Problems - Experimental Methods in Engineering.ipynb

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   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Problems - Experimental Methods in Engineering"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Marcos Duarte"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Problem 1"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Consider the following experimental data collected in an experiment to measure the acceleration of gravity in Santo Andr\u00e9 (in m/s$^2$): [9.1, 9.6, 11.5, 10.9, 10.0, 8.1, 9.4, 7.2, 8.6, 10.4, 11.2, 10.0] with an experimental uncertainty in each measure of 0.5 m/s$^2$.   \n",
      "\u00a0\u00a0 a. Calculate mean, variance, standard deviation, standard error of the mean for these data.  \n",
      "\u00a0\u00a0 b. Make a histogram of these data and indicate the mean and standard deviation in the histogram.  \n",
      "\u00a0\u00a0 c. What are accuracy and precision? What values of item (a) can be used to describe something about accuracy and precision of the measurements above?   \n",
      "\u00a0\u00a0 d. What are confidence interval and prediction interval? Calculate these intervals with a probability of 95% for the above data (for simplification, consider that these data have normal distribution).   \n",
      "\u00a0\u00a0 e. What is the value (with uncertainty) that best represents these experimental measures?   "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**a.**   \n",
      "For a sample with normal distribution and $n$ data:\n",
      "\n",
      "$$ mean\\;(m) = \\frac{1}{n}\\sum_{i=1}^{n} x_i $$\n",
      "\n",
      "$$ variance\\;(s^2) = \\frac{1}{n-1}\\sum_{i=1}^{n} (x_i - \\bar{x})^2 $$\n",
      "\n",
      "$$ standard\\;deviation\\;(s) = \\sqrt{s^2} = \\sqrt{\\frac{1}{n-1}\\sum_{i=1}^{n}(x_i-\\bar{x})^2} $$\n",
      "\n",
      "$$ standard\\;error\\;of\\;the\\;mean\\;(se) = \\frac{s}{\\sqrt{n}} = \\sqrt{\\frac{1}{n(n-1)}\\sum_{i=1}^{n}(x_i-\\bar{x})^2} $$\n",
      "\n",
      "So, for the given data:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Import the necessary libraries\n",
      "from __future__ import division, print_function\n",
      "import numpy as np\n",
      "from IPython.display import display, Math\n",
      "\n",
      "data = np.array([9.1, 9.6, 11.5, 10.9, 10.0, 8.1, 9.4, 7.2, 8.6, 10.4, 11.2, 10.0])\n",
      "m  = np.mean(data);         display(Math('m = %.2f \\; m/s^2' %m))\n",
      "s2 = np.var(data, ddof=1);  display(Math('s^2 = %.2f \\; m^2/s^4' %s2))\n",
      "s  = np.sqrt(s2);           display(Math('s = %.2f \\; m/s^2' %s))\n",
      "se = s/np.sqrt(data.size);  display(Math('se = %.2f \\; m/s^2' %se))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$m = 9.67 \\; m/s^2$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0x439fa90>"
       ]
      },
      {
       "latex": [
        "$$s^2 = 1.64 \\; m^2/s^4$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0x439fa90>"
       ]
      },
      {
       "latex": [
        "$$s = 1.28 \\; m/s^2$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0x439fb70>"
       ]
      },
      {
       "latex": [
        "$$se = 0.37 \\; m/s^2$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0x439fb70>"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**b.**   \n",
      "From [Wikipedia](http://en.wikipedia.org/wiki/Histogram): Histogram is a graphical representation of the distribution of data. A histogram is a representation of tabulated frequencies, shown as adjacent rectangles, erected over discrete intervals (bins), with an area equal to the frequency of the observations in the interval. The height of a rectangle is also equal to the frequency density of the interval, i.e., the frequency divided by the width of the interval. The total area of the histogram is equal to the number of data.   \n",
      "So:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import matplotlib.pyplot as plt\n",
      "%matplotlib inline\n",
      "plt.xkcd()\n",
      "fig, ax = plt.subplots(1, 1, figsize=(6, 4))\n",
      "plt.hist(data, bins=6)\n",
      "ax.set_title('Histogram of the measurements')\n",
      "ax.set_xlabel('Gravity acceleration [m/s$^2$]')\n",
      "ax.set_ylabel('Frequency')\n",
      "ax.set_xlim(7, 12)\n",
