{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h1 style=\"text-align:center\">Mid-Term 1 Solutions</h1>\n",
    "<h3 style=\"text-align:center\">MCHE 513: Intermediate Dynamics &mdash; Fall 2018</h3> \n",
    "<p style=\"text-align:center\">Dr. Joshua Vaughan <br>\n",
    "<a href=\"mailto:joshua.vaughan@louisiana.edu\">joshua.vaughan@louisiana.edu</a><br>\n",
    "<a href=\"http://www.ucs.louisiana.edu/~jev9637/\">http://www.ucs.louisiana.edu/~jev9637/</a></p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# We'll first import the necessary modules from SymPy\n",
    "# Import the SymPy Module\n",
    "import sympy\n",
    "\n",
    "# Import the necessary sub-modules for kinematics\n",
    "# from sympy.vector import express\n",
    "\n",
    "# Import the necessary sub-modules and methods for dynamics\n",
    "from sympy.physics.mechanics import dynamicsymbols\n",
    "from sympy.physics.mechanics import inertia, Point, RigidBody, ReferenceFrame\n",
    "from sympy.physics.mechanics import linear_momentum, angular_momentum\n",
    "\n",
    "# initiate better printing of SymPy results\n",
    "sympy.init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 1\n",
    "In Figure 1, a disk of radius $R$ spins about link BC at a rate of $\\dot{\\phi}$. Link ABC rotates about axis $Z$ at a constant rate, $\\dot{\\psi}$. The angle from link AB from vertical, $\\theta$, is also variable. The angle of the bend in the ABC linkage, $\\beta$, is fixed.\n",
    "\n",
    "  a. Write the angular velocity and angular acceleration of the disk. For each, be sure to indicate how to resolve all the components into the same frame.    \n",
    "\n",
    "  b. What is the velocity of point D?\n",
    "\n",
    "  c. What is the acceleration of point D?\n",
    "  \n",
    "<p style=\"text-align:center\">\n",
    "\t<img src=\"http://shared.crawlab.org/disk_on_bent_rod.png\" alt=\"A Spinning disk on a Bent Linkage\" width=50%><br>\n",
    "    <strong>Figure 1: A Spinning disk on a Bent Linkage</strong>\n",
    "</p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "### Part a."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Define the genearlized coordinates and other dynamic symbols\n",
    "psi, theta, phi = dynamicsymbols('Psi theta phi')\n",
    "\n",
    "# Also define the first derivatives\n",
    "psi_dot, theta_dot, phi_dot = dynamicsymbols('Psi theta phi', 1)\n",
    "\n",
    "# Define the symbols for the other paramters\n",
    "L1, L2, R, beta = sympy.symbols('L_1 L_2 R beta')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Define the Newtonian reference frame\n",
    "N = ReferenceFrame('N')\n",
    "\n",
    "# Define a body-fixed frame along AB, but with Y parallel to the ground\n",
    "XYZ = N.orientnew('XYZ', 'Axis', [psi, N.z])\n",
    "\n",
    "# Define a new frame relative to the last, such that its z-axis is along AB \n",
    "prime = XYZ.orientnew('prime', 'Axis', [theta, XYZ.y])\n",
    "\n",
    "# Finally, define the last frame such that its z-axis is aligned with BC\n",
    "xyz = prime.orientnew('xyz', 'Axis', [beta, prime.y])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now easily get the rotaion matrices between these frames."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left[\\begin{matrix}\\cos{\\left (\\theta{\\left (t \\right )} \\right )} & 0 & - \\sin{\\left (\\theta{\\left (t \\right )} \\right )}\\\\0 & 1 & 0\\\\\\sin{\\left (\\theta{\\left (t \\right )} \\right )} & 0 & \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡cos(θ(t))  0  -sin(θ(t))⎤\n",
       "⎢                        ⎥\n",
       "⎢    0      1      0     ⎥\n",
       "⎢                        ⎥\n",
       "⎣sin(θ(t))  0  cos(θ(t)) ⎦"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The rotation matrix from XYZ to x'y'z'\n",
    "R_theta = prime.dcm(XYZ)\n",
    "R_theta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAM4AAABLCAMAAAAPvEidAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n2G+Rh03JoM+p2QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\cos{\\left (\\beta \\right )} & 0 & - \\sin{\\left (\\beta \\right )}\\\\0 & 1 & 0\\\\\\sin{\\left (\\beta \\right )} & 0 & \\cos{\\left (\\beta \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡cos(β)  0  -sin(β)⎤\n",
       "⎢                  ⎥\n",
       "⎢  0     1     0   ⎥\n",
       "⎢                  ⎥\n",
       "⎣sin(β)  0  cos(β) ⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The rotation matrix from x'y'z' to xyz\n",
    "R_beta = xyz.dcm(prime)\n",
    "R_beta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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8f4yThUbQxRc5MRh3VKWrEHDThC+zUQSe9MmQujFmiY3fbFU5liTkFtOvKozKvadkxnR5\nW6wbTvYlrRh7/hhMd4EQOnH7OTv9boxZYqcF0G9fpQIsdv9/YFqleMWGyf/AtHNm5jPFhvXRq1Hg\n6MxM478lOepzBMFEwgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\sin{\\left (\\beta \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} + \\cos{\\left (\\beta \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & 0 & - \\sin{\\left (\\beta \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} - \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\beta \\right )}\\\\0 & 1 & 0\\\\\\sin{\\left (\\beta \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\beta \\right )} & 0 & - \\sin{\\left (\\beta \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} + \\cos{\\left (\\beta \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t))  0  -sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β)\n",
       "⎢                                                                             \n",
       "⎢                 0                    1                   0                  \n",
       "⎢                                                                             \n",
       "⎣sin(β)⋅cos(θ(t)) + sin(θ(t))⋅cos(β)   0  -sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t))\n",
       "\n",
       "⎤\n",
       "⎥\n",
       "⎥\n",
       "⎥\n",
       "⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The total rotation matrix from XYZ to xyz = R_beta*R_theta\n",
    "R_total = xyz.dcm(XYZ)\n",
    "R_total"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Define a body-fixed frame on the disk and use it to set the angular velocity\n",
    "w_disk = ReferenceFrame('w_disk')\n",
    "\n",
    "# Express the angular velocity in the \"easiest\" coordinate frame to do so\n",
    "w_disk.set_ang_vel(N, psi_dot * N.z + theta_dot * XYZ.y + phi_dot * xyz.z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then, we can express it in the frame of our chosing, using the `.express(frame)` method on the resulting vector."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\phi}\\mathbf{\\hat{xyz}_x} + \\dot{\\theta}\\mathbf{\\hat{xyz}_y} + (\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\phi} + \\dot{\\Psi})\\mathbf{\\hat{xyz}_z}$$"
      ],
      "text/plain": [
       "                                      d                d                ⎛\n",
       "(sin(β)⋅cos(θ(t)) + sin(θ(t))⋅cos(β))⋅──(φ(t)) xyz_x + ──(θ(t)) xyz_y + ⎜(-sin\n",
       "                                      dt               dt               ⎝\n",
       "\n",
       "                                  d          d       ⎞\n",
       "(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅──(φ(t)) + ──(Ψ(t))⎟ xyz_z\n",
       "                                  dt         dt      ⎠"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Either XYZ\n",
    "w_disk.ang_vel_in(N).express(XYZ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA1wAAAAbCAMAAABfjn6/AAAAM1BMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADxgEwMAAAAEHRSTlMAIma7zZnddlTv\nRIkQMquPBBGH8AAAAAlwSFlzAAAOxAAADsQBlSsOGwAACohJREFUeAHtW4myrKoOdR7Rx/9/7UsC\nCSCD2Hbde+qetmqLimuxkjC3u2l+x88DPw/8PPCHeEDp4XtKvkr2PVl/EdNdBO7y/yJX/ROmTucX\nS/kq2Rd1/T1UdxG4y/97PPXvWqq+VfzXiL4l6MeT8cBNpNoug7t57HBt17XwMp1uQF4241WHAh6C\nPR65fEXI4Ka5WPNAWDsNN74WrXxhjOc7Tj8gAtkGzSlzVacOePFADQODvxZMjAMX/IEegDI+lPQg\nnFI8XmRwN5FqeyZJp8c6JzMcbl4a1R/w0olNrPZg/NarZQRQNfhO0GNCVMxqmiayJhC2nYPWw7ml\nrNzWYVqfuKBpWGtI9wGRGOAMCTnl7s59CQ8ItrkDsz2Bzxz8yZUzJIpIFQ3jr5JCaaV4cjE7XYQ4\nm3cXqeGmOhzakHNRnAquw3a1YRNVT1b6Fn+s0Nlr+KsG3wh6TogWFay5CltXBMSH0tAL7ZnM+HV8\nIlqD7A+InAFiSEDp3dy4D/roQjxvwGLP1Wde+bWXYkhJT4EsV78iabl4Crep/xEO8u8i1eGo8cEh\nODP6LhNy7OlBLkXP+AkHf708A3+dkNXYuURojVi1LTjrW9FUtaDk8KApIZ3C5+Hd4vnIGR+8U0fU\n+ExsAKcBX82NAD+JJ4OdPeIzLtsXy89KKVMmI1IC2jzGx5KctHQ8I6U2YA7nir+J1HQzcDmi8Epw\npuiTdtBUfb9t8R0hNLawB+BQirl7RVi0RoSpCdVS4xoezv+cYi92nvEuv/7KY2rYAE7rWeybAvwk\nnhbs2SM+YyG+WH5WSl/pAeK8JCctHc+r0g1Hcjgcztzfn1sacUrvtYclD15yOArGZuvakFyMBEhz\nw/gJB86DRq6mElwW9AEhrLjEC0lrRNiuwRfYuJTGzmTWcCzNAufe3AB8hzuYpWk9DaumXiM034ud\naP2IyB+52ABOwyKDu7L7wCh8OxfPMljsAQbxmS3cM9s+KSbOkKKeLAfjU5KctEs8DdtVqayKEPco\nUnN2VnjMXTcPsFDGenQM63jM80mWogbBbUc7zsNkZ0O70Klx30ecOql9nndsc5YQ0YLfNObMdG4c\nmF65nqoEZQjr1EClSlojwkgoNq7WLkRHjc1yOsn8VZP1PUZj6uHRpBMjuYudr/UDIr9xcTg4vfoO\n7qvcl/NAFdi3JwqmM9toexcRqFaf1y8Jp6l4fjxRW6D02NqxXczszuAeRKqX/U5js5wVbSJi1Rlo\nxjf10LBazZNIwc3QevZxtaNbx1sa7QoN64Dx7KDxGcCOEMuw+FOfcKzQ1cMhYLq7nhy+JChNWKkG\nmnnSGhG24BCLwdh4SBqg69ntfiv0atCiFO7QNOiQUeNAdz1c7AKtz4n8WsDh4PRaKMh6E886cGCP\n+MxKcWbTg5cRaSrxSUlOWhxPEOcrheEB16Bmc8Ti6iM1xety442N6giOVD01rt4tjvAFwZlBE4Y4\nOsxOAFwO+GCByjXRC4c+HCG+afET4ez3RALGF6LD4UuC0oSVamBOR8VerRFhNGBh4+rMWAvdzap7\nHraVpv1CGr7AW5tO7rO62AVanxP5tYDDwWnkPZjoYZv/MJ514MAe8ZmV4symBy8j8qp+OWlxPEGc\npxT7KvyBa6cmYHH1kVrNIkkNkxyD6W4n3c80Ttm6bJoBzwstzoQLK5Lx4WFTZWZIMHiZ5VQD9UwI\n8VWLp+rH7zAYutlYDk6yKgQlCWvV5KxxwrD+YOMyXQ0aAk1I+qcRp4G806HW6zeI+4DHtFIC4Qq0\nPiFqrkzsTk5T/qtyX84DVeDAHuezSCx47W1EavFJSU5aE8bzqrTDiRv2tyNeQF3G84OQSyMxOO+s\ndliUYy9sGxdNK7BrpoNxdivF7BP55dsXF9vFw4pfCJHA4BVVzNGuTFi8KSE6C74gKE24seyyGnCb\n6Veu1jhhB0wHoXExD2iENkRjAsptoaF11MnBzalhDn2Yvgsz+ZCOMdT6nMjvYjkcnHJhXlrjvqwH\nasChPc5nRoOYTbdvI1KJT0vypMXx9EeuHqs/Ni6zEGFcdciz84gWZxE0+0nWZcaZeVRjduIhNtTE\nUz3T6AjRvQavqOlx1Wcw5seHwxcEpQnjfi6phmeFkTWesLbf13Wn71GMxGHvoeOwx6mnyS6zOtop\nTPw2IrUs1No8JvIbF4eDUxbk0ir35TxQBQ7t8XxGIsRsunsbkUp8WpIvLYqn37hWmBXikqs1kxPG\nVUcquwI2zjiBOlmXGUeTRVh52I0OWSvSnBqENistQWAjyRGiey0ee9rdNsibDQ2HLwlKE1aqaTLW\niFWoXK0yUsHdeOKQY7sYmDjwVBAGMWxyA/jvcrhaFmh9TuQ3Lg4Hp5cy4bbOfRkP1IEDewKfSfmi\n62VEzJrrw/oVSgvjGSgdYPqFSy4aweDSVI/6SLltSbHbXCxUhXqkpo7ZnHmQkZ1WiuYx8exn5GrW\nYqtvdpgY0UbisDeOEAuw5Y6za5iNgI2Cy9nhS4LShJVqTJOPrQmFeZ/LqB33B6FJsf0DL8AGnCxC\n9v8m3EJcNW96cC1H4zyt90RLR/PPQ2MVNodrpuxOSfkVl9a5L+OBOrBnDxjHNSEWi09eRqQWn5R0\nkebFk7Q6t+JQBWbYtmVMqohUA11sj0txHuqsC1yy7d3cwe9TB6wdTkXnY+n1apcUFrcd3T7vO04h\n6bCbIXCtznGGtgVuhB8k8ENXJjQvWrwa5zEBNu+EZ8aXBWUI69TkrHFWNcGHnvh7MTgfPITrKzxo\nCQwptDe9wm/IsF3a6611DcJvXE5rFVFLv6l5w6arBRLGbDjF/WX3ZTzwyvfkGH+YpQfvIvKqfnnh\nDOMZKT3Grp9H/j0FcTWRUlAZdpq4yGcmhrn6bHCX7ol/F69gict9AE7xvyIsWlMpDJsXTqOvB/yU\nLPuJkOc1ieub9j5HdMIAyZ/M4as+E1vPaYY6/7jogTzM5MSlRj7zxd7RYf4rPYL3S2JJnPp5/nWg\n1HxpSdkJXDpSaoQdL+oE/Wj5RdxdG9ylcSXW7zmeuNwH4BTpK8KiNXXCFvg5+bD7noG+1m0n4nNl\nxrjgleAmS3TAgs6vxj4TW89pwFhzU/TAHUFcauQzX+wdHea/0iN4vySWxKmf518HSr1/44px2UjB\nivtQOCPDZdUnB+IU78wbgtRX+Vnqa7mPwCnWV4QFayqFdTB79iccIrGHPUS5qbjIEsEMk3dlYxq2\nntP4jZsnBQ/cICH7Wmqlz0rEr/QUJD2T5kaPBC4XKTXBgovaIv9TWcnQVB7i+INhm391cQomz67l\nPgILi3fxirBgTa2wc3JfhHqyoD1sbrPey8heZohoJZftCdl6TrPsuYyCB3IQ9/xaaq3PHEN09UoP\nsOUkPZPmpvkpXCZSPazElflVxJtWRhaWHkQ4bwgt4TgvxD8EM4mfviIMwUDLgjj1S6q/hj0+GN1h\nd8NfdtXDL2+afdLLQ3vLBnCafqvwNAI+sTwEP0FmFYWU8NpD1hDPYE6zxWYyHuDgo50BNollVzfD\n+Hv853ighUbKv1D/Oap+Sn4e+A94AP73aOZPq/4D5vxM+Hngz/FAO9A/iv05gn5Kbjzwf1FxfZy/\n6X95AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi}\\mathbf{\\hat{xyz}_x} + \\dot{\\theta}\\mathbf{\\hat{xyz}_y} + (\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} + \\dot{\\phi})\\mathbf{\\hat{xyz}_z}$$"
