{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Universidade Federal do Rio Grande do Sul (UFRGS)   \n",
    "Programa de Pós-Graduação em Engenharia Civil (PPGEC)   \n",
    "\n",
    "# PEC00025: Introduction to Vibration Theory\n",
    "\n",
    "\n",
    "### Class 16 - Test P2: multiple degrees of freedom and continuous systems\n",
    "\n",
    "[P2:2019](#P2_2019) - \n",
    "[Question 1](#P2_2019_1), \n",
    "[Question 2](#P2_2019_2), \n",
    "[Question 3](#P2_2019_3), \n",
    "[Question 4](#P2_2019_4).\n",
    "\n",
    "---\n",
    "_Prof. Marcelo M. Rocha, Dr.techn._ [(ORCID)](https://orcid.org/0000-0001-5640-1020)  \n",
    "_Porto Alegre, RS, Brazil_ \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Importing Python modules required for this notebook\n",
    "# (this cell must be executed with \"shift+enter\" before any other Python cell)\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.linalg as sc\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## P2:2019 <a name=\"P2_2019\"></a> \n",
    "\n",
    "_Note: this test is to be solved with the aid of a scientific calculator, which must be able to solve eigenproblems,\n",
    "linear systems, and integrals. The total available time for solving the test is 2h (two hours). The student is allowed\n",
    "to prepare am A4 sheet of paper (two sides) with informations to be consulted during the test._\n",
    "\n",
    "### Question 1  <a name=\"P2_2019_1\"></a> \n",
    "\n",
    "A structural system is modelled as a discrete two d.o.f. system, as shown in the figure. Each column has flexural rigidity $EI = 500{\\rm kNm^2}$, length $L = 4{\\rm m}$, and are assumed to have no relevant mass. The floor beams are assumed to be perfectly stiff and to have total (lumped) mass $m = 4{\\rm ton}$ each. The system is assumed to present a viscous damping with ratio of critical $\\zeta = 0.01$ in all vibration modes. \n",
    "\n",
    "1. Define the stiffness, the mass, and the damping system matrices (1 pts).  \n",
    "2. Determine and sketch the two natural vibration modes, indicating the associated vibration frequencies (2 pts). \n",
    "\n",
    "<img src=\"resources/tests/P2_2019_figs1e2.png\" alt=\"P2_2019_figs1e2\" width=\"240px\"/>\n",
    "\n",
    "**Answer:** The stiffness and mass matrices are:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Global stiffness matrix:\n",
      "\n",
      " [[ 187500. -187500.]\n",
      " [-187500.  375000.]]\n",
      "\n",
      "Global mass matrix:\n",
      "\n",
      " [[4000.    0.]\n",
      " [   0. 4000.]]\n"
     ]
    }
   ],
   "source": [
    "EI =  500000.          # single column flexural rigidity (Nm^2)\n",
    "m  =  4000.            # single floor mass (kg)\n",
    "L  =  4                # column length (m)\n",
    "\n",
    "k  =  12*EI/L**3       # single column stiffness\n",
    "\n",
    "KG = np.array([[ 2*k, -2*k],\n",
    "               [-2*k,  4*k]])   # global stiffness matrix\n",
    "\n",
    "MG = np.array([[   m,    0],\n",
    "               [   0,    m]])   # global mass matrix\n",
    "\n",
    "print('Global stiffness matrix:\\n\\n', KG)\n",
    "print('\\nGlobal mass matrix:\\n\\n', MG)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To specify the damping matrix, we must first calculate the vibration modes and frequencies.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 576x432 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Uses scipy to solve the standard eigenvalue problem\n",
    "w2, Phi = sc.eig(KG, MG)\n",
    "\n",
    "# Ensure ascending order of eigenvalues\n",
    "iw  = w2.argsort()\n",
    "w2  = w2[iw]\n",
    "Phi = Phi[:,iw]\n",
    "\n",
    "# Eigenvalues to vibration frequencies\n",
    "wk  = np.sqrt(np.real(w2)) \n",
    "fk  = wk/2/np.pi\n",
    "\n",
    "plt.figure(1, figsize=(8,6), clear=True)\n",
    "x = np.arange(0,12,4)\n",
    "\n",
    "for k in range(2):\n",
    "    pk = np.zeros(3)\n",
    "    pk[1:] = Phi[::-1,k]\n",
    "    pk /= np.max(np.abs(pk))   # adjust scale for unity amplitude\n",
    "    \n",
    "    plt.subplot(1,2,k+1)\n",
    "    plt.plot(pk, x, 'bo')\n",
    "    plt.plot(pk, x)\n",
    "    \n",
    "    plt.xlim(-1.5, 1.5);  plt.ylabel(str(k+1));\n",
    "    plt.ylim( 0.0, 10.);\n",
    "    plt.title('fk = {0:4.2f}Hz'.format(fk[k]));\n",
    "    plt.grid(True)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And now we can calculate the coefficients that multiply the stiffness and mass matrices\n",
    "to build a Rayleigh damping matrix that is also orthogonalized by the eigenvectors:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mass matrix coefficient a0:        0.06124\n",
      "Stiffness matrix coefficient a1:   0.00131\n",
      "\n",
      "Rayleigh damping matrix original:\n",
      "\n",
      " [[ 489.89794856 -244.94897428]\n",
      " [-244.94897428  734.84692283]]\n",
      "\n",
      "Rayleigh damping matrix orthogonalized:\n",
      "\n",
      " [[338.51115694   0.        ]\n",
      " [  0.         886.23371445]]\n"
     ]
    }
   ],
   "source": [
    "zeta  =  np.array([0.01, 0.01])         # damping for two modes i and j\n",
    "\n",
    "A     =  np.array([[1/wk[0], wk[0]], [1/wk[1], wk[1]]])/2\n",
    "alpha =  np.linalg.solve(A, zeta)\n",
    "\n",
    "CG    =  alpha[0]*MG + alpha[1]*KG      # Rayleigh viscous damping matrix\n",
    "\n",
    "print('Mass matrix coefficient a0:        {0:6.5f}'.format(alpha[0]))\n",
    "print('Stiffness matrix coefficient a1:   {0:6.5f}'.format(alpha[1]))\n",
    "\n",
    "print('\\nRayleigh damping matrix original:\\n\\n', CG)\n",
    "print('\\nRayleigh damping matrix orthogonalized:\\n\\n', np.dot(Phi.T, np.dot(CG, Phi)))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 2 <a name=\"P2_2019_2\"></a> \n",
    "\n",
    "The system is now subjected to an initial kinematic condition, which consists of an imposed displacement on the lower floor, $u_{20} = 1{\\rm cm}$, only, and it is then released to vibrate. Accounting for the two vibration modes, calculate the peak displacement and the peak acceleration at the system upper floor caused by this initial condition (2 pts). \n",
    "\n",
    "**Answer:** For the modal superposition we must firstly calculate the modal masses\n",
    "and the modal stiffnesses:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Modal masses:      [  4000    4000]\n",
      "Modal stiffnesses: [ 71619  490881]\n"
     ]
    }
   ],
   "source": [
    "Kk = np.diag(np.dot(Phi.T, np.dot(KG, Phi)))\n",
    "Mk = np.diag(np.dot(Phi.T, np.dot(MG, Phi)))\n",
    "\n",
    "print('Modal masses:      [{0:6.0f}  {1:6.0f}]'.format(*Mk))\n",
    "print('Modal stiffnesses: [{0:6.0f}  {1:6.0f}]'.format(*Kk))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The initial condition is of displacement type (no initial velocity), what implies a\n",
    "cosine type response. \n",
    "\n",
    "Recalling that $\\Phi$ is a orthogonal matrix, it means that its transpose is equal to its inverse:\n",
    "\n",
    "\\begin{align*}\n",
    "\\vec{u}(t)   &= {\\mathbf \\Phi} \\; \\vec{u}_k(t) \\\\\n",
    "\\vec{u}_k(t) &= {\\mathbf \\Phi}^{\\intercal} \\; \\vec{u}(t)\n",
    "\\end{align*}\n",
    "\n",
    "where $\\vec{u}(t)$ is the _nodal_ response and $\\vec{u}_k(t)$ is the _modal_ response.\n",
    "The initial modal displacements are simply given by:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial modal displacement at mode 1: 0.005257\n",
      "Initial modal displacement at mode 2: 0.008507 \n",
      "\n",
