Flapping Advance Ratio.ipynb

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  "name": "",
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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import sympy\n",
      "import sympy.physics.mechanics as mech\n",
      "import scipy.optimize\n",
      "\n",
      "sympy.init_printing()\n",
      "sympy.physics.vector.init_vprinting()\n",
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "alpha, beta, eta, P, Q, R, r, alpha_l, beta_l, mu = \\\n",
      "    sympy.symbols('alpha, beta, eta, P, Q, R, r, alpha_l, beta_l, mu', real=True, imaginary=False)\n",
      "V_T, s, rho, c, c_0 = sympy.symbols('V_T, s, rho, c, c_0', postive=True, real=True, imaginary=False)\n",
      "C_L_bar = sympy.Symbol(r'\\bar{C_L}', real=True, imaginary=False)\n",
      "C_D_bar = sympy.Symbol(r'\\bar{C_D}', real=True, imaginary=False)\n",
      "t = sympy.Symbol('t')\n",
      "\n",
      "sub_adv = {V_T(t) : mu*s*eta(t).diff(t)}\n",
      "mu_sub = {mu:V_T(t)/(s*eta(t).diff(t))}\n",
      "\n",
      "#inertial frame\n",
      "frame_i = mech.ReferenceFrame('i')\n",
      "\n",
      "# body frame\n",
      "frame_b = mech.ReferenceFrame('b')\n",
      "frame_b.set_ang_vel(frame_i, \n",
      "    P(t)*frame_b.x + Q(t)*frame_b.y + R(t)*frame_b.z)\n",
      "\n",
      "# wind frame\n",
      "frame_w = frame_b.orientnew(\n",
      "    'w', 'Body', (-alpha(t), beta(t), 0), '231')\n",
      "\n",
      "# wing frame\n",
      "frame_l = frame_b.orientnew(\n",
      "    'l', 'Axis', (eta(t), frame_b.x))\n",
      "\n",
      "# local wind frame\n",
      "frame_m = frame_l.orientnew(\n",
      "    'm', 'Body', (-alpha_l(t), beta_l(t), 0), '231')\n",
      "\n",
      "# center of mass\n",
      "point_cm = mech.Point('cm')\n",
      "point_cm.set_vel(frame_b, 0)\n",
      "point_cm.set_vel(frame_i, V_T(t)*frame_w.x)\n",
      "\n",
      "# point on wing\n",
      "point_p = point_cm.locatenew('p', r*s*frame_l.y)\n",
      "point_p.set_vel(frame_l, 0)\n",
      "point_p.v2pt_theory(point_cm, frame_i, frame_l)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$V_{T}\\mathbf{\\hat{w}_x} -  r s \\left(- Q \\operatorname{sin}\\left(\\eta\\right) + R \\operatorname{cos}\\left(\\eta\\right)\\right)\\mathbf{\\hat{l}_x} + r s \\left(P + \\dot{\\eta}\\right)\\mathbf{\\hat{l}_z}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAdUAAAAbBAMAAADYAgi0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
       "prompt_number": 2,
       "text": [
        "V_T w_x - r\u22c5s\u22c5(-Q\u22c5sin(\u03b7) + R\u22c5cos(\u03b7)) l_x + r\u22c5s\u22c5(P + \u03b7\u0307) l_z"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Local angle of attack w.r.t. wing."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "alpha_l_expr = sympy.atan(point_p.vel(frame_i).dot(frame_l.z)/\n",
      "    point_p.vel(frame_i).dot(frame_l.x)).simplify()\n",
      "alpha_l_expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\operatorname{atan}\\left(\\frac{r s \\left(P + \\dot{\\eta}\\right) + \\left(\\operatorname{sin}\\left(\\alpha\\right) \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\eta\\right) - \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\eta\\right)\\right) V_{T}}{r s \\left(Q \\operatorname{sin}\\left(\\eta\\right) - R \\operatorname{cos}\\left(\\eta\\right)\\right) + V_{T} \\operatorname{cos}\\left(\\alpha\\right) \\operatorname{cos}\\left(\\beta\\right)}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
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       "prompt_number": 3,
       "text": [
        "    \u239br\u22c5s\u22c5(P + \u03b7\u0307) + (sin(\u03b1)\u22c5cos(\u03b2)\u22c5cos(\u03b7) - sin(\u03b2)\u22c5sin(\u03b7))\u22c5V_T\u239e\n",
        "atan\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\n",