      ],
      "text/plain": [
       "                                       d                d                ⎛\n",
       "(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅──(Ψ(t)) xyz_x + ──(θ(t)) xyz_y + ⎜(-si\n",
       "                                       dt               dt               ⎝\n",
       "\n",
       "                                   d          d       ⎞\n",
       "n(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅──(Ψ(t)) + ──(φ(t))⎟ xyz_z\n",
       "                                   dt         dt      ⎠"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# or xyz\n",
    "w_disk.ang_vel_in(N).express(xyz)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Once we've properly defined the angular velocity and motion variables, we can get the angular acceleration using the `.ang_acc_in()` method on the frame (or body) of interest. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAvEAAAAbBAMAAAAHVs0fAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nta4svKuo938ABjQEQuA7AAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\ddot{\\Psi}\\mathbf{\\hat{n}_z} -  \\dot{\\Psi} \\dot{\\theta}\\mathbf{\\hat{xyz}_x} + \\ddot{\\theta}\\mathbf{\\hat{xyz}_y} + \\dot{\\phi} \\dot{\\theta}\\mathbf{\\hat{xyz}_x} -  \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} \\dot{\\phi}\\mathbf{\\hat{xyz}_y} + \\ddot{\\phi}\\mathbf{\\hat{xyz}_z}$$"
      ],
      "text/plain": [
       "  2                                          2\n",
       " d               d        d                 d                d        d\n",
       "───(Ψ(t)) n_z + -──(Ψ(t))⋅──(θ(t)) xyz_x + ───(θ(t)) xyz_y + ──(φ(t))⋅──(θ(t))\n",
       "  2              dt       dt                 2               dt       dt\n",
       "dt                                         dt\n",
       "\n",
       "                                                                             2\n",
       "                                                 d        d                 d\n",
       " xyz_x + -(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅──(Ψ(t))⋅──(φ(t)) xyz_y + ───\n",
       "                                                 dt       dt                 2\n",
       "                                                                           dt\n",
       "\n",
       "\n",
       "\n",
       "(φ(t)) xyz_z\n",
       "\n"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_disk.ang_acc_in(N)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As with the angular velocity, we can express it in any frame we like using the `.express(frame)` method on the resulting vector."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\phi} \\dot{\\theta} + \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\ddot{\\phi} - \\dot{\\Psi} \\dot{\\theta})\\mathbf{\\hat{xyz}_x} + (- \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} \\dot{\\phi} + \\ddot{\\theta})\\mathbf{\\hat{xyz}_y} + (\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\ddot{\\phi} + \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\phi} \\dot{\\theta} + \\ddot{\\Psi})\\mathbf{\\hat{xyz}_z}$$"
      ],
      "text/plain": [
       "⎛\n",
       "⎜                                       d        d\n",
       "⎜(-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅──(φ(t))⋅──(θ(t)) + (sin(β)⋅cos(θ(t))\n",
       "⎜                                       dt       dt\n",
       "⎝\n",
       "\n",
       "                      2                          ⎞         ⎛\n",
       "                     d          d        d       ⎟         ⎜\n",
       "+ sin(θ(t))⋅cos(β))⋅───(φ(t)) - ──(Ψ(t))⋅──(θ(t))⎟ xyz_x + ⎜- (-sin(β)⋅cos(θ(t\n",
       "                      2         dt       dt      ⎟         ⎜\n",
       "                    dt                           ⎠         ⎝\n",
       "\n",
       "                                             2      ⎞         ⎛\n",
       "                       d        d           d       ⎟         ⎜\n",
       ")) - sin(θ(t))⋅cos(β))⋅──(Ψ(t))⋅──(φ(t)) + ───(θ(t))⎟ xyz_y + ⎜(-sin(β)⋅sin(θ(\n",
       "                       dt       dt           2      ⎟         ⎜\n",
       "                                           dt       ⎠         ⎝\n",
       "\n",
       "                          2\n",
       "                         d                                                 d\n",
       "t)) + cos(β)⋅cos(θ(t)))⋅───(φ(t)) + (-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅──(\n",
       "                          2                                                dt\n",
       "                        dt\n",
       "\n",
       "                   2      ⎞\n",
       "      d           d       ⎟\n",
       "φ(t))⋅──(θ(t)) + ───(Ψ(t))⎟ xyz_z\n",
       "      dt           2      ⎟\n",
       "                 dt       ⎠"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Express in the XYZ frame\n",
    "w_disk.ang_acc_in(N).express(XYZ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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OPVxM1j6AE2aPR16ZH+eQjJrCTSak2keXEQOj4YiAazCKj5wQ4xtRI08DyLFX4eQFnGnS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\n9i7WvD6ojJ4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$(- \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} \\dot{\\theta} + \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\ddot{\\Psi} + \\dot{\\phi} \\dot{\\theta})\\mathbf{\\hat{xyz}_x} + (- \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} \\dot{\\phi} + \\ddot{\\theta})\\mathbf{\\hat{xyz}_y} + (\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\ddot{\\Psi} - \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} \\dot{\\theta} + \\ddot{\\phi})\\mathbf{\\hat{xyz}_z}$$"
      ],
      "text/plain": [
       "⎛\n",
       "⎜                                         d        d\n",
       "⎜- (-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅──(Ψ(t))⋅──(θ(t)) + (-sin(β)⋅cos(θ(t\n",
       "⎜                                         dt       dt\n",
       "⎝\n",
       "\n",
       "                         2                          ⎞         ⎛\n",
       "                        d          d        d       ⎟         ⎜\n",
       ")) - sin(θ(t))⋅cos(β))⋅───(Ψ(t)) + ──(φ(t))⋅──(θ(t))⎟ xyz_x + ⎜- (-sin(β)⋅cos(\n",
       "                         2         dt       dt      ⎟         ⎜\n",
       "                       dt                           ⎠         ⎝\n",
       "\n",
       "                                                2      ⎞         ⎛\n",
       "                          d        d           d       ⎟         ⎜\n",
       "θ(t)) - sin(θ(t))⋅cos(β))⋅──(Ψ(t))⋅──(φ(t)) + ───(θ(t))⎟ xyz_y + ⎜(-sin(β)⋅sin\n",
       "                          dt       dt           2      ⎟         ⎜\n",
       "                                              dt       ⎠         ⎝\n",
       "\n",
       "                             2\n",
       "                            d                                                d\n",
       "(θ(t)) + cos(β)⋅cos(θ(t)))⋅───(Ψ(t)) - (sin(β)⋅cos(θ(t)) + sin(θ(t))⋅cos(β))⋅─\n",
       "                             2                                               d\n",
       "                           dt\n",
       "\n",
       "                     2      ⎞\n",
       "        d           d       ⎟\n",
       "─(Ψ(t))⋅──(θ(t)) + ───(φ(t))⎟ xyz_z\n",
       "t       dt           2      ⎟\n",
       "                   dt       ⎠"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# or the xyz frame\n",
    "w_disk.ang_acc_in(N).express(xyz)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part b."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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s3pUum50zuNOmboPVKQSI1MpWiyxFrVTFbmJaeMyqLtTQYGQYjXHPfV00YpuNsj5vrR1b\n6K1ITvePHqFN2vP2Lup/kbz8U2DeC+KEo13zxPjXONV13f3DLmx0a2dwKHpHX1vBNzf0jt7dRSnM\nMn6A08NoCwAGO9y/cGnzjL3dUhBsfH/iJ2PBRuKUsM0ypn24Z54iIKMsC3p/Q9/GAlypYTuWlAI1\nxzKLI7LRYGQKhCVgRx+jvQOHP9jHnTBo++81ulipXYrOziLL8dIn//JthOfapg3zv69/jYxFtLgs\nGB/v/1njBaEUxtSMqwsyy7qfkA6eKe/os9pLD/PZ2Eh+p2nERC2E5xmXf6hY0tlqXfuFBx/rOyVc\nZYy3a+0a/Xm++Qq+0LR1+IszTZPbMj8RAuVd6e6idMvfBh+G6S7iUMVwU1BkKWql6qBZLoeUVV2U\nsAYjwyiMV9uw3fxylmxL9qeSjKg0Iup+Q56/tyI53b9dGA2/owFjmMcChZ7Fm8P4PP4BFjl8Bc9x\nwnh3koauwh/XdXvaLYTt9A9/jbTxhRtW0Y62uItSmA5+KjRlP2M32OEuSks/A39HiykINleFGhuJ\no40LtlnG9NXDg4tNjICMFBajeB+g99K0cjarhGNZ7+AwNsdSwRHZaDCdIEUd+wL2fc39eFeBVXov\nP30cRtt4NM3GSu1SdHb51ztm2bP0aOvbpg0b7QAsfQrf4X6ejEW8uCx8AJIkNV4QSmFMzSA01QVJ\nTj/YYtLBM/xvRrCRBr4p2o8GD/PZuEi80zRiAge0cetxvWNLGhk1H1xhXhiY2uhifQqfR5HQi/hS\n7wCsw1+caZrclvmJ1DthV7psDvYB7kKIPURHqqWgbGA3IRg3RfWghrpxOnHF833EMd7dh8mTgHQl\n25L9qSQjKo2YIktqmncb9DYkp/v3OIZpHMNby4q7i17ehkdbD8Ho8sdX4BocxAnbFqnK4Vb8cV13\n/3ALobGIQ5Pz+MJt+z4YmcIHUX4WpTC7oPVhPGxbxm81EDa+ncf/0cagpbeYu5cJgldcFYLCRuJo\n44JtljF9ZHT14vcxAjJSWOC2/m4XR2mlFtuypBSoOZYKjshGgxFSGCwP+9xZWD2DyKOn8GVuX/Pg\n/XeeHT+x4mOldik6u/wdx/g/pty0YbVPwuTU8334VRMQ6cRlgRdcKyaUwpiacWKQ5PSDLSYdPLOf\nnZEGvinaCw/z2bhIvNM0YgIHtHFvmilpYvSgp+UqY2JhEt5MF80numhJxt+P7t/XOt3FGXaFzE1u\ny/xEyGfelS6bO3FH0XPNb+MPGeWbgrKB3eRh1J0T1FA3TlLxCeM5/BVzsrUUsy3Zn0oycelvobc+\nuYx/S4O3D5boPdS359YOzeFnT/ZZtPlP903BZU99Df7mDW/qonA4oXnL+0nBd+GP61rPeaF1yr6t\noYnYJv72ri+1Dq/95+G171//WTxHlIn/fuIpuh2M9sFg/3DwzA8H//qx0z37Vueb99CgCYIHvosq\nbCQOKOOCbYbxtrm1Nmy7bhEjICPKMGHxyJvu+3umoWCDZUkpUHMsFRyRjQIjUzBYDDs5eAUOD/BT\nZvyzBLQGp/cP+iNnLu+GWKldis4uf8sx/oc9Nm8YfPuWR6F265e7FJDoxGVBV2wrIZTAUM2wGGQW\n/QDEpIVnQKVEoUNTtBce5rOxkfzzSpofERM4uEni/GmcTbMljYmIf42E6/d/fvqePpHF70fSNMXf\n6QXYRzPsCpEbm2W2ZX4i5DPvSnPfaD49OPT0oMePNQKxbFeW7CZi6ZqielBDg2GTiW2GMbxn8BWA\nd9+L4JJtyf4sloSIumdRzbsNesvJZf370QuT+BFcY4rimWbvonzGRz+htsCX8MhVYi7V583hNjGe\ndhFl0l3L/McUeN0v5SARvgArwQnpEBCDmfUpos8roCOLZXsWreQJYQWnkGK6mZlsGMEckxQYC8d8\nt2bfNdtld37aHG2swMH90rNz/GsY92B+LHTO37BY6ICc6ZUQQqlcXWQMy0DhBfxOR1Sy2TnFHno2\n3mDeaSlSMY6oWNI5q7WvjEsscP2snCb93XHGupHxlxmp29InEseOsnm9QQgzGTAccaxoV67b5LB/\ns0pgtCRMNCdivDXeekkSbJ/3Zr31xIV/zfk9+EvQbSiKoNoVJoQ/dODcZrj54jMd/QXO/gMNdNQa\nhmG/nlDGreE4wEEifLGgBCdmy2BmfYooEmd8ZOHuotFKHg4rOIUU083MZMMI5pikwFg45rt7es+H\nJdPmUcUpHjjodoVxDxaQfO/8DYuF9sDZTgkhlMrVBUmuyi4xn6STACmHXL/YQ7/UR/o3BYMuFeOI\nYieds1r7yrjC4k/uldOkv62X7H3OrvAE7TJ6VbdlmBfFltlMLNOXDArVQpSiXRnWermIkNJ8xUds\neGISJpojGeP8kBcvFsdiT8JSDpBge6RinHxvGRekf7XBj1bAfPZp8beT09l2jbv0jeyQvdL65fvf\nSr2RqbwZeP0a+A872uwos/zS3CC8qBgnpENAxWCcFyNTAs1Fe6av5BVqCgEmCJEHI6QQWKF79TP8\nhhNR33JCQJfatR6OW2BYLLQkmPaLCaHxToxywxDZ/L1EaJAGK/WQ2ejOCLiyWuCKJdOCcQLAdZft\nsdGT0yJ/r6fvhPjGBP0F0Lclz8uP/Z37P7ZCKDwzIIbeRnZTWJXtSTWyo/Hmz2dMK4vYlnjCS4sD\nYIwSHL9/CUiAha7078gp+hVzKXGnVrt5zd1D7Dm/8gQmydf9cXwwOEYntRl/Kdu51D6v5txr/dLc\nIIxYjBPSIaBiMM6LkSkB89s7dyWv4JoJK+NeSTZRCgIrdHfc0fWIO3oHpK7MIc8uHg9gHsl3tsCw\nWGiPrHSKCaFU5n2MNavYMAQ/2qYIDEn9tJV4yEtLI5Xg+GInnYu0bluCRDxMk/7C6gsyBybor+Vs\nS54XQP0K19k/GJgez0zH6TwqRWUCry2Tiys+h00UJmeOi84RFTJl+5OXFgdA4E16G3Clf9cvH1lC\nyDYUNzehjuVS0txnQOqsWttddm9xkkn88VFpkBIccOPEtoSxRwLfOIGclbxCT8GjuP+3iVwCDGP+\n3x4ElugGrPr+xZ2nl8K5XxwuRT0eV8GimXjC+abX6bzWxhdqOUi1Ng2WiRxwcmCYwjoMw2gjPYpZ\na+NLXpOA2Tm1trmWY7CcX4zjtaO0clKTaHuwHnOmjczIibW2PMvv87wcULGQZ4pLvuvHcmDceLlc\nTq1iGCtBzhzHyTNy59Gh2BNeWhyAAItx/DgBCTDRJRDbJgY3DLp8Uh0rBYZAgfreISCpUXyPdpGu\n1RYb/byx6nqlQKVApcBWK4D/Kd5wtk/m0Z4895G8oep6pUClQKXA1itw99ZD/jgQx3KfoWtvX/5x\nEKhiVApUClQKWAXG+0OpxHh7KGlXpCsFKgUuPgXqnaHMaUgfoYdS64p0pUClQLEC7ygefm2O2n8i\n4rXJrWJVKVAp8P9MgSP9IUz4yNIQkq4oVwpUClycCtRuG8K8dg0h54pypUClwMWqwOuGL7Fmd/g4\nV4w3oMD/AaGZIndYwp+vAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$(L_{1} \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta} + L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\theta} + R \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\theta})\\mathbf{\\hat{xyz}_x} + (L_{1} \\operatorname{sin}\\left(\\theta\\right) \\dot{\\Psi} - L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} - R \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} + \\dot{\\phi}\\right))\\mathbf{\\hat{xyz}_y} + (- L_{1} \\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta} + L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\theta} + R \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\theta})\\mathbf{\\hat{xyz}_z}$$"
