LoadCellModelingTemco.ipynb

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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
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   "source": [
    "import numpy as np\n",
    "from math import pi\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Load Cell Response Modeling\n",
    "\n",
    "J.R. Leeman\n",
    "\n",
    "Here I show the predicted response of load cells based on linear-elastic theory and simple circuit analysis. Analysis is completed for the two load cell designs (hollow and solid) that are used in the lab. The capability to solve the full 3D problem is here if anyone needs to expand this for modeling the response of internal load cells or load cells in non uni-axial loading environments. \n",
    "\n",
    "## Assumed Material Properties\n",
    "Berrylium-copper:\n",
    "- G (Shear modulus) = 77.2 GPa [(source)](http://www.engineeringtoolbox.com/modulus-rigidity-d_946.html)\n",
    "- E (Young's modulus) = 100 GPa [(source)](http://www.azom.com/properties.aspx?ArticleID=863)\n",
    "- Minimum Yield Strength = 30 ksi = 207 MPa [(source)](http://www.azom.com/properties.aspx?ArticleID=863)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# These are the \"Tableau 20\" colors as RGB.  \n",
    "tableau20 = [(31, 119, 180), (174, 199, 232), (255, 127, 14), (255, 187, 120),  \n",
    "             (44, 160, 44), (152, 223, 138), (214, 39, 40), (255, 152, 150),  \n",
    "             (148, 103, 189), (197, 176, 213), (140, 86, 75), (196, 156, 148),  \n",
    "             (227, 119, 194), (247, 182, 210), (127, 127, 127), (199, 199, 199),  \n",
    "             (188, 189, 34), (219, 219, 141), (23, 190, 207), (158, 218, 229)]  \n",
    "  \n",
    "# Scale the RGB values to the [0, 1] range, which is the format matplotlib accepts.  \n",
    "for i in range(len(tableau20)):  \n",
    "    r, g, b = tableau20[i]  \n",
    "    tableau20[i] = (r / 255., g / 255., b / 255.) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "class LoadCell:\n",
    "    \n",
    "    def __init__(self):\n",
    "        self.inner_diameter = None\n",
    "        self.outer_diameter = None\n",
    "        self.area = None\n",
    "        self.shear_modulus = None\n",
    "        self.youngs_modulus = None\n",
    "        self.yield_strength = None\n",
    "        self.lame_parameter = None\n",
    "        self.gauge_factor = None\n",
    "        self.gauge_r = None\n",
    "        self.amplification_gain = None\n",
    "        self.excitation_voltage = None\n",
    "        \n",
    "    def set_size(self,inner_diameter,outer_diameter):\n",
    "        self.inner_diameter = inner_diameter\n",
    "        self.outer_diameter = outer_diameter\n",
    "        self.area = (pi*(self.outer_diameter/2.)**2) - (pi*(self.inner_diameter/2.)**2)\n",
    "        \n",
    "    def set_moduli(self,shear_modulus,youngs_modulus):\n",
    "        self.shear_modulus = shear_modulus\n",
    "        self.youngs_modulus = youngs_modulus\n",
    "        self.lame_parameter = (self.shear_modulus *(self.youngs_modulus - 2 * self.shear_modulus ))\\\n",
    "                              /(3 * self.shear_modulus - self.youngs_modulus)\n",
    "    \n",
    "    def model(self,min_force, max_force, step):\n",
    "        self.model_forces = np.arange(min_force, max_force, step)\n",
    "        \n",
    "        # Convert the force to a stress\n",
    "        self.model_stresses = self.model_forces/self.area\n",
    "        \n",
    "        # Calculate strains based on the given information\n",
    "        self.model_strains = self.get_hooke_strains(self.model_stresses)\n",
    "        \n",
    "        # Given the strains, now calculate the output of the load cell\n",
    "        axial_gauges = self.gauge_resistance(self.model_strains[:,0])\n",
    "        radial_gauges = self.gauge_resistance(self.model_strains[:,1])\n",
    "        resistances = [axial_gauges,axial_gauges,radial_gauges,radial_gauges,\n",
    "                       axial_gauges,axial_gauges,radial_gauges,radial_gauges]\n",
    "        self.bridge_voltage = self.bridge_output(resistances)\n",
    "        self.output_voltage = self.bridge_voltage * self.amplification_gain\n",
    "        self.calibration = ((self.output_voltage[-1] - self.output_voltage[0]) /\\\n",
    "                           (self.model_forces[-1] - self.model_forces[0]))*1e6 \n",
    "        \n",
    "    def gauge_resistance(self,strain):\n",
    "        return self.gauge_r + self.gauge_r * strain * self.gauge_factor\n",
    "\n",
    "    def bridge_output(self, resistances):\n",
    "        if len(resistances) != 8:\n",
    "            return -999\n",
    "        return self.excitation_voltage*((resistances[4] + resistances[5])/(resistances[4]+resistances[5]+resistances[6]+resistances[7]) - \n",
    "                    (resistances[2] + resistances[3])/(resistances[0]+resistances[1]+resistances[2]+resistances[3]))\n",
    "\n",
