAAAXqLt+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABNUlEQVQoFZWPP0vDUBTFfyk1rWlrH/EDGCoI\n4mDBsRSyqLhYdwcz+WcRRyfBfgHr5BrwA1RQcart5thCQehQ7OTgZKyLotT7kiEKWTzwLifnvHty\nHiQg95IgxlLzI+YJrDZOEGMp78f8/2zyK9yeX11w/kZsxLbhZerZBmRWHm2X7fGw4lIQe6pd7d+A\nqczAcklfYNzLpRHmKLQ5gUMwmL6S6G4HLqFQxgoiuw9NJU7RkfEs5wGyDoVGZNdhIKJeVKlAM5jr\nsKhC2/gidQ6WWialdDmNndJSWX6hq72VrrXgt1mn2MMa3MEBHEfbZoPMKcyW7L0jsh7MONCSfJ/J\n5JVcD+NT1kNIH2pbIELTDZWij/UemTKl55MiH5DWbxB0FZteyPQwq7trsP89rEiG/j4b9m/hB4x3\nTmubPT7VAAAAAElFTkSuQmCC\n",
       "prompt_number": 23,
       "text": [
        "x_GB \u001b[94m\u001b[1mb_x\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "La **velocidad angular de la pala** $\\omega_B$ es:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "omega_B = B.ang_vel_in(A).subs(psi.diff(), Omega)\n",
      "omega_B"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\dot{\\theta}\\mathbf{i}_{A3} + \\dot{\\zeta\\left(t\\right)}\\mathbf{k}_{A2} -  \\dot{\\beta}\\mathbf{j}_{A1} + \\Omega\\mathbf{k}_{A}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQ0AAAAaBAMAAABfmBSsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nr1IyYLMZ+ltBtn+zsAHr/wGcX1I2F5UwDgAAAABJRU5ErkJggg==\n",
       "prompt_number": 24,
       "text": [
        "\u03b8\u0307 \u001b[94m\u001b[1ma3_x\u001b[0;0m\u001b[0;0m + \u03b6\u0307 \u001b[94m\u001b[1ma2_z\u001b[0;0m\u001b[0;0m - \u03b2\u0307 \u001b[94m\u001b[1ma1_y\u001b[0;0m\u001b[0;0m + \u03a9 \u001b[94m\u001b[1ma_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Expresada en ejes pala por componentes:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "omega_B"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\dot{\\theta}\\mathbf{i}_{A3} + \\dot{\\zeta\\left(t\\right)}\\mathbf{k}_{A2} -  \\dot{\\beta}\\mathbf{j}_{A1} + \\Omega\\mathbf{k}_{A}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQ0AAAAaBAMAAABfmBSsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nr1IyYLMZ+ltBtn+zsAHr/wGcX1I2F5UwDgAAAABJRU5ErkJggg==\n",
       "prompt_number": 25,
       "text": [
        "\u03b8\u0307 \u001b[94m\u001b[1ma3_x\u001b[0;0m\u001b[0;0m + \u03b6\u0307 \u001b[94m\u001b[1ma2_z\u001b[0;0m\u001b[0;0m - \u03b2\u0307 \u001b[94m\u001b[1ma1_y\u001b[0;0m\u001b[0;0m + \u03a9 \u001b[94m\u001b[1ma_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "omega_B.to_matrix(B)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\Omega \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} - \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\frac{d}{d t} \\beta{\\left (t \\right )} + \\frac{d}{d t} \\theta{\\left (t \\right )}\\\\\\Omega \\left(- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\right) + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\frac{d}{d t} \\zeta\\left(t\\right) - \\cos{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} \\frac{d}{d t} \\beta{\\left (t \\right )}\\\\\\Omega \\left(\\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} + \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\right) + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} \\frac{d}{d t} \\beta{\\left (t \\right )} + \\cos{\\left (\\theta{\\left (t \\right )} \\right )} \\frac{d}{d t} \\zeta\\left(t\\right)\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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       "prompt_number": 26,
       "text": [
        "\u23a1                                                             d          d    \n",
        "\u23a2                           \u03a9\u22c5sin(\u03b2(t))\u22c5cos(\u03b6(t)) - sin(\u03b6(t))\u22c5\u2500\u2500(\u03b2(t)) + \u2500\u2500(\u03b8(\n",
        "\u23a2                                                             dt         dt   \n",
        "\u23a2                                                                             \n",
        "\u23a2                                                                     d       \n",
        "\u23a2\u03a9\u22c5(-sin(\u03b2(t))\u22c5sin(\u03b6(t))\u22c5cos(\u03b8(t)) + sin(\u03b8(t))\u22c5cos(\u03b2(t))) + sin(\u03b8(t))\u22c5\u2500\u2500(\u03b6(t))\n",
        "\u23a2                                                                     dt      \n",
        "\u23a2                                                                             \n",
        "\u23a2                                                                             \n",
        "\u23a2\u03a9\u22c5(sin(\u03b2(t))\u22c5sin(\u03b8(t))\u22c5sin(\u03b6(t)) + cos(\u03b2(t))\u22c5cos(\u03b8(t))) + sin(\u03b8(t))\u22c5cos(\u03b6(t))\n",
        "\u23a3                                                                             \n",
        "\n",
        "                               \u23a4\n",
        "t))                            \u23a5\n",
        "                               \u23a5\n",
        "                               \u23a5\n",
        "                       d       \u23a5\n",
        " - cos(\u03b8(t))\u22c5cos(\u03b6(t))\u22c5\u2500\u2500(\u03b2(t))\u23a5\n",
        "                       dt      \u23a5\n",
        "                               \u23a5\n",
        " d                    d        \u23a5\n",
        "\u22c5\u2500\u2500(\u03b2(t)) + cos(\u03b8(t))\u22c5\u2500\u2500(\u03b6(t)) \u23a5\n",
        " dt                   dt       \u23a6"
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "T\u00e9rminos inerciales"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "El tensor de inercia en ejes pala, asumiendo que estos son ejes principales de inercia, ser\u00e1:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy.physics.mechanics import inertia"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "I_theta, I_beta, I_zeta = symbols('I_theta, I_beta, I_zeta')\n",
      "I_B = inertia(B, I_theta, I_beta, I_zeta)\n",
      "I_B"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$I_{\\theta}\\mathbf{i}_{B}\\otimes \\mathbf{i}_{B} + I_{\\beta}\\mathbf{j}_{B}\\otimes \\mathbf{j}_{B} + I_{\\zeta}\\mathbf{k}_{B}\\otimes \\mathbf{k}_{B}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAUIAAAAVBAMAAADBf/pUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 28,
       "text": [
        "I_\u03b8 \u001b[94m\u001b[1mb_x\u001b[0;0m\u001b[0;0m\u2297\u001b[94m\u001b[1mb_x\u001b[0;0m\u001b[0;0m + I\u1d66 \u001b[94m\u001b[1mb_y\u001b[0;0m\u001b[0;0m\u2297\u001b[94m\u001b[1mb_y\u001b[0;0m\u001b[0;0m + I_\u03b6 \u001b[94m\u001b[1mb_z\u001b[0;0m\u001b[0;0m\u2297\u001b[94m\u001b[1mb_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "I_B.to_matrix(B)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}I_{\\theta} & 0 & 0\\\\0 & I_{\\beta} & 0\\\\0 & 0 & I_{\\zeta}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHMAAABLCAMAAAB0tEgUAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n5v2/iajcDxWMWVPfnukPNPxvIoxR/wFuaC75rySFMgAAAABJRU5ErkJggg==\n",
       "prompt_number": 29,
       "text": [
        "\u23a1I_\u03b8  0    0 \u23a4\n",
        "\u23a2            \u23a5\n",
        "\u23a2 0   I\u1d66   0 \u23a5\n",
        "\u23a2            \u23a5\n",
        "\u23a3 0   0   I_\u03b6\u23a6"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "El **momento cin\u00e9tico** $h_B$ ser\u00e1, por tanto:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "h_B = I_B.dot(omega_B).subs(psi.diff(), Omega)\n",
      "h_B.to_matrix(B)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}I_{\\theta} \\left(\\Omega \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} - \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\frac{d}{d t} \\beta{\\left (t \\right )} + \\frac{d}{d t} \\theta{\\left (t \\right )}\\right)\\\\I_{\\beta} \\left(\\Omega \\left(- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\right) + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\frac{d}{d t} \\zeta\\left(t\\right) - \\cos{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} \\frac{d}{d t} \\beta{\\left (t \\right )}\\right)\\\\I_{\\zeta} \\left(\\Omega \\left(\\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} + \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\right) + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} \\frac{d}{d t} \\beta{\\left (t \\right )} + \\cos{\\left (\\theta{\\left (t \\right )} \\right )} \\frac{d}{d t} \\zeta\\left(t\\right)\\right)\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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mOMLyGvNcOnmQhmUCHfW6IG8WLdcRwpebhQTfCZKE5yJYREgzjppHBosA2kdDkw76cpd5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\nNg38D+/ovu7WA74sAAAAAElFTkSuQmCC\n",