        "    \u239d      r\u22c5s\u22c5(Q\u22c5sin(\u03b7) - R\u22c5cos(\u03b7)) + V_T\u22c5cos(\u03b1)\u22c5cos(\u03b2)      \u23a0"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Local side-slip angle w.r.t. wing."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "beta_l_expr = sympy.asin(point_p.vel(frame_i).dot(frame_l.y)/\n",
      "    point_p.vel(frame_i).magnitude())"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#beta_l_expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "q_expr = (rho*point_p.vel(frame_i).magnitude()**2/2).collect(r).simplify()\n",
      "q_expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{\\rho}{2} \\left(r^{2} s^{2} \\left(\\left(Q \\operatorname{sin}\\left(\\eta\\right) - R \\operatorname{cos}\\left(\\eta\\right)\\right)^{2} + \\left(P + \\dot{\\eta}\\right)^{2}\\right) + 2 r s \\left(\\left(Q \\operatorname{sin}\\left(\\eta\\right) - R \\operatorname{cos}\\left(\\eta\\right)\\right) \\operatorname{cos}\\left(\\alpha\\right) \\operatorname{cos}\\left(\\beta\\right) + \\left(\\operatorname{sin}\\left(\\alpha\\right) \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\eta\\right) - \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\eta\\right)\\right) \\left(P + \\dot{\\eta}\\right)\\right) V_{T} + V^{2}_{T}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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yfM+mtgeBicRH451al0zJ5LdV3vGzwOMs2VVRXJ20o/AAANtKSOWfd5YgyGdWbpG8KMms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\nZH60Aue/Av8HyOb8jheujC8AAAAASUVORK5CYII=\n",
       "prompt_number": 6,
       "text": [
        "  \u239b 2  2 \u239b                     2           2\u239e                                 \n",
        "\u03c1\u22c5\u239dr \u22c5s \u22c5\u239d(Q\u22c5sin(\u03b7) - R\u22c5cos(\u03b7))  + (P + \u03b7\u0307) \u23a0 + 2\u22c5r\u22c5s\u22c5((Q\u22c5sin(\u03b7) - R\u22c5cos(\u03b7))\u22c5c\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                                                            2 \n",
        "\n",
        "                                                                         2\u239e\n",
        "os(\u03b1)\u22c5cos(\u03b2) + (sin(\u03b1)\u22c5cos(\u03b2)\u22c5cos(\u03b7) - sin(\u03b2)\u22c5sin(\u03b7))\u22c5(P + \u03b7\u0307))\u22c5V_T + V_T \u23a0\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                                                           "
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Lp_expr = -C_L_bar*rho*q_expr*c(r)/2*frame_m.z\n",
      "Lp_expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-  \\frac{\\bar{C_{L}} \\rho^{2}}{4} \\left(r^{2} s^{2} \\left(\\left(Q \\operatorname{sin}\\left(\\eta\\right) - R \\operatorname{cos}\\left(\\eta\\right)\\right)^{2} + \\left(P + \\dot{\\eta}\\right)^{2}\\right) + 2 r s \\left(\\left(Q \\operatorname{sin}\\left(\\eta\\right) - R \\operatorname{cos}\\left(\\eta\\right)\\right) \\operatorname{cos}\\left(\\alpha\\right) \\operatorname{cos}\\left(\\beta\\right) + \\left(\\operatorname{sin}\\left(\\alpha\\right) \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\eta\\right) - \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\eta\\right)\\right) \\left(P + \\dot{\\eta}\\right)\\right) V_{T} + V^{2}_{T}\\right) \\operatorname{c}\\left(r\\right)\\mathbf{\\hat{m}_z}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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       "prompt_number": 7,
       "text": [
        "            2 \u239b 2  2 \u239b                     2           2\u239e                     \n",