    "    def get_hooke_strains(self,stresses):\n",
    "        strains = []\n",
    "        lame = self.lame_parameter\n",
    "        G = self.shear_modulus\n",
    "        for stress in stresses:\n",
    "            sigma_1 = stress; sigma_2 = 0; sigma_3 = 0\n",
    "            e_1 = sigma_1*((lame+G)/(G*(3*lame+2*G))) - sigma_2*(lame/(2*G*(3*lame+2*G)))\\\n",
    "                  - sigma_3*(lame/(2*G*(3*lame+2*G)))\n",
    "            \n",
    "            e_2 = -1*sigma_1*(lame/(2*G*(3*lame+2*G))) + sigma_2*((lame+G)/(G*(3*lame+2*G)))\\\n",
    "                  - sigma_3*(lame/(2*G*(3*lame+2*G)))\n",
    "            \n",
    "            e_3 = -1*sigma_1*(lame/(2*G*(3*lame+2*G))) - sigma_2*(lame/(2*G*(3*lame+2*G)))\\\n",
    "                  + sigma_3*((lame+G)/(G*(3*lame+2*G)))\n",
    "                \n",
    "            strains.append([e_1,e_2,e_3])\n",
    "            \n",
    "        return np.array(strains)\n",
    "    \n",
    "    def plot_strain(self):\n",
    "        # Setup figure and axes\n",
    "        # Generally plots is ~1.33x width to height (10,7.5 or 12,9)\n",
    "        fig = plt.figure(figsize=(12,9))\n",
    "        ax1 = plt.subplot(111)\n",
    "\n",
    "        # Set labels and tick sizes\n",
    "        ax1.set_xlabel(r'Force [kN]',fontsize=18)\n",
    "        ax1.set_ylabel(r'Milli-Strain',fontsize=18)\n",
    "        ax1.tick_params(axis='both', which='major', labelsize=16)\n",
    "        ax1.grid(True)\n",
    "\n",
    "        # Plotting\n",
    "        ax1.plot(self.model_forces/1.e3,self.model_strains[:,0]*1.e3, color=tableau20[0], linewidth=2, label = r'$\\epsilon_1$')\n",
    "        ax1.plot(self.model_forces/1.e3,self.model_strains[:,1]*1.e3, color=tableau20[2], linewidth=2, label = r'$\\epsilon_2$')\n",
    "        ax1.plot(self.model_forces/1.e3,self.model_strains[:,2]*1.e3, color=tableau20[4], linewidth=2, label = r'$\\epsilon_3$')\n",
    "\n",
    "        plt.legend(fontsize=20, frameon=False, loc = 'upper left')\n",
    "        \n",
    "    def plot_bridge(self):\n",
    "        # Setup figure and axes\n",
    "        # Generally plots is ~1.33x width to height (10,7.5 or 12,9)\n",
    "        fig = plt.figure(figsize=(12,9))\n",
    "        ax1 = plt.subplot(111)\n",
    "\n",
    "        # Set labels and tick sizes\n",
    "        ax1.set_xlabel(r'Force [kN]',fontsize=18)\n",
    "        ax1.set_ylabel(r'Bridge Voltage [mV]',fontsize=18)\n",
    "        ax1.tick_params(axis='both', which='major', labelsize=16)\n",
    "        ax1.grid(True)\n",
    "\n",
    "        # Plotting\n",
    "        ax1.plot(self.model_forces/1.e3,self.bridge_voltage*1.e3, color='k', linewidth=2, label = r'$\\epsilon_1$')\n",
    "        \n",
    "    def plot_output(self):\n",
    "        # Setup figure and axes\n",
    "        # Generally plots is ~1.33x width to height (10,7.5 or 12,9)\n",
    "        fig = plt.figure(figsize=(12,9))\n",
    "        ax1 = plt.subplot(111)\n",
    "\n",
    "        # Set labels and tick sizes\n",
    "        ax1.set_xlabel(r'Force [kN]',fontsize=18)\n",
    "        ax1.set_ylabel(r'Output Voltage [V]',fontsize=18)\n",
    "        ax1.tick_params(axis='both', which='major', labelsize=16)\n",
    "        ax1.grid(True)\n",
    "\n",
    "        # Plotting\n",
    "        ax1.plot(self.model_forces/1.e3,self.output_voltage, color='k', linewidth=2, label = r'$\\epsilon_1$')\n",
    "\n",
    "    def print_stats(self):\n",
    "        machine_max = 1e6/self.area # Pa\n",
    "        print \"Load cell loading area: %.1f mm^2\" %(self.area*1000*1000)\n",
    "        print \"Load cell predicted calibration: %.4f mV/kN\" %(self.calibration)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Temco Internal Load Cell"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Load cell loading area: 189.9 mm^2\n",
      "Load cell predicted calibration: 36.6349 mV/kN\n"
     ]
    }
   ],
   "source": [
    "TemcoCell = LoadCell()\n",
    "TemcoCell.set_size(0.01935,0.02641-0.0015875)\n",
    "# shear mod, youngs mod\n",
    "TemcoCell.set_moduli(77.2e9,200e9)\n",
    "TemcoCell.gauge_factor = 2.15\n",
    "TemcoCell.gauge_r = 350.\n",
    "TemcoCell.amplification_gain = 100.\n",
    "TemcoCell.excitation_voltage = 10.\n",
    "TemcoCell.model(1,50670.7,100)\n",
    "TemcoCell.print_stats()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
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   "outputs": [
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       "AABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109350a10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "TemcoCell.plot_strain()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1095ef0d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "TemcoCell.plot_bridge()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109857050>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "TemcoCell.plot_output()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}