Lift Example.ipynb

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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Examples using the lift equation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Brian R Taylor"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<img src=\"https://farm4.staticflickr.com/3824/9615082251_72a747c4f3_n.jpg\" width=\"320\" height=\"213\" alt=\"DSC_5802\">\n",
      "\n",
      "One of the [UAV Laboratories](http://www.uav.aem.umn.edu/) aircraft, named Thor, has the following characteristics:\n",
      "   * Wing area: 3.36 $ft^2$\n",
      "   * Weight 5.2 lbs\n",
      "   * Symmetric airfoil with a lift curve slope of 0.1097 per degree\n",
      "   * Drag coefficient: 0.04\n",
      "\n",
      "**a) How much lift is needed for Thor to maintain level flight?**"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\"Weight, lbs\"\n",
      "W = 5.2;\n",
      "\"In steady level flight, the amount of lift needed is equal to the weight\"\n",
      "L = W;\n",
      "print \"Thor needs %f lbs of lift to maintain level flight\" % L"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Thor needs 5.200000 lbs of lift to maintain level flight\n"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**b) If Thor is flying at Anoka County RC Field (elevation 1,000 ft above sea level, air density 2.3081E-3 $slug/ft^3$) and a speed of 56 ft/s, what is Thor's angle of attack?**"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can recall that the equation for lift is:\n",
      "\n",
      "$$L = W = 1/2(\\rho)V^2SC_L$$\n",
      "\n",
      "$$C_L = C_{L0} + C_{L\\alpha} (\\alpha)$$\n",
      "\n",
      "Because the airfoil is symmetric, $C_{L0} = 0$\n",
      "\n",
      "So:\n",
      "\n",
      "$$\\alpha = \\dfrac{2W}{\\rho V^2SC_{L\\alpha}}$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\"Air density, slug/ft^3\"\n",
      "ro = 2.3081*10**(-3);\n",
      "\"Velocity, ft/s\"\n",
      "V = 56;\n",
      "\"Wing area, ft^2\"\n",
      "S = 3.36;\n",
      "\"Lift curve slope, per degree\"\n",
      "CLa = 0.1097;\n",
      "\"Solving for alpha, degrees\"\n",
      "a = (2*W)/(ro*(V**2)*S*CLa);\n",
      "print \"Angle of attack is %f degrees\" % a"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Angle of attack is 3.898134 degrees\n"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**c) For the problem in part B, what is Thor's drag?**"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can recall that the equation for drag is:\n",
      "\n",
      "$$ D = 1/2(\\rho)V^2SC_D$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\"Drag coefficient\"\n",
      "CD = 0.04;\n",
      "\"Solving for drag, lbs\"\n",
      "D = (.5)*ro*(V**2)*S*CD;\n",
      "print \"Drag is %f lbs\" % D"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Drag is 0.486407 lbs\n"
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**d) For the problem in part C, what is Thor's thrust?**"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can recall that thrust is equal to drag in steady level flight."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\"Solving for thrust, lbs\"\n",
      "T = D;\n",
      "print \"Thrust is %f lbs\" % T"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Thrust is 0.486407 lbs\n"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**e) Assume Thor's maximum angle of attack is 10 degrees, at what speed will Thor stall?**"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Start with the lift equation again:\n",
      "\n",
      "$$L = W = 1/2(\\rho)V^2SC_L$$\n",
      "\n",
      "And solve for velocity:\n",
      "\n",
      "$$V = \\sqrt{\\dfrac{2W}{\\rho SC_L}}$$\n",
      "\n",
      "$$C_L = C_{L0} + C_{L\\alpha} (\\alpha)$$\n",
      "\n",
      "Because the airfoil is symmetric, $C_{L0} = 0$\n",
      "\n",
      "So:\n",
      "\n",
      "$$V = \\sqrt{\\dfrac{2W}{\\rho SC_{L\\alpha}(\\alpha)}}$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\"Angle of attack, degrees\"\n",
      "a = 10;\n",
      "\"Solving for velocity, fps\"\n",
      "V = sqrt((2*W)/(ro*S*CLa*a));\n",
      "print \"Stall speed is %f feet per second\" % V"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Stall speed is 34.963621 feet per second\n"
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**f) What is the coefficient of lift in part E?**"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$C_L = C_{L0} + C_{L\\alpha} (\\alpha)$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\"Solving for lift coefficient, ND\"\n",
      "CL = CLa*a;\n",
      "print \"Lift coefficient is %f\" % CL"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Lift coefficient is 1.097000\n"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**g) If Thor's wing area remains unchanged with the deployment of flaps and Thor is landing at Anoka County RC Field (elevation 1,000 ft above sea level, air density 2.3081E-3 $slug/ft^3$) with a stall speed of 30 ft/s, what is Thor's lift coefficient?**"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$L = W = 1/2(\\rho)V^2SC_L$$\n",
      "\n",
      "$$C_L = \\dfrac{2W}{\\rho V^2 S}$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\"Velocity, ft/s\"\n",
      "V = 30;\n",
      "\n",
      "\"Solving for lift coefficient, ND\"\n",
      "CL = (2*W)/(ro*V**2*S)\n",
      "print \"Lift coefficient is %f\" % CL"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Lift coefficient is 1.490037\n"
       ]
      }
     ],
     "prompt_number": 36
    }
   ],
   "metadata": {}
  }
 ]
}