        "-\\bar{C_L}\u22c5\u03c1 \u22c5\u239dr \u22c5s \u22c5\u239d(Q\u22c5sin(\u03b7) - R\u22c5cos(\u03b7))  + (P + \u03b7\u0307) \u23a0 + 2\u22c5r\u22c5s\u22c5((Q\u22c5sin(\u03b7) -\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        " R\u22c5cos(\u03b7))\u22c5cos(\u03b1)\u22c5cos(\u03b2) + (sin(\u03b1)\u22c5cos(\u03b2)\u22c5cos(\u03b7) - sin(\u03b2)\u22c5sin(\u03b7))\u22c5(P + \u03b7\u0307))\u22c5V_\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "       4                                                                      \n",
        "\n",
        "       2\u239e      \n",
        "T + V_T \u23a0\u22c5c(r) \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                m_z"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Dp_expr = -C_D_bar*rho*q_expr*c(r)/2*frame_m.x\n",
      "Dp_expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-  \\frac{\\bar{C_{D}} \\rho^{2}}{4} \\left(r^{2} s^{2} \\left(\\left(Q \\operatorname{sin}\\left(\\eta\\right) - R \\operatorname{cos}\\left(\\eta\\right)\\right)^{2} + \\left(P + \\dot{\\eta}\\right)^{2}\\right) + 2 r s \\left(\\left(Q \\operatorname{sin}\\left(\\eta\\right) - R \\operatorname{cos}\\left(\\eta\\right)\\right) \\operatorname{cos}\\left(\\alpha\\right) \\operatorname{cos}\\left(\\beta\\right) + \\left(\\operatorname{sin}\\left(\\alpha\\right) \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\eta\\right) - \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\eta\\right)\\right) \\left(P + \\dot{\\eta}\\right)\\right) V_{T} + V^{2}_{T}\\right) \\operatorname{c}\\left(r\\right)\\mathbf{\\hat{m}_x}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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       "prompt_number": 8,
       "text": [
        "            2 \u239b 2  2 \u239b                     2           2\u239e                     \n",
        "-\\bar{C_D}\u22c5\u03c1 \u22c5\u239dr \u22c5s \u22c5\u239d(Q\u22c5sin(\u03b7) - R\u22c5cos(\u03b7))  + (P + \u03b7\u0307) \u23a0 + 2\u22c5r\u22c5s\u22c5((Q\u22c5sin(\u03b7) -\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        " R\u22c5cos(\u03b7))\u22c5cos(\u03b1)\u22c5cos(\u03b2) + (sin(\u03b1)\u22c5cos(\u03b2)\u22c5cos(\u03b7) - sin(\u03b2)\u22c5sin(\u03b7))\u22c5(P + \u03b7\u0307))\u22c5V_\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "       4                                                                      \n",
        "\n",
        "       2\u239e      \n",
        "T + V_T \u23a0\u22c5c(r) \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                m_x"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "assump_subs = {P(t):0, Q(t):0, R(t):0, alpha(t):0, beta(t):0, c(r):c_0}\n",
      "assump_subs.update({\n",
      "    beta_l(t): beta_l_expr.subs(assump_subs),\n",
      "    alpha_l(t): alpha_l_expr.subs(assump_subs)})"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "assump_subs"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left \\{ P : 0, \\quad Q : 0, \\quad R : 0, \\quad \\alpha : 0, \\quad \\alpha_{l} : \\operatorname{atan}\\left(\\frac{r s}{V_{T}} \\dot{\\eta}\\right), \\quad \\beta : 0, \\quad \\beta_{l} : 0, \\quad \\operatorname{c}\\left(r\\right) : c_{0}\\right \\}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAtUAAAAyBAMAAABhQzpLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
       "prompt_number": 10,
       "text": [
        "\u23a7                                 \u239br\u22c5s\u22c5\u03b7\u0307\u239e                        \u23ab\n",
        "\u23a8P: 0, Q: 0, R: 0, \u03b1: 0, \u03b1_l: atan\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u239f, \u03b2: 0, \u03b2_l: 0, c(r): c\u2080\u23ac\n",
        "\u23a9                                 \u239d V_T  \u23a0                        \u23ad"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Lp_assump = Lp_expr.subs(assump_subs).to_matrix(frame_l).subs(assump_subs).subs(sub_adv).simplify()\n",