       "prompt_number": 30,
       "text": [
        "\u23a1                               \u239b                                  d          \n",
        "\u23a2                           I_\u03b8\u22c5\u239c\u03a9\u22c5sin(\u03b2(t))\u22c5cos(\u03b6(t)) - sin(\u03b6(t))\u22c5\u2500\u2500(\u03b2(t)) + \n",
        "\u23a2                               \u239d                                  dt         \n",
        "\u23a2                                                                             \n",
        "\u23a2   \u239b                                                                     d   \n",
        "\u23a2I\u1d66\u22c5\u239c\u03a9\u22c5(-sin(\u03b2(t))\u22c5sin(\u03b6(t))\u22c5cos(\u03b8(t)) + sin(\u03b8(t))\u22c5cos(\u03b2(t))) + sin(\u03b8(t))\u22c5\u2500\u2500(\u03b6\n",
        "\u23a2   \u239d                                                                     dt  \n",
        "\u23a2                                                                             \n",
        "\u23a2    \u239b                                                                        \n",
        "\u23a2I_\u03b6\u22c5\u239c\u03a9\u22c5(sin(\u03b2(t))\u22c5sin(\u03b8(t))\u22c5sin(\u03b6(t)) + cos(\u03b2(t))\u22c5cos(\u03b8(t))) + sin(\u03b8(t))\u22c5cos(\n",
        "\u23a3    \u239d                                                                        \n",
        "\n",
        "d       \u239e                           \u23a4\n",
        "\u2500\u2500(\u03b8(t))\u239f                           \u23a5\n",
        "dt      \u23a0                           \u23a5\n",
        "                                    \u23a5\n",
        "                           d       \u239e\u23a5\n",
        "(t)) - cos(\u03b8(t))\u22c5cos(\u03b6(t))\u22c5\u2500\u2500(\u03b2(t))\u239f\u23a5\n",
        "                           dt      \u23a0\u23a5\n",
        "                                    \u23a5\n",
        "      d                    d       \u239e\u23a5\n",
        "\u03b6(t))\u22c5\u2500\u2500(\u03b2(t)) + cos(\u03b8(t))\u22c5\u2500\u2500(\u03b6(t))\u239f\u23a5\n",
        "      dt                   dt      \u23a0\u23a6"
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Para la derivada del momento cin\u00e9tico tendremos simplemente:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "dh_B_dt = h_B.dt(A)\n",
      "dh_B_dt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$(- I_{\\beta} \\left(\\Omega \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\theta\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) + \\operatorname{sin}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)} - \\operatorname{cos}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta}\\right) \\left(\\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\psi} + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\operatorname{cos}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)}\\right) + I_{\\theta} \\left(- \\Omega \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\zeta\\left(t\\right)} + \\Omega \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} - \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\ddot{\\beta} - \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} \\dot{\\zeta\\left(t\\right)} + \\ddot{\\theta}\\right) + I_{\\zeta} \\left(\\Omega \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\operatorname{cos}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)}\\right) \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\theta\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\psi} + \\operatorname{sin}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)} - \\operatorname{cos}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta}\\right))\\mathbf{i}_{B} + (I_{\\beta} \\left(\\Omega \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\theta} - \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\dot{\\beta} - \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\zeta\\left(t\\right)} - \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\beta} + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta}\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} \\dot{\\theta} + \\operatorname{sin}\\left(\\theta\\right) \\ddot{\\zeta\\left(t\\right)} + \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\beta} \\dot{\\zeta\\left(t\\right)} - \\operatorname{cos}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\ddot{\\beta} + \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta} \\dot{\\zeta\\left(t\\right)}\\right) + I_{\\theta} \\left(\\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\psi} + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\operatorname{cos}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)}\\right) \\left(\\Omega \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) - \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\dot{\\theta}\\right) - I_{\\zeta} \\left(\\Omega \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\operatorname{cos}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)}\\right) \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\psi} - \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\dot{\\theta}\\right))\\mathbf{j}_{B} + (I_{\\beta} \\left(\\Omega \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\theta\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) + \\operatorname{sin}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)} - \\operatorname{cos}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta}\\right) \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\psi} - \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\dot{\\theta}\\right) - I_{\\theta} \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\theta\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\psi} + \\operatorname{sin}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)} - \\operatorname{cos}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta}\\right) \\left(\\Omega \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) - \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\dot{\\theta}\\right) + I_{\\zeta} \\left(\\Omega \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\zeta\\left(t\\right)} + \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta} - \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\beta} + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\beta\\right) \\dot{\\beta} - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right) \\dot{\\theta}\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} \\dot{\\zeta\\left(t\\right)} + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\ddot{\\beta} - \\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta} \\dot{\\zeta\\left(t\\right)} + \\operatorname{cos}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} \\dot{\\theta} + \\operatorname{cos}\\left(\\theta\\right) \\ddot{\\zeta\\left(t\\right)}\\right))\\mathbf{k}_{B}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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hEx6/l568n/bu534b8t93Q+ci7b+VdoBfDwb3BmEkc0yAQiwyurciK2jh8TX8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\noMZcO6kiqk/FXvh0YVgGHO/MUU8E60iPBLrNddI19r6O1Gj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       "prompt_number": 31,
       "text": [