      ],
      "text/plain": [
       "⎛             d                                                    d\n",
       "⎜L₁⋅cos(θ(t))⋅──(θ(t)) + L₂⋅(-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅──(θ(t)) +\n",
       "⎝             dt                                                   dt\n",
       "\n",
       "                                        d       ⎞         ⎛             d\n",
       "R⋅(sin(β)⋅cos(θ(t)) + sin(θ(t))⋅cos(β))⋅──(θ(t))⎟ xyz_x + ⎜L₁⋅sin(θ(t))⋅──(Ψ(t\n",
       "                                        dt      ⎠         ⎝             dt\n",
       "\n",
       "                                               d            ⎛\n",
       ")) - L₂⋅(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅──(Ψ(t)) - R⋅⎜(-sin(β)⋅sin(θ(t)\n",
       "                                               dt           ⎝\n",
       "\n",
       "                      d          d       ⎞⎞         ⎛               d\n",
       ") + cos(β)⋅cos(θ(t)))⋅──(Ψ(t)) + ──(φ(t))⎟⎟ xyz_y + ⎜- L₁⋅sin(θ(t))⋅──(θ(t)) +\n",
       "                      dt         dt      ⎠⎠         ⎝               dt\n",
       "\n",
       "                                           d\n",
       " L₂⋅(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅──(θ(t)) + R⋅(-sin(β)⋅sin(θ(t)) + c\n",
       "                                           dt\n",
       "\n",
       "                 d       ⎞\n",
       "os(β)⋅cos(θ(t)))⋅──(θ(t))⎟ xyz_z\n",
       "                 dt      ⎠"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Define point A and its velocity\n",
    "A = Point('A')\n",
    "A.set_vel(N, 0 * N.x)\n",
    "\n",
    "# Define point B and set its velocity. Locate it relative to A\n",
    "B = A.locatenew('B', L1 * prime.z)\n",
    "B.v2pt_theory(A, N, prime)\n",
    "\n",
    "# Define point C and set its velocity. Locate it relative to C\n",
    "C = B.locatenew('C', L2 * xyz.z)\n",
    "C.v2pt_theory(B, N, xyz)\n",
    "\n",
    "# Finally, we can define point D, locate it relative to C, get its velocity, and express it in the XYZ frame\n",
    "D = C.locatenew('D', -R * xyz.x)\n",
    "D.v2pt_theory(C, N, w_disk).express(XYZ)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part c."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Once we have the velocity (and if we have define the motion variables properly), the acceleration is easy to get. As before, we can express it in any frame of our choosing."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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VvbRUkuNbaiHFMeYaMvelMgXbVqmEkhqzTCp+uFweRhHCIolLSG8p6bxxk9iA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\n",
      "text/latex": [
       "$$(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\left(L_{2} \\ddot{\\theta} + R \\dot{\\theta}^{2} - \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\left(- L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} - R \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} + \\dot{\\phi}\\right)\\right) \\dot{\\Psi}\\right) \\operatorname{cos}\\left(\\Psi\\right) + \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\left(- L_{2} \\dot{\\theta}^{2} + R \\ddot{\\theta} + \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\left(- L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} - R \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} + \\dot{\\phi}\\right)\\right) \\dot{\\Psi}\\right) \\operatorname{cos}\\left(\\Psi\\right) - \\left(L_{1} \\operatorname{sin}\\left(\\theta\\right) \\ddot{\\Psi} + 2 L_{1} \\operatorname{cos}\\left(\\theta\\right) \\dot{\\Psi} \\dot{\\theta}\\right) \\operatorname{sin}\\left(\\Psi\\right) + \\left(- L_{1} \\operatorname{sin}^{2}\\left(\\theta\\right) \\dot{\\Psi}^{2} - L_{1} \\dot{\\theta}^{2}\\right) \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\Psi\\right) + \\left(- L_{1} \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\Psi}^{2} + L_{1} \\ddot{\\theta}\\right) \\operatorname{cos}\\left(\\Psi\\right) \\operatorname{cos}\\left(\\theta\\right) - \\left(L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} \\dot{\\theta} - L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\ddot{\\Psi} - L_{2} \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta} - \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta}\\right) \\dot{\\Psi} - R \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} \\dot{\\theta} - R \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\ddot{\\Psi} + \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta} - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right) \\dot{\\theta}\\right) \\dot{\\Psi} + \\ddot{\\phi}\\right)\\right) \\operatorname{sin}\\left(\\Psi\\right))\\mathbf{\\hat{n}_x} + (\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\left(L_{2} \\ddot{\\theta} + R \\dot{\\theta}^{2} - \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\left(- L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} - R \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} + \\dot{\\phi}\\right)\\right) \\dot{\\Psi}\\right) \\operatorname{sin}\\left(\\Psi\\right) + \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\left(- L_{2} \\dot{\\theta}^{2} + R \\ddot{\\theta} + \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\left(- L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} - R \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} + \\dot{\\phi}\\right)\\right) \\dot{\\Psi}\\right) \\operatorname{sin}\\left(\\Psi\\right) + \\left(L_{1} \\operatorname{sin}\\left(\\theta\\right) \\ddot{\\Psi} + 2 L_{1} \\operatorname{cos}\\left(\\theta\\right) \\dot{\\Psi} \\dot{\\theta}\\right) \\operatorname{cos}\\left(\\Psi\\right) + \\left(- L_{1} \\operatorname{sin}^{2}\\left(\\theta\\right) \\dot{\\Psi}^{2} - L_{1} \\dot{\\theta}^{2}\\right) \\operatorname{sin}\\left(\\Psi\\right) \\operatorname{sin}\\left(\\theta\\right) + \\left(- L_{1} \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\Psi}^{2} + L_{1} \\ddot{\\theta}\\right) \\operatorname{sin}\\left(\\Psi\\right) \\operatorname{cos}\\left(\\theta\\right) + \\left(L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} \\dot{\\theta} - L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\ddot{\\Psi} - L_{2} \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta} - \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta}\\right) \\dot{\\Psi} - R \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} \\dot{\\theta} - R \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\ddot{\\Psi} + \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta} - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right) \\dot{\\theta}\\right) \\dot{\\Psi} + \\ddot{\\phi}\\right)\\right) \\operatorname{cos}\\left(\\Psi\\right))\\mathbf{\\hat{n}_y} + (\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\left(- L_{2} \\dot{\\theta}^{2} + R \\ddot{\\theta} + \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\left(- L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} - R \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} + \\dot{\\phi}\\right)\\right) \\dot{\\Psi}\\right) + \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\left(L_{2} \\ddot{\\theta} + R \\dot{\\theta}^{2} - \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\left(- L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} - R \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} + \\dot{\\phi}\\right)\\right) \\dot{\\Psi}\\right) + \\left(- L_{1} \\operatorname{sin}^{2}\\left(\\theta\\right) \\dot{\\Psi}^{2} - L_{1} \\dot{\\theta}^{2}\\right) \\operatorname{cos}\\left(\\theta\\right) - \\left(- L_{1} \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\Psi}^{2} + L_{1} \\ddot{\\theta}\\right) \\operatorname{sin}\\left(\\theta\\right))\\mathbf{\\hat{n}_z}$$"
      ],
      "text/plain": [
       "⎛                                       ⎛     2                     2\n",
       "⎜                                       ⎜    d            ⎛d       ⎞\n",
       "⎜(-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅⎜L₂⋅───(θ(t)) + R⋅⎜──(θ(t))⎟  - (-sin(\n",
       "⎜                                       ⎜     2           ⎝dt      ⎠\n",
       "⎝                                       ⎝   dt\n",
       "\n",
       "\n",
       "                                 ⎛\n",
       "β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅⎜- L₂⋅(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅\n",
       "                                 ⎝\n",
       "\n",
       "\n",
       "\n",
       "d            ⎛                                       d          d       ⎞⎞ d\n",
       "──(Ψ(t)) - R⋅⎜(-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅──(Ψ(t)) + ──(φ(t))⎟⎟⋅──(\n",
       "dt           ⎝                                       dt         dt      ⎠⎠ dt\n",
       "\n",
       "\n",
       "     ⎞                                                   ⎛               2\n",
       "     ⎟                                                   ⎜     ⎛d       ⎞\n",
       "Ψ(t))⎟⋅cos(Ψ(t)) + (sin(β)⋅cos(θ(t)) + sin(θ(t))⋅cos(β))⋅⎜- L₂⋅⎜──(θ(t))⎟  + R\n",
       "     ⎟                                                   ⎜     ⎝dt      ⎠\n",
       "     ⎠                                                   ⎝\n",
       "\n",
       "   2\n",
       "  d                                                 ⎛\n",
       "⋅───(θ(t)) + (-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅⎜- L₂⋅(-sin(β)⋅cos(θ(t)) -\n",
       "   2                                                ⎝\n",
       " dt\n",
       "\n",
       "\n",
       "                   d            ⎛                                       d\n",
       " sin(θ(t))⋅cos(β))⋅──(Ψ(t)) - R⋅⎜(-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅──(Ψ(t\n",
       "                   dt           ⎝                                       dt\n",
       "\n",
       "\n",
       "                        ⎞             ⎛               2\n",
       "     d       ⎞⎞ d       ⎟             ⎜              d\n",