      "Lp_assump"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\frac{\\bar{C_{L}} c_{0} r \\rho^{2}}{4 \\mu} s^{2} \\sqrt{\\mu^{2} + r^{2}} \\left\\lvert{\\mu}\\right\\rvert \\left(\\dot{\\eta}\\right)^{2}\\\\0\\\\- \\frac{\\bar{C_{L}} c_{0}}{4} \\rho^{2} s^{2} \\sqrt{\\mu^{2} + r^{2}} \\left\\lvert{\\mu}\\right\\rvert \\left(\\dot{\\eta}\\right)^{2}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQ0AAABXCAMAAAApruAfAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nXee2eb/Ulq932/2XZvhXIP8DIx1+QZfBfsUAAAAASUVORK5CYII=\n",
       "prompt_number": 11,
       "text": [
        "\u23a1                        _________        \u23a4\n",
        "\u23a2                2  2   \u2571  2    2        2\u23a5\n",
        "\u23a2\\bar{C_L}\u22c5c\u2080\u22c5r\u22c5\u03c1 \u22c5s \u22c5\u2572\u2571  \u03bc  + r  \u22c5\u2502\u03bc\u2502\u22c5\u03b7\u0307 \u23a5\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a2                   4\u22c5\u03bc                   \u23a5\n",
        "\u23a2                                         \u23a5\n",
        "\u23a2                    0                    \u23a5\n",
        "\u23a2                                         \u23a5\n",
        "\u23a2                       _________         \u23a5\n",
        "\u23a2               2  2   \u2571  2    2        2 \u23a5\n",
        "\u23a2-\\bar{C_L}\u22c5c\u2080\u22c5\u03c1 \u22c5s \u22c5\u2572\u2571  \u03bc  + r  \u22c5\u2502\u03bc\u2502\u22c5\u03b7\u0307  \u23a5\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a3                    4                    \u23a6"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Dp_assump = Dp_expr.subs(assump_subs).to_matrix(frame_l).subs(assump_subs).subs(sub_adv).simplify()\n",
      "Dp_assump"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}- \\frac{\\bar{C_{D}} c_{0}}{4} \\rho^{2} s^{2} \\sqrt{\\mu^{2} + r^{2}} \\left\\lvert{\\mu}\\right\\rvert \\left(\\dot{\\eta}\\right)^{2}\\\\0\\\\- \\frac{\\bar{C_{D}} c_{0} r \\rho^{2}}{4 \\mu} s^{2} \\sqrt{\\mu^{2} + r^{2}} \\left\\lvert{\\mu}\\right\\rvert \\left(\\dot{\\eta}\\right)^{2}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAARQAAABYCAMAAAADQ/joAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nn123Le4x/PXB/PE3/OXP/71dgVWXu4j4AAAAAElFTkSuQmCC\n",
       "prompt_number": 12,
       "text": [
        "\u23a1                        _________          \u23a4\n",
        "\u23a2                2  2   \u2571  2    2        2  \u23a5\n",
        "\u23a2 -\\bar{C_D}\u22c5c\u2080\u22c5\u03c1 \u22c5s \u22c5\u2572\u2571  \u03bc  + r  \u22c5\u2502\u03bc\u2502\u22c5\u03b7\u0307   \u23a5\n",
        "\u23a2 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u23a5\n",
        "\u23a2                     4                     \u23a5\n",
        "\u23a2                                           \u23a5\n",
        "\u23a2                     0                     \u23a5\n",
        "\u23a2                                           \u23a5\n",
        "\u23a2                         _________         \u23a5\n",
        "\u23a2                 2  2   \u2571  2    2        2 \u23a5\n",
        "\u23a2-\\bar{C_D}\u22c5c\u2080\u22c5r\u22c5\u03c1 \u22c5s \u22c5\u2572\u2571  \u03bc  + r  \u22c5\u2502\u03bc\u2502\u22c5\u03b7\u0307  \u23a5\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a3                    4\u22c5\u03bc                    \u23a6"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "L_l_x = sympy.integrate(Lp_assump[0], (r, 0, 1)).simplify()\n",
      "L_l_x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{\\bar{C_{L}} c_{0} \\rho^{2} s^{2}}{12 \\mu} \\left(- \\mu^{4} + \\left(\\mu^{2} + 1\\right)^{\\frac{3}{2}} \\left\\lvert{\\mu}\\right\\rvert\\right) \\left(\\dot{\\eta}\\right)^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAU8AAAAzBAMAAADldJfPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 13,
