Pendulum.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0,0.5,'Theta')"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "#Definitons\n",
    "def SmallAngleApprox(g,L,Theta):\n",
    "    ATheta = -(g/L)*Theta\n",
    "    return ATheta\n",
    "def Pendulum(g,L,q,Theta,VTheta):\n",
    "    ATheta = -(g/L)*np.sin(Theta)-q*VTheta\n",
    "    return ATheta\n",
    "\n",
    "#Initial Conditions\n",
    "L = 100\n",
    "g = 9.8\n",
    "q = 0.1\n",
    "to = 0.0\n",
    "ti = 0\n",
    "tf = 100\n",
    "Steps = 10000\n",
    "\n",
    "#Initial Position for Pendulum\n",
    "Theta = np.deg2rad(85)\n",
    "VTheta = 0.0\n",
    "ATheta = -g/L\n",
    "\n",
    "#Initial Position for Small Angle Approximation\n",
    "Theta2 = np.deg2rad(85)\n",
    "VTheta2 = 0.0\n",
    "ATheta2 = -g/L\n",
    "\n",
    "#Initializing arrays to store data\n",
    "Theta_array = np.array([Theta])\n",
    "VTheta_array = np.array([VTheta])\n",
    "Theta_array2 = np.array([Theta2])\n",
    "VTheta_array2 = np.array([VTheta2])\n",
    "t_array = np.array([to])\n",
    "T_array = np.array([0.0])\n",
    "\n",
    "\n",
    "#ThetaI_array = np.array([0])\n",
    "#Frequency_array = np.array([0.0])\n",
    "\n",
    "#setting up time incriments\n",
    "t = np.linspace(ti,tf,Steps+1)\n",
    "dt = (tf - ti)/Steps\n",
    "\n",
    "#for j in range(1,89):\n",
    "#    Theta = np.deg2rad(i)\n",
    "#    ThetaI_array = np.append(ThetaI_array,Theta)\n",
    "#Loop to compute derivitive\n",
    "for i in range(1,Steps):\n",
    "    #Numerically Computing Pendulum Derivitives\n",
    "    ATheta = Pendulum(g,L,q,Theta,VTheta)\n",
    "    Theta = Theta + VTheta*dt\n",
    "    VTheta = VTheta + ATheta*dt\n",
    "\n",
    "    #Appending Values to arrays\n",
    "    Theta_array = np.append(Theta_array,Theta)\n",
    "    VTheta_array = np.append(VTheta_array,VTheta)\n",
    "\n",
    "    #Numerically Computing Small Angle Aprrox. Derivitives\n",
    "    ATheta2 = SmallAngleApprox(g,L,Theta)\n",
    "    Theta2 = Theta2 + VTheta2*dt\n",
    "    VTheta2 = VTheta2 + ATheta2*dt\n",
    "\n",
    "    #Appending values to arrays\n",
    "    Theta_array2 = np.append(Theta_array2,Theta2)\n",
    "    VTheta_array2 = np.append(VTheta_array2,VTheta2)\n",
    "    t_array = np.append(t_array,(i)*dt)\n",
    "\n",
    "    #Find the Period\n",
    "    val1 = Theta_array[i]\n",
    "    val2 = Theta_array[i-1]\n",
    "    if(val1<=0.0 and val2>=0.0):\n",
    "        T_array = np.append(T_array,(i)*dt)\n",
    "            \n",
    "#Period = T_array[2]-T_array[1]\n",
    "#Frequency = 1/Period\n",
    "#Frequency_array = np.append(Frequency_array,Frequency)\n",
    "\n",
    "#Plotting solutions on graph\n",
    "plt.figure()\n",
    "plt.plot(t_array,Theta_array,'r-')\n",
    "#plt.plot(t_array,Theta_array2,'b-')\n",
    "plt.title('Pendulum')\n",
    "plt.xlabel('Time')\n",
    "plt.ylabel('Theta')\n",
    "\n",
    "\n",
    "\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}