       ")) + ──(φ(t))⎟⎟⋅──(Ψ(t))⎟⋅cos(Ψ(t)) - ⎜L₁⋅sin(θ(t))⋅───(Ψ(t)) + 2⋅L₁⋅cos(θ(t))\n",
       "     dt      ⎠⎠ dt      ⎟             ⎜               2\n",
       "                        ⎠             ⎝             dt\n",
       "\n",
       "                  ⎞             ⎛                          2                2⎞\n",
       " d        d       ⎟             ⎜        2       ⎛d       ⎞       ⎛d       ⎞ ⎟\n",
       "⋅──(Ψ(t))⋅──(θ(t))⎟⋅sin(Ψ(t)) + ⎜- L₁⋅sin (θ(t))⋅⎜──(Ψ(t))⎟  - L₁⋅⎜──(θ(t))⎟ ⎟\n",
       " dt       dt      ⎟             ⎝                ⎝dt      ⎠       ⎝dt      ⎠ ⎠\n",
       "                  ⎠\n",
       "\n",
       "                       ⎛                                   2        2      ⎞\n",
       "                       ⎜                         ⎛d       ⎞        d       ⎟\n",
       "⋅sin(θ(t))⋅cos(Ψ(t)) + ⎜- L₁⋅sin(θ(t))⋅cos(θ(t))⋅⎜──(Ψ(t))⎟  + L₁⋅───(θ(t))⎟⋅c\n",
       "                       ⎜                         ⎝dt      ⎠         2      ⎟\n",
       "                       ⎝                                          dt       ⎠\n",
       "\n",
       "                     ⎛\n",
       "                     ⎜                                          d        d\n",
       "os(Ψ(t))⋅cos(θ(t)) - ⎜L₂⋅(-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅──(Ψ(t))⋅──(θ(\n",
       "                     ⎜                                          dt       dt\n",
       "                     ⎝\n",
       "\n",
       "                                                  2\n",
       "                                                 d             ⎛\n",
       "t)) - L₂⋅(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅───(Ψ(t)) - L₂⋅⎜sin(β)⋅sin(θ(t\n",
       "                                                  2            ⎝\n",
       "                                                dt\n",
       "\n",
       "\n",
       "   d                           d       ⎞ d\n",
       "))⋅──(θ(t)) - cos(β)⋅cos(θ(t))⋅──(θ(t))⎟⋅──(Ψ(t)) - R⋅(-sin(β)⋅cos(θ(t)) - sin\n",
       "   dt                          dt      ⎠ dt\n",
       "\n",
       "\n",
       "                                     ⎛\n",
       "               d        d            ⎜\n",
       "(θ(t))⋅cos(β))⋅──(Ψ(t))⋅──(θ(t)) - R⋅⎜(-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅─\n",
       "               dt       dt           ⎜\n",
       "                                     ⎝                                       d\n",
       "\n",
       " 2\n",
       "d          ⎛                   d                           d       ⎞ d\n",
       "──(Ψ(t)) + ⎜- sin(β)⋅cos(θ(t))⋅──(θ(t)) - sin(θ(t))⋅cos(β)⋅──(θ(t))⎟⋅──(Ψ(t))\n",
       " 2         ⎝                   dt                          dt      ⎠ dt\n",
       "t\n",
       "\n",
       "    2      ⎞⎞          ⎞       ⎛                                       ⎛     2\n",
       "   d       ⎟⎟          ⎟       ⎜                                       ⎜    d\n",
       "+ ───(φ(t))⎟⎟⋅sin(Ψ(t))⎟ n_x + ⎜(-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅⎜L₂⋅───\n",
       "    2      ⎟⎟          ⎟       ⎜                                       ⎜     2\n",
       "  dt       ⎠⎠          ⎠       ⎝                                       ⎝   dt\n",
       "\n",
       "                     2\n",
       "           ⎛d       ⎞                                           ⎛\n",
       "(θ(t)) + R⋅⎜──(θ(t))⎟  - (-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅⎜- L₂⋅(-sin(β)\n",
       "           ⎝dt      ⎠                                           ⎝\n",
       "\n",
       "\n",
       "\n",
       "                               d            ⎛\n",
       "⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅──(Ψ(t)) - R⋅⎜(-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ\n",
       "                               dt           ⎝\n",
       "\n",
       "\n",
       "                                    ⎞\n",
       "      d          d       ⎞⎞ d       ⎟\n",
       "(t)))⋅──(Ψ(t)) + ──(φ(t))⎟⎟⋅──(Ψ(t))⎟⋅sin(Ψ(t)) + (sin(β)⋅cos(θ(t)) + sin(θ(t)\n",
       "      dt         dt      ⎠⎠ dt      ⎟\n",
       "                                    ⎠\n",
       "\n",
       "          ⎛               2       2\n",
       "          ⎜     ⎛d       ⎞       d\n",
       ")⋅cos(β))⋅⎜- L₂⋅⎜──(θ(t))⎟  + R⋅───(θ(t)) + (-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos\n",
       "          ⎜     ⎝dt      ⎠        2\n",
       "          ⎝                     dt\n",
       "\n",
       "\n",
       "     ⎛                                            d            ⎛\n",
       "(β))⋅⎜- L₂⋅(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅──(Ψ(t)) - R⋅⎜(-sin(β)⋅sin(θ\n",
       "     ⎝                                            dt           ⎝\n",
       "\n",
       "\n",
       "                                                       ⎞             ⎛\n",
       "                         d          d       ⎞⎞ d       ⎟             ⎜\n",
       "(t)) + cos(β)⋅cos(θ(t)))⋅──(Ψ(t)) + ──(φ(t))⎟⎟⋅──(Ψ(t))⎟⋅sin(Ψ(t)) + ⎜L₁⋅sin(θ\n",
       "                         dt         dt      ⎠⎠ dt      ⎟             ⎜\n",
       "                                                       ⎠             ⎝\n",
       "\n",
       "       2                                         ⎞             ⎛\n",
       "      d                         d        d       ⎟             ⎜        2\n",
       "(t))⋅───(Ψ(t)) + 2⋅L₁⋅cos(θ(t))⋅──(Ψ(t))⋅──(θ(t))⎟⋅cos(Ψ(t)) + ⎜- L₁⋅sin (θ(t)\n",
       "       2                        dt       dt      ⎟             ⎝\n",
       "     dt                                          ⎠\n",
       "\n",
       "            2                2⎞                       ⎛\n",
       "  ⎛d       ⎞       ⎛d       ⎞ ⎟                       ⎜\n",
       ")⋅⎜──(Ψ(t))⎟  - L₁⋅⎜──(θ(t))⎟ ⎟⋅sin(Ψ(t))⋅sin(θ(t)) + ⎜- L₁⋅sin(θ(t))⋅cos(θ(t)\n",
       "  ⎝dt      ⎠       ⎝dt      ⎠ ⎠                       ⎜\n",
       "                                                      ⎝\n",
       "\n",
       "            2        2      ⎞                       ⎛\n",
       "  ⎛d       ⎞        d       ⎟                       ⎜\n",
       ")⋅⎜──(Ψ(t))⎟  + L₁⋅───(θ(t))⎟⋅sin(Ψ(t))⋅cos(θ(t)) + ⎜L₂⋅(-sin(β)⋅sin(θ(t)) + c\n",
       "  ⎝dt      ⎠         2      ⎟                       ⎜\n",
       "                   dt       ⎠                       ⎝\n",
       "\n",
       "\n",
       "                 d        d\n",
       "os(β)⋅cos(θ(t)))⋅──(Ψ(t))⋅──(θ(t)) - L₂⋅(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))\n",
       "                 dt       dt\n",
       "\n",
       "\n",
       "   2\n",
       "  d             ⎛                 d                           d       ⎞ d\n",
       "⋅───(Ψ(t)) - L₂⋅⎜sin(β)⋅sin(θ(t))⋅──(θ(t)) - cos(β)⋅cos(θ(t))⋅──(θ(t))⎟⋅──(Ψ(t\n",
       "   2            ⎝                 dt                          dt      ⎠ dt\n",
       " dt\n",
       "\n",
       "                                                                    ⎛\n",
       "                                              d        d            ⎜\n",
       ")) - R⋅(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅──(Ψ(t))⋅──(θ(t)) - R⋅⎜(-sin(β)⋅\n",
       "                                              dt       dt           ⎜\n",
       "                                                                    ⎝\n",
       "\n",
       "                                2\n",
       "                               d          ⎛                   d\n",
       "sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅───(Ψ(t)) + ⎜- sin(β)⋅cos(θ(t))⋅──(θ(t)) - sin(θ\n",
       "                                2         ⎝                   dt\n",
       "                              dt\n",
       "\n",
       "                                   2      ⎞⎞          ⎞       ⎛\n",
       "            d       ⎞ d           d       ⎟⎟          ⎟       ⎜\n",
       "(t))⋅cos(β)⋅──(θ(t))⎟⋅──(Ψ(t)) + ───(φ(t))⎟⎟⋅cos(Ψ(t))⎟ n_y + ⎜(-sin(β)⋅sin(θ(\n",
       "            dt      ⎠ dt           2      ⎟⎟          ⎟       ⎜\n",
       "                                 dt       ⎠⎠          ⎠       ⎝\n",
       "\n",
       "                        ⎛               2       2\n",
       "                        ⎜     ⎛d       ⎞       d\n",
       "t)) + cos(β)⋅cos(θ(t)))⋅⎜- L₂⋅⎜──(θ(t))⎟  + R⋅───(θ(t)) + (-sin(β)⋅cos(θ(t)) -\n",
       "                        ⎜     ⎝dt      ⎠        2\n",
       "                        ⎝                     dt\n",
       "\n",
       "\n",
       "                   ⎛                                            d            ⎛\n",
       " sin(θ(t))⋅cos(β))⋅⎜- L₂⋅(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅──(Ψ(t)) - R⋅⎜\n",
       "                   ⎝                                            dt           ⎝\n",
       "\n",
       "\n",
       "                                                                     ⎞\n",
       "                                       d          d       ⎞⎞ d       ⎟\n",
       "(-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅──(Ψ(t)) + ──(φ(t))⎟⎟⋅──(Ψ(t))⎟ + (-sin\n",
       "                                       dt         dt      ⎠⎠ dt      ⎟\n",
       "                                                                     ⎠\n",
       "\n",
       "                                  ⎛     2                     2\n",
       "                                  ⎜    d            ⎛d       ⎞\n",
       "(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅⎜L₂⋅───(θ(t)) + R⋅⎜──(θ(t))⎟  - (-sin(β)⋅sin\n",
       "                                  ⎜     2           ⎝dt      ⎠\n",
       "                                  ⎝   dt\n",
       "\n",
       "\n",
       "                           ⎛                                            d\n",
       "(θ(t)) + cos(β)⋅cos(θ(t)))⋅⎜- L₂⋅(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅──(Ψ(t\n",
       "                           ⎝                                            dt\n",
       "\n",
       "\n",
       "                                                                             ⎞\n",
       "       ⎛                                       d          d       ⎞⎞ d       ⎟\n",
       ")) - R⋅⎜(-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅──(Ψ(t)) + ──(φ(t))⎟⎟⋅──(Ψ(t))⎟\n",
       "       ⎝                                       dt         dt      ⎠⎠ dt      ⎟\n",
       "                                                                             ⎠\n",
       "\n",
       "   ⎛                          2                2⎞             ⎛\n",
       "   ⎜        2       ⎛d       ⎞       ⎛d       ⎞ ⎟             ⎜\n",
       " + ⎜- L₁⋅sin (θ(t))⋅⎜──(Ψ(t))⎟  - L₁⋅⎜──(θ(t))⎟ ⎟⋅cos(θ(t)) - ⎜- L₁⋅sin(θ(t))⋅\n",
       "   ⎝                ⎝dt      ⎠       ⎝dt      ⎠ ⎠             ⎜\n",
       "                                                              ⎝\n",
       "\n",
       "                    2        2      ⎞          ⎞\n",
       "          ⎛d       ⎞        d       ⎟          ⎟\n",
       "cos(θ(t))⋅⎜──(Ψ(t))⎟  + L₁⋅───(θ(t))⎟⋅sin(θ(t))⎟ n_z\n",
       "          ⎝dt      ⎠         2      ⎟          ⎟\n",
       "                           dt       ⎠          ⎠"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# In the fixed frame\n",
    "D.acc(N).express(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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oeFIVteXDCqZcUNqruBinJJvnNw4pbAInaMiIR6wt+wBLtC22nK3tu/AkFz4k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\nbH6ZcsGoLcLCdb+MsnDc/+lbMP8HqoCJJYR84r4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\left(L_{2} \\ddot{\\theta} + R \\dot{\\theta}^{2} - \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\left(- L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} - R \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} + \\dot{\\phi}\\right)\\right) \\dot{\\Psi}\\right) + \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\left(- L_{2} \\dot{\\theta}^{2} + R \\ddot{\\theta} + \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\left(- L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} - R \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} + \\dot{\\phi}\\right)\\right) \\dot{\\Psi}\\right) + \\left(- L_{1} \\operatorname{sin}^{2}\\left(\\theta\\right) \\dot{\\Psi}^{2} - L_{1} \\dot{\\theta}^{2}\\right) \\operatorname{sin}\\left(\\theta\\right) + \\left(- L_{1} \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\Psi}^{2} + L_{1} \\ddot{\\theta}\\right) \\operatorname{cos}\\left(\\theta\\right))\\mathbf{\\hat{xyz}_x} + (L_{1} \\operatorname{sin}\\left(\\theta\\right) \\ddot{\\Psi} + 2 L_{1} \\operatorname{cos}\\left(\\theta\\right) \\dot{\\Psi} \\dot{\\theta} + L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} \\dot{\\theta} - L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\ddot{\\Psi} - L_{2} \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta} - \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta}\\right) \\dot{\\Psi} - R \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} \\dot{\\theta} - R \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\ddot{\\Psi} + \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta} - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right) \\dot{\\theta}\\right) \\dot{\\Psi} + \\ddot{\\phi}\\right))\\mathbf{\\hat{xyz}_y} + (\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\left(- L_{2} \\dot{\\theta}^{2} + R \\ddot{\\theta} + \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\left(- L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} - R \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} + \\dot{\\phi}\\right)\\right) \\dot{\\Psi}\\right) + \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\left(L_{2} \\ddot{\\theta} + R \\dot{\\theta}^{2} - \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\left(- L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} - R \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} + \\dot{\\phi}\\right)\\right) \\dot{\\Psi}\\right) + \\left(- L_{1} \\operatorname{sin}^{2}\\left(\\theta\\right) \\dot{\\Psi}^{2} - L_{1} \\dot{\\theta}^{2}\\right) \\operatorname{cos}\\left(\\theta\\right) - \\left(- L_{1} \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\Psi}^{2} + L_{1} \\ddot{\\theta}\\right) \\operatorname{sin}\\left(\\theta\\right))\\mathbf{\\hat{xyz}_z}$$"
      ],
      "text/plain": [
       "⎛                                       ⎛     2                     2\n",
       "⎜                                       ⎜    d            ⎛d       ⎞\n",
       "⎜(-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅⎜L₂⋅───(θ(t)) + R⋅⎜──(θ(t))⎟  - (-sin(\n",
       "⎜                                       ⎜     2           ⎝dt      ⎠\n",
       "⎝                                       ⎝   dt\n",
       "\n",
       "\n",
       "                                 ⎛\n",
       "β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅⎜- L₂⋅(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅\n",
       "                                 ⎝\n",
       "\n",
       "\n",
       "\n",
       "d            ⎛                                       d          d       ⎞⎞ d\n",
       "──(Ψ(t)) - R⋅⎜(-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅──(Ψ(t)) + ──(φ(t))⎟⎟⋅──(\n",
       "dt           ⎝                                       dt         dt      ⎠⎠ dt\n",
       "\n",
       "\n",
       "     ⎞                                         ⎛               2       2\n",
       "     ⎟                                         ⎜     ⎛d       ⎞       d\n",
       "Ψ(t))⎟ + (sin(β)⋅cos(θ(t)) + sin(θ(t))⋅cos(β))⋅⎜- L₂⋅⎜──(θ(t))⎟  + R⋅───(θ(t))\n",
       "     ⎟                                         ⎜     ⎝dt      ⎠        2\n",
       "     ⎠                                         ⎝                     dt\n",
       "\n",
       "\n",
       "                                          ⎛\n",
       " + (-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅⎜- L₂⋅(-sin(β)⋅cos(θ(t)) - sin(θ(t))\n",
       "                                          ⎝\n",
       "\n",
       "\n",
       "\n",
       "         d            ⎛                                       d          d\n",
       "⋅cos(β))⋅──(Ψ(t)) - R⋅⎜(-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅──(Ψ(t)) + ──(φ(\n",
       "         dt           ⎝                                       dt         dt\n",
       "\n",
       "\n",
       "              ⎞   ⎛                          2                2⎞             ⎛\n",
       "   ⎞⎞ d       ⎟   ⎜        2       ⎛d       ⎞       ⎛d       ⎞ ⎟             ⎜\n",
       "t))⎟⎟⋅──(Ψ(t))⎟ + ⎜- L₁⋅sin (θ(t))⋅⎜──(Ψ(t))⎟  - L₁⋅⎜──(θ(t))⎟ ⎟⋅sin(θ(t)) + ⎜\n",
       "   ⎠⎠ dt      ⎟   ⎝                ⎝dt      ⎠       ⎝dt      ⎠ ⎠             ⎜\n",
       "              ⎠                                                              ⎝\n",
       "\n",
       "                                   2        2      ⎞          ⎞         ⎛\n",
       "                         ⎛d       ⎞        d       ⎟          ⎟         ⎜\n",
       "- L₁⋅sin(θ(t))⋅cos(θ(t))⋅⎜──(Ψ(t))⎟  + L₁⋅───(θ(t))⎟⋅cos(θ(t))⎟ xyz_x + ⎜L₁⋅si\n",
       "                         ⎝dt      ⎠         2      ⎟          ⎟         ⎜\n",
       "                                          dt       ⎠          ⎠         ⎝\n",
       "\n",
       "          2\n",
       "         d                         d        d\n",
       "n(θ(t))⋅───(Ψ(t)) + 2⋅L₁⋅cos(θ(t))⋅──(Ψ(t))⋅──(θ(t)) + L₂⋅(-sin(β)⋅sin(θ(t)) +\n",
       "          2                        dt       dt\n",
       "        dt\n",
       "\n",
       "\n",
       "                   d        d\n",
       " cos(β)⋅cos(θ(t)))⋅──(Ψ(t))⋅──(θ(t)) - L₂⋅(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β\n",
       "                   dt       dt\n",
       "\n",
       "\n",
       "     2\n",
       "    d             ⎛                 d                           d       ⎞ d\n",
       "))⋅───(Ψ(t)) - L₂⋅⎜sin(β)⋅sin(θ(t))⋅──(θ(t)) - cos(β)⋅cos(θ(t))⋅──(θ(t))⎟⋅──(Ψ\n",
       "     2            ⎝                 dt                          dt      ⎠ dt\n",
       "   dt\n",
       "\n",
       "                                                                      ⎛\n",
       "                                                d        d            ⎜\n",
       "(t)) - R⋅(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅──(Ψ(t))⋅──(θ(t)) - R⋅⎜(-sin(β\n",
       "                                                dt       dt           ⎜\n",
       "                                                                      ⎝\n",
       "\n",
       "                                  2\n",
       "                                 d          ⎛                   d\n",
       ")⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅───(Ψ(t)) + ⎜- sin(β)⋅cos(θ(t))⋅──(θ(t)) - sin\n",
       "                                  2         ⎝                   dt\n",
       "                                dt\n",
       "\n",
       "                                     2      ⎞⎞         ⎛\n",
       "              d       ⎞ d           d       ⎟⎟         ⎜\n",
       "(θ(t))⋅cos(β)⋅──(θ(t))⎟⋅──(Ψ(t)) + ───(φ(t))⎟⎟ xyz_y + ⎜(-sin(β)⋅sin(θ(t)) + c\n",
       "              dt      ⎠ dt           2      ⎟⎟         ⎜\n",
       "                                   dt       ⎠⎠         ⎝\n",
       "\n",
       "                 ⎛               2       2\n",
       "                 ⎜     ⎛d       ⎞       d\n",
       "os(β)⋅cos(θ(t)))⋅⎜- L₂⋅⎜──(θ(t))⎟  + R⋅───(θ(t)) + (-sin(β)⋅cos(θ(t)) - sin(θ(\n",
       "                 ⎜     ⎝dt      ⎠        2\n",
       "                 ⎝                     dt\n",
       "\n",
       "\n",
       "            ⎛                                            d            ⎛\n",
       "t))⋅cos(β))⋅⎜- L₂⋅(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅──(Ψ(t)) - R⋅⎜(-sin(β\n",
       "            ⎝                                            dt           ⎝\n",
       "\n",
       "\n",
       "                                                              ⎞\n",
       "                                d          d       ⎞⎞ d       ⎟\n",
       ")⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅──(Ψ(t)) + ──(φ(t))⎟⎟⋅──(Ψ(t))⎟ + (-sin(β)⋅cos\n",
       "                                dt         dt      ⎠⎠ dt      ⎟\n",
       "                                                              ⎠\n",
       "\n",
       "                           ⎛     2                     2\n",
       "                           ⎜    d            ⎛d       ⎞\n",
       "(θ(t)) - sin(θ(t))⋅cos(β))⋅⎜L₂⋅───(θ(t)) + R⋅⎜──(θ(t))⎟  - (-sin(β)⋅sin(θ(t))\n",
       "                           ⎜     2           ⎝dt      ⎠\n",
       "                           ⎝   dt\n",
       "\n",
       "\n",
       "                    ⎛                                            d\n",
       "+ cos(β)⋅cos(θ(t)))⋅⎜- L₂⋅(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅──(Ψ(t)) - R⋅\n",
       "                    ⎝                                            dt\n",
       "\n",
       "\n",
       "                                                                      ⎞   ⎛\n",
       "⎛                                       d          d       ⎞⎞ d       ⎟   ⎜\n",
       "⎜(-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅──(Ψ(t)) + ──(φ(t))⎟⎟⋅──(Ψ(t))⎟ + ⎜- L\n",
       "⎝                                       dt         dt      ⎠⎠ dt      ⎟   ⎝\n",
       "                                                                      ⎠\n",
       "\n",
       "                       2                2⎞             ⎛\n",
       "     2       ⎛d       ⎞       ⎛d       ⎞ ⎟             ⎜\n",
       "₁⋅sin (θ(t))⋅⎜──(Ψ(t))⎟  - L₁⋅⎜──(θ(t))⎟ ⎟⋅cos(θ(t)) - ⎜- L₁⋅sin(θ(t))⋅cos(θ(t\n",
       "             ⎝dt      ⎠       ⎝dt      ⎠ ⎠             ⎜\n",
       "                                                       ⎝\n",
       "\n",
       "             2        2      ⎞          ⎞\n",
       "   ⎛d       ⎞        d       ⎟          ⎟\n",
       "))⋅⎜──(Ψ(t))⎟  + L₁⋅───(θ(t))⎟⋅sin(θ(t))⎟ xyz_z\n",
       "   ⎝dt      ⎠         2      ⎟          ⎟\n",
       "                    dt       ⎠          ⎠"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# In the XYZ frame\n",