       "text": [
        "                   \u239b               3/2    \u239e    \n",
        "              2  2 \u239c   4   \u239b 2    \u239e       \u239f   2\n",
        "\\bar{C_L}\u22c5c\u2080\u22c5\u03c1 \u22c5s \u22c5\u239d- \u03bc  + \u239d\u03bc  + 1\u23a0   \u22c5\u2502\u03bc\u2502\u23a0\u22c5\u03b7\u0307 \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                      12\u22c5\u03bc                     "
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "L_l_z = sympy.integrate(Lp_assump[2], (r, 0, 1))\n",
      "L_l_z"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{\\bar{C_{L}} c_{0}}{4} \\rho^{2} s^{2} \\left(\\frac{\\mu}{2} \\sqrt{1 + \\frac{1}{\\mu^{2}}} + \\frac{\\mu \\left\\lvert{\\mu}\\right\\rvert}{2} \\operatorname{asinh}\\left(\\frac{1}{\\left\\lvert{\\mu}\\right\\rvert}\\right)\\right) \\left\\lvert{\\mu}\\right\\rvert \\left(\\dot{\\eta}\\right)^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAeoAAAA0CAMAAABy8IvjAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nE2fT65Y/sgAAAABJRU5ErkJggg==\n",
       "prompt_number": 14,
       "text": [
        "                    \u239b       ________                   \u239e         \n",
        "                    \u239c      \u2571     1                     \u239f         \n",
        "                    \u239c\u03bc\u22c5   \u2571  1 + \u2500\u2500               \u239b 1 \u239e\u239f         \n",
        "                    \u239c    \u2571        2    \u03bc\u22c5\u2502\u03bc\u2502\u22c5asinh\u239c\u2500\u2500\u2500\u239f\u239f         \n",
        "               2  2 \u239c  \u2572\u2571        \u03bc                \u239d\u2502\u03bc\u2502\u23a0\u239f       2 \n",
        "-\\bar{C_L}\u22c5c\u2080\u22c5\u03c1 \u22c5s \u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u2502\u03bc\u2502\u22c5\u03b7\u0307  \n",
        "                    \u239d       2                 2        \u23a0         \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                4                                "
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "D_l_x = sympy.integrate(Dp_assump[0], (r, 0, 1)).simplify()\n",
      "D_l_x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{\\bar{C_{D}} \\mu}{8} c_{0} \\rho^{2} s^{2} \\left(\\mu^{2} \\operatorname{asinh}\\left(\\frac{1}{\\left\\lvert{\\mu}\\right\\rvert}\\right) + \\sqrt{\\mu^{2} + 1}\\right) \\left(\\dot{\\eta}\\right)^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAbAAAAAzCAMAAADBw63aAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nDFE7zH/vFQAAAABJRU5ErkJggg==\n",
       "prompt_number": 15,
       "text": [
        "                      \u239b                   ________\u239e     \n",
        "                 2  2 \u239c 2      \u239b 1 \u239e     \u2571  2     \u239f   2 \n",
        "-\\bar{C_D}\u22c5c\u2080\u22c5\u03bc\u22c5\u03c1 \u22c5s \u22c5\u239c\u03bc \u22c5asinh\u239c\u2500\u2500\u2500\u239f + \u2572\u2571  \u03bc  + 1 \u239f\u22c5\u03b7\u0307  \n",
        "                      \u239d        \u239d\u2502\u03bc\u2502\u23a0              \u23a0     \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                           8                            "
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "D_l_z = sympy.integrate(Dp_assump[2], (r, 0, 1)).simplify()\n",
      "D_l_z"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{\\bar{C_{D}} c_{0} \\rho^{2} s^{2}}{12 \\mu} \\left(\\mu^{4} - \\left(\\mu^{2} + 1\\right)^{\\frac{3}{2}} \\left\\lvert{\\mu}\\right\\rvert\\right) \\left(\\dot{\\eta}\\right)^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAUEAAAAzBAMAAAD7vad8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 16,
       "text": [