      "ax.plot([m, m], [0, 4], 'r', linewidth=4, label='$m$')\n",
      "ax.plot([m-s, m-s], [0, 4], 'r--', linewidth=4)\n",
      "ax.plot([m+s, m+s], [0, 4], 'r--', linewidth=4, label='$m\\pm s$')\n",
      "ax.legend(loc='best', framealpha=.5)\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
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1MnA4HMjNzQUA/PLLL3A6nbj22mtx9913o2XLltizZw/WrFkDsZhVf6WlpQCABx54gH/e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\nTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x41ffcf8>"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**c.**   \n",
      "**Accuracy** of measurement is the closeness of the agreement between the result of a measurement and a true value of the measurand. Accuracy is a qualitative concept.   \n",
      "**Precision** of measurement is the closeness of the agreement between the results of measurements of the same measurand under similar conditions. Precision is a qualitative concept.\n",
      "\n",
      "The mean of the measurements should be used to describe the degree of accuracy of the measurements (if we adopt g=9.8 m/s$^2$ as the true value, we could say that the degree of accuracy (mean minus true value) of the measurements was -0.13 m/s$^2$.   \n",
      "The variance, standard deviation, and standard error of the mean can be used to describe the degree of precision of the measurements. Using the standard error of the mean (preferred in this case), the degree of precision of the measurements was 0.37 m/s$^2$.\n",
      "\n",
      "**d.**  \n",
      "**Confidence interval** for the mean is an estimate of an interval with a certain probability that the **mean of new values** will be inside this interval.   \n",
      "**Prediction interval** is an estimate of an interval with a certain probability that a **new value** will be inside this interval.\n",
      "\n",
      "For a probability of 95% and considering the data with normal distribution:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "display(Math('Confidence\\;interval:\\;\\;[%.2f \\; , \\; %.2f]  \\; m/s^2' %(m-1.96*se, m+1.96*se)))\n",
      "display(Math('Prediction\\;interval:\\;\\;[%.2f \\; , \\; %.2f] \\; m/s^2' %(m-1.96*s, m+1.96*s)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$Confidence\\;interval:\\;\\;[8.94 \\; , \\; 10.39]  \\; m/s^2$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0x439fb00>"
       ]
      },
      {
       "latex": [
        "$$Prediction\\;interval:\\;\\;[7.15 \\; , \\; 12.18] \\; m/s^2$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0x439f860>"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**e.**   \n",
      "The value that best represents these experimental measures is the mean value and the correspondent uncertainty of the measurement. For a set of values with instrumental uncertainty, the total uncertainty is the square-root of the sum of squares of the statistical uncertainty (type A) and the instrumental uncertainty (type B):\n",
      "\n",
      "$$ u_g = \\sqrt{u_A^2 + u_B^2} $$\n",
      "\n",
      "The statistical uncertainty can be described by the standard error of the mean, so:\n",
      "\n",
      "$$ u_g = \\sqrt{0.37^2 + 0.5^2} $$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ug = np.sqrt(se**2 + 0.5**2)\n",
      "\n",
      "display(Math('g= %.2f \\pm %.2f \\; m/s^2' %(m, ug)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$g= 9.67 \\pm 0.62 \\; m/s^2$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0x439f9b0>"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Problem 2"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Consider an experiment to measure the mass of an object using a scale with a tension spring. Without any mass on the scale, one reads 20 cm on the scale of the balance. Putting on the weighing plate an object with mass equals to 50 g, the reading on the scale is 25 cm. When you place a cube on the empty plate the reading is 40 cm. Adopt $g=10/;m/s^2$ (no uncertainty). Assuming that the two cases obeyed Hooke's Law,   \n",
      "\u00a0\u00a0 a. How much is the mass of the cube?  \n",
      "\u00a0\u00a0 b. Assuming that the uncertainties in the measurements of mass and displacement are 1%, estimate the uncertainty associated with the value of the mass determined in (a). "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**a.**   \n",