        "(-I\u1d66\u22c5(\u03a9\u22c5(-sin(\u03b2)\u22c5sin(\u03b6)\u22c5cos(\u03b8) + sin(\u03b8)\u22c5cos(\u03b2)) + sin(\u03b8)\u22c5\u03b6\u0307 - cos(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307)\u22c5((sin(\u03b2)\u22c5sin(\u03b8)\u22c5sin(\u03b6) + cos(\u03b2)\u22c5cos(\u03b8))\u22c5\u03c8\u0307 + sin(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307 + cos(\u03b8)\u22c5\u03b6\u0307) + I_\u03b8\u22c5(-\u03a9\u22c5sin(\u03b2)\u22c5sin(\u03b6)\u22c5\u03b6\u0307 + \u03a9\u22c5cos(\u03b2)\u22c5cos(\u03b6)\u22c5\u03b2\u0307 - sin(\u03b6)\u22c5\u03b2\u0308 - cos(\u03b6)\u22c5\u03b2\u0307\u22c5\u03b6\u0307 + \u03b8\u0308) + I_\u03b6\u22c5(\u03a9\u22c5(sin(\u03b2)\u22c5sin(\u03b8)\u22c5sin(\u03b6) + cos(\u03b2)\u22c5cos(\u03b8)) + sin(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307 + cos(\u03b8)\u22c5\u03b6\u0307)\u22c5((-sin(\u03b2)\u22c5sin(\u03b6)\u22c5cos(\u03b8) + sin(\u03b8)\u22c5cos(\u03b2))\u22c5\u03c8\u0307 + sin(\u03b8)\u22c5\u03b6\u0307 - cos(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307)) \u001b[94m\u001b[1mb_x\u001b[0;0m\u001b[0;0m + (I\u1d66\u22c5(\u03a9\u22c5(sin(\u03b2)\u22c5sin(\u03b8)\u22c5sin(\u03b6)\u22c5\u03b8\u0307 - sin(\u03b2)\u22c5sin(\u03b8)\u22c5\u03b2\u0307 - sin(\u03b2)\u22c5cos(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b6\u0307 - sin(\u03b6)\u22c5cos(\u03b2)\u22c5cos(\u03b8)\u22c5\u03b2\u0307 + cos(\u03b2)\u22c5cos(\u03b8)\u22c5\u03b8\u0307) + sin(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307\u22c5\u03b8\u0307 + sin(\u03b8)\u22c5\u03b6\u0308 + sin(\u03b6)\u22c5cos(\u03b8)\u22c5\u03b2\u0307\u22c5\u03b6\u0307 - cos(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0308 + cos(\u03b8)\u22c5\u03b8\u0307\u22c5\u03b6\u0307) + I_\u03b8\u22c5((sin(\u03b2)\u22c5sin(\u03b8)\u22c5sin(\u03b6) + cos(\u03b2)\u22c5cos(\u03b8))\u22c5\u03c8\u0307 + sin(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307 + cos(\u03b8)\u22c5\u03b6\u0307)\u22c5(\u03a9\u22c5sin(\u03b2)\u22c5cos(\u03b6) - sin(\u03b6)\u22c5\u03b2\u0307 + \u03b8\u0307) - I_\u03b6\u22c5(\u03a9\u22c5(sin(\u03b2)\u22c5sin(\u03b8)\u22c5sin(\u03b6) + cos(\u03b2)\u22c5cos(\u03b8)) + sin(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307 + cos(\u03b8)\u22c5\u03b6\u0307)\u22c5(sin(\u03b2)\u22c5cos(\u03b6)\u22c5\u03c8\u0307 - sin(\u03b6)\u22c5\u03b2\u0307 + \u03b8\u0307)) \u001b[94m\u001b[1mb_y\u001b[0;0m\u001b[0;0m + (I\u1d66\u22c5(\u03a9\u22c5(-sin(\u03b2)\u22c5sin(\u03b6)\u22c5cos(\u03b8) + sin(\u03b8)\u22c5cos(\u03b2)) + sin(\u03b8)\u22c5\u03b6\u0307 - cos(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307)\u22c5(sin(\u03b2)\u22c5cos(\u03b6)\u22c5\u03c8\u0307 - sin(\u03b6)\u22c5\u03b2\u0307 + \u03b8\u0307) - I_\u03b8\u22c5((-sin(\u03b2)\u22c5sin(\u03b6)\u22c5cos(\u03b8) + sin(\u03b8)\u22c5cos(\u03b2))\u22c5\u03c8\u0307 + sin(\u03b8)\u22c5\u03b6\u0307 - cos(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307)\u22c5(\u03a9\u22c5sin(\u03b2)\u22c5cos(\u03b6) - sin(\u03b6)\u22c5\u03b2\u0307 + \u03b8\u0307) + I_\u03b6\u22c5(\u03a9\u22c5(sin(\u03b2)\u22c5sin(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b6\u0307 + sin(\u03b2)\u22c5sin(\u03b6)\u22c5cos(\u03b8)\u22c5\u03b8\u0307 - sin(\u03b2)\u22c5cos(\u03b8)\u22c5\u03b2\u0307 + sin(\u03b8)\u22c5sin(\u03b6)\u22c5cos(\u03b2)\u22c5\u03b2\u0307 - sin(\u03b8)\u22c5cos(\u03b2)\u22c5\u03b8\u0307) - sin(\u03b8)\u22c5sin(\u03b6)\u22c5\u03b2\u0307\u22c5\u03b6\u0307 + sin(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0308 - sin(\u03b8)\u22c5\u03b8\u0307\u22c5\u03b6\u0307 + cos(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307\u22c5\u03b8\u0307 + cos(\u03b8)\u22c5\u03b6\u0308)) \u001b[94m\u001b[1mb_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Aparece una expresi\u00f3n bastante compleja que no merece la pena analizar. Para la aceleraci\u00f3n del acoplamiento:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "E.set_vel(A1, 0)\n",
      "v_E = E.v1pt_theory(O, A, A1)\n",
      "v_E"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$e \\dot{\\psi}\\mathbf{j}_{A1}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADIAAAAYBAMAAABKJrPSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABKUlEQVQoFWNggIFqGAOdZvyLLkIjPhfQ3BWF\n2AxnAgru/4xLxv4bNpkYoCBnAjaZo9gEwWIKDAxs/z8hSzMqmwQwqhgwKDAXMISjyKQHsCowcX5g\nncD8kYEPLMMXwMCwjoGBtZNBZEEkSwGPAEMDVOZ8AsOS7QwM3D3GBxkEOBSA3nGAyDD6H2BgmM7A\nwL8BZGl+ABMDowFEhkncASKjAJJRZ4hkYBKAyCzmewCW4QDKGDD4MRxkmMYAkdnAUgCWYb7AIBLA\nkMFgwJMAkeGZwPwBLMNgYnMAGCyeb0wZIDLiLn4fITJAASDgmsDAwLgA5B9XBq5fyDLMAkBn/uL4\nycB5gIHhDwPDDLB6EAGKHf5v6wsY3j9giPt+IKR+M0yKGchgu14C1AkDADkaRU8053F0AAAAAElF\nTkSuQmCC\n",
       "prompt_number": 32,
       "text": [
        "e\u22c5\u03c8\u0307 \u001b[94m\u001b[1ma1_y\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a_E = v_E.dt(A).subs({psi.diff(): Omega, psi.diff().diff(): 0})\n",
      "a_E"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-  \\Omega^{2} e\\mathbf{i}_{A1}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAXBAMAAABT8LQsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n06mWH/4AUOtgLzNiJBYAAAAASUVORK5CYII=\n",
       "prompt_number": 33,
       "text": [
        "  2  \n",
        "-\u03a9 \u22c5e \u001b[94m\u001b[1ma1_x\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a_E.to_matrix(B).simplify()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}- \\Omega^{2} e \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )}\\\\\\Omega^{2} e \\left(\\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\right)\\\\\\Omega^{2} e \\left(\\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} - \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\right)\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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55EiwMm1gRVqgPkBPgpjQU+bMX2H9JldH/C0hkBNC5lzpMtUfTiWxypIMeFGdSciRYGXO\nwIo8Q/VtMwnhWang0fKCnpJ3CGgoq6CugTNspasT/E0hEFMUFix0jR5OJbH6xsozCceJlQUblw5l\nqL5hJiE8K0tg2fk5PSXvENBQVkFdA2fYSlcn+JtC4IVUEtfhETswkTBtihWVE6caJahyOJEQnpUp\niEKWnJ8eScRAQ5k2W6gnCLFVIiQcKihd620hEPV8ax/vgRAC9v8K2P8S8PGd+XZgpQfOfuDtO8qu\n9Z/4U5srgb7NP9UD1zDy5v9fAfJPk28sSAAAAABJRU5ErkJggg==\n",
       "prompt_number": 34,
       "text": [
        "\u23a1                  2                                       \u23a4\n",
        "\u23a2                -\u03a9 \u22c5e\u22c5cos(\u03b2(t))\u22c5cos(\u03b6(t))                 \u23a5\n",
        "\u23a2                                                          \u23a5\n",
        "\u23a2 2                                                        \u23a5\n",
        "\u23a2\u03a9 \u22c5e\u22c5(sin(\u03b2(t))\u22c5sin(\u03b8(t)) + sin(\u03b6(t))\u22c5cos(\u03b2(t))\u22c5cos(\u03b8(t)))\u23a5\n",
        "\u23a2                                                          \u23a5\n",
        "\u23a2 2                                                        \u23a5\n",
        "\u23a3\u03a9 \u22c5e\u22c5(sin(\u03b2(t))\u22c5cos(\u03b8(t)) - sin(\u03b8(t))\u22c5sin(\u03b6(t))\u22c5cos(\u03b2(t)))\u23a6"
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "EGB = GB.pos_from(E)\n",
      "EGB.cross(M_b * a_E).to_matrix(B).simplify()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}0\\\\M_{b} \\Omega^{2} e x_{GB} \\left(- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\right)\\\\M_{b} \\Omega^{2} e x_{GB} \\left(\\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\right)\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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PY6coNcsBB6zqeHzZSrgp2I7fPEaxR67moBpQsCF8JzyGEq/CcmIZOisxNUBkeFfD0Hbi\nrrM42QrRGS4JwMO8ihYGPHTqVuBqhFKTAw5Y1fF4s+WlYFOV05Su8F4MqUvBhvCdtYhXZjnPqVx0\nTmX2ze8MYBmtc/GuhqEVohNOnWyF6MSXXMxReZhX0fLTqVhytUKpyQEHrOp4vNmSAKQ6Owo2YetP\nBA2J1/Xap9OwJtAx7HQ62dIQfRToeTL/4EmENbSW6nHqFkLRsWIHs5W2tlq17H2X157OjoIFyels\ndw2J1xe0R6eRyqKDyU6WZEkO/kqgEU6rh07VWh54C6HoWINWq5bDs3WWFGzzxGuP7Vyc01iNU0V0\nei7ac8LQLyTL0ndIataoodV8KGRicfG5Psvh2ZIYmqyscgDYPPFqlJ0GX3016oBh6WBo9Gpb2iNS\ns0YNreZD0XwrLHqtuiyPnxX7KrFuktpdWSVzJoNpG9MbJUSMsgG2wLvig3OJbe5xCrd90kIo/ElO\nidcqW3fkXciWNvUBOHK22v+XblM3Wv1fuitzuU83Bf+Ag/B/6ebo/yAqDwDVtay4AAAAAElFTkSu\nQmCC\n",
       "prompt_number": 35,
       "text": [
        "\u23a1                                 0                                  \u23a4\n",
        "\u23a2                                                                    \u23a5\n",
        "\u23a2     2                                                              \u23a5\n",
        "\u23a2M_b\u22c5\u03a9 \u22c5e\u22c5x_GB\u22c5(-sin(\u03b2(t))\u22c5cos(\u03b8(t)) + sin(\u03b8(t))\u22c5sin(\u03b6(t))\u22c5cos(\u03b2(t)))\u23a5\n",
        "\u23a2                                                                    \u23a5\n",
        "\u23a2     2                                                              \u23a5\n",
        "\u23a3M_b\u22c5\u03a9 \u22c5e\u22c5x_GB\u22c5(sin(\u03b2(t))\u22c5sin(\u03b8(t)) + sin(\u03b6(t))\u22c5cos(\u03b2(t))\u22c5cos(\u03b8(t))) \u23a6"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Y ahora el momento del peso, teniendo en cuenta que la gravedad ser\u00e1 $\\mathbf{g} = -g \\mathbf{k}_{A1}$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "EGB.cross(M_b * (-g * A1.z)).to_matrix(B).simplify()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}0\\\\M_{b} g x_{GB} \\left(\\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} + \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\right)\\\\M_{b} g x_{GB} \\left(\\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} - \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\right)\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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G8jNRe5fxlsNwADSU4cxmLYZQxUjiWF7ZxNo6GaFCLRIBHyVUdsoIIWhGQodDBSVrf3+4\nBJta7cd7QMPlzR/9LcIfP6I2gO/xwF2ChH6wg35VmT/xprPv6bGh/mAPzBol3X+Qwd2N9Uv3GgAA\nAABJRU5ErkJggg==\n",