      "Initial modal displacement at mode 1: 0.005257\n",
      "Initial modal displacement at mode 2: 0.008507\n"
     ]
    }
   ],
   "source": [
    "u0  = np.array([0.00, 0.01])   # initial displacements in nodal coordinates\n",
    "u0k = np.dot(Phi.T, u0)        # initial displacements in modal coordinates\n",
    "\n",
    "print('Initial modal displacement at mode 1: {0:8.6f}'.format(u0k[0]))\n",
    "print('Initial modal displacement at mode 2: {0:8.6f}'.format(u0k[1]), '\\n')\n",
    "\n",
    "# The most general way (as shown in classroom), considering phase is pi/2\n",
    "\n",
    "u01  = np.dot(Phi[:,0], np.dot(MG, u0))/(np.sin(np.pi/2)*Mk[0])\n",
    "u02  = np.dot(Phi[:,1], np.dot(MG, u0))/(np.sin(np.pi/2)*Mk[1])\n",
    "\n",
    "print('Initial modal displacement at mode 1: {0:8.6f}'.format(u01))\n",
    "print('Initial modal displacement at mode 2: {0:8.6f}'.format(u02))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The total response is a superposition of modal responses, which are cosine functions with\n",
    "the respective frequencies and amplitudes:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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0/rVuTb91UVgIDBtGKr5DB5qO/O03aqQZdbZuBTp3Brp2rdp2112AEOx6cjNQWEij1XvvBSzTq8XG0v7UVPeU7Wbhww+BvXudOt3fn4x3S5aQ58PKlcC5c741WEjGit27gbAwICCg+va4OKBLF+Crr9xRqnpHYSHw3XdOTDSsWgX4+JDFW2HkSGDHDtZdDri1xXe/fkBpKQlwptaYPRsIDCSL9759pJ8XL6YZdc3Ex9Pc79tvA3v2AH//O1kE2fVEnfJyumejRpHYVmjThqymLL7rPj/8QFY+c5cThYED6TcPPm2zbx/w3HPkM3z+vO7Tz58n9/pHH6X41gceAP73P3qd1q51uJYbY5SKChowxcZa7/PwIOv3li0UeMzUKq+9RvU+PBzYvFnnyVKSxW3UqOqDqFGjgKIibrsccGuL77596bcuFcjoYf9+Sq4xcyZp5eho4N136f/z5um40JYtdNIzz1AD3LgxMHo0pWBj/2VrkpMpmNjc5URhwgR6MGfO6LrkpUvASy8BTz3Ft9wlfPcdWfnU8gZ37gy0bUsDUUadd96hTv/aNcqUobPSzplDtpk//alqW8eOwNixwLp1bVFaaqBMpaUkHtltwjbp6TToVBPfAE0/SAls2uTactUzTp2imR/FfnP77cAjj+iIlTx8GDh+nPy9zRk2jPpwdj2xy60tvrt1A5o2BQ4ccHdJbllmzyad/Ic/VG1r1oxm0hcv1jGdtXUrMGgQZetQuO8+4PRpcsxkqqP4e48YYb1vwgT6/dNPmi6Vl0fpwbt2pTjlL76g2USmlklKouwmHirNsBBk/WbrkTrHj9Os2NNPk7l63Trgs880n15aSjFhY8ZQiIk5Tz4J5OU10Pr6VOff/wYWLACefRYoKTFwgXqAUqdtie/gYPIF4riVWuXVV+n3/Pnk3fbKKxTr+re/abyA0kko2WoUmjShgHGlj2JUubXFtxBk/WbLd62g9H9PPUUC3Jzp08m48cMPGi6Um0uWXPPId4Beam9vdj1RY+tWihBrpzI93rMnEBSkyfWkrIxint54gyx+KSlASAg1zBzrWotcugScPQtERNg+JjaWFiDhBXesef99wNOTgrKffZZU9AsvkFVVAz/8QLf/2Wet991xB9CqVRG++EJnmfbsoVztkZE067Rkic4L1BN27waaNwe6d1ff7+FBfhAsvmuNlBSKjXzuOVpM19cX+Ne/KO576VLyGnHIqlXUebRvb71v1ChyC8vPr/Gy3yrc2uIbIPF96BCMzSHWE9auNZTW7L33qP977jnrfXFxZEnV5HoSH0+/LcV3s2b0ErPrSXVKSoDt263vl4IQZP3essXhQi0bNgDZ2aQTli4F+vQBXn+dwiSWLauFst+CnDhBBmxd9yslhX7bE9+K37cTAYW3JLm51LA8+CAFZwtBAXr+/uR+ooGPPqLx6Z13Wu/z8gLGjfsNGzcCv/6qsUzXr1O6pw4dqD3r04cs8jyCtWbPHqrb5rEqlkRE0DvC7X6t8OKL5BRgaeWeOpWMZmvWOLjAmTM0I23pcqIwciRZdnbsqJHy3orc+uK7Xz+Kuj182N0lqZusX09CbcoUXdHJubk0XfXQQ+oDXw8P6gfj40nc2WXrVuo4o6Ot9913H5nYk5M1l+2WZ+9e4MYNdX9vhYkTSaT//LPdS82fT9lp7r23atv991Migtde40x3jpCSXK527KBX6D//0agXtIjv/v1pdMuuJ9X5/HOq/3/5S9W2du3I8r1nD3Dxot3TDx6k2O4//lHd4wcAxo49Bw8PYO5cjWX6y1+oofvmG1I1L75IVnhDviu3MFeuAEeO2HY5UejTh1Tg6dOuKVc9YvNm6hZeecU6ydLIkdSfL1zo4CLKrKot8T14MGVBYdcTm9QP8Q2w64ka6ekU3NKmDS3W8t13mk9duZL6PzWrt8Kjj1Ln5jBr1NatFKTh7W29b+JEEiDselLFli10Y4cPt33M4MHkP2/H8nDhAmmDRx6pfus9PGgKMjOT1ktgbDNvHnVm779PVqNXXqF4O4fuvsnJ1Mu1amX7mIYNSZxz0GUVRUVktr7jDhohmjNkCP12MFhZvJhiux97zPYxrVoVY9w4Gpw6nDTdu5cGBC+8UPVO3n8/BdO+9RZbb81RZnEciW9lUKoMUjWwbRsbWrXwzjsUWPzHP1rv8/SkCaV16xx4u61aRVNHvXqp7/fzoxguDrq0ya0vvoOCKCKegy6rk5tLPtV+ftS59+pFfiQaO4pt28hias9wFxhIrpgLFlBmPFXOnAEyMmy7ULRsST4sy5dzJ6awdSu5U1maLczx8gJ697abZnPRIrJsq4mQiRNp3Pr66xw3ZoucHNJbcXHkO/zNN8CsWcDXX1McoF1SUuy/PAqxsSRYbL5A9Yx16yhH4J//bL2vXz8aRe7aZfcS27YBAwaQgdoeDz9MH+Ww6/j5Z3KhePnlqm1eXmQN37OHXMQYwtbiOpaEh9NvDX7fZWWUpWn4cBqTnTpVA+W8RSkpofp/9900AFVj6lS6pzbd6Coq6B0bM8a+69CoUWRkyM11uty3Ire++Pbw4KBLS8rLyTJz+jSZsDt1Ap5/ngRBQoKmS2zbRn6u9t49gFxPzpyxc1llZOzIhSIzk1tVgKYb9uyxPVgxJzzc5iItUpLVduBAdeOFEMA//0n+zJwy3BopKStGWRnw5ZfUzAhBrjpPPUXW1StXbJxcUkKzTlrE98CB5Ld/5EiNlv+mJSmJzHNDh1rv8/Ojtt6O+M7Pp0sMG+b4o5SPcGhN3bmTrPCWav6xx8h48N//Ov6w+oKtxXUsCQigbGUOLN9nz1LX8dZbJBqlpGy1t7ydRkpyx1y1Spd1ZP9+6kLUspsqRERQ12HT9eTkSXqRHLVfI0dSOTVqivrGrS++AWqQU1LYgVVh714Sve++WzX999BDNAX+7rsOTz99mt4/LR3YmDEkTOyK7+bNycfPFlFR9Jv99ukelJZWBePZIyyMTHcqPrD79pGeU1vfReH224FGjariYZkqli0jI+wbb9DkmjmPP07hEzY9pdLT6RlqtXwD7PetkJJC2Xx8baxAOXgwBYLZECS7dpHtwZ7HlkLbtpT1zq74Li+nwfDgwdb7GjYk37uNG3m1P8D+4jpq9Olj1/J99SqFRezfT0Jx4ULK9Lh27S3spXjyJPXVnTpRw3P33ZSwXiNKW+6o/k+dStX62DGVnYpBR5mdsEV0NMVysd+3KvVDfPfrRwmnNaShqqig7F63NIoPqXmUnZ8fzZWvWUNWZjsos6hqxidLAgJo7PPLLyo7paQXc8QI25FPALlPACy+AXLRAYAePRwfqzSOKq4n8+eTNrj/ftune3uTplB9dvbIy6NO00GmlZuZRYsom4/5Ai0K/fvT4/nmGxsnawm2VOjWjayn7PdNJCfbv2+DBpFf+MGDqru3bSOPEK36b8gQEt82LamHD9MiP4MGqe+PiSGjD7ddjhfXsSQigtTfjRuqu5csAX77jbx+pk6lbc8+S939s8/qWCzmZuLVVylP5uDBwCefUJJ6HWmWEhJoTNOihf3jHnyQZvJUY34U8W0Zc2GJtzf5d+3fr7l89Yn6I74Bh64nUgK//z2lH/34YxeUy13s2UPBQG3bVt/+zDMUofzBB3ZP37aN8nrbM1abM3w4GTyscodmZ5MZ3Z7LCUC+ze3acQcGAEeP0rS7pblVDaVxtBDfN27QYgqTJ1vnZ7ckLo5uu65U0/fcQx1nQAAF844ZQ6nYbhFKS6kTGzOGHoUlQlAQ6/btNlLVpaSQ5dZWnmPLi8XGsvgGyHc0J4fyaNtCEXY2XE+2baPuoFEjbR85ZAh9rDLmtWLnTvqtZvkGqsrK2ZqqrNhKf+yIPn3IGmaj3Z83j5oZ81vv5UUZai5dooQztxTKqp93303JEZ55htaG37GDRiEOKC6m6mrP5UQhMJC8Rr79VmXnoUNkeXDkOgTQLFVGRj3wA9JP/RDf3btTa+sgcubllykzR7dutHaDjuQfNxdKnlVL2rShKa0FC+xaLbdvpwZPTXioMWwYzQJbpStW/L21+C/37s1+rwCJ76AgGiQ5om1bMnFY+H2vW0cue48+6vgSyvSk5jTwZ8+Sqfyhh4A336SWfuNGSglSl3CiM9i7l16P226zfcxDD9mxHKWk0KyEl5e2D+zThyyA9X2tAmXGwJ74bt+eDAuKKDajsJDcrbS4yykoCVRsup7s2kXvWdeu6vuDgqjvYfFNM6pCkC+PFuxkPElOpu58+nTruKOoKGDmTBLht9QsdloaZSW7/faqbZMnU1umYTW7xER6B9QWRVZj/HhqdqyyPaamOnY5UQgNpZmhc+e0HV+PqB/i29OT3kg7lu8PPySt8MQTVMeHDCHr1aZNLiynKzhzht4mWz7D991HplEbA5VLl0gDa3E5UVACM60E3P795O8dEuL4Ior4ru+LVmRkWK+HbQshyPptIb43byajhSIs7NG/P3kkaXY9WbmSfv/977SCw8KF5Pe3YYPGC7iA9etpNmXxYkOnb95MXlL2xoydOlEn9803FjpfSu2ZThRCQsh14cQJQ+W9ZVAErKN7N3gwiW+LAda+fWQE0OLvrRASQl4/NsX3zp3kcmIr8tzDg8rL4pvari5dbPvrW9K1q81l5ufNo2wdDz2kfqqSbeiW6r+VL2Muvnv1IuuyBif3hASqploHn8p7Uq3tLy6mQZRW8a24Rx49qu34ekT9EN8AOR4fPKiasmv5chop33038OmnJDZWr6Y6PWmSTffBmxPF/GxLfCsL3SQmqu5WOiE91qNmzehdtRLfhw+TOHSUMgWgRub69fqd8aS8nBo+Lf7eCuHhNJo0EyKbN1PDqsXw6uND2kKz+P7hByqfkkLFx4fcijZsqDtTj++9R76nU6ei89df6y7Xpk00KLGX6RGgwXtWloXHyG+/0QhWr/gGHMZi3PJoyY0OUIU9d46C08z45Rdqamx5iKghRJXftxW//UZ+RY4uGBlJZa/vhoPMTG2GFgVlmXkLy3dREY2b77mHbDdqdOsGdO7s5ITbuXN1y91r40ZqWzt2rL598mTqXM+ft3t6fDw1O7bumSV9+lAbVy1ZQno69UNafU4VQ5FNv636S/0R3xERZNG1aJArKigda1QUBXAorhRNm5JeaNCAktLfMuzeTV9KySBiSatW1Grt26e6e/t2Or1/f30fO3w4zdBWzpxLSaJQCaZ0BAddkuWzpES/+C4oqKz3J0+Sq709lwlLhg8n45PDAKZLl0jhmAfyApR898SJuiEeT56kHvlvfwOmTUPXBQsoobPGbBTXrtH4Vcv9u+ceCmqtFnip1XprDotvIiXFvsuJghL8aOF6sm0b3XZH+b0tGTKE3hkrt1rFr9xWsKVCZCT5ed2iMxfl5eSKbxcp9YtvoCrjidkAeeVKaovsZWoSgt7RrVsNpsg/fpyCBYcM0fDlXEBREbWto0db77vvPhIyyqyjCsXFVF21+HsreHiQka2a+FZmIbRavjt0ILcrtnxb4VbxLYS4QwiRIYTIEkL8TWX/o0KIi0KIZNPP7w1/mI0O7JdfqD/+85/J4m1Ou3bAXXeRj6xTWQrrksVjzx6aBbDnMxwTY9PyrSxQYStBvy2GDbPwZjlzhpSMo4hpBRbfVQ2YHvFtEXSpZH1yFONqzvDh1Pc5XCtk1Srq6e65p/r2MWPod11wPVGWW33ySeCrr3B8+nQyo2nMxfzLL/QVzWd+bREQQLfiu+/MMt8pVjytliOATFUtWtRv8V1cTG5nWgYt4eHkrmAWdFlSQn/qmbFTUNyzrNzId+4kF4q+fe1f4BYOuty+nSZLO3Uiw6xNzp2jAYhWlzmFiAhKmG/meDxvHnmkOPJdvu02OlX3Eh/Z2aRS8/Op7547V+cFaoGdO0mAqzU8YWF0X+24nigJD7T6eyvExdHtqBx/pKZS568lWBwgBR8aypZvFdwmvoUQngA+AXAngF4AHhBCqK1VulRKGWn6+dLwB9oQ3199RRkfJk1SP238eBplO1g0zTaHD1NAjs28Yy6ktJT8rB3liI6OJiuNRX7oggJywTHSgSk+4pWuJ4qI1mr5btqUppzrc9Cl0oDp6cAU8W3y+/rAgvgAACAASURBVN6yheJqtd52gMZiDRpocD1ZsYJ8Oi1nVbp2pTK7W3yXl9MLf9ttVE4hcGrqVMqQsX69pkts2kTWbK3Z0iZPJgFQKdxSUuizmzTRV/aQkPotvo8cIQuIFsu3pye1cWaNdlISBZsZabuiosgwY+V6smsXtZWOgp/DwqhMt5D4PnOGEm0MG0YTXt260ZpCeXk2TlDqrhHLN1BpcT1+nNqwxx6zn50WqIrJ0OV6kpVFivPGDfLTuPNOEt/uDnbeuJFS96mZroUg63d8vOqaDkCVv7eeWC1Axe87NZX8cbUGiwPU9rPl2wp3Wr5jAGRJKY9LKUsAfAdgYq19WqtW1OGZdWDXrtFg8Xe/ow5VjdGjqc7/9JPBz501i16Ip5/WlGe8Vjl0iIa/WsQ3YJWfc/du0i96X2CABF+PHmYvsZL+To8K7N2bLd8tWzpO0mpO48ZklkpNhZQ0DTtypDY3ewVfX6oydsX31avUy917r/rF77iDeoDCQu0fXNNs2UIxA5bz1aNGkZvV1asOL7F5MwkOrTM/I0ZQP/Xzz6YNeoMtFTSK77o0yVaj6MmNDpAryKFDZL1E1aDfiPj28aEBaDXxXVhIit6RywlAyr1Hj1sqeOj++4EffwT+8Q9qlpYtAy5coG5ONYTCqPhW3BtMz3/JEmpetGRqat2aqotm8S0lMGEC9ZFbt9LJTz9N/karVukrd02zaRON+P391fdPnkwv/48/qu6Oj6dBpKM4FUv69CG7V6XriZ5MJwo9epB7gTvb/jqIjuFLjdMBgHkSmxwAA1SOu1cIMQxAJoDnpZSWiW8AAEKIJwA8AQCtWrVCgsqSin3btUPZ3r04ZNq3dm1bFBb2QJ8+SUhIuGazoBERfbB0qS/GjVP3g7ZFo+xsRC9fjrPjxqHljh0ouesuJH32GSq0pImrBdqvXIkQALulRLGdJV89b9zAECFwYtkynDTzxVm4sAs8PDqjrGwHEhL0O9IFB4cgPr41tmzZgV6bN6NFs2bYpbIAjM3zGzdGu+3bsX3rVsdmDwsKCgpU68TNROTevUC7dkjW+T3C27dHg7178f2CfTh3LgYdOhxFQoK+1E+dO3fBokWdsWbNDvj7Wz/71ps2oVdpKZK6dsU1lfI1b98efYqKkPLxx7isDO5cTK8330Szxo2xq1kzSFMZCwoKkNyiBSIrKpD68cfItRM8d/FiA6SnxyIuLgsJCdr9QMPCIrBihRfGjtiFoZmZOBkTgxM6n2EnLy90y8nBtvXrUWHpH2fiyy+7YvXq9pg69STuvvsMfHzqSIBrDRC8Zg3a+fpie06OppzGzRo1QkRFBZLnzMGVfv3www/h6NzZF4cPq7vTWWLZXnTs2BVLlnTCunU70LBhOZqkpCCqtBSpjRsjV8Oz7NmuHZrs3Ys9daANEuXlEOXlhvuhnBw/7No1AE8+mY0RI05Xhgc9+mgnfPllNwQHH8Htt1+odk7Q5s1o7+OD7dnZNpLf22ZAu3bI37wZRwYPxpIlfREaCmRnJyE72/G5oaFBWLmyAzZs2AFfX/sjU98zZzAwPR2ZM2bgbF4eKU4/Pwxs0waFb7yBlJYt3dKPeF++jMEHD+L49Ok4ZeuzpURMhw648dVXSLNwCSkpEdi5cyjuvvsMEhI03DQLevUKw/r1DbFj1SYMOXsW2Y0a4bSOe9CqrAy9pUTikiW4rmV9ivqClNItPwAmA/jS7O+HAXxkcUwLAA1M/38KwFYt1w4JCZGqTJ0qZadOlX8OGSJlaKiUFRXqhyt8+KGUgJSZmfaPs2LSJCmbNJEyL0/KNWvoIn/8o86L1CBTp0rZvr3jLyyllD17SjluXLVNo0ZJ2bev8Y9fvJhuQVKSlDI6mi6ohzlz6ALHj+v+7Pj4eN3n1Dlat5Zy+nT95734opTe3vKjd0skIOWvv+q/xJYtdOvXrrVxwKRJVLfKy9X337ghpa+vlM8/r//Da4JLl6T08ZFyxoxqm+Pj46UsKpLSz89qnyULFtA9SEnR99FvvknnXdySQv9ZulRn4aWU339P5yYnq+7euJF2d+pEv4OCpFyxQturflMwfLiUAwdqP/7cOboR778vKyqkbNZMyj/8Qfvplu3F+vV0uU2bTBsqH+pFbRecPVvf8TXNhQtSzpsn5eTJUjZtShWlpMTQpWbNklIIKXNyqm8vK5Ny0CDq8k6etDhp/Hgpw8ONlX3sWCkjI+WFC/S5r7+u/VTluW3cqOHguXPp4PT06tuVZ33kiHv6kSVL6PP37rV/3LRpUrZta7U5MZFO//57Yx//7rt0/oVl8fSfDRv0XSA52Xi7d5MAYL/UqYHd6XaSA8A8Z04ggLPmB0gpc6WUShqCuQA0Lo1lg5AQmnYuLMSxYzSN+Oijjqfgx4+n37pcTw4epOjj55+nuZ5x4yif4SefuG8Ka/du8h/Q4nOgBF2a5hClpFlWvVlOzFGmfH+JryAfTj0uJ0DV8e7y+37kEWDsWPes1piXR/O6eoItFcLDgdJSpK/ORFAQuRzrZeBAcr9SdT25cYP8uSdNsj0j4edHDoTu8vtetIii7tRSJDRoQL5UDuanN22iqWytMcIKSrzp4R8M+Owr2Ml4cvEiMG0auWKmp5P7eoMG5AGkIf2vbZKTqb7160duQ9Omucd1wkhu9NatyUUrLQ1ZWRS3ExNjvAixsdRsVrqR791blQRcC4qvusqCMbVOeTl9+enT6QvExFA/qCxypgMp6VWKi6NEFuZ4elJa/5IS4I03LE40kulEweRytX6dhJTUlWpl6FBqt5RAc7ts3UrxWZbv5+OPk+/R55/rKnaNsXEjaQhHK4P27UuBrRYzQ8orayvBmSMUN/MTa0zrReh1O1Es8e4Kujx+nPqe3bvd8/k2cKf4TgTQXQjRVQjhA+B3AFabHyCEaGf25wQAzjlNKy9/Vha+/pp0wiOPOD6tSxfqcHWJ71mzyFlq5syqbW+9RZ3Z7Nk6LlRDXLxIYcuO/L0VoqNJ7JmizE+epA7MUWC/PQIDKfYuY+NJErB6VYySO9odft/p6dSzrF9PKXBu3HDt5ysNl1HxDaBwb6quLCfmNGxIbb9q2tvkZPLnc5QC5I476D5apPt0CT/8QOLNVpaRUaNoUGfHpSE+ng7T6fGEiAjSgue3m4Sz1hX+zFHOsRDfUpKmys2lpaAbNqTbrMR1OpWoYd48Crxu25Y+YNky4N//duKCBjl1iqJWtQRbKigLTKWlVSZuckZ8N2lCmqwyDCY1Vd9gQDnWHUGX+/fTc/zsM2rPV6+mWBADSzgnJlJM4tSp6vu7daPYv+++M3PxLS2lvscZ8X3jBnYtP4u2bfWJyEaNaODk0O/bXkBM69b0pRYsgIc7/JYTEqhcjpaUVjpni/QuSUlUf7t1M/bxffrQ+YX7UinzUrt2jk8yp2FDSl/sjqDL/Hzy49+2jdLL1iHcJr6llGUA/gTgZ5CoXialPCyE+KcQYoLpsBlCiMNCiBQAMwA86tSHml7+8qOZ+Pprski1b6/t1PHjKa2Sw1zHAOXTW70aeOGF6lkNGjSgYfv+/Wa5x1yEo8V1LFH8ck0Ofcr77Iz4Big+qWCvzkwnCkrGE3eI7w8/pOf3wQf0Ik+Y4NoAEiOZThRCQyE9PRFUaFx8A9Tpqa4VolQOR5aZO+6g35XRhy5CsZzaS1Gi3Bgb1sBz54CzZ40JOA8PCtyWGRmQHTuSItBLw4Y0erUQ359/TkaBt96qrgW9vEggbdlC5dZNRQUNWO68E1i7llTXtGlkhdOYE73G0LKsvBqmAO19eyX8/KrG7kbp39+UKvX6dbKm6TEetGpFpmJ3iO/166kSTp5MwrJBA5qlWrlS97NcvLhqVsUWjz5KscuVE7wnTlCmGiNtF1DZb5+Jz8TYsfoHv7fdRk1Ubq6dg9LTaZEaW8vWPvkkcO0aWrjaelpQQPdPy4gjIoKer8Xs1MGD9OroCbI3x9OTZq0bnzhEStzIhXr0cL34rqigNRyOHqWsGtu22Vmq1vW4Nc+3lHKdlDJEShkkpfyPads/pJSrTf9/SUrZW0oZIaUcIaV07umZpj9Ob85ETg49F62MH0+zd5pmzb/6ijrYGTOs98XGUoPn6unbxER6ixwJJIWICJqvM5mNkpLodL0zTpYMGgR0uGIg04mCOzKe5OVRqsiHHqJn+tVXJNLuucep9BKlpaSp09PpK9ldvPPoUXoeXbvq/6AGDXCpRSjCkKY7z6s5UVGUIchqrZCkpCpxYY/QUMrUUpns3UWcPElqwJ6lMjKSrDo25qeV19Xo4HPMGKBzcSby2xm0/gFWGU9u3ABeeonExXPPWR/+8MNUPZcsMfBZe/eSajdXWePHkxhwddBgcjJ1+Hobn7AwID8fp3aeRr9++rKjqdG/P92Si78coQGd3vIoK126mvXradRoniVpyhR6J3QMhMvKyKJ91132M2XGxVGCpQULTBuMZjpRMPXb7a9n4q679J9+2230uOLj7Ryk7LQlvk1+dwHHjukvgDMogrVnT8fHBgTQvTKzfJeVUdIfZ41mccMqEFyUhoKuBgWAkuvblascz5pFI8B33qFZvJYtVfyh3Ef9WeESoDQ97dvjwo4M+Pjo8x2LiSF9ocn1ZNcuelkbN7bep1jfDCcON8jRoyTcbOVUtKRBAxrlmonv3r0p7ZwzxMYCvXEY15sH6s91DFAh0tNdm1Ptyy9J6SgK55FHgLffppGYjcWItDBjRtVK7GFhNDNnUygdPUoNq0EFkYYw9PVJc7gytz0U44vVuPHgQWrdHVlEhKhyTHYlWtLUeXpSXsDNm1U7CKU/02t8VRh9u0QoMnBMGLT+AVbie8UK0k8vv6xuDQwJoQWxFi408FkrVtBgz1ztjBxJvvuG864a5NAhcrvRO2NgGtyXHzqMmkiwo9gtzvxsMh7odZuLiqK678oZs4sXqY26887q22+7jQabOlxPtmwhT8SHHrJ/nIcHTZJs2kT5wCtn7YyK78BAlHr5ItTjmK6VeRWio6nbs7tI2Nat1ADbMm74+ABhYfDPytJfAGdQ2kot4hugdthMfGdkUHUz6u+tMDrkBPxxHYc9DIrvHj1oxujMGecKopX164F//Yv89WfMoArw/PO0vY6k/NQsvoUQzYQQvYUQ3YQQN61ol6Gh8MzOxO23q2tjW3h6Uvu1aZODwdv169RZ2Jribt+eXnJXT18ZCXiJiQH274csr8CBA86PngEyFvXxSMNJf50dl0Lv3iSEXeU3XFYGfPwxmXPM/YWVHsigFfDMGRqM33sv9X9Ll5Jl7YUXKlMTV+foUWP+3qDbteNST3Qo+ZVy2BpEWSukWttVXEw527W27r16kZnflRaQlBRtltNRo8gnVqWDPXiQ9J+eNsOc1h6X0AxXsPOSk5bvvLzK+fP588mP017u6ocfpuZIV5yflCS+b7ut+lrsfn7k1//TT659fseOGXNZMInvkNI0p/y9FSIjSVje2JdGVgi9TrSRkTR9qiO9qtMoHZbi8qXg7U1+zKtXa45fWbSIqsPYsY6PnTaN7CMLF4L6nhYt9K1PYI6HB457dkds80wEBOg/3cuLbr1NzVVRQZZvW1ZvhchIEt+urPvp6fQFtMaJREVR32hqI5wNtlToWUGzzfEXDfbbyvvrqqDLRYtocZFPP60yCj3zDDXgb77pmjI4wK6IFkI0EUL8XQiRCmAPgC8ALANwUgjxvRDCiUls93CpeQg6F2darYCthYEDaTUvs5Vurdm/nxpYe77Vgwa51vItJXVgesV3dDSQn4+LOzJw4ULNiG8vUY6eSMeBYgMuJ4Drl5lfuZIeuOW8fps2JCTtzmXa5r33qM2fPZtmgO+/nzT+uXMqM2NKwJJBn8mdO4EjFaHwgFQVllrx9SUDTLWZ87Q0GqBorRy9epGAtLESW62QkgIEBdleoEJBMaupuJ4cPOhkB2bqdDb+GoJrtpcUsI9ZxpPsbBr3Pf64fR/YKVOo79Zl/T54kHyL1Bx7x48n/yjTiqm1jjTVWSNBqs2b43rT9ghDzYhvf3+q/75ZqVSPHQXAWaK0Xa70fV2/nqbb1dJUTZlCxqK1ax1eprCQmsLJk7UtMBUURJlGFiwApDOZTkBVMbU4BKEexld4tRmvAlD7cPmyY/EdFQWfK1c05ZmvMdLTqe57e2s7XmmHTar74EFqtw3abSrxzCQL/PLDGi3wligF0FD3y8udDCuRkvy7hw+vXlmbNgX++EdKAVUHlrt3ZMFeDloIZ6iUMlRKOURK2V9K2RHAWwAmCiFUcnfVXRKvhqAlcjFxiL3oC3WU9sti4cfqKOkg7Inv2FgyfdpV8TXIb79RI2tEfAM4s4pcK2pCfCM7Gz4VxYi/EGYsY58SNeWqdIMffEBTkUq+SXPi4iiAQ+fSw3l5FCj3u99Vn+UcMIA8Wt5910IjHz9OAtdgC7plC5Dlqb3xs0dUlIUFSW8krqufH6A9TV1wMNCxo1VqhCtX6BE4Vf9N7iJHKkKNZHgjzMT3/Pkkuh2t9NeyJVkqlyyhTk0TK1aQsJyosuCw4qvnKteTc+fIMmtEfAM40bA3IjwPGwqVUKN/f6B9XhqkkeCXrl3pobnKdaGigny6x4xRH6ENH05GBA2uJ4mJ1IVMmODw0EoefZQ0TmlqhlPie+1aIBMhaJaXTe2gAaKiaEbx+HGVnYoBxVFAjOJz5kq//fR07S4ngJVvYFISTdg6G++Ao0dRENAWB7KbGgvgbteOfNJt9D/vv0+vR9OmVNbGjWnGzhAnTwI5OepTgjNnkiB/+22DF6857IpvKeXtUsqFUsorKvsOSClnSinn1V7xap41GdQItMjTHzgRHk4Vw6743r2bfHPtTbEpSxK7yvptNOClZ0/A1xfFe5IhhLFVsa0wWaxTZW/799EWTZqQotCyvJmzZGeT2fiZZ9StXCNGUI+k84t88gmd9uKL1vveeovcC194wWyj0mA5Ib6bxoRUv5ZBIiMp6OyCsoCd3jxWSkfiKvGdn0/PUUvlFYKe6fbt1aaWlb7WWcu39PHB5YDOWgyN6nTpAnh5oeJoJhYsIE8CRzGuAA3ofvtNY65jxeVk+HD1HNbt2tGg3FXiWxGqBsV3UkkYesgjELJmYkSG9MxF24rfcDXQwPR7gwYUiegq8X3gAM0wWbqcKHh60pTbunVwNB2jxaZkyeTJQCu/AvhcOms80wlIfF9tEwJRVqYS7a0N5d21yMJHbN1K5XP0MiltiKt8hktKqK7oEd8tWpBba1ISpKwKx3Ga9HTIEOp/VNd6cIQQVUGX1pfGX/5CGU2nTQNee42s9a+/brCs27bRbzXx3bo1GdI2bTJ4cWoiX36ZZqp12t2qocfnu48QYoIQ4h7lx/jHuoejR4HNp20vVuEIX18S4Da1lpTUStlLaQbQUNTPz3V+38p3tVh21iGenkCvXvDNSkNoqONZe02Y/B2PoJfxrx8U5BrxraybbCvKZ/hw+q3D7/v6dTKm33WXugtyu3bAK6+QK+bGjaaNTqQZvHyZ+uDBoxuRVdfJ6TalE6s0/ij+GFrTT3XoQBYQVwVdKu4RWkeO/frRyOLcucpNSoftlPjOzIQIDsaYsZ746SeD8cLe3kC3bji/PRNnz6qvF6TGXXeRRWnxYg0Hp6dTHbGXS278eHo3zp/XVgBnUISq3rYL9K5ty+0N34pC3Uua2yK2MRkPjngY9H0NDnad+F6/nt5LZZUnNSZPpjgQB2Jkzx4qutY1hQB6zZ8cSUauki7GLN9Xr9KgMTDO9PwN9NsAefx4e6vo5tJSUpNa0kA1bozC9u1dZ/nOyqLpKj3iG6CGKikJv/5K989Zf29ICRw9ikbRPdG4sRPJjlTSDUpJcZD+/tTnffABJSmZOZMynRqyfm/bRosS2cqmNmAAWccrLUj6+PRTcg199ln6iB9/NHQZbeJbCDEfwHwA9wIYb/oxkPTHvaxcCfyKrpBeXoZFSP/+JL5VYy5OnKAOyZF5wNubghldKb59fSlPsF7Cw9E+N7VmRs8AWb67dkXH0EbGDf+uEt8HDpC1ytZL3KoVRSHq8Pv+8kuKhbGX73/mTJqCqxz5Z2VRr2cgO0xCAtXVUaNQI7lWlZnXgwdBU8ApKfpadyHI9cRVlm8tmU7MUZlaPniQxgytWztRjgyaep8wgZoIQ7M+ABASguK0TLRqBc1p1xo0oDzjmjrNFSvoGU2aZPuY8eOpUhk24esgK4umGzt10n3qwYPAIWkSyTUU5Ni9iAZzO686Ib5dla5uwwbqsOylOIqOpvtrJ/2nlFWLI+tlch/qZ3/5zZj4XrOGDMADHzGdb/DeKc24lfhOSqL0mRpzsBYEB7vO8q0304lC375AZiZSd1HkvtPi+/x54MoVePTsgaFDnRDfoaHkamvmb7puHXlGzZpVvZrOnEmuJ//8p4HP2b4dGDLEdjCMkvrIQKayzEzgr3+l8ezq1WSftNdU2kOr5Xugydd7mpTyMdPP48Y+0n388APQN8Ybols3wyPo/v3Jmqg6+6XMzTmyfCvHJCW5Ju1UZiZZjvSuTgCgoGs4Wlecw+DQSzVTlsOHgbAwDBpEDbqhwPGgIAr6srNQ0cWLlGHIqfVc9u8nMWYv2CUujlxTNCyaJCX5tg0ZAgwebPu4Bg3I02XXLpOXTna24Wn3LVsoQ9uAAagS305E6zdrRrOaBw+CrlVUpH9e09Xiu2lT7eJNyWhjlh4kKcnJqdvychKRoaG4805qsFevdnyaGjcCQ9D66jE8MrUCPj7az4uNpVfGob/mpk3UyNlbxS4igmZRXOF6kpVFI1EDTqv79tEMG4AaC9D2yUzDNc+m2Jqhwd9Hje7dqQPJy6uR8tgkL49ytdtyOVHw9SVVakd8nz5NE0FGxHeYTyYqIPBlgrH2a/lyGvj2G9OS3mOD/TZQaRCu3vwp31tLnw0gv3t3ao8NR03rQBHfet0NTY3VhU0pNbI2h3mu8bg4egSGYk6V2SuT4aykhKzePXpQHKQ5zZpRjoMVK3Rav8+dowLaSwHVty9pIWVmWyNlZZQ9yteXMk2NH08Tq59+qusylWhVY7uFEE6uD+ZeTp0iLXXPPbDKl6sHu0GXu3dTPkkt+V8HDaKn6YoFR5yINs/woTc31r8Gshsoq8r07o3YWMocY2gGNiiI5u1V0g1KSfFDvXoBH31E2bQMTXJUVNCzUcsSYM6IERQQpmEUfewYDdq0LO40bRpp/i+/BDVWQUGaim3Jli3UDvn4gFq5ggKDSx5WoWQOMLzsac+e1EhqWi7WSZRgS61uMU2bkm+1yfJ94wb1PU5Zj06coLofEoJmzSgLhFHxvfNiCBqiEE+M05cvV9EWdifbKiroeztKii0E5V3durX2064ZzXQC6lubdwqg0WJNpfdLS8O5lmHYf0AY++rKd6lt15Pt2+l5jh7t+Nh+/VRUaRV6F0c2xyMrE1cCOmLlBj/d442CAjLe33MP4OEpnOq3AWqmLl60aP5SUkjpaZwVLlDaYcPRgDpITyejgd789qbGqjwxCb16Ob82h7kFPi6O/mvI71u5dybx/dFH1Ce++666fWvmTHJd+te/dHyGkszdnvj296cBp07x/eabdMpnn1WtjO7lBTz9tK7LVKJVfH8NEuAZQohDQohUIYQLal/N8f339HvSJNBLfOyYIcfLsDASMqrie8+eqmk8RygtWW0HXZaVUWU3KL53F5D4Di2tAfF97BiJkN69K2NODXneWLzECvn51FA/8AAZy37+mRqee+8FCgt1Wv0zM6n1dyS+lZdcw1yc4p2iNGD2aNUKuPtu4LuviyFPnzYkvs+cIeFYuaR8DeVajYqi21OyN4liF/T6oisZT2rb77uigkwTeiOFzVYiPHSILuOsvzeAyvs0YQIVS68bspTA4kR6j0OkvmcYFUUzKnbft+xseom0DKaU5U7tLsvqJEqKVIPiOzHRNI4IC6sZ8S0lkJaG0pAwXLpk8Ku7Snwr4lDLqlB9+5IlJCdHdfeePdSOmi9zoJljx+DdsztKS6v6YK2sW0cTa/fdZ9rgpPhWXSQsOVnX2usFyvNzheuJ3kwnCu3aAW3aoNmvSc67nCjl8PcHOnRAZCSM+32b9dtXr5JLydix1us/KTRvTtbv5ct1ZDbdto0GK46+eEwMKWmNI+gjR6i8DzxAGTprAq2KZD6AhwHcgSp/b5Xca3UTKYG5c0nvhoSA/iksNLTako8PNUJW4ruwkF5IjdNXaNWKpmFq2+/75EkS4AbF9/ZjbZHn0QINs2pAfCsdYFgYevYkF2ZDYw8b4vutt2g12f/9j647ejSlWDtyBHjvvRB9lirlATsS3y1bUoXQ4PedkEDtotbYsT/8AWh8+QRERYUh8a1kt6gU3zpyrdojMpLeqcKdSSRs9eY7dlW6wexs8i/UK74jIqiTv37d6WXlAVit8Keka9PrtbF3L7DllKnz1xnz4ONDBk67zY2eFTkUJVab+b4vXqTBgAHxfekSpZWLiQFZuTIynEtNAFB/ceUKGsWSQcLQpGW3biT0alt8p6XRZ2mJkleW7rTxhfbsoWZQa6rpamRlwT+qO3r31r/S6ooVFGdR6aIXEkIjHoOumsoEWKVuLi/XPTgvadmS+u7aDrqsqKB22oj4FgJFvfuiZ1FSzcRqKQu8CQEvLxj3+27WjH6ys7F4MY3dZ82yf8rzz5MzwWefafyM7dtJgzmqrDEx5Jql0QLy3Xf0SD74QGM5NKBVfJ+SUq6WUv4qpTyp/NRcMWqXX36htvepp0wbzPLlGqF/f2qnqom5pCQSuXrm5mJjSSXW5tSt0UwnJpIOCpxtEV4znezhw+Rr1aMHPDzoVhkae7RtSxZXMwFy+XKVm8lf/1o1+XD77fSCb9rUFnPn6viM/fvpM7T428XF0XO0szKAlNRgjRih3QNi1ChgUGvTdzQovpWxv+RUwQAAIABJREFUAQCaK/P3r5Fc3wIV8MtINqZKO3eme1vb4ltvsKWCMrpIS0NSEllgOnZ0ohyZmdTpmFJFBAXR+EOv68m8ecBlvw6Qvr6GxFtsLLVbNsMTDh6kF8dWgLE5imudE9PvFRUOco87kWZQcZWIiQGVVUnb5gwm40H728Pg7W0waFYJfHeF+Nbi/ghQA+HhoZqHr6SE6owRlxPk5QF5eRAh3TF1KoXGqObZVqGwkOJ577nHbGxv4TOsF39/ukTl1zx2jD5Iy+yAghAOlsusIZRBhhHxDSCnVV/0whH061UDMWXp6dX6weHDSU8Z8vsOCoI8fhyff05dhyPbVvPm1L1qSpN6+TK1R/ZcThQU1zqNridr1tAg0F7ssl60iu+jQoglQogHbsZUg59/Tq6c999v2qCIbycynly9atEGKCpSTys1YACluzFggdeM0RzfoO+YlQUUBodTY24oP5oZaWnUkZqc0AYPJk2vO/ZICLLqmLXkH35IRrJXXrE+/JVXgP798/CXv+hYOWv/flKZWlyIRoyghtLOi5yRQW7OWlxOFDw8gCnRVMlOeOoT31JSgzVihFmcrRDUiDrpdhIYCPRvmg2fIo0uCpaYBmAuEd+entrEpDlKZ5ySUpknV+uASRVTphPzi4wfT0aBq1e1XaKggKwvk6d4QAQFGRbfxcV2dENSEgk2LUsYNm5MvvEGxHdFBa3+3Lkz1U+b76QT4nv79qqkUpUi1NmgS5P49onqjbAwQ8kSiNrOeFJcTHVOq/hu2JBGgyqW75QUupwh8a18x+BgPPgg/XfJEm2n/vwzTVpVy3jppNEMoHe5sv4bHZxHRVFd0hBkbxijmU5MpIhIeKEckT5OtrH5+eSOZFYOpQ9T0mnrIigIxYezkZoKPPmktnZ11Ch65Da8oqrYuZM6Pi3iOyyMdIgG8X3mDNUZrdmltKJVfPsBKAYwGjdZqsHz5ynLyaOPkrENAFkAGzUy/BIrs3TVLB979pCjcZs22i/kCt/XzEwaeehJ0GpC+X4NY8KpJTS4wEElhw9XE0K33VYlEnVjlm7w2jXKInL33ep+iR4ewH335SA/H9pWFywvp7fN0bBcQXnZlWAPFZRpOj3iGwCGtM1GARrhix911CuQ5e/MGfKpq0ZoqNOWbyGACYEGgy0VevWqfZ/vlBT6vnojjjp3Bpo0QXlSMlJTayBVV2amlV/8hAk0UbZhg7ZLfP89CfDp02E4V7TdoEtlRQ49X7ZPH90zYjt3kpB7+GEKptq+nTIdqE7+ZWXRy9uli67PAOi6/fqRrkSPHnQdZ/2+09LIb6xFC8TEkPg2ZI/o3r12Ld8ZGdSG6Ulz0bevquXbyOI6lZjlaO/UiSymixZpm+hdvpzWi1GWUlCuA8Bpv++TJ00Gn+RkGqH10plLIjKShLeT7ahdnBTf8bnUEfpnpzg40gGKocbM8h0VRe+u0X7b6+xJNGlUhgce0HaK4jbpsO/eto3862JiHF/U25vqvAbxrWRUdYv4Nksv+NjNlmrwq6/I1e/JJ802Cucip3v3JuNQNfG9d6/+FqqGfHDtomQ6MWC6Uyw7gXeaGnFnXE+KisgSYmaNiY4mv29D6QCDgsjyLSU+/piWAFezeitERV2Gv7/GhPhHj1KaC63iu3lzGnjZsQImJFDKLL1GPP/z2bgU0A1fLRC6DC3ffEODTau1Unr0oClNs1yrRhjU8CBK4I3S7gaTIPXqReXIz3eqHHbRuqy8JablXIt2J6OkxEl/7+vXyWRjMfM0YABNYa5ape0y8+bRJQYPRpX41qn82ren5Amq4vvsWfKx1vNlw8Opc9Y4nXT4MImpM2eAr78mLfvKK/TdPvlE5YSsLBoI6cmpCJqESkwk31QA9CIEBTnvOpeaWiloBw2iWQtDkzfBweSUfsVq4WgA9Fi//trOehKOMIut0Uy/fjQ1Z5EJac8emunSspKqFceO0bvUtSsAYOpUqi52bBQAqDr99BMwcaKF625AAA1+aiDoMjkZ1D707Km7flVf7KCWSE8ng5kBoxkAbDgWhCLPhtVSphrCLM2ggpcXBUn++CMZEPRwvW0QvGQZ/jThFAICtJ0THk63waHY376dREWlldUBMTFV7sJ2WLOGxv8Gx0E20brIztdCiKZmfzczLbxTp6moAL74gqY2rVx3nRDf3t70/lWK79xc6mD1msjatCGrdG1bvg0GWyYmUp/VZJDJWu1M55WRQQ/EzPLt5UXW740bDXQyQUHAjRu4nn0O775LFl5lRkINHx+JsWNJ7DjULFqDLc0JD7cpvhV/77g4A2Og7Gz4hgfj/Hlg6VJtpxQXk4vCpEmwbuCUF8HJae8eRSlIR08c/VWDi4IaSktWWwPPK1dI3BsR3wAQGQmfjEMQqNDlEmqFcp8tLN+enrS44IoVjrNmZGSQxfjxx031JziYBrMGUkbGxtoQ30aW8ezThyysGtuvzz+n733wIC157+FBC0mNH09pxayCuAymGUxMJIPLkCEWZXVGiJSXk9I2CVolCNBQwHiw/aDZd9+lmdroaGpWZs+mLkYzqanUuOpp95VBl4X1e88eg1ZvgJ5fp06VM08PPkgDwP/7P/vt/dy5NJupahmtoYwnSUmoynSil5AQEni1GXRpNNMJyPU5+4Qnctvb7pN0lcPLy+o9nDKFxup6Ay83HCP3yUeHavfb9/AgDbdli516U15Oz0OL1VshOppG6nbc0QoLgc2byertlOuhClrdTvpIKSuH6VLKywBqIolNrbJxI3lKVAZamhMSQpGuBv22+venF7iiAlUVXG9Hr/jg1pYAKSyknt0J8R0dDVJwXbs6J75tWGNGj6ZFHHTfAlMA4ur3spGbC7z6quNT7r6b3JCUYCyb7N9P0Tl67lufPtQpFBVZ7Tp6lD5Xr8sJKiqA48fRJjYIvXoB77yjbZCybh01wI88orKzhmZbWp1NRjIijRt/ajvjidJBGzVXRETAu/g6ejfINhqrTFhkOjHnxRfpeb7xhv1LfPYZidZp00wbnEhXFxtL75tVmMnBg5UWf80oPl4aOvgbNyjbxb33Vl8p1MODXBG6d6fBSDVjsEHxrVhWqy1kFRFRlUrRCMeO0bttsnx360bfY+dOA9ey8/z27wdeeonaqi++INf6//s/sjRqNlCkpdF7rseiq6TbMxPfFy7Q5OKAAdovUw2LNJENGwL//je1v7bSDubmAv/4B7kaVGZpMqd7d6cMBy1bUvB09u4LFDFoZHDu6Un9WG3l+pbSKfGtjAkqwiJowOlMQoejR6mvtcgecued1EUuW6b9UlICn2ygfjtY6AuaHTWK2iybjz47u9r7qQlFqNtxPYmPJxlV0y4ngHbx7SGEaKb8IYRoDkD/kmMupKSEUs61bk0NmRUhISRuDEZO9+tH7XhmJowHbgD0gum0fKekaAg+AKr53Onl/HnqpCvX2wh3MuPJ4cM0grYoi7IGxMaNOq9nEt+J32Vj5Eht1pmxY6kNceh6sn8/WYL0pNALD7dpBVSsAxpXMK7i7FmguBgiOAh//jM9dy0+6998Q7Ozqp1XcDB1ss6I7wsX4H3pHI54RRg3/igNem2Lb6PK2WQRuyswWXcmxWocPUr3W6UcnToBv/89rZZmK5zi0CHg44/JEtq2rWmjk+IbULF+HzxIbaKW1HQKwcHkf6dBhCxbRm4a1dz/TDRuTAL80iW6FwDIKffyZcPiu3dv8hmuRGmbjbZhFmkYhSBx71SqVAslkZ9P1t62bckV54kn6PqffEKGEEfuGpXoyXSi4O9PszNmQZfOLK4DgOqnRb1/5BEas/3tb+reSq+9RvXk/fdtWBpDQmhU4MQCXVFRQEmiqc82Oq3Vo0ftBc1evEj136D4Vqpq02F96D5pEgs2sDEI8POjuJUVK7Rn8NyxA0g41h5lXg10666RI+m3TdcTxbinR3wHBVEWKjvie80aCg+sFntQQ2gV3+8A2CWE+JcQ4p8AdgH4n7MfLoS4w7RwT5YQ4m8q+xsIIZaa9u8VQnTRcl0pBe6+m0Yt//ynDQOAk5HTyixdcjKo82ndWl+wpUKPHuRrZ8P/z5IDB0j4d+5MlX/NGjvpupTGwYDlW/H3ria+MzN1pAuxIC2NGneLh9GlCxVPt993586QHh5ompeNP/1J2ylNmpAAXrnSjjGgtJQeqh6XE8CuFTA+nvwmu3XTd8lKcRUUhIceoir2zjv2T7l0iQJEHnzQRqIWX1+axXBGfJsGmwXBTli+vbyoPtSWy9WxY2RW1X3TCdmzF0rhhSEBTvpMHj5MZbDhh/j3v5PI+M9/rPeVl5MAa96cDAmVdOxIAxcD4jsykqqAlfhOStLvNqekJdQgaL/4gh63rUQE/fqRj/ZHH5naM4OZTsrLSbBW+nsrKOLbqOtJcjK1XWZCZNAgKub58zqv1bAhOVFbPL8//YkszYsX0zNXeOwxGki8/76Ga+fn00hOr/gGrIIuExPJ/mAo5sGUZtBSfHt6khvNr79aL8udmkqzPE8/baf4NZCkIDISaJ7jhMEMoE7r9Gma0qlpVPys9ZCURNUrYIiTdb60lNpRG+l2p0yhR6w18PLtt4FmzT3gEdRNt/gODqZmz6bxKTWVGlI9wbNCVC22o4KU1JfedlsNrBKqgtaAy28A3AvgPICLAO6RUupMmV8dIYQngE8A3AmgF4AHVJawnw7gspQyGMB7AP6r5dqnT/vh55/Jd0zN0gLAafHdsyf1fykpMB7YpVwI0CSGiotp6rlNG5qKTEwkf8nBg20IcCesf4mJpF0qG147ll1NWGQ6MWfMGLIO69L1Pj4436ATwv2yMV7Hck93303tic3bfeQITV/pFd+KFdBCiDjr7w0ACAqCry91zuvX28+YtnQptZmqLicKzqYbNJm7fQeQ5dvwrGbPnrVr+e7cWVvaPBXO5vniKHqgd6mTfp0OrJCBgSSwv/rKOgfy55+T9fHdd6uLMXh6kqA3IL5VF9vJzSX3NCNpXezEOigcOkS+w088Yf8deO450o0//QTD4vvQIdKf1fy9AZpmaNrUOfHdu3c144FTq/RaZKz54QeasXr1VesBip8f9WM//qghT7bSOOixACr060dW0gsXANBX7tHDlDFGL2ZpBi0ZPZra/H/9qyrNrJTk99+kCcUB2KQG3NUiIoAImYyS1oEW0yM6UPSDDut3bi5Ziu3mtje/pkF30cqkRTrcwlQ5fpyCEW0MAsaMoVkrLa4nqam0rsFzzwEewfrFtxA0kxsfbyNmKzWVLNl6K2tMDLXRKgkI0tKoWawNlxPAgfgWQlTOQUopj0gpP5ZSfiSlPKJ2jE5iAGRJKY9LKUsAfAdgosUxE0FL2wPAcgCjhHAsYYqKPLF0KU3p2qRpUzIlGhTfihEk9WAZNXiG1t6FLh/cf/6TPmrOHODNN6livPsuddDffqtyQmYm+R9oDSs2IzGR2rlGjUwblMbcyLTt9ev0ItsQIaNHk1/Vjh3aL5mdDaQVBiG6RbamVNwKyuqCNl1PjARbAlVWQIuGLj2dZhF1u5wA9CW9vEg4gCxCfn70zG3xzTfUuditjqGhVQGwRkhJAQIDERLbAleuUOouQ4SEkNpyMG+ZlEQdta5Ml8eOGe68ABIeyYhEuwtOiO+SEiqHgzzjL71Ej/nf/67advYsbb/tNuChh1ROMphuECA3ggMHzAa7iu+QERNnnz40c3fxos1D5syhMVClz7oNJk6kqv7BB6DvpuTz14HimmFl+RbCeNClkobRwkWhXz/qBwz7fZs9v7feoq7AVsamZ56hMddHHzm4rpFMJwoWQZdG4xEBOHR5nD2b3Eu6daOuJS6OrJqvv+5AD9fAAl0REUAEUnCujUGDGaDbeLdqFTUD991HsR52ycqiBsHAyl43bpCU6NsXpIy7djU+4FQMbTYs3w0akDFr5UrHoXNvvUWeTX/6E6rSBOu02owcSQMY1a+TlmZswBkVRf2gSn1as4Z+W6XrrSEcyZZVQohkAKsAHJBSXgcAIUQ3ACMA3A9gLkgY66UDgNNmf+cAsAztqDxGSlkmhLgKoAWAS5YXE0I8AeAJAGjSpCNatkxwGIkb2aYNsG8fkg2tlQq0adMDl/eeA4qLke7jg/MGriPKyzHU2xs5P/+M43by2R49GoC33uqLO+44h0aNMiq/W0QEEBTUH3/7mwfatUuEp2dVhY5KTIRs3Vr395MS2LVrEGJjc5GQQBZSUVZG5Vy3Dsd1NgoBGRnoByBNSlxSKYunpye8vAZj7twceHpqWwLt88+7YSS6YejV75Gg4fsVFBRUHtejR18sXAjExlrntQ1euxbtfH2xXTUqzT6hrVuj+b592G1WnrVr2wEIhbf3XiQk6FttrNfu3Qho3Rp7zUYlt9/eHQsXtsO4cXvQvHn1Fu/kyYbYty8GTz+dhYQE235+7YRAaGEhdi9bhuJKR2LtRO/ciaLAQFRUHADQDwsXpmHoUKtX0iFty8rQo7wce5cuRWFgoOoxV6964fe/j8alSw0wa5bEwIG5uOeeM+jXz47Pp5QYkp6Oc2PGIMtB3TCvF+asXNkJTRCJhy8uws5Vq1DapImOb0Y0+vVXRJeV4YgQuOCgHHfdFYyvvgrEzz8Xon37Qly96o3i4oZ49NH9+OUX63oT7OuLdhkZ2B4fr3tKpVGjVigp6Y2vv96PkJACdPz+ewQB2HH9Osp0thXNpEQEgOSFC3FFRbwXFXlgwYJBGDo0F6mpjmfN7rijI+bMCcKx4j3o2KoV9iiJpjXyww+90KZNYxw/vsfKShzcogXarV+P7Vu3mq08pY55vfC5dAmDLl7EMX9/nLG4P927R2H9emDcOH3+V508PNDt/HlsX7cOqSfaIDGxH557LhM7dtjOYBMX1xNz5rTAbbftRqNG6ubT4PXrqf06ccJxGh0LPAsKMBTA8eXLkVrsj9OnhyAgIBsJCacdnmtJl02b0FkIbD99GhU2/HJef70lEhObITe3Ac6f98HAgaXo2TMNCQn2RVm/wECUbt+OQwb7bRSVYAjSsbJkBI7rvIZSLzwLC+le/fwzTtlZ9jA/3wsffRSMTZvaIiioAOHh1/HOO21QXp6JiRPVn3Wv3bvh37Yt9umxRpk4cqQxKir6wts7DQkJl9C7Qwc02r0b+wzcq07r1qEbgO0XL6Lcxvk9ejTHN9/0wTvvHEJsrPpqeWfO+OK77wZg8uTTOHToODpUVKD79evYuXIlSqtN6dnHz88HwCDMnZuF+++v6t88iosx9NgxnBwwACd0fk+/ggIMAHB0+XKcs7B+f/99BIKCvJGZud+ZBDu2kVLa/QEwFsBiACcAXAOQC/L5fhlAW0fn27nuZABfmv39MICPLI45DCDQ7O9sAC0cXTskJERq4vHHpWzbVtuxKrz9tpRT8K2UgJTJyYavI3v3lnLCBJu7Cwul7NVLyg4dpLx82Xr/qlVUhHnzLHa0bi3l73+vuzi//krX+/RTix0REVLeeafu68kFC+iCGRk2D4mLo8tr4cYNKZs3l3Jh2Ft03atXHZ4THx9f+f///IdOO3NG5cBRo6SMjtZWEEveeYcufOFC5aannpKySRMpy8sNXK9fPynHjKm2KTNTSi8vKfv3r/Yx8uRJKYODpfT3l/K33xxcd/t2KueaNfrLVFgopaenlC+/LK9fl9LDQ8pXX9V/mWrlWLdOdXdFhZT33COlt7eUq1dL+fe/S9mqFZ3y7bd2rnvuHB304YcOi2BeL8y5/34pp7bdRNfZtEnDl1Hhu+80tw3Xrkn5+utSTplCz7ZVKynfe8/OCR9/TNd2+LCtycy0aC8eeEDK/2/vzOOjKq///3mSEPZdkH1LCElYQlgFRFlkk82CG+6tFm2/WK12sbW1rX5dWvu1VVErWpf+qtaVskQEgSCyQ4BAFmRRkEVBUGRfQs7vj8/cZDK5M3PvncwkhPN+vfJKMsudOzPPfZ7znPM557Rr5/o4IlL6WQc5WevSX7rU2eEOHRKpXVtkW/MBIkOHujqV4mJO5zfeGOQBL7/Mk9m2LeyxyoyLrKygb+KXvxRJTORl4Yp33+UxN2yQm24SqV+fYyAUa9eG/KjJsGEi/fq5PBk/kpNFJk2SRYv4WgsWeDzOjTeKtG/v/TxCcfPNIm3aeH9+To4IIL9Pfcf1U8uMi9atRW65JeTjR47kfP2HP4icPi1y9qzI2LGcN4NMeyKZmd7WWeGaDYjs3Om74aGH+GLHj7s/2E038T2G4PRpkUaN+JUEY+pUkZo1Rfbt890wdy5Pcvly16eUmmrz0axfz+O94/77lKIintz995e7uX59kZ/8xNlhAKwTlzZwWM23iHwoIjeKSAcRaSAiTUVkoIg8KiJfR2D37wHg70JtAyBwK1jyGGNMAoCGANw2Iw9OSgpDpkeOeHp6jx4MXxXHJ0RWgT1MxZP/9/8YFZkxg2qZQMaPZ2Lkww/7hX+OHKF2z2OdXMAv2dKiZ0+GJN2KfPPyGKOysvxtGDWK4aSvHYyod96hVrDnZN/xXOrHLA3XwoU2dxYUuG9HbmFpPfykOVbhlDCONnt27Cj3mXXuTH1oXh51rTt3UtkweDAj/wsW+FXFCEZGBr2lXrIlCwooWszIQJ06jEh6rnhihaSDyCf++U++10cf5Rh/9FHmOHXrRolGUNVMpJVOwLFoevkXBfZAXh61AgE1vu2oX58l1v7zH15/Bw5QAxuUCCqeJCUxBFzy9bvtbOnPxRdTvhdEVzp3LnXt5TTYQWjShN0vGxzYjlNt3M1dO3Zw/gj6Wl6TLq0BbpPXM3Ag51zXQ8T3/R1euw1vv81qNuHUgX36ML/nmWdC6Ia9VDrxp2dPYNOmUG/ZGQFlBiuU9HRq0z2u29abm7unZ0RV+NClS0jZSUEB5+NHHgH++EdKlBISeI336AFce61N01URzyU2AV7KTZqUKBX5BRYXh04UCkZOTlgpWmIi+0l88IH9R7F3L/Daa+xR0LKl78Ykb+s2QPnmp58GjH9rvfUiO4mP5yIW8Pls2cLcEc9lNh3gxSSoKNYC6GyM6WiMSQRwPYDZAY+ZDcBSCl4NYLFvl1ExRJh0mZEB9MAmHGruoUuWP6mp1EQHyTh87z2O1zFj7J9uDA3vXbtosAAoHdgeje/ERJux3K8fU/tdhjORn88NRoiabaNG8fd7DgRMzz/Pj6zrBN9FHDYLqSzdulEOVy5R6rvvWPvVbbthC+sD8xkiZ87wz1DNf4Ly7besgGOzYRk/Hvj4YxpogwYxQevECSajWKXkQlK/PseFF+M7YFXOzIyg0Vvz5rQCbZKWPvuMyTnDhwP33196e82a1Ezm57OeuS0RJiydOMFDdOrblOV4/MqvuSI/n59zNFLlrevaQ7mzuDh+fRs2gPkYn33m3fgGgpYhLS7mmBw+3J0y5t4ffo/m+AYrvnE3dwXVe1t07co379b43rCB12GDBuXuspIuXeu+fd/f+ne24+xZ4H/+x9nTfvpTVgqxLdBw4AB/IjG+e/QAduxA4dpjaNWqbE12V9iUGawwIq14kpuLszXrYuOxJO/5KgDnl88+C+qMeu45zleBuWf16nFTWrOmTX+Kb76h1efR+LaKFpVcb143nMeO0QJ1sHj9/vfMcxw7ltW2LESo4T93DvjlL/2e0LEjT9CD8d2/P0+tTL2AzZv5YXrd7HXrVs74tspsVkvjW0SKAEwDMB9AIYB3RCTfGPOwMcaXFod/AmhqjNkO4D4A5coRRoTlkfJofDdvDmTG5WJr7QgSNwAapufO2XqxvvuOiShXXx16ARs1igvBo4/6+rx4rBYA0Pju0cOmUIRVlD5sl5oA8vLCepN79qQh+fjjTL4Mxty5XHimTQNMsrcddFwcL6pyxreVdOHV+A7wAubn0wB3m7sJoEyZQTsuvRRYupR/x8Xxb1f2k1erOTeXWbi+8+rZk06og+4l3xzQNomDIvQE1qrFNtuBUYPrrmPe1Z+D1T7aupWliErcP+7Iy6PhmJEBLj5ePd8hKvxETPv2dKN5TLrMzORXWbwpjx94JG08e/Tghxbgjs3LY4KU22TjtERez2+tSXZcQxigl7Fp0xBByNq1Oed78XwH+XyaN+cQdl3vu149SIsWOLBiO0aOdBQcAcB53hhuvsvhpdZxIBkZgAhOrtnsfUhYZQaj6fkGvCddrl+PUykZEMRF1n09JYULtE370SNHmPx+/fX2HeJbtwZuv52Vfco0qo1g3T57lnZoGWd1x4609t2+Uas5j4Mk7I4dmVC6ezcTME+d4h5i3DhWnPvpT/mYEmrWZDjMg/FtraUl3cUBjvu0tCC1dR3QtStP3i+Ssno1K+9EkLMflsr0fFuSlhQRSRKRR323PSQis31/nxKRa0QkWUT6iYg7F2c4kpI4k3lV0x86hFbFe7HmlMdKJxYhKp7Mns1qP5Mnhz6EMdxl7t3rK/3jV6bODcXFdPSVk5wApRa5G+P7yJFSrUAIrFrH+/axxJodp07RG5qWxrJlaNCAM5uHi3jAAE5UZRreWZN5JAaTnxfQcph68nw7+P66d+cpFxR4UD1lZtKF5rC+fAkbN/KFfVEMy+CPSHoSYECuX8/SdA8/zEUqkBo16A1ftixIhZytW/m5eZyMy/TM6tWL5/f99+4OcuoUnxct4zshgV75CIzvY8eAbxb55CJeqzVZzz11qtx1aNXkdV3px/eeVn+bjA8+cPaUw4dZwei668JIvDIy3BkiR4/yfEJYolazHbcx2YONktHyxHbcfbfz5zRtyvnEtilZJJVOLHzjoMHOTVGrdBIxHTtyHfIipTh3Dli/HrUG9YYx3guBAAgZOf/Xv3h9hYpo/PjHPJ2SaDUQkfFdUEBnTxknTFyco3Kg5XC5eA0YwPe8fDkN8IwM1v+ePt1XvSiQpCTXEWuAplLduqXSWABcbyPZcFpztN94Wr1azlS5AAAgAElEQVSavkZPclGHOD60MaaxMaarMaaTMaZSjfYKo2ZNLmBejW/fgF54IMOVh6YcltvDJoz23nt04Dnxng4fzmv21VfBi/jii911rAM/iqNHgxjfiYk0RkJ0hCqHFY52YIRcfjlLqz3+OCeuQJ58ktfrs8/6dbu1yha5ZMAAbjTKXMT5+YyfefSYAijjBVy3jrtnl/sfYr2nMKXWGjbkj2u8WM0iXK38VmXrT8/Sk+RkbgKKikpueu01Xpo33BD8abffzn2Xrfd727aI9d7163NqKFl83L5Bq5RjJIZQOCIoN2h9/UeWbeKbbd/e+3lY7zFAepKdzbHv+nLyvSfpmIRnnnH2lLffpv3/wx+GeWBGBvV5TjedlnUWIqw0cCDVHm6XkXXfJaNL/PagcsJgjBjBzWk5yfPmzbTOvTR7s2jfHkX1GqJbca5349uSQkXL+LZ0ul483599Bpw4gRqX9EZycnSMbxFKTvr1C7KO+khO5nr30kt+QaPt22nxhah8FoyAJqylWBtON7vDnBwmD7Vq5fgp114LPPYYG+Y1bEgD9n/+J0jE3uO6HR/PKblk3f7uO3ocI5lnA4zv48d5KUVTcgKEr/Pd0BjzW2PMZgCrALwI4B0Au4wx7xpjvFQvrlpYui0v+IzvnKIeETUMRN26XKECDnLkCD0ckyc700waw3D9kiXAyTxvSRuWXR100ujfnxem092GJYZ0OJIfeYQhq8B6tjt38sK+5pqAtunJyZ50r9bplJGeFBQwpBnJdtfPC2h1I3XdXAfgxNSypccOFw6wZmg3RuWXX9ID7JeF1bQpy9FGZHwXFZUUCz99GnjzTXpPGjcO/rQ6dYC776YMqUzSUnExF7AI4oW5ufwa4+JQGnZ1q/u2Tipanm+g1Pj2kAaTnk7neUK+/5v1iBW583MenDsHfPJJaVtoV2zfDrRsiTvuqYsVKwJCzEF47TV+1GEdddbYdeoJtDanISzR0aP520m+isWmTcCy/clocW4f4k+Vb/ARipEj+fmWq6pWWMgPwdOE48MYHGzVAz2wKbJkS2MCtAYVTHq6N+Pbz6PrNghSjg4deBEFGN+LF3Mpd6Ljv/NOBoY/+sh3w/bttAU85JBt2EBTotyeJyOD87