      "From Hooke's law:\n",
      "\n",
      "$$ F = P = mg = k \\Delta x $$\n",
      "\n",
      "Using the first case (known mass), we can obtain the stiffness constant of the spring: \n",
      "\n",
      "$$ k = \\frac{mg}{\\Delta x} = \\frac{0.05 \\times 10}{0.05} = 10 \\; N/m  $$\n",
      "\n",
      "In the second case the mass is unknown but we already know the spring stiffness:  \n",
      "\n",
      "$$ m_{cube} = \\frac{k \\Delta x}{g} = \\frac{10 \\times 0.2}{10} = 0.2 \\; kg  $$\n",
      "\n",
      "So, the mass of the cube is 0.2 kg.\n",
      "\n",
      "**b.**   \n",
      "The formula we used to estimate the mass of the cube was:\n",
      "\n",
      "$$ m_{cube} = \\frac{k\\Delta x}{g} $$\n",
      "\n",
      "The uncertainty of the mass of the cube can be estimated using error propagation:  \n",
      "\n",
      "$$ u_m^2 = \\left(\\frac{\\partial m}{\\partial k} u_k \\right)^2 + \\left(\\frac{\\partial m}{\\partial x} u_x \\right)^2 $$\n",
      "\n",
      "$$ u_m^2 =  \\left(\\frac{\\Delta x}{g} u_k \\right)^2 +  \\left(\\frac{k}{g} u_x \\right)^2 $$\n",
      "\n",
      "To estimate the uncertainty of $k$, we have to go back to the first case:\n",
      "\n",
      "$$ k = \\frac{mg}{\\Delta x}  $$\n",
      "\n",
      "$$ u_k^2 = \\left(\\frac{\\partial k}{\\partial m} u_m \\right)^2 + \\left(\\frac{\\partial k}{\\partial \\Delta x} u_x \\right)^2  $$\n",
      "\n",
      "$$ u_k^2 = \\left(\\frac{g}{\\Delta x} u_m \\right)^2 + \\left(-\\frac{mg}{\\Delta x^2}u_x\\right)^2 $$\n",
      "\n",
      "Where $g=10\\;m/s^2$, $m=50 \\; g$, $u_m = 0.01 \\times m = 0.5 \\; g$, $ \\Delta x = 5 \\; cm $, $u_x= 0.01 \\times \\Delta x = 0.05 \\; cm$, then:\n",
      "\n",
      "$$ u_k^2 = \\left(\\frac{10}{0.05} 0.0005 \\right)^2 + \\left(-\\frac{0.05 \\times 10}{0.05^2} 0.0005 \\right)^2 = 0.01 + 0.01 $$\n",
      "\n",
      "$$ u_k \\approx 0.1414 \\; N/m $$\n",
      "\n",
      "Back to the formula:\n",
      "\n",
      "$$ u_m^2 =  \\left(\\frac{\\Delta x}{g} u_k \\right)^2 +  \\left(\\frac{k}{g} u_x \\right)^2 $$\n",
      "\n",
      "Where now $\\Delta x = 20\\;cm, u_x = 0.01 \\times \\Delta x = 0.2 \\;cm$, then:\n",
      "\n",
      "$$ u_m^2= \\left(\\frac{0.20}{10} 0.1414\\right)^2 + \\left(\\frac{10}{10} 0.002 \\right)^2 \\approx 0.002828^2 + 0.002^2 $$\n",
      "\n",
      "$$ u_m \\approx 3.5.10^{-3} kg $$\n",
      "\n",
      "Then:\n",
      "\n",
      "$$ m_{cube} = (200.0 \\pm 3.5) \\times 10^{-3} \\; kg $$"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Problem 3"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For the experiment of the previous question, consider the following experimental data to determine the elastic spring constant traction, $\\Delta x$ (cm): [0, 0.5, 2.5, 3, 3.5, 5.5], m (g): [0, 10, 20, 30, 40, 50]. The experimental uncertainties of these measures are not known (ignore them) and adopt g = 10 m/s$^2$ (no uncertainty). Consider the curve that best describes the relationship between the force on the spring versus its deformation is a straight line passing through the origin.  \n",
      "\u00a0\u00a0 a. Describe a method to determine the best straight line fit for experimental data.   \n",
      "\u00a0\u00a0 b. Derive mathematically straight line fit passing through the origin.   \n",
      "\u00a0\u00a0 c. Determine the equation of the straight line fit passing through the origin using the experimental data above."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**a.**  \n",
      "A classical solution to find the best fit is to minimize the sum of the squared differences between data points and the straight line function (y=bx), which is known as the least-squares fit.\n",
      "\n",
      "**b.**   \n",
      "The analytical expression for the least-square fit when the model function is a straight line passing trough zero is:\n",
      "\n",
      "$$ y(x) = y(x | b) = bx $$ \n",
      "\n",
      "We want to find $b$ such that minimizes the sum of the squared differences between data points and the y=bx function:\n",
      "\n",
      "$$ \\chi^2(b) = \\sum_{i=1}^{n} \\left[ y_i - bx_i \\right]^2 $$\n",
      "\n",