       "prompt_number": 36,
       "text": [
        "\u23a1                               0                                \u23a4\n",
        "\u23a2                                                                \u23a5\n",
        "\u23a2M_b\u22c5g\u22c5x_GB\u22c5(sin(\u03b2(t))\u22c5sin(\u03b8(t))\u22c5sin(\u03b6(t)) + cos(\u03b2(t))\u22c5cos(\u03b8(t)))\u23a5\n",
        "\u23a2                                                                \u23a5\n",
        "\u23a3M_b\u22c5g\u22c5x_GB\u22c5(sin(\u03b2(t))\u22c5sin(\u03b6(t))\u22c5cos(\u03b8(t)) - sin(\u03b8(t))\u22c5cos(\u03b2(t)))\u23a6"
       ]
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Acoplamientos inerciales"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Vamos a estudiar cada uno de los grados de libertad desacoplados entre s\u00ed en vac\u00edo, es decir, sin acciones aerodin\u00e1micas: $\\mathbf{M}_b^{a, E} = 0$. Adem\u00e1s, se cumplir\u00e1 la relaci\u00f3n $I_{\\zeta} = I_{\\theta} + I_{\\beta}$."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Din\u00e1mica de batimiento en vac\u00edo"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "En este caso tenemos $\\beta(t) \\neq 0$, $\\zeta(t) = \\theta(t) = 0$. Las magnitudes analizadas anteriormente se simplifican:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "omega_B.express(B).subs({theta: 0, zeta: 0})"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$(\\Omega \\operatorname{sin}\\left(\\beta\\right) + 0)\\mathbf{i}_{B} -  \\dot{\\beta}\\mathbf{j}_{B} + (\\Omega \\operatorname{cos}\\left(\\beta\\right) + 0)\\mathbf{k}_{B}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAYQAAAAZBAMAAADDHETJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
       "prompt_number": 37,
       "text": [
        "\u239b           d    \u239e\n",
        "\u239c\u03a9\u22c5sin(\u03b2) + \u2500\u2500(0)\u239f\n",
        "\u239d           dt   \u23a0 \u001b[94m\u001b[1mb_x\u001b[0;0m\u001b[0;0m - \u03b2\u0307 \u001b[94m\u001b[1mb_y\u001b[0;0m\u001b[0;0m + \u239b           d    \u239e\n",
        "\u239c\u03a9\u22c5cos(\u03b2) + \u2500\u2500(0)\u239f\n",
        "\u239d           dt   \u23a0 \u001b[94m\u001b[1mb_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "_.doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\Omega \\operatorname{sin}\\left(\\beta\\right)\\mathbf{i}_{B} -  \\dot{\\beta}\\mathbf{j}_{B} + \\Omega \\operatorname{cos}\\left(\\beta\\right)\\mathbf{k}_{B}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAASYAAAAZBAMAAABusZ4lAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nxDXY/wWLXFvqrQlYSQAAAABJRU5ErkJggg==\n",
       "prompt_number": 38,
       "text": [
        "\u03a9\u22c5sin(\u03b2) \u001b[94m\u001b[1mb_x\u001b[0;0m\u001b[0;0m - \u03b2\u0307 \u001b[94m\u001b[1mb_y\u001b[0;0m\u001b[0;0m + \u03a9\u22c5cos(\u03b2) \u001b[94m\u001b[1mb_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "h_B.express(B).subs({theta: 0, zeta: 0}).doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$I_{\\theta} \\Omega \\operatorname{sin}\\left(\\beta\\right)\\mathbf{i}_{B} -  I_{\\beta} \\dot{\\beta}\\mathbf{j}_{B} + I_{\\zeta} \\Omega \\operatorname{cos}\\left(\\beta\\right)\\mathbf{k}_{B}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAV8AAAAaBAMAAADieULZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 39,
       "text": [
        "I_\u03b8\u22c5\u03a9\u22c5sin(\u03b2) \u001b[94m\u001b[1mb_x\u001b[0;0m\u001b[0;0m - I\u1d66\u22c5\u03b2\u0307 \u001b[94m\u001b[1mb_y\u001b[0;0m\u001b[0;0m + I_\u03b6\u22c5\u03a9\u22c5cos(\u03b2) \u001b[94m\u001b[1mb_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "dh_B_dt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$(- I_{\\beta} \\left(\\Omega \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\theta\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) + \\operatorname{sin}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)} - \\operatorname{cos}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta}\\right) \\left(\\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\psi} + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\operatorname{cos}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)}\\right) + I_{\\theta} \\left(- \\Omega \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\zeta\\left(t\\right)} + \\Omega \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} - \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\ddot{\\beta} - \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} \\dot{\\zeta\\left(t\\right)} + \\ddot{\\theta}\\right) + I_{\\zeta} \\left(\\Omega \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\operatorname{cos}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)}\\right) \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\theta\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\psi} + \\operatorname{sin}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)} - \\operatorname{cos}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta}\\right))\\mathbf{i}_{B} + (I_{\\beta} \\left(\\Omega \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\theta} - \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\dot{\\beta} - \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\zeta\\left(t\\right)} - \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\beta} + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta}\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} \\dot{\\theta} + \\operatorname{sin}\\left(\\theta\\right) \\ddot{\\zeta\\left(t\\right)} + \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\beta} \\dot{\\zeta\\left(t\\right)} - \\operatorname{cos}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\ddot{\\beta} + \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta} \\dot{\\zeta\\left(t\\right)}\\right) + I_{\\theta} \\left(\\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\psi} + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\operatorname{cos}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)}\\right) \\left(\\Omega \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) - \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\dot{\\theta}\\right) - I_{\\zeta} \\left(\\Omega \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\operatorname{cos}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)}\\right) \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\psi} - \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\dot{\\theta}\\right))\\mathbf{j}_{B} + (I_{\\beta} \\left(\\Omega \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\theta\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) + \\operatorname{sin}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)} - \\operatorname{cos}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta}\\right) \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\psi} - \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\dot{\\theta}\\right) - I_{\\theta} \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\theta\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\psi} + \\operatorname{sin}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)} - \\operatorname{cos}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta}\\right) \\left(\\Omega \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) - \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\dot{\\theta}\\right) + I_{\\zeta} \\left(\\Omega \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\zeta\\left(t\\right)} + \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta} - \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\beta} + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\beta\\right) \\dot{\\beta} - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right) \\dot{\\theta}\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} \\dot{\\zeta\\left(t\\right)} + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\ddot{\\beta} - \\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta} \\dot{\\zeta\\left(t\\right)} + \\operatorname{cos}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} \\dot{\\theta} + \\operatorname{cos}\\left(\\theta\\right) \\ddot{\\zeta\\left(t\\right)}\\right))\\mathbf{k}_{B}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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hEx6/l568n/bu534b8t93Q+ci7b+VdoBfDwb3BmEkc0yAQiwyurciK2jh8TX8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\noMZcO6kiqk/FXvh0YVgGHO/MUU8E60iPBLrNddI19r6O1Gj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       "prompt_number": 40,