        "                   \u239b             3/2    \u239e    \n",
        "              2  2 \u239c 4   \u239b 2    \u239e       \u239f   2\n",
        "\\bar{C_D}\u22c5c\u2080\u22c5\u03c1 \u22c5s \u22c5\u239d\u03bc  - \u239d\u03bc  + 1\u23a0   \u22c5\u2502\u03bc\u2502\u23a0\u22c5\u03b7\u0307 \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                     12\u22c5\u03bc                    "
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "F_A_l = sympy.Matrix([(L_l_x + D_l_x).simplify(), 0, (L_l_z + D_l_z).simplify()])\n",
      "F_A_l"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}- \\frac{c_{0} \\rho^{2} s^{2} \\left(\\dot{\\eta}\\right)^{2}}{24 \\mu} \\left(3 \\bar{C_{D}} \\mu^{2} \\left(\\mu^{2} \\operatorname{asinh}\\left(\\frac{1}{\\left\\lvert{\\mu}\\right\\rvert}\\right) + \\sqrt{\\mu^{2} + 1}\\right) + 2 \\bar{C_{L}} \\left(\\mu^{4} - \\left(\\mu^{2} + 1\\right)^{\\frac{3}{2}} \\left\\lvert{\\mu}\\right\\rvert\\right)\\right)\\\\0\\\\\\frac{c_{0} \\rho^{2} s^{2} \\left(\\dot{\\eta}\\right)^{2}}{24 \\mu} \\left(2 \\bar{C_{D}} \\left(\\mu^{4} - \\left(\\mu^{2} + 1\\right)^{\\frac{3}{2}} \\left\\lvert{\\mu}\\right\\rvert\\right) - 3 \\bar{C_{L}} \\mu^{2} \\left(\\mu^{2} \\operatorname{asinh}\\left(\\frac{1}{\\left\\lvert{\\mu}\\right\\rvert}\\right) + \\sqrt{\\mu^{2} + 1}\\right)\\right)\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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+BomSLSR+r+aF8BLsqFTlCVWxlDpfV2qHL64zbUR4vtxW80RLeMMsrx5g9GKe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\n",
       "prompt_number": 17,
       "text": [
        "\u23a1          \u239b               \u239b                   ________\u239e               \u239b      \n",
        "\u23a2     2  2 \u239c             2 \u239c 2      \u239b 1 \u239e     \u2571  2     \u239f               \u239c 4   \u239b\n",
        "\u23a2-c\u2080\u22c5\u03c1 \u22c5s \u22c5\u239c3\u22c5\\bar{C_D}\u22c5\u03bc \u22c5\u239c\u03bc \u22c5asinh\u239c\u2500\u2500\u2500\u239f + \u2572\u2571  \u03bc  + 1 \u239f + 2\u22c5\\bar{C_L}\u22c5\u239d\u03bc  - \u239d\n",
        "\u23a2          \u239d               \u239d        \u239d\u2502\u03bc\u2502\u23a0              \u23a0                      \n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "\u23a2                                               24\u22c5\u03bc                          \n",
        "\u23a2                                                                             \n",
        "\u23a2                                                0                            \n",
        "\u23a2                                                                             \n",
        "\u23a2          \u239b            \u239b             3/2    \u239e                  \u239b             \n",
        "\u23a2     2  2 \u239c            \u239c 4   \u239b 2    \u239e       \u239f                2 \u239c 2      \u239b 1 \u239e\n",
        "\u23a2 c\u2080\u22c5\u03c1 \u22c5s \u22c5\u239c2\u22c5\\bar{C_D}\u22c5\u239d\u03bc  - \u239d\u03bc  + 1\u23a0   \u22c5\u2502\u03bc\u2502\u23a0 - 3\u22c5\\bar{C_L}\u22c5\u03bc \u22c5\u239c\u03bc \u22c5asinh\u239c\u2500\u2500\u2500\u239f\n",
        "\u23a2          \u239d                                                    \u239d        \u239d\u2502\u03bc\u2502\u23a0\n",
        "\u23a2 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "\u23a3                                               24\u22c5\u03bc                          \n",
        "\n",
        "       3/2    \u239e\u239e     \u23a4\n",
        " 2    \u239e       \u239f\u239f   2 \u23a5\n",
        "\u03bc  + 1\u23a0   \u22c5\u2502\u03bc\u2502\u23a0\u239f\u22c5\u03b7\u0307  \u23a5\n",
        "               \u23a0     \u23a5\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "                     \u23a5\n",