    "D.acc(N).express(XYZ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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qeB28LDiSlom0fv4SvFAulBs2Qrr9QvVEmx0OWIiYKdSkYmiardEEruNEZexr\nhNRHxVA1ECg45q5b53fMdqiLGGGaYqhLshjqoripTT042FzRjwzhXkN0wmyvTHYA0snHHP7Rki/K\nd6xCMwK8MWuxvAW41s05pRV7u+GboovpXPRdwTD0sFztLlw8Qph304/yX/FPPDZMtSBnpRsjZ+hp\nQKb70DY9WFBB6kLS+YDPIKT8+S48AHu9tMlkvKdZzULFoLiPZzFIsQdiOPd34XH75BaQjp530vYW\nrFN7iGEkAtKdv+jCoIhuljGs+CaF7s+4BEfomTCOirw8SRAIk4exyC4DGmTF1wsGw4ong6DAZ6CD\nORM0ha79UFCvcYBF/1qMe+bSxvqDa4+6k0Am+QEGfmX5zJ67f8f9BE7Gg7gZLr907hv0RuZsfCPz\nEfrzJPS4u/Z7k/P4Rub54Y2MWCFWBFsr+MnSbbuAhnbwGxmlx1r3yFARxW09/UTmYZgnv+CMTyLx\naEPuMBXVuKpzFPEOjzzk5S30Oj048DB3cYwLA7C70jUVJVTG74kLHzGnKwf0xz6whRbrgMErQtwl\ndlbrTpyHNzI3gxn9Rmb9ocxJdguk6Lj+LNBlg36ck0eU1lPTrtDm1YDNoEg4xvrPw6lMlXZXaJgx\nKCq0FHgjGIu7tkHD44gA4w9HxEw0LzDLj3Lwn2qCz7BOb4XwcuRyhVl66I3MFx/bh81nHn0uvF43\n2fganPigXOAOFZGGA6QMqPYB9AAOTuUpjJoOQIMS2kdpF95DXYQEiUzgS8lU/Msv8DdPDqFvED8F\nuUeokQ8pwa44hQAYnFSUstToAJUliIoT0FRmHcQvs6DKPIsI6O/gZQEaUq58CmXJ9/tIOY4J/LIJ\n/zMyUD3x6HDAQKQsRjUW4bx/Vr9Drqp+qGskeOGVz2+3hl6qA2QQUsegSC6kSYW6JEuZOSVuadP8\nDlEuueJbS7jXhDcya3feAf6MJj+3B6f7HlzfhRMdvCOhVayUBLjXnT76GNtL3waESmDBudQjjlus\nRA9Uk6Qe4B8SRCe37Ne6ixswk3hcmmpCTktX1MO51NOCrAqJ4NK9Iq980h7Kn1k5ALtxL/FJ5QjJ\nPa2rfVnrirK6CXA8PFvcC+CNzOIezDFZc8ryHTV96lF3/qI1ofIY/tIsrseDmIyPEJIGXkIQXUUm\ngekPDbISu2ZXvNRDhYfrHxP9jSoBTUErb0c53+0ax4/vyNnzS7ujx9ce5/ci/pNY5NNt5939u/6b\nfx4L/Hx5JwyhbezDN1Lwq2X+jQyqvABPVnugecu5v+0WHldvZGCrWPl3rMRzIbx3/Abt4E9FlR5r\n3SNDRWgq+2pZ9AsN6Q8qOyB3mMg4/pkAABaESURBVEJt/qg6h3AsNToyBNk99RO//r2iTs4+8HDJ\nH/kNwG64pqKEBphKBTjM6Sos3wv/Q0CLdcDgFSHuErtn6PJ4NB7hJwX3Q+5AzbfwtXAy/SQTH0VO\n8hfacrrQ7tR0qY2E8CUqEo7REkEpU6XdlTcy2VctosIGkTEW4ScTP4LVAFYjZqZ5nmbpgExSLGgJ\nL0cuV5ilB1PkfujGs/B63Td/Fl5veEj9q02UCyrSXg6QMjf67g9+7AA0CUN+3o02VESblNA+Srtg\nD5MiJEhkAl4MpurqKPMlOWVj8tU1i6haj7WuKkITWZAzwgT5vLSPFDSVWRlITXwjC6rM04hgXDp4\nWUQbUu7zhpGWFPq2ECnHIYfb6Wm6nemGl7eEvBIMRMoiK6bXuqvAKAu4jpOuEfHC657jbp3dMTtC\nXcQI0qRCXZKlpJ4St7QlhesfRgyu+OYZ7jUkgvxa/Bg+1Ln7fuoAfiIyGcd/6oJ3aFqpSkmAs2T2\n6FNuv7JQIUh/VGNUuwvrGHH3su6mb3RndsGC9ngo5LR0BS+ch0FW9UpwqX3nlU/aQ/kbXy0rberq\nrJc2R0juaV3ty1pXlNVNgOPhmeje757izuxAOdDzTkFZfjhK88QEpDt/cS9E5TH8pVk27l+joqQx\nEYiuIpPA9IcGjPL1gonsar5Fnv1jon/Tgd4LaMJrJDLnu13jzwMw8DfS3KkxPMo9AW/gf5+iw/k5\nBQv3bK8t765u0Wx4OTEOQ3oCvG6s3sigyk34uuoG9BI4A/I/Um9kcKu38mxHh3DhJlC1fB7+JALt\n4F/4hP2iB75SW6x7ZKhoAUTx3/tUh/ILZhGWHGCyBbnDlGiBc9W5hU1YNRDryJAMfAKOP0RKDx94\nmLwfnvEHYTdcU1FCZZ5KOThMzCI8XuKH8hjQfB3i5xUtbPpwJ9i92sXthQ34SZm8BZM3BpmT7BZI\n0XHPgXNv9W9ksojSemoapjJtJIQvSpEwGaepSjPFubv+uThlkFbYzvXipnM7vwxvpX4bqQU2I2YM\nJhwZZsgTHZ6k+EZG6ozg5gpzvKD15N4XsW2uTzYhbZOz7vtIIb5QLnBHPwco31A9H9zHzX77+nl3\naiPZ36CE9lHcYA+TIiRIbGOm+obPjiBnhAnyeWkfKWgqszbx6/1iYTNSUnIGLQJmNdFzXuZtA9e5\nFhdgp7R+aQuBcrAEx60HbvURByyE6olH2wETkbIY9dRdRfpawBdVP0QZqRHdXODeybdJWJ27bp3d\nMa0ep2OQxwjTFEOdlg9Gy8hcFC8ijto21Q3fR9ngim+e4V5DIguw9yfX7rwAWi7il8rOPOwWx3Cm\ng3doWqlKSYCzZPboU263fOsfKoEF52qMandhHaOFTdRk3k3dbdjNtceWqRbkrHTREh/DIKMab2Zh\nExQ8G/6XQqJheAnlz3fhAdghP/kDnU/qwiarB8x4dLUva11RtsDOxHDuJfCtiheAfiZrQVm2nubJ\nExDv/PjwkeCHKRfD3wgZCipFvmRw1kO5qXxq1gTyN/uMQkpjAQ2y4kOLweiIV5lnnwcDNAUtDwRy\nYDP2hAUY2zV+NwguPQAPPMD6pW33CnqOpcDCwiIs3LD9lye38c2EPlbPq6vTd7gTG/BxjHwigyqf\n59bfCKeTF+DPae6778LvlcF/+DYDt3orr2QlwoVnjN1H1z/kFi/QDv4bwEqPM9Y9MlSEpvwXrFlt\n4hdMISw5OiB3mBItcK46R3AMxDoyDNkt364U8tAHHi7g64hwDMBuuKaihMo8lXJwmBionT/fBREM\naL4OGLwixF1i92pXtlapov33xeWPg2dOsltghY63AC2e8G9ksojSemoapjJtrAVelSJhMq5d3IGX\nTHHurvyAH98axEMptBSwEYzF6P1udeOrB+6HKP6gIGL2NM8wC+t98WE7kzojuLnCHO8r3do9++uT\nXTB1bgdK9bH9Lz36nsn+nW+GCU4x7tARqTtA+QbhD+JeOCiVT//k3Z9O9zcooX0UN9hD1Vy4+NkE\nvBpM1dVh5Ova9Q1mlCBniAnyISXYHSeJQdVHKi+jAnWAyhIEtSloKjPDiKpzIwu6zIvW38XLHDSy\nMJaJtH5pC4FyHJFz8BOSR9Z30oZn8VY5UJReZlFibRLOu2r1u7LgSBPViG4u4TYJy3PXrbM7Zleo\n8+TifSkmNy0fjJZBvShulYnmd4MrvnmGew3RCerhS0+4i4+D3cVH4eXcHWv3/Nydd6w/tgsXvAPx\nyqEqJQHOktmjT7nd8k2zsh0qAQHnaoxqd2EdI+4B5t3U/TQa0R5bplqQq4U0DLIqJILr7xVys1Gh\nCOXPvyg3AHu7i6EJ/3jS1b6sdUXZArtnInxmcuEE3m+5sRXtja2nefIExDt/cS+EKeNJ2QgZCipF\nujExCIuH+qnaB6Y/NMiK6pod8QJ0GWifBwM04TUSmfG9qPGdyfdPdvBDqy+fu3TvOfhKKJL+T96+\nj6Hhuj957tLYnXzh9torXkWT8UU+r6OZlX//1j9Yv+/S/77v0l/e+BswAypX/u/HP4mKFg+c+/B3\nfOfuX00+91eT//bux/b4O0neystAAg7hwtp/fueGe9onP0s7YD7T44x1jwwVoSnn0L4c2i+YA3Xh\n6IDcYSqoaThHcAzEOjIM+XfPff2zSiMOJfAwXN7DiQHYDddUlFCZp1IODnP6ke98539AEQxovg4Y\nvCLEXWIXhv7Z33k7rMNnE7v4Cn9yHI/cSXaLltzapyb3fmqy59/IZHQhkdR0qY31pIqEybhGUDLF\nubtCQ82gJjJUwEYoFl9+xUfd6JV/uIvxR5sRM9M8j4BnvZAU25nUGUcuU5jjfRn8m28XrptMNvAX\nDeG92uSDkyfdjb/5a2ibUww7ejpA+XYrn5rcvBu3n5tMvpnub1BC+yhukIfiH99R0YlwGEzV1WHk\n69r1Dc6OICeIKfIhJdgZpxCCqo9EKaMCdYDKEkx7OVihMjOMqDo3sqDKPIkIw+7gZQ4aU67KxKfQ\nt4VIOdL9o5M/cu7l74AxVE882g6A02npZRajnrqrVr9DNSpOSY14L1j1XHfr5I5p3s90DPLkYppi\nckuyGJmL4laZaH7Lw4jBFXlkkHsNiSwerE6+5e6bPAjvhiF365PH7pwcnNlyd2CaeIeilS7vBDhJ\nSk5lpdh+haFCSP6oxqh2F9Yx4h5g3k3x/cOwQsqzWy2kYZBVIRFculdklU+xiOXPrFTZ6gp3vbQ5\nQuGpx2oWmuHWuqJsgV2YuP6PX/hqdILJWlCWH450nsJjN72RyfGTqi0prqT34FJyCJNRkW5MDCJP\nKgRVE0gC0x8aKPBVQcHoaL5Fnv1jogGa8OaByOBaNf7XD67C90mX4EHIHwvEfL6Q/Mhafn5tPqGu\nQeWqvyz+bVaYD1tHWyxVs9WhJ0InRWwK/jRacXi/lKeFyDBTxfY4EZyzPEfEOjKmTNRFoxObeLoC\n7GGrhFveyJB6/QLYL/B1ENXLQRH947AldnkjA18WpeOGfTwtPMFX2Su7lU2+J7sOl6npMF0d5IoY\nSiaeu+tpaDGI/wxBtj8SuYwFiAbMZjDlzhp0UjvzVyZcNKfSE7VuwDc35Vh6bIeGnAvYIUceEZoP\n61EZy4dUynY7AB0+JnXtizCYDJrjACKmqyMu8ChsFbSJfiXdoSdDbSZPaXNz2T486BOb2hM/hkAq\nIuUCgbf8b0Dny+V1W53q9zrSlbYQtEcUYSoOLsti3B5GwYgwKqzQIDVjy6Q73IlNnAh6s1W/1iJ5\n2BrMlbXIWlN0haWgqKtbm/0uqBtkJeyqDkAdH5oLModxs2JjcCWUvr/XkEjQOeIv95Le6x7nUuAd\nMSrBqK+UcC0/0IoTOKpuj2JXK1QhRoEC0QZxS8eIfT6xqUX8+NvpbHgcZTsgd9zhEj18YUJWdy6E\na8tEZTQiVl4B9rhVWHGZdZTGoMQemDgaE3Ama0FZtm7m6fQWbTRe2kwoNuSKChC0AyKjCSRvZPpD\nC6Etg4EGAmgJfIYzy4MCbeLN4AalqsbXzt8GP6tQdfxh/PMA/ljbkJF95iqx10ClhOrjhkTYKr+J\nV7PVoSdCJ0Vs6hOGQe+X8rQQGmaq2B4ngnOW54hYR8aUibpotHo7nq4Ae9gq4Xb/PTMhl4DdP+AE\nUVnCc1BEv1ReYg9qn8m7OBWMX+uhsTltPm2jeGq6UFZM5IosVrjcXU9DUzZXSBY7c824zGDCTzXS\nClMF7UwIGd6g9fSldaIIGRvRnzFzPsUBoP+wK49TWA/KvERIZdhhBiDmJVdA+xIPfREGk0FzHECW\ndXXEBR6FrWIs0a+kO/RkqEsiK1U4nMv24UGbZQaBbNT50FrLeJkFDy5D3nSkTWBqb0ShJmXYdqBi\nUTarczAijFJrMEzN2DLpDsduBb3ZKl7CWovkYWswV9Yia03RFZaCoq5ubTcb0TfIimyqn0EdH5oL\nMleLjcGVUPr+XkMiQedte18NOte/zg9evCNGJQj4SgnXeWPmher2uO9qhSrEKFAg2ij4wz4bEXIr\nF/BP7sVmp5XIuAOyXbqyWZ3bkFUhIVzTLaWMhsRKI1tRro09bhVWXGYdpTEosQcmwl9dxYNTUSSE\nrRfTuEPf+fE6Hh1hjYI8yhWZ1soiGwothLYMBuIIoCXwGcwsDwq0iRes6X4ZlOoaH03+et/B76/4\nY/0H3vsiGXeeT3DabLnvcv+TF9Y2DYGw9fPGYjLV1hOhoyJv6jbwKDuCX8HTTAAvh5gytocpcc70\nPDNTkQm6aAB/gAGPK8AuWzvDfWJjbZuMOVtUFCHuFvYLrORNdFra46vs1buVzv5ieqmutGk1XRvm\nihhKJl1x12AQfI8520uXnbnmTXYwc4Wnb48zJlz4PTSdnqB14dLnD8LWE/dthTH8xtNGuGg6UMl3\n2AwDc3+kZUCj9+ixFKGCpJd53C7CsLXTWFtPirpFZEIlyK+kBMPeLujiYwVV4lhFJgFtlllKpCIN\nA2st42WhLrAwwbu0VwomM4IimfQXHQ7YFi1FYsROS2rGlsm0zmG3NvtdcCuNQZgOA4lgktywWgyG\n8DtsXtoLw2Lg7zUoEiHc8Dn60gsL33iv3iR4w1wo7zDTGhTblfDVCpXEqNKVrR5glflX3vvufUR3\nJZAFSgyt8lcNRa4COUSG9PQqJGblFWCXrV3Qh7WvBnb/uL20h3FZ2oOX8rDy5PSdP93SEdZUGN4R\nqUcIXFvag5fySAgUlgdAk9DawRDQnYFn0wr00l5AowYJ3KhU1/gXHsX3VE+VTcuTyQMy7jyPzjZE\nniof7YhTiWzYKgFJVvVFW0+Ejoq8qfvHWgGNg1/B00IEVflPo0zIqSlje5gS56pqlJmKTNBFg7WH\n6HQF2GVrZ7hHZ+lnOGDPFhVFiLuFfayBGwnBZe8WScrL097yGRnmZ206Xyuvc0WWMfiI23bXApwr\nZJOduWYxO5gZ6tEtl6AW+bDhZnij1k//oGyEP5298nC8gB1y0Xagkm/ZDGd7f/+ycFKEEZLSLsN2\nEYat0XXZmJ3belLULSKTWkEeG2VmDS+HWDS2hynxsYIqMVORIV0C2qRShfgCYlitVetI1AUWJnit\nKgs7cCAokkl/0eGAbdFSJEZsRqVmbJlMq4+36M1W6TLJYSkgWzvNpejqijDsSegz0XYmhljJFFuX\nQ/gd9rcQjlkKRaKT171+l6fx9eKDcWzQSipFC9XHkh1L4mqFSmJU6coJf7zPVpnfOZkQyiuBLFBi\naC3HsQX4eZu1ITKkx5bJFHPOrwC7bO2CDtjbTz3iWxf2BfaAgVcoa+VJ3/mzIIjpChNyafUIQUsV\nEAmBgo4B0CS0diIFdGfgybT23sabwI1KdY3feOELO0DCsbucw3851dw6Gvtp/8FvJiTfa4UO2z5G\nY79u63F+nT7U8yIn9vwW4+TFjRUVhR6mTAUy6Z1rq8F/AZH/l2318/fQ0mhclwhrtlG/7D/6rKtx\n8sst+G9TWodX1BP7ab6B3Gbrwr/HPeDQpgds86IeSr5RKJy622JQpqFXrlPtmQbzsgI3SY+t9TmP\nuMmvKp3ioZpKhn7dVpZIVi5GY1oYoKAFyWurVUZvtB16nF8n1HbNOOPwu4yVGW4fZplJDuy0iZ+9\nI9NWF9pKoq/WFkJ0BUWY0IPLsaj3y3g0ppEdhwCc+mFNRlT58/x1645+1xHq0ZgdT5LrY2GdvLqK\n+Gjs9yTrnVyBf+wO7jLJnmj7xNk4VoWqJ/uPR+OG7NUKlddTYVyAgP6Kz2aZM9Qgz5fpawdkueWI\nmXSzumpDFjUEt+KWUoZDZuVojOPK0cY+GvO2TugBXA2Yt4OKaiJsil6ZrDXKNvKkdIRhR1iDnD2o\ngBiNvXgamv7QvIJaMFS8vKGeJxuvt8ZcTyF7vSuTF092e9qYG7GF2+cGak+gz+kpN2tiJzcJ0Y9W\ncE3TLQ+lgiSbPnwGDYKboT++PI6AjsA0y0zb7RrX2kLXvllfn9V41+N2+P2ujo1XenClKjLaXjro\n0n8E1uePdh1Bn31WVhxgJtb4ONU81UDYyKcKzYQwDK+p4uhMruEfkD9Sx/078+nO8phwv7+Cfppu\neSgVJNn04TNoENwM/fHlcQR0BKZZZtpu17jWFrr2zfr6rMa7HrfD73d1bLzSgytVkdUn39Wl/iis\nzx/tOqI++6ysOMBMrPFxqnmqgbCRTxWaCWEYXlPFEZp82xHyhVw5tTmfHv0rRl/7hGyabjGUvmE8\ndAYNg9vXrWO5v4kRmGaZDYjvqVpbGKBjJkVnNN6tWB16v2uBg7UeXKmLjL7/Qof6I7E8h7TriPus\ns7ICn5lY5eM081QFYUOfJjQTwUC8po4jNLl8cIScIVc+NJ8O8fvr+qcLU3Rr2Fv9Q2fQMLjzyY5j\n1FOKwBTLbIBH9bYwQMlMis5mvFuhOvR+1wIHaz240kOkw8i8L88f7ToiPuusrMBnJtb5OMU81UHY\n2KcIzQQwFK+p5OhMLmweHV/Yk2fPpUP+DX79ByvTc2vgzxoWNg834APhHi7YY+szHoHpldmQQNTb\nwhAtsyg7m/FuRWphs7V6+Gs9uNJD5PD9uKYI5o92HeFY2OwQmM1lZmKdj1PMUx2EHbopQjMBDMVr\nKjlCky85Qr6QK4vb8+jRMzYQ9eh9VezTc4uhVIEUC4fMoKFwC/zHE8cRCBGYXpkFk92DRlvo3jzb\nEjMZ73bIDrnftcG1biGy8wjTSVzsOs8h7Tpcmm1WVsAzExt8nF6eGiBs8NODZtofjNfUcoQmv3Bw\nhJxBV9Z+dg4dWvsZAv2FnSr2qbnloVSBFAuHy6DBcAv8xxPHEQgRmFqZBYs9Bo220GP3TIvMZLzb\nETvcftfG5lwPrvQQ6bIy7+tzSLuOkM82KyvgmYkNPk4vTw0QNvjpQTPtD8ZrajlCk6PXHiFnyJWn\n04cb8+XViTHhfV4D9bTc8lAaSLKlw2XQYLgZ+uPL4wjoCEyrzLTNrnGrLXTtnfX1WYx3O2aH2+/a\n2JzrwZUeIl1W5n59/mjXEfLZZmUFPDOxxcep5akFwkY/NWim+eF4TTVHaPLbjpAv5MroYP48+l8E\neW23gXxabjGUBpBi6VAZNBxugf944jgCIQLTKrNgsHvQbAvd22dbYgbj3RWwQ+13HeB6cKWHSIeR\nI7A8h7TriPoss7ICnZnY5OO08tQEYcOfFjTT+mXgTfT8f1cHsZYPAh/RAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$(L_{2} \\ddot{\\theta} + R \\dot{\\theta}^{2} - \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\left(- L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} - R \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} + \\dot{\\phi}\\right)\\right) \\dot{\\Psi} - \\left(- L_{1} \\operatorname{sin}^{2}\\left(\\theta\\right) \\dot{\\Psi}^{2} - L_{1} \\dot{\\theta}^{2}\\right) \\operatorname{sin}\\left(\\beta\\right) + \\left(- L_{1} \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\Psi}^{2} + L_{1} \\ddot{\\theta}\\right) \\operatorname{cos}\\left(\\beta\\right))\\mathbf{\\hat{xyz}_x} + (L_{1} \\operatorname{sin}\\left(\\theta\\right) \\ddot{\\Psi} + 2 L_{1} \\operatorname{cos}\\left(\\theta\\right) \\dot{\\Psi} \\dot{\\theta} + L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} \\dot{\\theta} - L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\ddot{\\Psi} - L_{2} \\left(\\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta} - \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta}\\right) \\dot{\\Psi} - R \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} \\dot{\\theta} - R \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\ddot{\\Psi} + \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta} - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right) \\dot{\\theta}\\right) \\dot{\\Psi} + \\ddot{\\phi}\\right))\\mathbf{\\hat{xyz}_y} + (- L_{2} \\dot{\\theta}^{2} + R \\ddot{\\theta} + \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\left(- L_{2} \\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right) - \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\beta\\right)\\right) \\dot{\\Psi} - R \\left(\\left(- \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\theta\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) \\dot{\\Psi} + \\dot{\\phi}\\right)\\right) \\dot{\\Psi} + \\left(- L_{1} \\operatorname{sin}^{2}\\left(\\theta\\right) \\dot{\\Psi}^{2} - L_{1} \\dot{\\theta}^{2}\\right) \\operatorname{cos}\\left(\\beta\\right) + \\left(- L_{1} \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\Psi}^{2} + L_{1} \\ddot{\\theta}\\right) \\operatorname{sin}\\left(\\beta\\right))\\mathbf{\\hat{xyz}_z}$$"