       "text": [
        "(-I\u1d66\u22c5(\u03a9\u22c5(-sin(\u03b2)\u22c5sin(\u03b6)\u22c5cos(\u03b8) + sin(\u03b8)\u22c5cos(\u03b2)) + sin(\u03b8)\u22c5\u03b6\u0307 - cos(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307)\u22c5((sin(\u03b2)\u22c5sin(\u03b8)\u22c5sin(\u03b6) + cos(\u03b2)\u22c5cos(\u03b8))\u22c5\u03c8\u0307 + sin(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307 + cos(\u03b8)\u22c5\u03b6\u0307) + I_\u03b8\u22c5(-\u03a9\u22c5sin(\u03b2)\u22c5sin(\u03b6)\u22c5\u03b6\u0307 + \u03a9\u22c5cos(\u03b2)\u22c5cos(\u03b6)\u22c5\u03b2\u0307 - sin(\u03b6)\u22c5\u03b2\u0308 - cos(\u03b6)\u22c5\u03b2\u0307\u22c5\u03b6\u0307 + \u03b8\u0308) + I_\u03b6\u22c5(\u03a9\u22c5(sin(\u03b2)\u22c5sin(\u03b8)\u22c5sin(\u03b6) + cos(\u03b2)\u22c5cos(\u03b8)) + sin(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307 + cos(\u03b8)\u22c5\u03b6\u0307)\u22c5((-sin(\u03b2)\u22c5sin(\u03b6)\u22c5cos(\u03b8) + sin(\u03b8)\u22c5cos(\u03b2))\u22c5\u03c8\u0307 + sin(\u03b8)\u22c5\u03b6\u0307 - cos(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307)) \u001b[94m\u001b[1mb_x\u001b[0;0m\u001b[0;0m + (I\u1d66\u22c5(\u03a9\u22c5(sin(\u03b2)\u22c5sin(\u03b8)\u22c5sin(\u03b6)\u22c5\u03b8\u0307 - sin(\u03b2)\u22c5sin(\u03b8)\u22c5\u03b2\u0307 - sin(\u03b2)\u22c5cos(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b6\u0307 - sin(\u03b6)\u22c5cos(\u03b2)\u22c5cos(\u03b8)\u22c5\u03b2\u0307 + cos(\u03b2)\u22c5cos(\u03b8)\u22c5\u03b8\u0307) + sin(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307\u22c5\u03b8\u0307 + sin(\u03b8)\u22c5\u03b6\u0308 + sin(\u03b6)\u22c5cos(\u03b8)\u22c5\u03b2\u0307\u22c5\u03b6\u0307 - cos(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0308 + cos(\u03b8)\u22c5\u03b8\u0307\u22c5\u03b6\u0307) + I_\u03b8\u22c5((sin(\u03b2)\u22c5sin(\u03b8)\u22c5sin(\u03b6) + cos(\u03b2)\u22c5cos(\u03b8))\u22c5\u03c8\u0307 + sin(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307 + cos(\u03b8)\u22c5\u03b6\u0307)\u22c5(\u03a9\u22c5sin(\u03b2)\u22c5cos(\u03b6) - sin(\u03b6)\u22c5\u03b2\u0307 + \u03b8\u0307) - I_\u03b6\u22c5(\u03a9\u22c5(sin(\u03b2)\u22c5sin(\u03b8)\u22c5sin(\u03b6) + cos(\u03b2)\u22c5cos(\u03b8)) + sin(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307 + cos(\u03b8)\u22c5\u03b6\u0307)\u22c5(sin(\u03b2)\u22c5cos(\u03b6)\u22c5\u03c8\u0307 - sin(\u03b6)\u22c5\u03b2\u0307 + \u03b8\u0307)) \u001b[94m\u001b[1mb_y\u001b[0;0m\u001b[0;0m + (I\u1d66\u22c5(\u03a9\u22c5(-sin(\u03b2)\u22c5sin(\u03b6)\u22c5cos(\u03b8) + sin(\u03b8)\u22c5cos(\u03b2)) + sin(\u03b8)\u22c5\u03b6\u0307 - cos(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307)\u22c5(sin(\u03b2)\u22c5cos(\u03b6)\u22c5\u03c8\u0307 - sin(\u03b6)\u22c5\u03b2\u0307 + \u03b8\u0307) - I_\u03b8\u22c5((-sin(\u03b2)\u22c5sin(\u03b6)\u22c5cos(\u03b8) + sin(\u03b8)\u22c5cos(\u03b2))\u22c5\u03c8\u0307 + sin(\u03b8)\u22c5\u03b6\u0307 - cos(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307)\u22c5(\u03a9\u22c5sin(\u03b2)\u22c5cos(\u03b6) - sin(\u03b6)\u22c5\u03b2\u0307 + \u03b8\u0307) + I_\u03b6\u22c5(\u03a9\u22c5(sin(\u03b2)\u22c5sin(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b6\u0307 + sin(\u03b2)\u22c5sin(\u03b6)\u22c5cos(\u03b8)\u22c5\u03b8\u0307 - sin(\u03b2)\u22c5cos(\u03b8)\u22c5\u03b2\u0307 + sin(\u03b8)\u22c5sin(\u03b6)\u22c5cos(\u03b2)\u22c5\u03b2\u0307 - sin(\u03b8)\u22c5cos(\u03b2)\u22c5\u03b8\u0307) - sin(\u03b8)\u22c5sin(\u03b6)\u22c5\u03b2\u0307\u22c5\u03b6\u0307 + sin(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0308 - sin(\u03b8)\u22c5\u03b8\u0307\u22c5\u03b6\u0307 + cos(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307\u22c5\u03b8\u0307 + cos(\u03b8)\u22c5\u03b6\u0308)) \u001b[94m\u001b[1mb_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(dh_B_dt.express(B)\n",
      "    .subs({\n",
      "           theta: 0,\n",
      "           zeta: 0,\n",
      "           psi.diff(): Omega,\n",
      "           psi.diff().diff(): 0,\n",
      "           I_zeta: I_theta + I_beta})\n",
      "    .doit()\n",
      "    .to_matrix(B)\n",
      "    .simplify()\n",
      "    .expand(trig=True))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}0\\\\- I_{\\beta} \\Omega^{2} \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} - I_{\\beta} \\frac{d^{2}}{d t^{2}}  \\beta{\\left (t \\right )}\\\\- 2 I_{\\beta} \\Omega \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\frac{d}{d t} \\beta{\\left (t \\right )}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAWcAAABNCAMAAABNJlVbAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 41,
       "text": [
        "\u23a1                    0                     \u23a4\n",
        "\u23a2                                          \u23a5\n",
        "\u23a2                                   2      \u23a5\n",
        "\u23a2      2                           d       \u23a5\n",
        "\u23a2- I\u1d66\u22c5\u03a9 \u22c5sin(\u03b2(t))\u22c5cos(\u03b2(t)) - I\u1d66\u22c5\u2500\u2500\u2500(\u03b2(t))\u23a5\n",
        "\u23a2                                   2      \u23a5\n",
        "\u23a2                                 dt       \u23a5\n",
        "\u23a2                                          \u23a5\n",
        "\u23a2                          d               \u23a5\n",
        "\u23a2        -2\u22c5I\u1d66\u22c5\u03a9\u22c5sin(\u03b2(t))\u22c5\u2500\u2500(\u03b2(t))        \u23a5\n",
        "\u23a3                          dt              \u23a6"
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "EGB.cross(M_b * a_E).express(B).subs({theta: 0, zeta: 0}).doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-  M_{b} \\Omega^{2} e x_{GB} \\operatorname{sin}\\left(\\beta\\right)\\mathbf{j}_{B}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMkAAAAZBAMAAABp1hcZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMmarVHbvu4lE\nIpkEsbzRAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAD6UlEQVRIDa1UW2gcVRj+ZndmL5lMdigisUK7\nBaUiBVPqQ32QDF6KT2arPiglZgNtoX3ojgjeomSI2C4Gk4CXlmp1FRQrphkrgqWQLmletJQMwiLe\nmjEPARFjQ2Kt2jp+/+y622w3W6X7wTn/7fvPf+ac/wzQFJPHzKbxlgR1L5ZuyUJNF4l66uWmhJYE\nk0XjUksWusYi8cVrEK4v/PUBTxaIOv9nmfuFXOqFZomivnp162i53yU0etgWMfWLMWdRTnI0hXqo\nFjYK1BP2Wgs7xRkLHBErEfuLtporilf7jXRW1XyxmkGVrAoisvdH0F5ARDyJoChiJUpv0NZyobMn\nTXGHhzOYDu3/Ng2TZmQQ70LClYznG6T1vkfn9sNhJCesU7668el3GzBXcx1hIMGTykIZW43j/8Sd\nnL0Yhrttir5CMgjOr0av+I37qoT2Lqqs8oUHbKQ6E7jVWFXxR8nRlkN7IkMxmi3HlM9HPHTu/37Y\nKdvVWT2QT397Afe8/cB+Cek+p+lzQ3dSSNcY3VdXMTKDFp6IncdUmgV8snJSijjtGa7ialujad7b\n7p87PXx5YWGvB6wFuvABcLtjyC+iTfjOzZOvUTzEgQkX7RP75sdFr0DzZm3FThbCLd3URe9d5Yhx\nEOss3dQX4x6MW6EcZa0x6GM8GhOuVKGHPYmozckFhiielFxWwWPACWDNFsEtQAdSTgKzDiIka9yb\nGr4fXubr+e1QECkwr8cCniI3C/kpRP98BVKFbSPXmSxycngfFD0cYZV5YNAUo4wOJP0X0Gch5dFx\nG4+5UA6ksqFMORRvcvCB84/QkQaU44EvVT6qVWm3wu3XqmwF5pj0L0qIbMhgD7eS74f3FZfypByQ\nckPBzzATlb/aegtD3OAP0C5dUUVOTCd3N8dpyeGJKctIbBC9gl50HAL4XF40R+DGTJyTMyaSFE7c\nHEXClPsXzORfzlIMALuuqCK3H2FrHWGoevsX8/zYGvhgl6HyfD8kZddAEQPP+GE0kcY6b8aeQAkp\nH/G594HjwMMm8KyJcTmuyonp2bDP7i0ybUFS+S16Gtpm0cvYs+TEnfjRpX68hXmcCjLoC5xyaOSl\nDG7Md36WQZSflaT3GLDeBs48uqO4LejfFux87o8sIK9yesfwJ5K1SSZWafOhVLpIPDUs4mDNWKnJ\nlc8WAeYNeitDYt0dNrJoxhhwg1RJ2YhfFk89NmuFelfVZsf8akJdhCHdVo/Hq1V0Bwisd2z0mPjG\nreeJ/eDJRt6yT9938jvg078X9hYbkNo8xSm711AE00vQuxfmP27AvB6XUZDnIhjnmNryY6i3fCpV\nVtTqLu0fbhwY0PbIAcUAAAAASUVORK5CYII=\n",
       "prompt_number": 42,
       "text": [
        "      2              \n",
        "-M_b\u22c5\u03a9 \u22c5e\u22c5x_GB\u22c5sin(\u03b2) \u001b[94m\u001b[1mb_y\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "EGB.cross(M_b * (-g * A1.z)).express(B).subs({theta: 0, zeta: 0}).doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$M_{b} g x_{GB} \\operatorname{cos}\\left(\\beta\\right)\\mathbf{j}_{B}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKYAAAAVBAMAAAA++IifAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nEPHpyg+/mBntn2jHVDan8Zf1E1jT4iAoOQr3AAAAAElFTkSuQmCC\n",
       "prompt_number": 43,
       "text": [
        "M_b\u22c5g\u22c5x_GB\u22c5cos(\u03b2) \u001b[94m\u001b[1mb_y\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "* **Nota1:** Si primero se sustituye y luego se cambia de sistema de referencia, las sustituciones no se actualizan.\n",