      "Using the property that at the minimum of $\\chi^2$ its derivative is zero:\n",
      "\n",
      "$$ \\frac{\\partial \\chi^2}{\\partial b} = -2 \\sum_{i=1}^{n} x_i(y_i - bx_i) = 0 $$\n",
      "\n",
      "So, the angular coefficient of the best straight line passing trough zero, $ y(x) = bx $, is:\n",
      "\n",
      "$$ b = \\frac{\\sum_{i=1}^{n} x_iy_i}{\\sum_{i=1}^{n} x_i^2} $$\n",
      "\n",
      "**c.**  \n",
      "For the given data:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.array([0, 0.5, 2.5, 3, 3.5, 5.5])/100  # in m\n",
      "m = np.array([0, 10, 20, 30, 40, 50])/1000    # in kg\n",
      "b = np.sum(x*m*10)/np.sum(x*x)                # angular coefficient\n",
      "display(Math('F\\:[N] = %.2f \\; x\\:[m]' %b))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$F\\:[N] = 9.66 \\; x\\:[m]$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0x4380cf8>"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Problem 4"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Consider the following experimental data from a measurement of the time interval between the instant of the first and second impacts of an object with a surface (in ms): [723, 722, 715, 719, 716]. The uncertainty of each of these measures is 10 ms, the object was dropped from an initial height of 1 m (no uncertainty) and adopt g=10 m/s$^2$ (no uncertainty).   \n",
      "Determine the experimental value that best describes the coefficient of restitution using these data."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The coefficient of restitution is given by the following formula:\n",
      "\n",
      "$$ C_R = \\frac{velocity_{after}}{velocity_{before}}$$\n",
      "\n",
      "For a free falling object with zero initial velocity, if the time of fall is known, the velocity at the ground is given by:\n",
      "\n",
      "$$velocity = gt $$\n",
      "\n",
      "If the height of fall is known, the time of fall is given by:\n",
      "\n",
      "$$ t = \\sqrt{\\frac{2h}{g}} $$\n",
      "\n",
      "So, for the problem above, one solution is to determine the time of the first fall of the object and then determine its velocity at the first impact (with no uncertainty because the height of release and g have no uncertainties):"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "h   = 1                                         # in m\n",
      "t   = np.array([723, 722, 715, 719, 716])/1000  # in s\n",
      "t0  = np.sqrt(2*h/10)                    # time of the first fall \n",
      "v0  = 10*t0                              # velocity before first impact"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "With the five measurements of the time between first and second impacts, we determine the mean flight time (the time of fall will be half of that). The uncertainty of the time of fall will be the statistical uncertainty (type A) plus the instrumental uncertainty (type B). The standard error of the mean will be used as the statistical uncertainty:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "tm  = np.mean(t)                         # mean flight time after first impact\n",
      "uta = np.std(t, ddof=1)/np.sqrt(t.size)  # standard error of the flight time (u type A)\n",
      "utb = 0.01                               # instrumental uncertainty (u type B) \n",
      "ut  = np.sqrt(uta**2 + utb**2)           # total uncertainty of the flight time"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "With this time of fall, we can determine the velocity the ball had just after the fist impact and finally calculate the coefficient of restitution:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "v   = 10*tm/2                            # velocity after first impact\n",
      "uv  = (ut/tm)*v                          # velocity uncertainty by error propagation\n",
      "Cr  = v/v0                               # coefficient of restitution\n",
      "uCr = (uv/v)*Cr                          # uncertainty (because v0 has no uncertainty)\n",
      "\n",
      "display(Math(r' C_R = %.3f \\pm %.3f' %(Cr, uCr)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$ C_R = 0.804 \\pm 0.011$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math at 0x439fe10>"
       ]
      }
     ],
     "prompt_number": 8
    }
   ],
   "metadata": {}
  }
 ]
}