      ],
      "text/plain": [
       "⎛     2                     2\n",
       "⎜    d            ⎛d       ⎞                                           ⎛\n",
       "⎜L₂⋅───(θ(t)) + R⋅⎜──(θ(t))⎟  - (-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅⎜- L₂⋅(\n",
       "⎜     2           ⎝dt      ⎠                                           ⎝\n",
       "⎝   dt\n",
       "\n",
       "\n",
       "                                      d            ⎛\n",
       "-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅──(Ψ(t)) - R⋅⎜(-sin(β)⋅sin(θ(t)) + cos(β\n",
       "                                      dt           ⎝\n",
       "\n",
       "\n",
       "                                              ⎛                          2\n",
       "             d          d       ⎞⎞ d          ⎜        2       ⎛d       ⎞\n",
       ")⋅cos(θ(t)))⋅──(Ψ(t)) + ──(φ(t))⎟⎟⋅──(Ψ(t)) - ⎜- L₁⋅sin (θ(t))⋅⎜──(Ψ(t))⎟  - L\n",
       "             dt         dt      ⎠⎠ dt         ⎝                ⎝dt      ⎠\n",
       "\n",
       "\n",
       "            2⎞          ⎛                                   2        2      ⎞\n",
       "  ⎛d       ⎞ ⎟          ⎜                         ⎛d       ⎞        d       ⎟\n",
       "₁⋅⎜──(θ(t))⎟ ⎟⋅sin(β) + ⎜- L₁⋅sin(θ(t))⋅cos(θ(t))⋅⎜──(Ψ(t))⎟  + L₁⋅───(θ(t))⎟⋅\n",
       "  ⎝dt      ⎠ ⎠          ⎜                         ⎝dt      ⎠         2      ⎟\n",
       "                        ⎝                                          dt       ⎠\n",
       "\n",
       "      ⎞         ⎛               2\n",
       "      ⎟         ⎜              d                         d        d\n",
       "cos(β)⎟ xyz_x + ⎜L₁⋅sin(θ(t))⋅───(Ψ(t)) + 2⋅L₁⋅cos(θ(t))⋅──(Ψ(t))⋅──(θ(t)) + L\n",
       "      ⎟         ⎜               2                        dt       dt\n",
       "      ⎠         ⎝             dt\n",
       "\n",
       "\n",
       "                                         d        d\n",
       "₂⋅(-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅──(Ψ(t))⋅──(θ(t)) - L₂⋅(-sin(β)⋅cos(θ\n",
       "                                         dt       dt\n",
       "\n",
       "\n",
       "                           2\n",
       "                          d             ⎛                 d\n",
       "(t)) - sin(θ(t))⋅cos(β))⋅───(Ψ(t)) - L₂⋅⎜sin(β)⋅sin(θ(t))⋅──(θ(t)) - cos(β)⋅co\n",
       "                           2            ⎝                 dt\n",
       "                         dt\n",
       "\n",
       "\n",
       "        d       ⎞ d                                                   d\n",
       "s(θ(t))⋅──(θ(t))⎟⋅──(Ψ(t)) - R⋅(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅──(Ψ(t))\n",
       "        dt      ⎠ dt                                                  dt\n",
       "\n",
       "\n",
       "              ⎛                                         2\n",
       " d            ⎜                                        d          ⎛\n",
       "⋅──(θ(t)) - R⋅⎜(-sin(β)⋅sin(θ(t)) + cos(β)⋅cos(θ(t)))⋅───(Ψ(t)) + ⎜- sin(β)⋅co\n",
       " dt           ⎜                                         2         ⎝\n",
       "              ⎝                                       dt\n",
       "\n",
       "                                                           2      ⎞⎞         ⎛\n",
       "        d                           d       ⎞ d           d       ⎟⎟         ⎜\n",
       "s(θ(t))⋅──(θ(t)) - sin(θ(t))⋅cos(β)⋅──(θ(t))⎟⋅──(Ψ(t)) + ───(φ(t))⎟⎟ xyz_y + ⎜\n",
       "        dt                          dt      ⎠ dt           2      ⎟⎟         ⎜\n",
       "                                                         dt       ⎠⎠         ⎝\n",
       "\n",
       "               2       2\n",
       "     ⎛d       ⎞       d                                                 ⎛\n",
       "- L₂⋅⎜──(θ(t))⎟  + R⋅───(θ(t)) + (-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅⎜- L₂⋅\n",
       "     ⎝dt      ⎠        2                                                ⎝\n",
       "                     dt\n",
       "\n",
       "\n",
       "                                       d            ⎛\n",
       "(-sin(β)⋅cos(θ(t)) - sin(θ(t))⋅cos(β))⋅──(Ψ(t)) - R⋅⎜(-sin(β)⋅sin(θ(t)) + cos(\n",
       "                                       dt           ⎝\n",
       "\n",
       "\n",
       "                                               ⎛                          2\n",
       "              d          d       ⎞⎞ d          ⎜        2       ⎛d       ⎞\n",
       "β)⋅cos(θ(t)))⋅──(Ψ(t)) + ──(φ(t))⎟⎟⋅──(Ψ(t)) + ⎜- L₁⋅sin (θ(t))⋅⎜──(Ψ(t))⎟  -\n",
       "              dt         dt      ⎠⎠ dt         ⎝                ⎝dt      ⎠\n",
       "\n",
       "\n",
       "             2⎞          ⎛                                   2        2      ⎞\n",
       "   ⎛d       ⎞ ⎟          ⎜                         ⎛d       ⎞        d       ⎟\n",
       "L₁⋅⎜──(θ(t))⎟ ⎟⋅cos(β) + ⎜- L₁⋅sin(θ(t))⋅cos(θ(t))⋅⎜──(Ψ(t))⎟  + L₁⋅───(θ(t))⎟\n",
       "   ⎝dt      ⎠ ⎠          ⎜                         ⎝dt      ⎠         2      ⎟\n",
       "                         ⎝                                          dt       ⎠\n",
       "\n",
       "       ⎞\n",
       "       ⎟\n",
       "⋅sin(β)⎟ xyz_z\n",
       "       ⎟\n",
       "       ⎠"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Or in the xyz frame\n",
    "D.acc(N).express(xyz)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 2\n",
    "Figure 2 shows an instantaneous view of a disk rotating at a rate $\\omega_2$ about point Q. An ant is relaxing on the disk (*i.e.* its location on the disk is fixed) at a distance $R$ from its center. When, $\\theta = 0$, the ant reaches her topmost position. The center of the disk is located on the T-bar at a distance $L$ from the axis about which the bar is spinning, $Z$. The rate of rotation of the T-bar is described by $\\omega_1$. \n",
    "\n",
    "a. What is the velocity of the ant, $\\bar{v}_A$?\n",
    "\n",
    "b. What is the acceleration of the ant, $\\bar{a}_A$?\n",
    "\n",
    "c. What does the ant observe as the velocity and acceleration of point P, the top of the T-bar?\n",
    "\n",
    "<p style=\"text-align:center\">\n",
    "\t<img src=\"http://shared.crawlab.org/disk_on_Tbar.png\" alt=\"An Ant Riding on a Spinning Disk on a T-bar\" width=50%><br>\n",
    "    <strong>Figure 2: An Ant Riding on a Spinning Disk on a T-bar</strong>\n",
    "</p>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Define the genearlized coordinates and other dynamic symbols\n",
    "theta_1, theta = dynamicsymbols('theta_1 theta')\n",
    "\n",
    "# Also define the first derivatives\n",
    "w1, w2 = dynamicsymbols('theta_1 theta', 1)\n",
    "\n",
    "# Define the symbols for the other paramters\n",
    "L, R = sympy.symbols('L R')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Define the Newtonian reference frame\n",
    "N = ReferenceFrame('N')\n",
    "\n",
    "# Define a body-fixed frame to the T-bar\n",
    "prime = N.orientnew('prime', 'Axis', [theta_1, N.z])\n",
    "\n",
    "# Finally, define the last frame such that its z-axis is aligned with BC\n",
    "xyz = prime.orientnew('xyz', 'Axis', [theta, prime.y])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part a."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$R \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta}\\mathbf{\\hat{prime}_x} + (L \\dot{\\theta}_{1} + R \\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta}_{1})\\mathbf{\\hat{prime}_y} -  R \\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta}\\mathbf{\\hat{prime}_z}$$"
      ],
      "text/plain": [
       "            d                  ⎛  d                       d        ⎞\n",
       "R⋅cos(θ(t))⋅──(θ(t)) prime_x + ⎜L⋅──(θ₁(t)) + R⋅sin(θ(t))⋅──(θ₁(t))⎟ prime_y +\n",
       "            dt                 ⎝  dt                      dt       ⎠\n",
       "\n",
       "              d\n",
       " -R⋅sin(θ(t))⋅──(θ(t)) prime_z\n",
       "              dt"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Define point A and its velocity\n",
    "O = Point('O')\n",
    "O.set_vel(N, 0 * N.x)\n",
    "\n",
    "# Define point Q and set its velocity. Locate it relative to O\n",
    "Q = O.locatenew('B', L * prime.x)\n",
    "Q.v2pt_theory(O, N, prime)\n",
    "\n",
    "# Define point A and set its velocity. Locate it relative to Q\n",
    "A = Q.locatenew('Q', R * xyz.z)\n",
    "A.v2pt_theory(Q, N, xyz)\n",
    "\n",
    "# Now, we can express the velocity of the ant in any frame we chose. Here, we'll use the x'y'z' frame.\n",
    "A.v2pt_theory(Q, N, xyz).express(prime)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part b."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Once we have the velocity (and if we have define the motion variables properly), the acceleration is easy to get. As before, we can express it in any frame of our choosing."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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WTtT67yvh6ZNwyT/tOzV1tK3HguGCxLCrQE8+BGM1vFCqa2VKjHlVXvrkwZ9r7mlDZ8I0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\nm+TwvSiBSAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$(- L \\dot{\\theta}_{1}^{2} + \\left(- R \\operatorname{sin}^{2}\\left(\\theta\\right) \\dot{\\theta}_{1}^{2} - R \\dot{\\theta}^{2}\\right) \\operatorname{sin}\\left(\\theta\\right) + \\left(- R \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta}_{1}^{2} + R \\ddot{\\theta}\\right) \\operatorname{cos}\\left(\\theta\\right))\\mathbf{\\hat{prime}_x} + (L \\ddot{\\theta}_{1} + R \\operatorname{sin}\\left(\\theta\\right) \\ddot{\\theta}_{1} + 2 R \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta} \\dot{\\theta}_{1})\\mathbf{\\hat{prime}_y} + (\\left(- R \\operatorname{sin}^{2}\\left(\\theta\\right) \\dot{\\theta}_{1}^{2} - R \\dot{\\theta}^{2}\\right) \\operatorname{cos}\\left(\\theta\\right) - \\left(- R \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta}_{1}^{2} + R \\ddot{\\theta}\\right) \\operatorname{sin}\\left(\\theta\\right))\\mathbf{\\hat{prime}_z}$$"
      ],
      "text/plain": [
       "⎛               2   ⎛                          2               2⎞\n",
       "⎜    ⎛d        ⎞    ⎜       2       ⎛d        ⎞      ⎛d       ⎞ ⎟\n",
       "⎜- L⋅⎜──(θ₁(t))⎟  + ⎜- R⋅sin (θ(t))⋅⎜──(θ₁(t))⎟  - R⋅⎜──(θ(t))⎟ ⎟⋅sin(θ(t)) +\n",
       "⎜    ⎝dt       ⎠    ⎝               ⎝dt       ⎠      ⎝dt      ⎠ ⎠\n",
       "⎝\n",
       "\n",
       "⎛                                   2       2      ⎞          ⎞           ⎛\n",
       "⎜                        ⎛d        ⎞       d       ⎟          ⎟           ⎜\n",
       "⎜- R⋅sin(θ(t))⋅cos(θ(t))⋅⎜──(θ₁(t))⎟  + R⋅───(θ(t))⎟⋅cos(θ(t))⎟ prime_x + ⎜L⋅─\n",
       "⎜                        ⎝dt       ⎠        2      ⎟          ⎟           ⎜\n",
       "⎝                                         dt       ⎠          ⎠           ⎝  d\n",
       "\n",
       " 2                        2                                          ⎞\n",
       "d                        d                         d        d        ⎟\n",