      "* **Nota2:** La \u00faltima llamada a `.doit()` es para que se simplifiquen expresiones del tipo `Derivative(0, t)`."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "La ecuaci\u00f3n de momentos para la pala quedar\u00eda, por tanto:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "MrxB, MrzB = symbols('M^r_xB, M^r_zB')\n",
      "k_beta = Symbol('k_beta')\n",
      "Mr = MrxB * B.x + k_beta * beta * B.y + MrzB * B.z\n",
      "Mr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$M^{r}_{xB}\\mathbf{i}_{B} + k_{\\beta} \\beta\\mathbf{j}_{B} + M^{r}_{zB}\\mathbf{k}_{B}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPEAAAAWBAMAAADjpGcAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZqtUdu+7iTJEIs0Q\nmd0e9zMQAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADxUlEQVRIDa1VXWgUVxT+Znd1Zmd3kxgqgtC6\niChoxRRKwZdmaIPQP5qnWhTNWim0QslCU3ywtNu/l1JwfagIbclasBAoTQp9KWvb8RfxxZg2VLAh\nUuKDUGraXWG10ek5d+6de2ecJT54Hvaec+53vm++u3d2AR3W5Uc26+phZQ/Cao3Xbyi91wdU1n11\nVzXg7PW7A8SOZnWCdhzbH7BK7tb+b3dEG9lgKsrN5Pxeo3q2+AYygWd0jDSN9ZOEMrZVaeLD28CG\naNINylEeS37SlVXDUeSCuu7EshTWQkLZXV2mkf5F4FU9+pFOObuoypZKgOPA08B23eAsAqaxJpWt\nb+hkrXMVuCTeJZ6U/fySBnjAel3JTAGRxppULpWGgYlJD/lKRNSbOG1FaNNDqqCBXRgPqqoWqwIi\njZWUB03iUo68eutqJoP7S9ksoQgLVWtWAvOf/vZzDe7NbsoprKR87nGD93i2g6y/zehQOl8Gjv47\nt7ERtpXypS/f/bwODF2ml6F4kB9itIr86NaZAyEuekSksRba7k7CRfi1bgtn8IoclQsrW8Owh8Na\nKQ9+zLXt9/iwSXxKKGM18E6I08pprIX2eB/jFH4abXhuW45Ko6xcKiMrb5RSnv+RYWuxwkeOHn4x\nVJ4BBn3egFaOsUqXhf+CMoMUfhp7sg2rwy0KaZSVezyUyBUONZvXm03xJi98zY0qCxwTWD5tLABX\nqW0CEWcNXRburfuHcRLv9qF5CplhblFIo6x8qYYXfO5FVoqdYsuFu/2HCssjUxaeix3kTjKKQh1O\ngjV0WWjlggag8FYDsx5WVHiOQhpl5fGhHfQpQhJai8XbNrINvA2Q+RON8Ia1h14LYVo5wRq6LLQw\neo2+JInvAV6u4bAHjF15pqaMsvIh4DEfIqRypoKOzXfrJWAfHP5/odOml9z6K8RFnjXrdycnlEtS\nnrxLFzTEZ0YWCP7UyJ9+zt/SU5dGe4MAeIsOfCCmvLIPj3p8t35H/oOxg36ovHIaRfWrqh4xYsWv\na+gnR7h0gmDq/eAOTLygd/A9rYZRYhtsiK3IiqjO8i88Gae4wJ57B5C9I0p92rLk5UzkUjZNvGzx\nuDbqLEG8+Lx5RCLE8hVKdMQiDWojAzjs4wTdNxExoOi4dSRcmvhwyll0AG30vbtzG2nq/nh+iC4Y\nGacIjt2DfXNu5s37Uaozgc9ip5KCn7Sv2TCNqtnk6nGDXmeKL/7+Q6zdP9xNT1Rip5ICPf3c2Ivo\natQY8Ix82TTn7PaXOZVlORTASf0K1O4Dr/8Dfx5HZf5Zz88AAAAASUVORK5CYII=\n",
       "prompt_number": 44,
       "text": [
        "M_xB__r \u001b[94m\u001b[1mb_x\u001b[0;0m\u001b[0;0m + k\u1d66\u22c5\u03b2 \u001b[94m\u001b[1mb_y\u001b[0;0m\u001b[0;0m + M_zB__r \u001b[94m\u001b[1mb_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "lhs = Mr + EGB.cross(M_b * (-g * A1.z)).express(B).subs({theta: 0, zeta: 0}).doit()\n",
      "rhs = (dh_B_dt.express(B).subs({theta: 0, zeta: 0, psi.diff(): Omega, psi.diff().diff(): 0, I_zeta: I_theta + I_beta}).doit()\n",
      "       + EGB.cross(M_b * a_E).express(B).subs({theta: 0, zeta: 0}).doit())\n",
      "lhs.simplify()\n",
      "rhs.simplify()\n",
      "eqxB = Eq((lhs & B.x) - (rhs & B.x))\n",
      "eqxB"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- I_{\\beta} \\Omega \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\frac{d}{d t} \\beta{\\left (t \\right )} - I_{\\theta} \\Omega \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\frac{d}{d t} \\beta{\\left (t \\right )} + M^{r}_{xB} + \\Omega \\left(I_{\\beta} + I_{\\theta}\\right) \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\frac{d}{d t} \\beta{\\left (t \\right )} = 0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAvIAAAAqBAMAAAAjR6IUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nXdLHNiYGrAAAAABJRU5ErkJggg==\n",
       "prompt_number": 45,
       "text": [
        "                 d                          d                                 \n",
        "- I\u1d66\u22c5\u03a9\u22c5cos(\u03b2(t))\u22c5\u2500\u2500(\u03b2(t)) - I_\u03b8\u22c5\u03a9\u22c5cos(\u03b2(t))\u22c5\u2500\u2500(\u03b2(t)) + M_xB__r + \u03a9\u22c5(I\u1d66 + I_\u03b8)\u22c5\n",
        "                 dt                         dt                                \n",
        "\n",
        "          d           \n",
        "cos(\u03b2(t))\u22c5\u2500\u2500(\u03b2(t)) = 0\n",
        "          dt          "
       ]
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "eqyB = Eq(expand_trig(collect((lhs & B.y) - (rhs & B.y), I_beta)))\n",
      "eqyB"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$I_{\\beta} \\frac{d^{2}}{d t^{2}}  \\beta{\\left (t \\right )} - I_{\\theta} \\Omega^{2} \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} + M_{b} \\Omega^{2} e x_{GB} \\sin{\\left (\\beta{\\left (t \\right )} \\right )} + M_{b} g x_{GB} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} + \\Omega^{2} \\left(I_{\\beta} + I_{\\theta}\\right) \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} + k_{\\beta} \\beta{\\left (t \\right )} = 0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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pxXpeyxhSk3hFO8eU2H2jzTUULKAGpP941pi/yZ125X2zPChC9iqSRhw9fV2PvUN+bb2X\nsV/dBk5amgIYpUmrs9U3GlNr5tc3rAhqvG79DQ3k7l+7+i9sPtYgdkKf2MaXfu7IuA+TO3/be4Cb\nDq+9oO6bQuclQGs9T5HUby/vhHBM5hO/zrOV2Ijj94vulzwvq1DsOYtvH6pSsDhaaBlTtVvC38xz\nNjevPVrGmmnq+TaTW8pxQ/frBAEMT6YojtX3VcSM4oby1iAxhFjxew/8Zqj6YpklQsNTJLUIppVY\npimfjGjaoEf1sgtl9e1DVQo2UC9M36k3WlbqTtGf5IzjRjnxAXbJiQ8pfuph7PFN4bPN99XEDKNG\ncrYg/Bz7p5GWDK2KyPNFbLITM/WkkgkpXnFC2Xz7UMMUeqBxvPJ1UyK/mdf3zUv6jlEcamIfQIFP\nDeDzM3Z5tRT0N/PlOSztO0Y/48Yk8KemAuY383xWHOA7RqdmCxJW/38K8Nmz0JC/JCUKnFCB0aa8\nbOZfkhIFTqjArP5mXt8Vn9A3cXitK8AXIPzNvL4rfq1LkbT/hAqY38zr++YT+iYOr3UFzG/m9X3z\na12KpP1WBf4Pit1TwbJYlUUAAAAASUVORK5CYII=\n",
       "prompt_number": 46,
       "text": [
        "     2                                                                        \n",
        "    d               2                            2                            \n",
        "I\u1d66\u22c5\u2500\u2500\u2500(\u03b2(t)) - I_\u03b8\u22c5\u03a9 \u22c5sin(\u03b2(t))\u22c5cos(\u03b2(t)) + M_b\u22c5\u03a9 \u22c5e\u22c5x_GB\u22c5sin(\u03b2(t)) + M_b\u22c5g\u22c5x_\n",
        "     2                                                                        \n",
        "   dt                                                                         \n",
        "\n",
        "                                                              \n",
        "                2                                             \n",
        "GB\u22c5cos(\u03b2(t)) + \u03a9 \u22c5(I\u1d66 + I_\u03b8)\u22c5sin(\u03b2(t))\u22c5cos(\u03b2(t)) + k\u1d66\u22c5\u03b2(t) = 0\n",
        "                                                              \n",
        "                                                              "
       ]
      }
     ],
     "prompt_number": 46
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "eqzB = Eq((lhs & B.z) - (rhs & B.z))\n",
      "eqzB"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$I_{\\beta} \\Omega \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\frac{d}{d t} \\beta{\\left (t \\right )} - I_{\\theta} \\Omega \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\frac{d}{d t} \\beta{\\left (t \\right )} + M^{r}_{zB} + \\Omega \\left(I_{\\beta} + I_{\\theta}\\right) \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\frac{d}{d t} \\beta{\\left (t \\right )} = 0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAtsAAAAqBAMAAACQD4oOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nlyN48hWyjUHbF+pyCfgfliLQ2C5ZoIoAAAAASUVORK5CYII=\n",
       "prompt_number": 47,
       "text": [
        "               d                          d                                   \n",
        "I\u1d66\u22c5\u03a9\u22c5sin(\u03b2(t))\u22c5\u2500\u2500(\u03b2(t)) - I_\u03b8\u22c5\u03a9\u22c5sin(\u03b2(t))\u22c5\u2500\u2500(\u03b2(t)) + M_zB__r + \u03a9\u22c5(I\u1d66 + I_\u03b8)\u22c5si\n",
        "               dt                         dt                                  \n",
        "\n",
        "        d           \n",
        "n(\u03b2(t))\u22c5\u2500\u2500(\u03b2(t)) = 0\n",
        "        dt          "
       ]
      }
     ],
     "prompt_number": 47
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Si tomamos la ecuaci\u00f3n en $y_B$ y linealizamos teniendo en cuenta $|\\beta(t)| \\ll 1$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "beta_ = Dummy('beta')\n",
      "eqyB_lin = Eq((eqyB.lhs\n",
      "               .subs(beta, beta_)\n",
      "               .series(beta_, n=2)\n",
      "               .removeO()\n",
      "               .subs(beta_, beta)\n",
      "               / I_beta)\n",