        "                     \u23a5\n",
        "                     \u23a5\n",
        "                     \u23a5\n",
        "      ________\u239e\u239e     \u23a5\n",
        "     \u2571  2     \u239f\u239f   2 \u23a5\n",
        " + \u2572\u2571  \u03bc  + 1 \u239f\u239f\u22c5\u03b7\u0307  \u23a5\n",
        "              \u23a0\u23a0     \u23a5\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u23a5\n",
        "                     \u23a6"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "const = {C_L_bar: 1.2, C_D_bar:0.5, rho:1.225, s:0.6, eta(t).diff(t):5, c_0:0.3}"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N_to_g = 1000/9.8\n",
      "two_wings = 2\n",
      "sympy.plot(two_wings*N_to_g*(F_A_l[0]).subs(const),\n",
      "           (mu,0,2), title='Thrust vs. $\\mu$', xlabel='$\\mu$', ylabel='Thrust, gm');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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MBC4xsyZmdiRQACx294+Az8ysKGpovwz4Uwyxiojsk2HDQmlk8OAwsj3fxNXF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\nRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x44be390>"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sympy.plot(-two_wings*N_to_g*(F_A_l[2]).subs(const),\n",
      "           (mu,0,2), title='Lift vs. $\\mu$', xlabel='$\\mu$', ylabel='Lift, gm');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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ZMu3jj8PYxTPPwPTpYcnzHj3CjO999427OhGJmcKjWD7LjqquDpdufe65cKZU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\nREQkZQoPERFJmcJDRERS9v8BB4LCtzo0X8gAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x4232bd0>"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "F_A_l.subs(const)[0]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{1}{\\mu} \\left(0.40516875 \\mu^{4} + 0.25323046875 \\mu^{2} \\left(\\mu^{2} \\operatorname{asinh}\\left(\\frac{1}{\\left\\lvert{\\mu}\\right\\rvert}\\right) + \\sqrt{\\mu^{2} + 1}\\right) - 0.40516875 \\left(\\mu^{2} + 1\\right)^{\\frac{3}{2}} \\left\\lvert{\\mu}\\right\\rvert\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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       "prompt_number": 21,
       "text": [
        "              \u239b                \u239b                   ________\u239e               3/2\n",
        "              \u239c     4        2 \u239c 2      \u239b 1 \u239e     \u2571  2     \u239f       \u239b 2    \u239e   \n",
        "-0.1688203125\u22c5\u239c2.4\u22c5\u03bc  + 1.5\u22c5\u03bc \u22c5\u239c\u03bc \u22c5asinh\u239c\u2500\u2500\u2500\u239f + \u2572\u2571  \u03bc  + 1 \u239f - 2.4\u22c5\u239d\u03bc  + 1\u23a0   \n",
        "              \u239d                \u239d        \u239d\u2502\u03bc\u2502\u23a0              \u23a0                  \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                         \u03bc                                    \n",
        "\n",
        "    \u239e \n",
        "    \u239f \n",
        "\u22c5\u2502\u03bc\u2502\u239f \n",
        "    \u23a0 \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "      "
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f_thrust_mu = sympy.lambdify(mu, F_A_l.subs(const)[0], default_array=True)\n",
      "f_lift_mu = sympy.lambdify(mu, -two_wings*N_to_g*F_A_l.subs(const)[2], default_array=True)\n",
      "mu_max = scipy.optimize.newton(f_thrust_mu, 0.7)\n",
      "lift_max = f_lift_mu(mu_max)\n",
      "print('max lift: {:0.2f} gm @ mu = {:0.2f} -> V_T = {:0.2f} m/s'.format(\n",
      "    lift_max, mu_max, mu_max*const[s]*const[eta(t).diff(t)]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "max lift: 501.13 gm @ mu = 1.25 -> V_T = 3.76 m/s\n"
       ]
      }
     ],
     "prompt_number": 22
    }
   ],
   "metadata": {}
  }
 ]
}