       "──(θ₁(t)) + R⋅sin(θ(t))⋅───(θ₁(t)) + 2⋅R⋅cos(θ(t))⋅──(θ(t))⋅──(θ₁(t))⎟ prime_y\n",
       " 2                        2                        dt       dt       ⎟\n",
       "t                       dt                                           ⎠\n",
       "\n",
       "   ⎛⎛                          2               2⎞             ⎛\n",
       "   ⎜⎜       2       ⎛d        ⎞      ⎛d       ⎞ ⎟             ⎜\n",
       " + ⎜⎜- R⋅sin (θ(t))⋅⎜──(θ₁(t))⎟  - R⋅⎜──(θ(t))⎟ ⎟⋅cos(θ(t)) - ⎜- R⋅sin(θ(t))⋅c\n",
       "   ⎜⎝               ⎝dt       ⎠      ⎝dt      ⎠ ⎠             ⎜\n",
       "   ⎝                                                          ⎝\n",
       "\n",
       "                    2       2      ⎞          ⎞\n",
       "         ⎛d        ⎞       d       ⎟          ⎟\n",
       "os(θ(t))⋅⎜──(θ₁(t))⎟  + R⋅───(θ(t))⎟⋅sin(θ(t))⎟ prime_z\n",
       "         ⎝dt       ⎠        2      ⎟          ⎟\n",
       "                          dt       ⎠          ⎠"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# In the x'y'z' frame\n",
    "A.acc(N).express(prime)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 3\n",
    "The sketch in Figure 3 shows a thin plate of mass, $m$, connected to a massless shaft. The corner of the plate is labeled as O, and the shaft-plate body rotates around $X$ at a constant rate, $\\omega$. For this system:\n",
    "\n",
    "a. Write the angular momentum about point O. You do *not* need to derive the inertial properties of the plate, but be sure properly identify what terms would be needed.\n",
    "\n",
    "b. What moment must be applied at point O for this motion to occur?\n",
    "\n",
    "<p style=\"text-align:center\">\n",
    "\t<img src=\"http://shared.crawlab.org/rotating_plate.png\" alt=\"A Rotating Thin Plate\" width=50%><br>\n",
    "    <strong>Figure 3: A Rotating Thin Plate</strong>\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Define the genearlized coordinate\n",
    "theta = dynamicsymbols('theta')\n",
    "\n",
    "# Also define the first derivative\n",
    "theta_dot = dynamicsymbols('theta', 1)\n",
    "\n",
    "# Define the symbols for the other paramters\n",
    "m, L, Ixx, Iyy, Izz, Ixy, t = sympy.symbols('m L I_xx, I_yy, I_zz, I_xy t')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Define the Newtonian reference frame\n",
    "N = ReferenceFrame('N')\n",
    "\n",
    "# Define the frame that is fixed to the plate\n",
    "xyz = N.orientnew('xyz', 'Axis', [theta, N.x])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEcAAAAvBAMAAABDOcobAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nBAs1AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{L \\dot{\\theta}}{2}\\mathbf{\\hat{xyz}_z}$$"
      ],
      "text/plain": [
       "  d\n",
       "L⋅──(θ(t))\n",
       "  dt\n",
       "────────── xyz_z\n",
       "    2"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Define the location of the points O and the plate COM\n",
    "O = Point('O')\n",
    "O.set_vel(N, 0 * N.x)\n",
    "\n",
    "# Define point Q and set its velocity. Locate it relative to O\n",
    "G = O.locatenew('G', L * xyz.x + L/2 * xyz.y)\n",
    "G.v2pt_theory(O, N, xyz)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Now, we need to define the rigid body element for the plate\n",
    "\n",
    "# But first, we need to define its inertia properties\n",
    "I = inertia(xyz, Ixx, Iyy, Izz, -Ixy)\n",
    "plate = RigidBody('plate', G, xyz, m, (I, O))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMQAAAAcBAMAAADM5Q0aAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdqvNmSJEiTK73RBm\nVO/lkYivAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADrklEQVRIDc1TQYgbVRj+MslkZmfSTChFDx52\nuj1UrDQ5iBUVsxUEQXCj3jzszq0IYmahIK1gppcem6CHohUMXRVtxX1YtRfdBPEk0kbwqDQgLcJi\nk03bbDdp8vzfZGbysl3XtVD0h3n/93////3v8b83wH9gtdw9bWre2LnMsHdeK1eelYP/E2b3dhjz\nn2Tf1sMKdelf7GH+FMrw6lMRDMBu/lhhzCXP/BIGbzx5NYRb+Fj2xnMS/dp3s0GU/P0Lqd+InHgL\np5gVnN1cw9tSj7uglZMofRX9IKxltKaUETC5LhOfw2oEMcO2Uy26ks6oYxCEJGJSRkCtKxGpJqy6\nFG8DS46UvATckcJNMN6WCGMW5Z85b1zhlWm+hhJHvvcMr0sVEcxHiMBH0AfTfLCLb8Q4z2i8HedL\nrWjTiZkWHUw7iR6MA0BrFlMuPmPGUG4WYfkKk13Eupir4ixD+iZwGsYZLDfD2mImROSzMzMtIG8f\nIzzdxwrDH5i7LRVEMLkWQUDpzBxsI97RKkCMM3MVaS/BvbCiFCFilgA63fzGecK1NRwFmknuhqWy\n1+SNjRxoGGrrfQaYNKhZpFEcP6OJma7TTOmR8QZ1SwwhfowVaSIXzgl7Qmw1cYWWizJJiv6rmmum\nC7RTKyfKfNsIAXlzgERTpFcpUPmvNrnsLd0ht9msisTUGniRwpUho3Vh8D2tCi+cDCrETLU9Jx7G\n4vO7PbMLK0MnPObvmz3nASle14i6y8QVCskRG+dRcyCu5tCHFVqV3qe0Xr6DxUAkZvqQkvskxvYZ\nDq7hOPHPmi2XXFmMaGED8Tr5zUZX6Evcvt5E2tGqNFl7Rcxf5zkxggqeDjRGG/Coie4zh1Ua/8U+\nyuJ/jIvR5lfN0m/L/XSPBYLALVMvkqjfVGMVJDLzBZglpnNHSBqkHbKLg0tDVr6Gl/MdF7AKlLlF\n31d7GDS+jiynKFUVRxq2uDfVnXcoOzZ1H99PERUprpKBefgF0H/qKpxTpx8pU+bX+W21hzQLRGV4\n0Nv6uAUhFUo9Ipb/jKAEhMRw0p5EAQU8GsVZ54cRVtvH4S5oVS1KCXDFPjWOF8QT22y+xPCKE/yu\nDt1NaOXcxyNo7j3yIPvygcW3wozvLx/w3/+Ie6c3kRsFviS5992JVKpzwYuIqY4d4S2AsuRErGIX\nK1EwCb7GB5PEI4+PY11czY4s1cpYw8bWpfuTua0TPvveNrkdp15/8+9LX4L4Ce+rlU7Y97U/NX/l\nNC1/ATLSI/ZlKi8LAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$I_{xx} \\dot{\\theta}\\mathbf{\\hat{xyz}_x} -  I_{xy} \\dot{\\theta}\\mathbf{\\hat{xyz}_y}$$"
      ],
      "text/plain": [
       "    d                      d\n",
       "Iₓₓ⋅──(θ(t)) xyz_x + -I_xy⋅──(θ(t)) xyz_y\n",
       "    dt                     dt"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "angular_momentum(O, N, plate).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The sum of moments is just the total time derivative of this. In this case, we know that it is just \n",
    "\n",
    "$$ \\quad \\sum \\bar{M}_O = \\bar{\\omega} \\times \\bar{H}_O $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGsAAAAaBAMAAABC779bAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$-  I_{xy} \\dot{\\theta}^{2}\\mathbf{\\hat{xyz}_z}$$"
      ],
      "text/plain": [
       "                2\n",
       "      ⎛d       ⎞\n",
       "-I_xy⋅⎜──(θ(t))⎟  xyz_z\n",
       "      ⎝dt      ⎠"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "((theta_dot * N.x).cross(angular_momentum(O, N, plate))).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr style=\"border: 0px;\n",
    "        height: 1px;\n",
    "        text-align: center;\n",
    "        background: #333;\n",
    "        background-image: -webkit-linear-gradient(left, #ccc, #333, #ccc); \n",
    "        background-image:    -moz-linear-gradient(left, #ccc, #333, #ccc); \n",
    "        background-image:     -ms-linear-gradient(left, #ccc, #333, #ccc); \n",
    "        background-image:      -o-linear-gradient(left, #ccc, #333, #ccc);\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Licenses\n",
    "Code is licensed under a 3-clause BSD style license. See the licenses/LICENSE.md file.\n",
    "\n",
    "Other content is provided under a [Creative Commons Attribution-NonCommercial 4.0 International License](http://creativecommons.org/licenses/by-nc/4.0/), CC-BY-NC 4.0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<script>\n",
       "    MathJax.Hub.Config({\n",
       "                        TeX: {\n",
       "                           extensions: [\"AMSmath.js\"]\n",
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       "                tex2jax: {\n",
       "                    inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
       "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
       "                },\n",
       "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
       "                \"HTML-CSS\": {\n",
       "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
       "                }\n",
       "        });\n",
       "</script>\n",
       "\n",
       "<style>\n",
       "    @font-face {\n",
       "        font-family: \"Computer Modern\";\n",
       "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf');\n",
       "    }\n",
       "    @font-face {\n",
       "        font-family: \"Computer Modern\";\n",
       "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunsx.otf');\n",
       "        font-weight: bold;\n",
       "    }\n",
       "    @font-face {\n",
       "        font-family: \"Computer Modern\";\n",
       "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunsi.otf');\n",
       "        font-style: oblique;\n",
       "    }\n",
       "    @font-face {\n",
       "        font-family: \"Computer Modern\";\n",
       "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunso.otf');\n",
       "        font-weight: bold;\n",
       "        font-style: oblique;\n",
       "    }\n",
       "\n",
       "    div.cell {\n",
       "        max-width: 1100px;\n",
       "    }\n",
       "    \n",
       "    h1 {\n",
       "        font-family: Computer Modern;\n",
       "    }\n",
       "    \n",
       "    h4 {\n",
       "        margin-top: 12px;\n",
       "        margin-bottom: 3px;\n",
       "    }\n",
       "\n",
       "    div.text_cell_render {\n",
       "        font-family: Computer Modern, \"Helvetica Neue\", Arial, Helvetica, Geneva, sans-serif;\n",
       "        line-height: 145%;\n",
       "        font-size: 130%;\n",
       "        width: 100%;\n",
       "        max-width: 1100px;\n",
       "    }\n",
       "    \n",
       "    .CodeMirror {\n",
       "        font-family: \"Source Code Pro\", source-code-pro, Consolas, monospace;\n",
       "    }\n",
       "    \n",
       "    .warning {\n",
       "        color: rgb( 240, 20, 20 )\n",
       "    }  \n",
       "  \n",
       "   \n",
       "    hr.style-end {\n",
       "        border: 0px !important;\n",
       "        height: 1px !important;\n",
       "        text-align: center !important;\n",
       "        background: #333 !important;\n",
       "        background-image: -webkit-linear-gradient(left, #ccc, #333, #ccc) !important; \n",
       "        background-image:    -moz-linear-gradient(left, #ccc, #333, #ccc) !important; \n",
       "        background-image:     -ms-linear-gradient(left, #ccc, #333, #ccc) !important; \n",
       "        background-image:      -o-linear-gradient(left, #ccc, #333, #ccc) !important; \n",
       "    }\n",
       "\n",
       "    hr.style-end:after {\n",
       "        content: &#x269C !important;\n",
       "        left: 50% !important;\n",
       "        position: absolute !important;\n",
       "        /* Controls the whitespace around the symbol */\n",
       "        padding: 0px !important;\n",
       "        background: #fff !important;\n",
       "    }\n",
       "    \n",
       "/*  Center figures, etc\n",
       "    .ui-wrapper {\n",
       "        margin-left: auto !important;\n",
       "        margin-right: auto !important;\n",
       "    }\n",
       "*/\n",
       "    \n",
       "</style>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# This cell will just improve the styling of the notebook\n",
    "# You can ignore it, if you are okay with the default sytling\n",
    "from IPython.core.display import HTML\n",
    "import urllib.request\n",
    "response = urllib.request.urlopen(\"https://cl.ly/1B1y452Z1d35\")\n",
    "HTML(response.read().decode(\"utf-8\"))"
   ]
  }
 ],
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 },
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