      "              .expand()\n",
      "              .collect(beta))\n",
      "eqyB_lin"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(\\Omega^{2} + \\frac{M_{b} e}{I_{\\beta}} \\Omega^{2} x_{GB} + \\frac{k_{\\beta}}{I_{\\beta}}\\right) \\beta{\\left (t \\right )} + \\frac{d^{2}}{d t^{2}}  \\beta{\\left (t \\right )} + \\frac{M_{b} g}{I_{\\beta}} x_{GB} = 0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAgMAAAA1BAMAAAAuSfX9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nP/a6LeWV7WOOAAAAAElFTkSuQmCC\n",
       "prompt_number": 48,
       "text": [
        "\u239b          2            \u239e          2                       \n",
        "\u239c 2   M_b\u22c5\u03a9 \u22c5e\u22c5x_GB   k\u1d66\u239f         d          M_b\u22c5g\u22c5x_GB    \n",
        "\u239c\u03a9  + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u239f\u22c5\u03b2(t) + \u2500\u2500\u2500(\u03b2(t)) + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 = 0\n",
        "\u239d           I\u1d66        I\u1d66\u23a0          2             I\u1d66        \n",
        "                                 dt                        "
       ]
      }
     ],
     "prompt_number": 48
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "En general, el t\u00e9rmino gravitatorio es despreciable frente al t\u00e9rmino centr\u00edfugo:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "assert eqyB_lin.lhs.is_Add\n",
      "# Filtramos t\u00e9rmino gravitatorio\n",
      "args = [add for add in eqyB_lin.lhs.args if not add.has(g)]\n",
      "# Reconstruimos la ecuaci\u00f3n\n",
      "eqyB_lin = Eq(eqyB_lin.lhs.func(*args))\n",
      "eqyB_lin"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(\\Omega^{2} + \\frac{M_{b} e}{I_{\\beta}} \\Omega^{2} x_{GB} + \\frac{k_{\\beta}}{I_{\\beta}}\\right) \\beta{\\left (t \\right )} + \\frac{d^{2}}{d t^{2}}  \\beta{\\left (t \\right )} = 0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAZcAAAA1BAMAAAB/67tvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiUSZq1TvELvdZiIy\nds1Wk1T5AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAJmUlEQVRoBc1aa2xcRxX+7tq71/veEKpSkJ1F\nRSgRUbxRqoJAdSxRNREieFvRQhWiuKUVgRa4pVEEP4JXahWkQInLq6ASsSiCUozqC2krZAmygl+k\nEl7LKSpUjhdCVRURx2lSOW2TmHNm7ux9P9blR460M+fxnXNn5s7j3LGBa5zWP2Jc4y1M3jzN7O8k\nR1/jyKxZeLu3JmrV3vBrR+/v1TVTzb/em8/63uDvAJ3q9OycXunJRRuNgufG3mDz4r56FCrCph+a\ntK3P2GwCbuPngWw7AdCG3BW9X/SfJ2hhrGo79MiVarbDQMXmE3A0jr1NG/3e6Kgz95E9NxYNirKW\n67ZVv9/m47nUCnKOkYh3wEAMfOq7FOSpfQkihUCGDYfhLgcfy2qTeD82x8IcgL3OZzn0iq3dDOQ/\ndEWJvddDTpdsxSnF8KVa4ZU7vxUDcpm/75L8Qm0RKOYuS4M+OG/6IdGaf+LYg11EOvmx8ezgbD2z\nurrc9Y1ncjHR8829LXymfxlbOhRsk5nvZWj58frrD32NlmVxvskS4sZOgLhIPYAjRldKxmQb0bic\neaKu1zMNaDWabvfj4VY03mdNX26z7no8IUzTSRt4oIkRX7AYxWzVAQhI6/pQbhdxoo1SHej/0sJT\nNlz7yIdtIZTru1olW87EFwRkQnQtFG0btht43JaScbS8uxSU1vUhU9uN6RbKJlAe7WKJeR8+5RRD\n+OzLjNKQlr0odUJwXvU+mqBeXZzMp4iioLRuBqVXm7gNGF7YgXJFQbl+ErPUwTiaaE3UTTy35SUJ\nLMasURVOP4/0sq6kZHXhLQcuKK2bQt+XATpm/m7MI0OdkQPMXvfiRNXhHcJOY6JawW5odQHIJ9zk\n9Tcw0NkdEjNE7Rkof1o3ieJlFKgFj+EZFDt42Pky4s4ofuhOlFsVVFCsySYkPX7P4unRSkirQ9QD\n49Lw3Iuikb607rY32+l2+tCbO+g93ATM/6PJ+PweOWZHpXNkeRi5W/h99lktO2Sh87/mJ55kKdfi\n0kP7517YVXXoZqZIKDRZ8xsuAoj2XKYtH8uPtKiOSOtW0M2rUmfM4/8ltFajIhHpbWRaErlBjBrw\nov4dGpWGUH5SmqLKYv3rLeD3AtJPnJv0rfOkmxDtydHKKVKKGpXWPZFrqADTTfomqAPHlCK2TtWw\n1QJNVwWTa2EjUDKAb5goxQagt5FqAK/SIfwk0PHiS3VWD7dZP8vWc6ad1ukmq13053VKLFwlrvQ2\nUpVUS+l8tTtC8Q9z1HlBs+KJmAHuBuZItZcGsmIZ3V6Wkqt8E2lavoRLL0s3hxH4JTBEkeqsHONg\nGxxpnXUouBy6gvjC0S7gr3d+1ugqvYw7QtE2y7nAm+KtwAOkH6HRHrfsbi/bCRSgfxQZE+jrAAcc\nFsE+Sj0xsNhi4Wd1KqYbdloXGpTRWZqQyF3CV1dXWQwmdwRHZ8qjwqHNyVdqkng+518ROhp26mMg\nUYAPmthENs67ORtx0UWaY3VsN1l5pEnF4igVFoUGZfsAn01F8mc6eOp3c0EtcEf4kwRzWW5wmXrh\nltdavIOkdl7aZW88bi8GWrT5+OmfCtgHjjza9u08+gWxYGRnFmvkM9ZUnhEjxJAiuaJfnrZ6JXc0\n26EXdevWgyY2Xty2U4xORIQMoSmGvu7mFgaacg3QESYptDPth9Y/CBwmFC0x5DoMH1piOkNc/jyt\n+wrGDFYfmKTiR8xZFBqU7foXqciOMwvN0FbSJvL/gU4HSG4cmtSHT5gMP4tyNT6Bs3W5BngvEBT6\n3ApwGvghgWiJ8V7gItUZoeSPmgKvBEUqaHnVT7S6TBoIageRjlKDqtkW8Glq2ChUDhEUgYBqFGgN\npJaRqco1gFlhst+n+7nLIoka5gRKLrECaZxEiRzvy2NSd5bGqqHM2tLSa19ZWqop2VeX2sB7lLZM\nAu6h30lqGB30HWKjIshXulm8Ru7MBP1kZyK8Ui2x/VJnRCrt7Qxo4tO+TF8NROYfqSGmyawkNa5K\ndtX5fIdS5rylo5diFK2buokWTouA9hi7PFnITnJZEVstT7Nhertim2J12HNpVvJOTtOM3qchd0Hc\n8BOmH7Dfj2m6GJDJRaXfwHF+gKKwoMKuYQfyNY5PTzcWUTR4C2AaXnhpVDDhzYJcMx3geVNsAEP4\nHCI2APkFWKJFTmfSYXrpnXTL2gCsJ3H1N7GW5DnziduruH1PzbZGdqb/F+9C+rp5gR6uH6HDvFxD\neoSeNAQ8bcgooREyHQYcRuF71LJR4CbK27CNdUR+L/kFSNveX6rAu3lZHqTC0VThl63rtNlNE4QO\n/9UmplcpqCJ/UGWhWlt9C7nVK0Kzf+Hge5viIMuQ+yO0BOoSGRpBnDOpk4PrDJozk5Q7LpDDv6RT\nQGfkF+Dmn89dT5BN9IIeo+eUqhZeVfrgqRYtG69amj1NObV6xlR+QTWv+hNVgDbEvRbQEcHtLdIZ\nOUf5Y1VQflzWrs7wTbP6AqxIe9YK/rwF91S0oQVRqunWXnWLPukoeCsprCB/j2VzRnB5z/ITVWeO\nSbSmWuH0oqHhVJiTLrWcczWJH5SVt7TSPq/aI6doH48k7ePrfgvccGHbzqof5/amLZSSIAs1YArm\nm34nmoMrIkMGDRQvK0HfFqXeEZWvkMvRp/YotJ7vSZwB3N5i/6RjRlC+ISrZRqnqlnTTzNNXo7fB\nx4wg+XFmDYGls6uBZZsP50qJUGH+bm/rmLbAM1znzCDXUk1sLNZMlIh8k+spKfhKddz5DC6F2IFc\nml4Et/e+ZK5808wZcuQXoCeUKyPz2GxRbBMLH7UVPXEu74RXTfKmmTLkyC9AbzPu8yqC5A0tmrtG\n1giyxetc3p67rTDvA+KmeQTRX4Be7+1eRZB8jpRTyKyxMy7vgcmgB/h0tNE/LjNknylCMVyNMCrT\nJWIq2KPEHmuXd7mdyFvcNPd82ZytxQdPXaE04o6zjXhkEMLtTcltApI3zSJDToDuQnJW9t5VBDAa\n/d0sbeK6AFMCldv73wk86JQUN80iQ06EVyCRASkhuM4uiwRE5snBkAityztF37RJSNw0iww5CbqL\n4dQnmp49d5HuvdG9k4xGe61u74EEs5ojiJtmkSF740XKdNAmIcqm6IthrdT1jvkPirXGV3556w/J\nSg6pb0RfJcSUQK284/6DIkGoaMjd0WbL+vLCGte/8FfeSWdZoiYFgfpGg7ReXdur6ElW3r/qyWst\n4F1JnFRzkmD9GMs71fGb/s+avmp8wEICTHgU5b3fCMe8c8v/ANubhMq6H+5RAAAAAElFTkSuQmCC\n",
       "prompt_number": 49,
       "text": [
        "\u239b          2            \u239e          2          \n",
        "\u239c 2   M_b\u22c5\u03a9 \u22c5e\u22c5x_GB   k\u1d66\u239f         d           \n",
        "\u239c\u03a9  + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u239f\u22c5\u03b2(t) + \u2500\u2500\u2500(\u03b2(t)) = 0\n",
        "\u239d           I\u1d66        I\u1d66\u23a0          2          \n",
        "                                 dt           "
       ]
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Vamos a manipular convenientemente el lado izquierdo de la ecuaci\u00f3n para sacar factor com\u00fan $\\Omega^2$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "eqyB_lin = Eq((eqyB_lin.lhs / Omega ** 2)\n",
      "              .expand()\n",
      "              .collect(beta)\n",
      "              * Omega ** 2)\n",
      "\n",
      "assert eqyB_lin.lhs.is_Mul\n",
      "for factor in eqyB_lin.lhs.args:\n",
      "    if factor.is_Add:\n",
      "        adds = factor.args\n",
      "    else:\n",
      "        mul = factor\n",
      "\n",
      "eqyB_lin = Eq(mul * adds[0] + mul * adds[1])\n",
      "eqyB_lin"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\Omega^{2} \\left(1 + \\frac{M_{b} e}{I_{\\beta}} x_{GB} + \\frac{k_{\\beta}}{I_{\\beta} \\Omega^{2}}\\right) \\beta{\\left (t \\right )} + \\frac{d^{2}}{d t^{2}}  \\beta{\\left (t \\right )} = 0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAacAAAA1BAMAAAAXGDlXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 50,
       "text": [
        "                                     2          \n",
        " 2 \u239b    M_b\u22c5e\u22c5x_GB     k\u1d66 \u239e         d           \n",
        "\u03a9 \u22c5\u239c1 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u03b2(t) + \u2500\u2500\u2500(\u03b2(t)) = 0\n",
        "   \u239c        I\u1d66           2\u239f          2          \n",
        "   \u239d                 I\u1d66\u22c5\u03a9 \u23a0        dt           "
       ]
      }
     ],
     "prompt_number": 50
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Y ahora sustituyo:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Omega_beta = Symbol('Omega_beta')\n",
      "eqyB_lin = Eq(eqyB_lin.lhs.subs(\n",
      "    Omega * sqrt(1 + M_b * e * x_GB / I_beta + k_beta / (I_beta * Omega ** 2)),\n",
      "    Omega_beta))\n",
      "eqyB_lin"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\Omega_{\\beta}^{2} \\beta{\\left (t \\right )} + \\frac{d^{2}}{d t^{2}}  \\beta{\\left (t \\right )} = 0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMMAAAAvBAMAAACs12NgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ncnC6pKpq9xT/AtfmDioYj8y1AAAAAElFTkSuQmCC\n",
       "prompt_number": 51,
       "text": [
        "             2          \n",
        "  2         d           \n",
        "\u03a9\u1d66 \u22c5\u03b2(t) + \u2500\u2500\u2500(\u03b2(t)) = 0\n",
        "             2          \n",
        "           dt           "
       ]
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Esta es la **ecuaci\u00f3n diferencial de batimiento en vac\u00edo**, donde $\\Omega_{\\beta} = \\Omega \\sqrt{1 + \\frac{M_{b} e}{I_{\\beta}} x_{GB} + \\frac{k_{\\beta}}{I_{\\beta} \\Omega^{2}}}$."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Ap\u00e9ndice: Mejoras en SymPy"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Nota:** No se pueden definir ecuaciones vectoriales."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "try:\n",
      "    Eq(Omega * A.z, 3 * A.z)\n",
      "except Exception as e:\n",
      "    print(e)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "SympifyError: Omega*A.z\n"
       ]
      }
     ],
     "prompt_number": 52
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Issue:** \u00bfFuncionamiento extra\u00f1o? Yo esperar\u00eda que `Eq(a, 1) + 1 == Eq(a + 1, 2)`. Issue #2646 https://github.com/sympy/sympy/issues/2646"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy.abc import a\n",
      "Eq(a, 1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$a = 1$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADAAAAAPBAMAAABQAFHMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIquJdjLdEETvu2aZ\nVM0GsGrEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAgklEQVQYGWNgwAqeYxVlYJD6jl2i6hAOCQYW\nvBKMyq7p6CaCdXQKcH0ASdi9A4E3YDUgCfbZDDwLwDwkAiTB852BzQFJDMwESXAuYKgPwCZRb8Bw\nn6EAKGOWBgJJcB31FxjW8QqgaQEZxefAncSMJg72ILuKqZAHmgSP5jdtNCEEFwCGsCO4LdYyMwAA\nAABJRU5ErkJggg==\n",
       "prompt_number": 53,
       "text": [
        "a = 1"
       ]
      }
     ],
     "prompt_number": 53
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Eq(a, 1) + 1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1 + a = 1$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAQBAMAAACYUF7xAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZlSJ\nRDLkM64aAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAvElEQVQoFWNgYBBiIAIwKgAVmXwiQiWzP1Cp\nShhMKZMBTj1MrvlApQzsMKXcBTiVMjDUU6zU7PhLkBkoALupvM0M/iBlPIJgEADWgl2pxgKGEBQT\nQRzsSuMVGKbjUwq0TbJRUBBkWwsD0z98SoFy0MBi+sPA/YEJyOeZCQYJYG1YHcD0m4FFYDNYHonA\nqpRBnGFlwgEkVWAmSClrxs8siDgsttRqHqdfQFMa1j9tAZIQTCmSEC4mL7I+LIoAB4gzBU00pKoA\nAAAASUVORK5CYII=\n",
       "prompt_number": 54,
       "text": [
        "1 + a = 1"
       ]
      }
     ],
     "prompt_number": 54
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Si utilizo un objeto `RigidBody` para definir la pala y calcular el momento cin\u00e9tico, los resultados no son tan \u00fatiles."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy.physics.mechanics import RigidBody"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 55
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Definimos la pala como un `RigidBody` de centro $GB$, sistema de referencia fijo $B$, masa $M_b$ e inercia $I_B$ alrededor de $E$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Pala = RigidBody('Pala', GB, B, M_b, [I_B, E])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 56
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "SymPy calcula el tensor central de inercia autom\u00e1ticamente:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Pala.central_inertia"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$I_{\\theta}\\mathbf{i}_{B}\\otimes \\mathbf{i}_{B} + (I_{\\beta} - M_{b} x_{GB}^{2})\\mathbf{j}_{B}\\otimes \\mathbf{j}_{B} + (I_{\\zeta} - M_{b} x_{GB}^{2})\\mathbf{k}_{B}\\otimes \\mathbf{k}_{B}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAhwAAAAaBAMAAAAOFph7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nxH3iNqvo+gAAAABJRU5ErkJggg==\n",
       "prompt_number": 57,
       "text": [
        "I_\u03b8 \u001b[94m\u001b[1mb_x\u001b[0;0m\u001b[0;0m\u2297\u001b[94m\u001b[1mb_x\u001b[0;0m\u001b[0;0m + \u239b             2\u239e\n",
        "\u239dI\u1d66 - M_b\u22c5x_GB \u23a0 \u001b[94m\u001b[1mb_y\u001b[0;0m\u001b[0;0m\u2297\u001b[94m\u001b[1mb_y\u001b[0;0m\u001b[0;0m + \u239b              2\u239e\n",
        "\u239dI_\u03b6 - M_b\u22c5x_GB \u23a0 \u001b[94m\u001b[1mb_z\u001b[0;0m\u001b[0;0m\u2297\u001b[94m\u001b[1mb_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 57
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "E.set_vel(A, E.v2pt_theory(O, A, A1))\n",
      "Pala.angular_momentum(E, A)  # \u00bfQu\u00e9 estoy intentando hallar?"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$I_{\\theta} \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\psi} - \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\dot{\\theta}\\right)\\mathbf{i}_{B} + (M_{b} e x_{GB} \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\psi} + \\left(I_{\\beta} - M_{b} x_{GB}^{2}\\right) \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\theta\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\psi} + \\operatorname{sin}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)} - \\operatorname{cos}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta}\\right))\\mathbf{j}_{B} + (M_{b} e x_{GB} \\operatorname{cos}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\psi} + \\left(I_{\\zeta} - M_{b} x_{GB}^{2}\\right) \\left(\\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\psi} + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\beta} + \\operatorname{cos}\\left(\\theta\\right) \\dot{\\zeta\\left(t\\right)}\\right))\\mathbf{k}_{B}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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MyuzMyqAsDcnOHWz3/f1Y1x96xyUX/fg4YjrfGzE8ITjCKID/aoRms8Y2CZUV\nnQHO9VSVtg5wwSGa+tIQbQ/EdLXTtxn5wS+qj0ku+nZ0ZBSrqyMDJQ6kf0ZtEioe1YHO9FSVZpYG\nuuawjH1yWIZHye6Y5GKUQtYflulR/0TUn1sprU1CpUIy0IGeqlJtfqBrDsnYeKDcqPM/GV5uNEoD\n0P/qAGyMg4lNQg01Sz2G98qhghiQcf3nYAZkblTNjEUuRjV4vePiv0XRu/gYS24SapjJ67EqfXoM\nHhNUPjHMQI2O7XHIxehEq28kn+70rTpmipuEGmLCeq1KhRuHCGJApvEv4fxEtHHIxTMgEdc8A3zo\nzYVNQvUWp76keq5Kr+vL/DFV+sdjutpxXGwMcnEcozOgpSutARkaAzObhBpekmJV6f8B76210/+1\nNTUAAAAASUVORK5CYII=\n",
       "prompt_number": 58,
       "text": [
        "I_\u03b8\u22c5(sin(\u03b2)\u22c5cos(\u03b6)\u22c5\u03c8\u0307 - sin(\u03b6)\u22c5\u03b2\u0307 + \u03b8\u0307) \u001b[94m\u001b[1mb_x\u001b[0;0m\u001b[0;0m + \u239b                              \u239b             2\u239e                                                                            \u239e\n",
        "\u239dM_b\u22c5e\u22c5x_GB\u22c5sin(\u03b8)\u22c5cos(\u03b6)\u22c5\u03c8\u0307 + \u239dI\u1d66 - M_b\u22c5x_GB \u23a0\u22c5((-sin(\u03b2)\u22c5sin(\u03b6)\u22c5cos(\u03b8) + sin(\u03b8)\u22c5cos(\u03b2))\u22c5\u03c8\u0307 + sin(\u03b8)\u22c5\u03b6\u0307 - cos(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307)\u23a0 \u001b[94m\u001b[1mb_y\u001b[0;0m\u001b[0;0m + \u239b                              \u239b              2\u239e                                                                           \u239e\n",
        "\u239dM_b\u22c5e\u22c5x_GB\u22c5cos(\u03b8)\u22c5cos(\u03b6)\u22c5\u03c8\u0307 + \u239dI_\u03b6 - M_b\u22c5x_GB \u23a0\u22c5((sin(\u03b2)\u22c5sin(\u03b8)\u22c5sin(\u03b6) + cos(\u03b2)\u22c5cos(\u03b8))\u22c5\u03c8\u0307 + sin(\u03b8)\u22c5cos(\u03b6)\u22c5\u03b2\u0307 + cos(\u03b8)\u22c5\u03b6\u0307)\u23a0 \u001b[94m\u001b[1mb_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 58
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Pero al calcular el momento cin\u00e9tico, SymPy no aprovecha el hecho de que el punto $E$ es fijo en el sistema $B$. Adem\u00e1s, he tenido que definir expl\u00edcitamente la velocidad del punto $E$ en el sistema de referencia $A$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Otros:\n",
      "\n",
      "* Si no se conoce la velocidad del punto, \u00bfhay alguna manera de calcularla autom\u00e1ticamente? As\u00ed no tengo que definir velocidades.\n",
      "* Para lo anterior, puede ser \u00fatil tener la noci\u00f3n de `Origin` de un `ReferenceFrame`.\n",
      "* Ahora mismo los `Point` est\u00e1n un poco desligados de los `ReferenceFrame` en s\u00ed, y esto choca con la intuici\u00f3n al principio. \u00bfNo ser\u00eda m\u00e1s apropiado definir la posici\u00f3n de un punto en un determinado sistema de referencia, en lugar de la velocidad?\n",
      "* Para el momento cin\u00e9tico (`angular_momentum`) hay expresiones m\u00e1s inteligentes si el s\u00f3lido tiene un punto fijo. \u00bfEsto se puede detectar, o al menos dar una forma de pedirlo expl\u00edcitamente?"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}