abXC0HyZZ2PPAAN93r1oVJ1k1KYv6YnZctDH378rI8cwYVI7Xq2JGSNJ/xnZPDZaRSjW+wfvduAINFpIuIXCoifUSkLdhtcqIx5vbonmKUSUnhxeMljzM3F0VNmmM/WkR2EQO2FU+ysjjAwklO/LnlFs57Zwq8Gd8LFtCjGFTG0K8fRdnl0rSD8Omn9Ow7zNy55BJqxf7yl7LOqfvuo31Qrr261eY3lFDchsaNOX+X0Wrm53vXe1v4JoGz6zd5T7YEbCudVCjNm9Or4cZqDmKIZGZWXMWTrCzKRcM1ZAG4uNWtG9B0aM8ebn48et2KiwOc+82acXfhVvedn8+FOVreP4DX95EjngT3NWsCXdMFzb7eFJnkBGB0rV27MvPXhg1c7z01l/Il6912Gw9tG7b2o7CQnuDbbnNgd7pNQNu4kRNiCA9gu3ZMev73v50vI9OnA7tq+MaGSw/ggAHcfJbRfYvwgwjSEMUNW2vR+E7q6LEJl2VARiPR2CI9nRt2t8ZbTg4NrdRUZGTwoy8jPXRDjRqcowOcd9Onc8hce234Q0ycyEDFiy/6btixw7NWPieHw7tcXQO3Y/7ECY4lD4uXMWyYVNKcLxjW2uZBetK3L500eXkojbZF4vmOi+N48hnfsUi2BMIY3yIyQkT+n4iUi9GJyDoRuVdE/mn33POGlBRewE5q3AWyaRPiMnsgMTHCHTTASXPLljL10957j85PRxUsfLRtC4wddhINj+5FcZK7i/jcOe7Ax4wJUZjEGpFOdN/FxfR8O60z5uORR2h4jxzJn/792UXrd7+zafqVksKFx6P0ZNUq34J5+DAF55Ea3z7P+cFFm3DmTATG99at0TXcAPdJl7m5nF0DvAyZmSXRXPcEVO14/XWO+REjwj+1aVNGQsp0WLO8UB493zt3cjEu47Xp1cu95zs/n+cQSRWkcERQbhAAhqfsRr2i7yM3voFyzgPPem+gpExdw4aUkbz9dujp+fXXOV/ddJODY7dsScvI6YRtlWEMY9XfeCOnbyeX07ff0lDvMtbb91ezJiV6ZXTfBw/ywJGUvPWx+mQGGuAo4nfv9HaAaJYZtLAiSm5Dzjk5/D4TEkqucS9Nm0uwnHc+9u9nntbttzvbe9SowcdmZdGH5LXM4MmTlGIMGmRzp5WjU0ZjGYLcXK7dnhcvB0RQbtCKyq9dC1rgDRsygTMSunYtcSiuXk1neIjeSRWCo1ijMWaRk9vOS7xWPCkqAvLyENczA127VoDxnZZG62UPOzcdPw7MmwdMmuQ+IvzT0dxN5p905zlds4aJ22PHhnhQx45cvJwY3wUF1GQFrf1lT8+eTKg7cYKGUMOGlBjcd5/NgyMoFzlgANerrVtRajhEKhOoXRvo3Bmn13FG91Tp5NtvOYtHuhEIR2Ym37fTqEFuLheGALdGz56cq93m9ACgrrBuXWD7duzfz0Xo5pud50pOmsR9U0kIPkK9aZlkS4vevUuTIZySnx9dvTcQtk56OC5vxDd7sFUFGN/p6WWcB9nZnNJcN1C1HCE+A+TuuznVBkvCPneOfRDGjHH4WsY4T7o8e5YLsgPx89VX05D697/DH/bVV3nJXXW/b3728P2NGMEhWaIkCNMK3CkiwLy9vvHg6YJGdMsMWnipeHLuHHdHPqPSaxW+MqSkcM7xjfu5c/nn9dc7P8SPf8zP/c3p3qvErFxJB8SQITZ31qvH95yd7exgEVUKcEiS97HfsSMdL2vXojTZMhKpFcB1/6uvgO++w+rV0fd6A+E137WMMU0AXORLuGzi++kAwLkSvyrj1XjbupWxjx49IteOAeUqnnz0ESdoN5ITi+HtOaDfWe/uIs7K4gZ55MgQDzKGI9NJ0mXYwrvB+etfue6tXEkPzzPP2JQ+BEoneQ+6fSuisHIlSjOdK8Lg7dEDdXfkolGjsPmS9lgLaQV4sUKSmckFyYmESITfuU3imVdZNIAy5QbffJOn40RyYjFiBG33mTN9N2zdypi8i0Qhf3JzudktYzf36sX371Rbc+IEw6nR1HsD1JzGxXk2vrvDl7NyJgK9pEVaGiesXbtw9iwvfU9674D+BJ07U4b217/aeygXLGDAKmyipT89e3LMB+mrUMKWLbRoHBjfTZrQafGf/4RogAPe99xznBK7X9qQ7jUP3581R5dITypozti5E1h9ohvEaymQ/ftpQDqtneiVpCTudtxUPNm6lV4tn1HZti2b1kVsfJ8+7XNbs3Rg27bumhN16MASkp++5r03R3Y21+6gQeahQzl/H3eQX5CTw3FpV2qqomjUiHobD7XajaFTa+0a4aRQEU4O3zEOfpKPPXuqgPEN4E4AOQBSfb+tn1kAnovuqcWItm1p1bk1vi1LIzMTGRnMdveiXCnBmjR9g/GDD+hg9mC3IvFLTuYvL0l2VUkuK4uZ+6ES3QBQ911YGL7D2Kef0giKZuJN/foMJXvwfKemcg5YuRL0oNSu7SnLvBwZGWh29AsMzjjibUNuTUix8HwDzmLl27YxKmOjI2jblj9W3XHX+Izv119nSNHN265VC7jyShrf586hNOTtMYEwJ4fraZk8V8sD5FT3XVhIYz3axndiIserx2zv1oc2YQc6IWdrmNaKTvCbv9au5RrvWe8NlDFAXniBVUUnTiwvb3/lFXrBrK61jhg6lHkBn3wS+nFWpqfDsh833kjnWSgH47x5LO4zbZrvBo8J4+npnFpLpCdbtnDQltPluWPjRuAE6uJ0m2Rvnu+IW2M6JCGBBr4bz3eAR9cKgnjOVwHKOO9OneJmaPx4947YW28F6h/wXmZwyRK+LZs+UGToUEZynBSkX78+gkoBLgjT2TsUffsCh/P3MrEkkmRLC99cvfsjbuYq3fgWkadFpCOAX4hIJxHp6PvJEJHp0T+9GBAXR/eKW8/pqlUM56Snu06gt6VZM1q9W7aguJgX8ZgxHksV79iBooZN8PXpxnj7bWdP2buXk9CVVzp4cP/+NC7CaciWLeNWPNoXcZg2v8Eo02zH6sJZAYU9z6ZzQIxp7XFAWBuBSMq/OaFDB+4+nBjfi3wqszKlZogx1KAuWeItbxnJySje8Tnycotw223unz5pEh1uq1aB48Cj3luEY8HqWlhCixbc4Dl17VveuGjLTgBvNXx9JBZuwo46PbxXqvHHMr4LCkr03rYh8HDYNJdq3Zqbq337mMR29iyN3GuvZV7Mbbe5lNYPG8bra3ZgQ+UA5syhhetQyjFuHI2fUNKT6dN5yB/8wHeDx3KRxrBM3aJFPsVDYSHnwQjnLyvyk9DHYzjXVrcVJdxWPFm3riTZ0qJnTzpPQ0UrQuInW128mEGv8ePdH2bCBCC9hm8cuAyXnjhBFWjI623QIBoT4aQnJ09y/oqm5MTCMr49LBp9+wJdi33zXkUY3+3aAfXq4eTaPNSoEVnDX6eEk51cCgAi8myQ+xsYY2KwwkSZ9HT3E82qVfQAx8dXjHbMGE4KhYXYvJnl9q64wuOxtm9HfEoSMjI42TsZ2/Pm8XdIvbeFk06Xu3YxFOfFde+WgKQXNwwc6OtomF9QYZ7mLbXoKetfy+OAKCjgWIhmhX+AY65nT+fGd9u2Qb0yQ4Yw+uPJCZucjLiis+gQt9tRhYBArrySxtd/3z1Lt6JHvem2bcx5sE1w7t3bnfGdmBjdajUWGRkc+y6r/eDkSWDrVhxuX0HGd9OmrKBTWIjsbJ5W06YejrN9O48T4MLr3x+YMYO2w7hxvDxmzwb+939ZgtQVtWtTtzFnTtDJMe7kSWr/XCTd1KpF7fcHH9h/HStWsAbyXXf59Sno3JkRJbffH/gWDh3yXb5btlSITG3jRk6nCb08lgLJzeU80aRJxOcSlvR0Xu9OM71zcjjf+Xm0MjL4dA95f6RFCzrhPvsMc+ZQAudl01mnDjCk7XbsNW1wOq62q+euWMENacjXrVePFms443vTJu5EPJQZdE1aGj3XHiQDffsCGfCtrxWRMG4MkJ6OWp/nIyMjuoV6LMLNKpONMSuMMQ8ZY8YaY/oZYy4zxvzIGPP/AMwF4G6kVEX692fmyldfOXv8iROcZC65BADnGS/VyMqRlgZs2RLKyeiM7dthkpNx3300LOfPD/+UrCy+B0fOusaNOUOH0n1HoPd2TUpKaba/SwYMAOrJEcTt3VNhMoGVX7bGITRB8jGP8czCwujrvS0yM0sn3GCcO8dJ+4orgkYxLr+cv8NF8u2QZBrL12Rut23FHI4GDXhqOe/vZHaeR8+3FZEt5/kGuBht2eJMM5mfT+vQa7tjN/ToQden225/+flAcTHievbAjh3uG3jakpaG4oJCrFzp0esNhKz2cMstTLpesICLb14e8OCDHgvKTJjAOT9I1KDpmjWekm5uvJH26vvvl739889Zuz4pKaD+s/VePZRcs5wzS7KO09lRAWUG16/3qWwsg8ZpSVmLjRsdy3QiJj2dmycnUeuAZEsLy8Z0WgikHMYAKSmQrVsxZw43RF4Nt261tmObJJU4wpyyZEkYvbfF0KF8o6E2VLFItrQIkNq6oWVL4JLaufimbnuPXebKU9y1G9p8nx8TyQkQXnbycwBjAXwF4BoAjwC4D0BnAC+KyGUi4nXYVh18RrTjtuk5ObyYreeBi4HnC9giNRXYvx+r5n2H1FSP+Q5nznAiTk7G9dfzGE8+Gfopp08DCxfSg+hYIdKvHz+vYG71ZctoFcUi9G4ZWx6b7XSFL3RZQZ7vnPUG+QkZqP+FB8/3sWM0CqKt97bIzKSREWoB27iRG5sQu8GkJI61co0/HLDpBA2QCWnuvz+LSZOAWnsiq3SyciVVOLY2TO/epUXAQ3HuHA8Ui7gl4L5pjIXv8U2HZnh6ui1paSjOL8TJk+K2umgpYUqtPfkkz/XjjyOsZjd2LCe7INKTi5YuZdKNyzdy+eUcPz/6EeuTi7Aaz9ix3BdmZQU4hSMoF3nxxT7n73xf1C/CDfv+/QxW9u0Lb6VArHkkFpIToHSOdLJBCEi2tOjWjU7hMs3W3JKSgjN5W7F3L/d0Xml0aAf21ErGG2+4e152NhMQ64dL3Rg6lPPTsmXBH7N+PUNWkXR5dkoExjcA9ErYhFxU3Fjb37QrmssBXJb2TYUdMxRh42ki8p2IvCQit4nIKBG5SkR+IyIhvsHzjMxMxgGdGt+rVvG33xapXz+Grjz0uyjFNxgPLtvi3eu9axeNhORkJCYC99zDmruhvPKffkqbz5HkxKJ/f4aLgnlsPv2UOrOgBcMrkAjKDTZoAFzR2md8V5Dne+1a4ECrnjCbN5dpm+4IS7cRS883EFp6YoViQpSvsHTfn3ziXsL3xuKWOIHayKzvrWoHwEWvi/F9/x6N7xUruJ+2VRk4Tbpcu5YbFavtYbTp1Imxbreat02bgDp10GUM9aURR+0AID0dCUcPowW+tq83HI6TJynBCGFVx8VVTGUxXHwx5zA74/v0aTRdtYquapfRi/h42jajRwP33ssk0cmTuTZ88IFNEZAIa7Vfdhlwcn3FlBm08kv79AGNr4YN3e3K8vNp3MXK+O7ShefoJNPb8ugG1H6Nj+cwcJKHGJSUFCTu3YlaOOUsZ8qOo0dh9u9Hg8xkzJkTvpaBxfHjDEA7ijQNHEg7J5T0JKK2zC5p1Yo7Bi/G98mTaH3sM6w4nuGqqEQoNp7l+j+ggcsookec1vl+yO4n2icXM2rXZqjMMqrDsWoVFz2/ro2WHR6R99s3eXY4VRiR3htAid506lSO71De76wsFnxxVRpswgQuTIF94AEKEQsKYiM5AVhNJT7eU7lBABjcJB8nUQvF7TpEfConT/pKovXIYEUFt974gor1woclNZVx0lBjf9Eink/LliEPdfnl3I+52QOJAG+/G4f99ZNRa49347tZM2Boiy34Pr6xp+4I339P28FWcgJwoWjRIrTXCGDyRFxcmHqdFYhljbp1XefmAt27o1WbOLRsWQFRO6Bkwzi8ZWG4oWKPtZGPdoMWiwkTaHHu3Vv29o8/RsKJE97qvIKOw1mzgL//nbLxxYuBl14KYiA1bkxXuIeoHUDju/2pLRCrcEAErFtHm6tXL/CPHj3cbeqsx8ZKdpKQwEXr44/D7/j9OlsGMnAgT91Dp3OSmgoDweQe25w2ci6PT3SeOi4Zp09zo+aE5cvp33FUWahOHXoXghnfe/dy8bJyuqKNT2ftyfjOz0ecFCMXGRUzdwHIPkDju81hl1IrjzjN6Dru93MOwBgAHaJ0TpXDJZdwBQqX9myVRPCTnAClm0Un5a+D0rEjiuITkYYtkWkmgZIFrGFD4M47gXffZQ3XQHbvZoe4K65w0BLWn3bt2FJuxgxmh/pjGSixMr4TE2mAe0y6TJcCbEEqPtseuZd+40ZOhk2G+RYgtx7JwsLStsWxICGhtECxXd3j06cZxXAQirHGrBvd95o1VNnEpXgrueZP7zqFyDuXhq3b3HttLAVV0G6yxjDTLysrdHLcvHncicci4czCMpKchhxEaKz7dL39+lWM8S2pNL5HtPUWRrYrMxhVLI3A3Lllb//gAxTVreuxUDkxhlHHdetYqSVk7XqPFU8AGt9pKMThJp2CNEJwztq13D/Vq+e7ISOD48Sv63JIcnP5ZE/NDTwyYgSjveHmDptkS4uBA/kWvV4DB5py3E9O9zjugZLvv/OYZHTqBMfSE0vv7TjSNHQow1x2SR7PP88PwlXR/AjxWm7Qt67mIiMym8uPjza3xpEaTWA2Rdq0xRmOjG8R+T+/n0cBDAEQxQrslUD//ozhhEtc2rOHiZkBxnf9+tzERTQQ4uOxq2YKLmlUiEaNPB5j+3ZOgH5b8Hvu4WLwt7+VfWhREROEzp6ll8Y1v/41vbtPP1329pkzaRB7au/okQgqnlx8MA/56BpZ6NGHNYF3uSqNRrRb47uggB6sknIIMWDqVOql7NwtK1fS2HQQiuncmc5hN7rvt9/mUGkxKJmeT881v4AWhwtRiLTShjsuWLGCTuSQTp9rr6V7LFgG8zff0NoaM8b9CURCRgY7yQZ6cIOxbx+lMX7G99atiDh8u+NkK3yPBuhT20X5N39ibXynp9NQnDOn9LazZ4FZs3Bw4ECPmZxl6dGD6pWQdO7s2fhu3RrokbgF2xIik6mJcOharbsBcP4+dsx5Mu/GjXzD0a7S5E+5bkM2nDljm2xpYUWtvc7/c7Z2QTEMBjX2OO6Bku/fJCfhhhsYLdm3L/zTsrP5nZVsmMIxZAgNbKsggsXJk8CLL1InFcvNU1oa7Sm3k09uLlC3LmqkdHKsFg7FsWNAfoHBN216Oav+VQF4vUrqAIjhNxQDLGM6nPTEuj/A+AZ4EYfKQQzH0aPA+pNpSDPemmYAYPgqObmMZqtNG+CGG9is4plnSs/vkUd4Db7wgsf1LjWVodnp00t30q+8Qlf6tGmxqddjEdDm1zEHD6LG13uwtXbPyJJufKxZQ4VC646JnFjcdnAoLIyd5MTiiisYOZgxo/x9ixZxMbXKmYTAGM7tTnXfxcWMyIweDdTs1pmL5J497s8fAA4dQvyhb3CsbZrjkK0/K1cy+SpokwqAXqOmTXnSdsyfzzcea+PbqkzhVHoSUIvZMrgsza9Xlq8wKEQa2p+MwPPdpImDLl8VhDEsyrxwYWkVm08+Ab79FgdjFbUDOPl++WX4jpt2FBWhU9FWrDqc5nrq82fPHiZclvGXuClhJMJxFSu9t0VSEueukm5DNixYQOtq1Cjbuxs35pTr1fie+VFt7E7oiGaHIvB8b9tGh1n9+iVRknAOsaNH6exx1cxqwABGSAKlJ2+8QbnoPfe4Ou2I8Zp06ZPN9bskLiKby2LdOp/p0CuT0pszZyI7oAOcar43G2M2+X7yAXwG4Olwzzuv6NSJ2e1OjO+aNW0nmX79OH6/+MLbKSxdChRKKpp+/zk9yl7Yvt1WsvC3v3HuueceqgzefZc1cm+9leoRz/zmNzS8n3+eArS77mIo8M9/juCgHujShSUgnbgL/PHtcou6Z1aI8b12rZ/3qGdPd57vU6e4eYpVsqVFXBzw4x/TZR2om1+0iG/IYTmnyy/nV+DEkbdyJRf9a69FqV7Vo27fSlRtPSwVa9a4s+GLi3lZB9V7WyQksDvK7Nn20pN587iAxqJGrj9WkwmnY80S9/qeZxlckYZvly0DdiSmoe6XERjfsfJ6W0yYQKO3WzdaMffeC9Spg2/LuICjTHIyrQcvC8cXX6BG8RlsOJXqtWgEgNKNV5m33aEDG305CWXt3MkswVgb3wC939nZjFrY8dZbtLBD5GEMHMj5yO0G5sQJTpHH2qTBRPIF+JWXTU4Grr8eeO658opOf156iYFCV019atWiRuXNNynXATj2nn6am3gHTpYKxYvx7bfR69+f/SWst+IVy3vebFQvGt5umjd5xKnnexyA8b6fkQBaVZsOlxbG0JvtxPju3ds2JGmFrL0uYgsXAjtqpMEUF3vTv547x9C9zQLWpAlthunTOU9dey0fNj3Sb7FXL7oun3qK9d7at6eWIBY1jv3xWvHEZ3w3GZ6JgoLIQu+HD/PlSxawjAyG1A4ccHYAy3Mfa883QJ1fQkJZ7/ehQxzMLkrvuNF9v/km97Hjx6P0PbutV23hm7z73MzJ/L//df7UggLaDWGNbyC49OTcOd42alRsw+4AN0YdOjj3fH/yCcemT9vmpGy/E5YvB051Sof5+mvKYNxSGcb35ZcDf/wjJ++zZ+kB/8lPUBzLqJ31nr3M+b5NZyHSPNXYt1i7lpd/OdvZaQmjWCdb+jNyJN3AdvqDEyeY/Xr11SFlRAMGcMi6XT4WLqTPpG7fND7ZbXUrgJ9tfn6Zalu//z339//3f/ZPOXYMeOIJBi2D5qkE46mneNLDh3N9ys5mucZ7741NlRN/OnbkIuDG+N69mw4/n/ENOC9UF4zVq+mzbHCZg+pfFYRTzfcuv5+9IuJhhJVijGlijPnYGLPN99s2zmiMOWeM2ej7CdMLuALo35+DIJgFduYMEzdsJCcAnSe1a3sfCAsXArUzfdnYXloF7t7NBSTIAmYMGzysWwdMmcLWzI61YqH47W+pGT51ihZ+rMLG/kRifLdrh8zhTJCL5CK2vEclumG3tXKt3XasPd8AxdpXXUXJ0KlTHEtDhnDQuKj60KULq7iFa6T2zTfAq68C113nk3o0b87Ik1ePw5YtQK1a6DikPdLTnVcLAErDzY4WsWDSk3XruFmJteTEwmllitOn+YYDMroj7VNw6BCnzgb9PIaRT5+m9CLWxnd8PPCHP9BhsGwZvc9//WtszyGScoO+z/lIy1RHFfeCsW4dAyHl9hxDhnBuD3dd5uZyrohFX4dAhg3jhtdO9z13LjdUU6aEPIS18XYrPZkzh/NXmxHpHMNeohf79nH37+d0SU2l93v6dPvyxZZX/OGH3b8cMjIYpfv6a1rvjz7KClFhPqOoEB/PtdvNfOEnm+vRg2O2IozvSy4BI7D16lVQ7dXQxNhFU8IDABaJSGcAi3z/23FSRHr6fiIoX+8Qy6gOtgrl5vICC2J816hBR7AXD9JXX3HzmTzWVwjWSwjLMjzDLGBdu9LraEWrI2bwYOAvf2EliMowHAEKrevUcS9b2LAByMxEv36cvyORnljDpkQ36db4LizkSXjs0BgxU6fSinr4YY7xL7/kJO1CRmEVBfngg9ANY59+mjb+b37jd2PXrpF5vrt0AeLjMWkSnXVOa+6vWMG1x1GBmWDSk1iXGAwkI4NjP5xcbc0aPibA+O7XjzaA05zNQCyjpf1oj8b3zp0l/V2yfjMAACAASURBVAkuOJo2ZRTCi/G9ZQvQogUyhzbC0qXetK9WsqVtfrxT3bfVl95VyawKolEjDmA73fdbb7FE6mWXhTxESgqjw26M7+Ji2vajRgEJ3SNoGBOkvOzvf0/H/VNPlX34kSNcbseM8eD1trjkEu4cPv+c2Z133RXbHC1/3FY8sdbT7t1RowaFCJEY33v2cO7r3x+cw3v2rDqe7ygwEcDrvr9fBxAuHzw29O1L6yGY9CREsqVF//7cNAWTnwXDqnY16gd1KN3w4vnevJm/K8P78Mtfuu4GV6FYdW7deL6PHePje/VC/fr82CKpeLJmDU+hxPF/0UUsR+DG892xI8MnlcHw4cx9ePxxeiSWL3dU5SSQBx7g+P/LX+zv//57enQmTw4ou5ueTuPbiwVRWFhysEmTuDAGaV5YjpUruYg5jrjaSU/mzaMB0LSpu/OuKJy2mV+yhG80IKHQitZ49X4vX07nQ/fxHRhGdhvBiHWlk6qEMd7LDfrG/WWXcbPrKxftis8/p+TCVubesSPQtm143XdlJFv6M2IEJ2B/udPhw8CHHzK8FqbZW1wcl3U3zpecHDqPx49HZN0arWs2oMlbWhqnmmefpU/E4plnWKzoT39y/1JlGDqUXpJhwxgSryzS0hgxCFXC1Z/cXHpKfC09+/fnd+E1R9Iy3Et6JmZmcjMZQeUtJ8RYmFvCxSLyFQCIyFfGmGCl6WsZY9YBKALwhIgEVXIaY6YCmAoAzZo1wxIvfa4B9G3fHqc+/BCbbbLde/zrX6jTrBlWbd8edJarW7cZTp3qildfXYeUFOdV+197rRsuvrgeDh5chUMXX4zEtWuR4/I9pC5YgMYXXYSVTtrtVkPSGzVCvdxcrLH53I4dO1ZuTDTIy0MvEWxOSMChJUvQoUNnLFp0MRYvXuZJtrts2QBkZBzGkiWlE3D3Nm1Qc/lyrHPwXfZZtw6nWrZEnsexWxE0v+EGXLxwIT67/36cOXjQW794ACNGdMHzzzfHpZeuRtOmZWfFN95oh++/74QRI9ZhyZLSa6RVjRpIOXIEK997D6ddNMqJO30ag3fuxM7LL8euJUsgArRo0R8vv3wcnTqFvhYKCrj/uuKK7ViyxFmWpomLw8AGDXD0scdwMDsb9bdvR4u1a7Hz1luxq5K+u9qnTqE/gC3vvIOvjx4N+riMmTORkJSEnAB9+JkzcYiPvxTvvrsbjRq5D51nZWWic2dg9boN6N2uHYqWLEGui8+i7dy5SAKw/MABnK3E8W9hN19Ek7QGDdAgLw+r3bymCAbl5eHAsGGoVWsNgH548cUtGDv2a1evvXhxcwDpAMpejxapqalosnAhVmRn2+5Q448dw+AvvsDnw4bhy0r67ho2a4bM4mLkPfssDvq83C3mzUPqmTPISUnBUQfn1bJlO3z4YSfMmbMM9evbK2v9x8Urr3RAXFx7NGiwHEs2FGHARRfhu+xsbPHrfO2ElIUL0axBAyzPzy+3aR01qg7eeacvBgw4hkGDDiI9/Qj+/Od0DBz4PY4fz/M6PZdSuzZd7IWFntu8R0ozEXQVwdo33sBxB5vvfqtW4XjHjsj3vfl69Zrh9OmuePXVHHTpEnzuC8a773ZCjRptcPjwp1iyRNCiTh2kHj+O1W+8gZPt2oV9fmOvZaJEJCo/ABYCyLP5mQjgcMBjvwtyjFa+350A7ASQ5OS1U1JSxDM/+pFI06YixcVlb1+5UgQQeeKJkE///HM+7IUXnL/kiRMitWuLTJvmu+Hee3nDuXPuzj0zU2TUKHfPqU48+KBIfLzI6dPl7srOzi7/+OnT+WXt3i0iIq+/zn83b3b/0nv38rl/+1vAHb/9rUhCgsipU6EPcPasSI0aIr/6lfsXr4Js386v4mc/K3v78eMizZqJjB5t86TsbH6I8+e7e7GNG/m8t98uuennPxdJTBQ5ciT0UydP3i01aogcOODuJWXqVL4mINKkicjIkSJffOHyIBVIUZFInToi99wT/DGnTonUqsX5xYZevUSGD3f/0idP8rO+/37fDXfeKdKwobv56/rrRdq2df/iUcJ2vogmv/udSFxc+HnCn127OP6mT5fiYpGLLhK55Rb3L33//SI1a4qcORPkAS+/zNcpKLC/f+lS3p+V5f7FK4ozZ0Tq1xe5+mqRo0d524gRIp06lV/Lg7B4Md/Ghx8Gf4z/uMjIEBk82O/O4cNF+vZ1f+6DBgUcqCwvvMBr05jSKWfDBvcvU2XZtIlv6q23wj/22DF+EH/8Y8lNO3eWXAaeuOwykf79/W6w1pM333R2gClTBMA6cWkjR012IiJXiEg3m59ZAPYbY1oCgO+3bTkIEdnn+/05gCUAMqN1viUMGMAYz6JFZW9/+GGGlMOEZzp0oH7Uje578WJGXMaN892QlsYbvvzS+UGKihi+smr+XoikpDBU5DT2umFDqTQEpfo5L7pvK1xfrklLz578bsJJT3bsoFajMiqdRIGkJJaxfPHFsjril19motCDD9o8yQq7utV9Wx4bv3yDSZMYhgxV9eTkSWDBgosxaZKHjvRPPMHe4bt3U1w+fz4v/soiPj58m/kgem+Lvn396t26YO1aftYlXfb69KG2yI0GIicnaBOUCwJLNmRJB53gVx/QGMqzFy92r9pau5bTVNC+XuF031a+Q2V+fzVqMEPxvfeY8X3ddVzDr7/esZ6sb1++jeXLwz/2yy85pZcp85eWRrmomy/AptJJIHfdxcvj0CEWbpk5s3KKykSNlBR+8E4875s38zPzkzi1a8ev3Ivuu6iIl1GZYEV6OivjOE26tDp6u6SyNN+zAVgNd28FMCvwAcaYxsaYmr6/LwIwCED0iy9OmcLBcNttpUKrNWs4wdx/f9jyIMbQAHMzEObM4WFL1sRUDxVPtm7lCnghG9/W5O/0w/clW1qTc3IybXGvxnd8vM2kePnlPP5HH4U+wOLF/O0yZFmV+d3vuBf605+Y0/Czn7Gq2+DBQdIDmjXjF+DF+LY0/z4GDuQc+uijwat/vf8+cPRoDUyd6u7lAFDYP2oUO1jFujxXMHr0CK1VDKL3tujXjzaz24p3H33EsV/S7MMSDzsVkFsveiEb39Z7z8lx/pycHCYA++b8MWOYPOa04iTAobJ+fRC9t0VSEh0UdhqH06e5ox4/nhZQZfKPf7BZxi23lDrPbrzR8dPr1eOl8dZb4TegVo5WGeM7PZ0lD91kLX/9NbXpDpwujRuzLH3YjqnnGzVrMtfIifFtDW4/49uY0gaHbsnPZ1JrmWW3Rg06MpwkXX75JR0wHqgs4/sJACOMMdsAjPD9D2NMH2PMy77HpAFYZ4zJBZANar6jb3zXrcur78AB4I47uMt6+GGmQk+b5ugQl15K6ZaT70SEF/LIkRyDALwlb1gekworYXIekpbGzHcnroszZ/iZZZYGU6xS716M7zVr+NHXqRNwR/PmtGqs2ToYWVmcgLp0cf/iVZSOHbmHfeklLlIvv8xJ7oUXQjzJS8WTLVv4Yn7Z+nFxNLw/+4zVE+148UWgdesT7jrEVWVGjmTCWbBuf0uWcNFq0sT2bq99CubNY9TIVzac32GtWs6Nb2uRu5CN744daV25Mb4D6gOOHcub58xxfojCQuYO21Y6sbDc6nb1vt99l6GsykzYs4iLo/X8wgvMPt2503Uk8c47mYAaGPgO5L//pbOmzHRtrdtuko2DVDq54Oja1Zmxm5PD2o4BUcb+/el/dNtewCqwUM7n1asXd6Xhohgevd5AJRnfInJIRIaLSGff7299t68TkTt8f68Qke4ikuH7/c+YnWCvXgwr//e/nFSysoD77ivJrg3H1Vfz99tvh3/sxo3cKJfZQV90ERdIN57vTZvoBSlTPuICIy6OsW8nF0RBAWUemWWVTIMG8WMPVSYvEKtUV1Dv0bhxNES+DpIIdfIkPd9jx1YdL2oF8dhj/Fm0iBn68+aFjLDyzoICd6Fbv+5w/kycyEn1j38sn0hfUMBhMm7cV9XnI58wgZu9F18sf9/p09xVhuhgl5ZG34Obiidff801s0x584QEXldOE5Esg/NCNr6N4brj1Pi2qQ/YogU3UG6M76ws/h42LMwDhwzhlx1YTeq55xgpdtGIKybUqMEqLS6ZNInL7z/+EfwxGzawpPgttwRM116cZkEqnVxwDBvGaj+ffx78MSJcQIYNK7dOWsazW8fBggUsLtepU8AdvXrRkg8n/V22zLFdGEhleb6rPvfey7DyCy/QpePQ6w1wR9y3L/Cf/4R/7Jw5HEdXXul3ozG8kN3soDdtouFd4j6/QLGs53BFnq1ddoDxbRkR8+Y5f8kNG3idlmheA7HE/MEOmp1N69ByXVUjmjVjLe9hwxyWke3alYVsnYZuz52jQWBjfBvDPfSePcDzz5e9b8YMrs+jR7urDFGlSUwEfvQjRlkCP7+1aznGgui9AUpHevd2V27TUlOV6y3Uty89R046/uXkUL7TPFjRqwuE3r0ZjTt9Ovxjd+7kbjbAZT1+PA2QYPv8QGbNop0R1k61wkNPPVW6MV6/nuV3f/rT2Hd1jRI1a7LZ76xZrP1sx5/+RJPg7rsD7mjenE4zN8Z3QQEjHpUt2alsLAMo1MK7eTPlBCXJcaVYVaLdSE/OnqVTaPRoG5+XZReE031/+qnD1sjlqR5XTDSIiwNee41xpYcfZgtnF1x/PdeUcPrJOXModSi37lhhD6ftajdvvrD13haWmDicBbFhA918fjphgB9h69YsD+uUuXN58QZtbpiRwYMGk55kZVGvEsIrecHgts38F1/QWAnS3GnIEKoxHn+c0mKAGr9//YterkaNXBbkr+rccQc3JK+8UvZ2S+8dptnIyJGct5zufebNo8e1XK5D3778oJ0YIhd6sqVF7960CJyUirWiCjbGN1Dq0Q7F/v20nSdOdHBuycns5TBjBi8mgF7vunWZWV2NmDrV/hICgG3b6mHWLODnP/eTWVlYTjO3xnd6erWLeLomOZk/oRZea/0s46kkDRpQgRWus7I/K1dSoj9qlM2dPXrQGxFKCvPdd7xWPfY3UeM7FC1a8EIqt8UNz3XX8XoK5f3et49zaBnJicWAAVy8nGTPfP89sGvXha33tujThy7NcLrvDRtoFAd4bKwoxIIFzov2Z2Ux3BvUcWcMvdp2BxXhAYYPr7wOY1UJK/zqNOpjSbNCyK0ee4y50yNH0khs0oTzpqdEy6pOUhIbjrz0Umni5YkT1OZ27x5U720xaRJ/zyqXAl+eoiIOaVvPkWUUhpOeHDnCyIUa3+6SLtetY6QjoKFajx70YjuRnsydy+nHkfENMIx0000sVfR//8c2yTfdZGOFnt8kJ/MSmjGjfO7y6693QKNGTB63xU3E2kGlkwuKMWNKo8B2ZGXxGmnZ0vbucePoiP72W2cv99FHVMjZSq5q1+aaEsrzbTn4giSwh0ON73B43JG2bl2aOR1Mvvruu/xtE0UpDWU4yf6zki3V882Lpk+f0Lrv4mKK7TPtK1eOHcsdsRPp+P79jOiHVYyMG8fMpqVLy95eWMiNUzWUnHiiWTP+OPV825QZDKR3b+AnP+F31aoV99KzZzvQuZ6v3Hknw7Pz57O04FVX0UPz0ENhn5qWxmDfBx+Ef5nVq1mowTbik5JCd1Q4AbkmW5bSqRMNWafGd0YGDXA/jOFU8/HH/OpDMWsW9a6Ol424OOCf/+Qu9he/4AtUhUTLKGBdQv4qiA0bgOXLL7L3elukp1PyGE72CLCow7ffarKlxZVX0vC2K2l58CBtIVtjiUycyM2S06j1/Pn0cQYVNQwcyPX6xAn7+5cto6MvZKmg4KjxHUWmTKFtYFe69fvvWY3hssuCdINv1447PCcCTMs7rsY3ufRSLk7BVp/t22kIBzG+hw/nmuYkdDtvHjdXYW1ny7MdeFDrf5tQ2gWL1WbeCYWFjFCF8b49/zxlsh9+CDz5ZJBoU3VhwgRqSKdPByZPpiX2yiv82wGTJlGlEs6DZJV3HjHC5k6r7nM441uTLUtxmnRZXMzHBClRMn487YVQIfjjxzksJk506V9KTGQt7YEDaQhV02jrhAmcVh5/nCVJV65kI8h69c4G93oD7pIutdJJWS6/nGukne7bWmhDGN99+tBkmj07/Evt30+n9ujRIR504430ws2caX//smWct8qVOHOGGt9RZPJkyobeeqv8fY89xs3c3/4WZPIzhhOcE+N782YaH75mMRc8gwZR3hEs5G1dTEGSz6ya606M76wsXvBB7PhS6tRh0lKg7jsrqzRWrBA3FU8KCi7sCj921KjBxMt587jbePFFV7rcH/yAHqRw0gWrxGDjxkEe0KcPO5GESiDMyeG8daEnnFlYSZehNG87dtB7E8T4HjqUUuxQ35/lGZ8wwcM51q9PwyNUB6vznBo1KC1ZsYLVywYO5FR99dV7Qu/zLUPaiW5fK52UpXZthiPtjO+sLO6GevUK+vS4OG48580Ln7P88cf8bav3thg8mCUN7WrVnjrFzGaPem9Aje+o0qwZcMUV1H372xGffw78/e9cD0OMJV7xO3eGr3u3aRMNuAs9acPCkuzY6b5FeDENGkR9bBDGjmWN6FBN+s6epebVcYXAcePodbfKdR0+zEVMJSdlcVrx5MQJui/KtRVVMHUqN3TPPuta3N6nD4uPBHP4AKymsX59iCRjgOHYs2dDd23UZMuy9O5NwzuU8RYk2dKiVi1GIyxNtx2zZtFfEyb/NjjG0LNUjfnNb6gM2biRe9g33gCmTAnTvKNtW3pjAuWFdhQUUPMQRMN8QTJmDKtUbN9eetvZsxRoX3ll2Ko6EyYwqB0u8fKjj2ifhXSaxcWxnuTChSyZ5U9ODq9TNb6rLlOm0H5+6qlSZ8avf02h/6OPhnmyk37nIlrpJJBmzShctRNtr13LkGAYT6BlD4fyfi9bRhvRse1sPfCHP6Rm8oEH6GJU47ssTtvML1/OibnadMmpQDp0YC6BixKpFsbQ+z1/PuUJdsyfz99hjW8geATq6FFNtgzESdLlunW0sEPIFcaPp2bZapzrz7lzNMyvvDJES3kFAJeSjAyO8xtuABITw7S+NIbe28WLw0fuCgo416nTrBS7koPLlzPSE0JyYjF8OKM+oaQnxcV0mo0a5aBC5i238Hv897/L3m7ZFh7LDAJqfEedyZPpXfjFL2gP/u53lMz9+tdM/gpJr17U2IUyvnft4iKmxndZLr2UF21gn+DXX+fCde21IZ+elMTvK5TxPXcuv54rrnB4Tu3bM0Hp4EGW6XrxRQ6CSy5xeIALBKflBhcv5i42Au9DtSaCRf0HP2Bk1arj7U9xMZuQtG5tU2LQn/btgaZNg+u+N2zgwqbGdylJSfSGhjO+MzM59oNw3XXM37zzzvIbqJUrOQU5rnKiuGPYMLrMQ1U9sSqdqN67LJ06MVnbP2vSxUJbqxaN6tmzg+99NmxgU9aQkhOLpCSuL6+9VvaAy5ZR7tismYOD2KPGd5SpV4/JSx9+yHXo0Ue5aP3iFw6eXLNm+K4XVrJlNU188cygQawn598l9PRpCvCvuspR3fYrr+R3F8z7l5VFbXi9ei7Oa/p06llOnGDsPi+v2odvXdOsGfV94ZL1srMpOXH1BShOGDyY85Wd9OSVV1gf+rHHwniOjKH3O9j3qMmW5QmXdHnuHPU+IfvB0/v3yiuUzf32t6W3nzgBPPIIbZmQyWaKd6xun6F61G/ezB1Qub7mCsaM4cJ7553ANdcAr77KhdZhJ8kJE6hYDFYl0IrajRzp8HxuvZVrttU+c+ZMSlE8a7aIGt8xwGrAsnZtaQ6U4wTZgQPp6QiWQWAZ37YlUy5gLG+of9miuXNpkDtMPhs7llIhOwNkxw5ej54VI8YwySxottoFzujRdLueDdIE58gRXhcqOYkKCQlcxObMKVtp6+BBRu0GDwZuvtnBgQYMoIfPX8NpsX49Iz8tWlTYeVcLevfmvG6XdLl1K0WtYYxvgMUjpk0DnnmGEuTvvqPB8fHHTAVo0CAK566U9iu30/xYzJ7NNaBal13yyJQpdDzOmsW5Iy0NuP9+x08fO5ZOAbteBWfOsDx9r14uGupecw1d6q++yhr3kyZRafDHPzo+J1tEpNr9pKSkSLXh/fdFAJGVK+3vHzxYJC0ttud0PlBcLJKaKlKnjsj770t2drbI+PEiLVuKFBU5OsTZsyK9eok0biyyc2fp7efOiUyZwq9lx47onP4Fz8yZ/IAXLrS/f+5c3r9oUUQvk52dHdHzqzOLF/Mj7t1bZNcu3nb77SIJCSKbNzs8yFdfidSuLXLTTWVv//prkQYNRK6/vkLPuaKo1HHx1lv84NevL3/fc8/xvvx8R4c6dkykUyeRpCSRbt1EEhNF3n23gs/3AsLxuLjjDpGGDYOvNX37ilxySYWdl1KWyy4T6d6dZoA/f/gDL5///tflAa0FH+B3e+pUmbsBrBOXdqp6vqs6VtKlnfRkyxa2dLrttpie0nmBMfQ8dO8OTJ6MTv/4B8MON93kWOaRkAC8/TY7+U2ZUuqEfeghqlcef5wODiUKjBhBb0OwzJnsbMbOretDqXCGDqX3aNs2OmP/+lf2WPn5z10E2lq0oPv1jTfKamAfeIANNf70p6ic+3mNVb0nsCzpmTPAX/7C+0M0lfLHX36ycyenwKuvrtjTVWwYNoxJgnbtyfftYxjcU51HxQk33URlz89+VirVXr+est+bb/aQ7zBtGuWQM2awe3DNmpGfpFtr/Xz4qVaebxGRDh1EJk8uf/t999EN9fXXsT+n84WTJ0Vuvrl015qX5/oQb7/Np/7qVyIvv8y/f/zj8rtqpYIZN06kfXv7D7pXL5EhQyJ+CfV8h2fLFgbXAJE2bUSOHnV5gG++EalXT+Saa/j/ypU82K9/XeHnWlFU+ri46iqRunVF9uwpve2FF/i5ffSR68PNnu1p6lMCcDwuvvqK39Wf/1z+vhdf9LwWKc4oLha5/35+zPfcQ0d19+4MfH/7bcW/Hjx4voOnSytVhwEDqF0WKa1gcPo0K3dcdZU2qAhFrVrA669jW6NG6JyY6KmhwbXX0on+l7/QaT5qFIuVaIWoKDNxIr1/mzax3pfFt9/SoxSp5k5xRJcubCX/pz/xK3Gd33rRRcC99wL/+7/83u6+m7WNH3wwKudbLXjqKXq3f/UrRg1OnaLbbuBAF5lipai0OMa0aMFKJosW8Tv0Z/Zshky10knUMIadjIuKgKef5teQl8flpKqkWans5Hxg4ECGqlatKr1t5kzg0CHXDTQuSIzB3kmTGDf3yN/+xtB7jx7AO+9ofdyYMH48Z9HAzJmlS7kR1WTLmFG/Pi+fwYM9HuC++1hhaOxYJso++aTj6gUXJB070mh7802WNXv5ZTb6ePhh3fWfLwwfTlmof+Ls8eOslDFhgn6PUcYYrtvTptHw/uEPq1ZLDTW+zwemTGHTjMmTSzstvfQSJ2irrJESVWrXZn3cNWu0SkDMuPhi1kAPNL6zs/mFaJmu84fGjVlf9auvWAb0hhsq+4yqPg88wI6J06axruNll1FLrJwfDBvGvIbVq0tv+/hjRq1V7x0TjGG1nwULGK2uSqjxfT7QtClrfh07Rm9gbi51EHfc4aBFk1JR1KgRsq+FEg0mTmSmzG6/ts6LF7OUZGJi5Z2X4p577gF++lN6cdXrF546dRhuyM3lpkW93ucXl19emvhvMXs20KiRNgaLIcYwf7927co+k7JUiuVmjLnGGJNvjCk2xgQtWGqMGW2M+cwYs90Y80Asz7HK0a0bS29s2sSLOj6ecRRFqc5YaemzZ7Pg9CuvMIaokpPzj/r16X5KTa3sMzl/uOYathu9+mrO+8r5Q+PGLCj97LPMb/jsM4qOx4xR3aJSaZ7vPACTACwN9gBjTDyA5wCMAZAOYIox5sLOUBgzBvj731nCaPx4Ji0pSnUmNZXthp94gq1hb7+diUo33VTZZ6Yo0ccY4P33mWiinH/MmEF53BNPcC775huVnCgAUDnVTkSkEABM6BBaPwDbReRz32P/A2AigIJQT6r2TJvGTGqtb6xcKNx4IytlTJ4M3HUXta8aflcuFHSsn7/06gVkZbFgwr//TQnRuHGVfVZKFcCIVYG8Ml7cmCUAfiEi62zuuxrAaBG5w/f/zQD6i8i0IMeaCmAqADRr1qz3O+opUPw4duwY6rmukaZUCYqLEXf2LIororFBADouFDt0XCh26LhQ7Bg6dGiOiASVUNsRNc+3MWYhgBY2dz0oIrNsbi93CJvbgu4URGQGgBkA0KVLFxkyZIiT01QuEJYsWQIdE0ogOi4UO3RcKHbouFAqiqgZ3yJyRYSH2AOgrd//bQDsi/CYiqIoiqIoilJpVOU6dWsBdDbGdDTGJAK4HsDsSj4nRVEURVEURfFMZZUa/IExZg+AAQCyjDHzfbe3MsZ8CAAiUgRgGoD5AAoBvCMi+ZVxvoqiKIqiKIpSEVRWtZOZAGba3L4PwJV+/38I4MMYnpqiKIqiKIqiRI2qLDtRFEVRFEVRlGqFGt+KoiiKoiiKEiPU+FYURVEURVGUGKHGt6IoiqIoiqLECDW+FUVRFEVRFCVGqPGtKIqiKIqiKDFCjW9FURRFURRFiRFqfCuKoiiKoihKjFDjW1EURVEURVFihBrfiqIoiqIoihIj1PhWFEVRFEVRlBihxreiKIqiKIqixAg1vhVFURRFURQlRqjxrSiKoiiKoigxQo1vRVEURVEURYkRanwriqIoiqIoSoxQ41tRFEVRFEVRYoQa34qiKIqiKIoSIyrF+DbGXGOMyTfGFBtj+oR43E5jzGZjzEZjzLpYnqOiKIqiKIqiVDQJlfS6eQAmAXjRwWOHisjBKJ+PoiiKoiiKokSdSjG+RaQQAIwxlfHyiqIoiqIoilIpVHXNtwBYYIzJMcZMGFEX0QAACSFJREFUreyTURRFURRFUZRIiJrn2xizEEALm7seFJFZDg8zSET2GWOaA/jYGLNFRJYGeb2pAKYCQLNmzbBkyRIvp61UU44dO6ZjQimHjgvFDh0Xih06LpSKImrGt4hcUQHH2Of7fcAYMxNAPwC2xreIzAAwAwC6dOkiQ4YMifTllWrEkiVLoGNCCUTHhWKHjgvFDh0XSkVRZWUnxpi6xpj61t8ARoKJmoqiKIqiKIpyXlJZpQZ/YIzZA2AAgCxjzHzf7a2MMR/6HnYxgGXGmFwAawBkichHlXG+iqIoiqIoilIRVFa1k5kAZtrcvg/Alb6/PweQEeNTUxRFURRFUZSoUWVlJ4qiKIqiKIpS3VDjW1EURVEURVFihBrfiqIoiqIoihIj1PhWFEVRFEVRlBihxreiKIqiKIqixAg1vhVFURRFURQlRqjxrSiKoiiKoigxQo1vRVEURVEURYkRanwriqIoiqIoSoxQ41tRFEVRFEVRYoQa34qiKIqiKIoSI9T4VhRFURRFUZQYoca3oiiKoiiKosQINb4VRVEURVEUJUao8a0oiqIoiqIoMUKNb0VRFEVRFEWJEWp8K4qiKIqiKEqMUONbURRFURRFUWKEGt+KoiiKoiiKEiMqxfg2xjxpjNlijNlkjJlpjGkU5HGjjTGfGWO2G2MeiPV5KoqiKIqiKEpFUlme748BdBORHgC2AvhN4AOMMfEAngMwBkA6gCnGmPSYnqWiKIqiKIqiVCCVYnyLyAIRKfL9uwpAG5uH9QOwXUQ+F5EzAP4DYGKszlFRFEVRFEVRKpqEyj4BAD8C8LbN7a0B7Pb7fw+A/sEOYoyZCmCq79/Txpi8CjtDpTpwEYCDlX0SSpVDx4Vih44LxQ4dF4odXdw+IWrGtzFmIYAWNnc9KCKzfI95EEARgDfsDmFzmwR7PRGZAWCG77jrRKSP65NWqi06JhQ7dFwodui4UOzQcaHYYYxZ5/Y5UTO+ReSKUPcbY24FMA7AcBGxM6r3AGjr938bAPsq7gwVRVEURVEUJbZUVrWT0QB+DWCCiJwI8rC1ADobYzoaYxIBXA9gdqzOUVEURVEURVEqmsqqdjIdQH0AHxtjNhpj/gEAxphWxpgPAcCXkDkNwHwAhQDeEZF8h8efEYVzVs5vdEwodui4UOzQcaHYoeNCscP1uDD2ig9FURRFURRFUSoa7XCpKIqiKIqiKDFCjW9FURRFURRFiRHVyvjWdvRKIMaYtsaYbGNMoTEm3xhzT2Wfk1J1MMbEG2M2GGPmVva5KFUDY0wjY8x7xpgtvnljQGWfk1L5GGN+7ltD8owxbxljalX2OSmxxxjzijHmgH8vGWNME2PMx8aYbb7fjcMdp9oY39qOXglCEYD7RSQNwCUA/kfHheLHPWBCt6JYPA3gIxFJBZABHR8XPMaY1gB+BqCPiHQDEA9WYFMuPF4DMDrgtgcALBKRzgAW+f4PSbUxvqHt6BUbROQrEVnv+/souJC2rtyzUqoCxpg2AMYCeLmyz0WpGhhjGgC4DMA/AUBEzojI4co9K6WKkACgtjEmAUAdaN+RCxIRWQrg24CbJwJ43ff36wCuCnec6mR827WjVyNLKcEY0wFAJoDVlXsmShXh7wB+BaC4sk9EqTJ0AvANgFd9cqSXjTF1K/uklMpFRPYC+CuALwF8BeB7EVlQuWelVCEuFpGvADr8ADQP94TqZHy7akevXFgYY+oBeB/AvSJypLLPR6lcjDHjABwQkZzKPhelSpEAoBeAF0QkE8BxOAghK9Ubn4Z3IoCOAFoBqGuMualyz0o5n6lOxre2o1dsMcbUAA3vN0Tkg8o+H6VKMAjABGPMTlCiNswY8+/KPSWlCrAHwB4RsaJj74HGuHJhcwWAL0TkGxE5C+ADAAMr+ZyUqsN+Y0xLAPD9PhDuCdXJ+NZ29Eo5jDEG1G8WishTlX0+StVARH4jIm1EpAM4VywWEfVkXeCIyNcAdhtjuvhuGg6goBJPSakafAngEmNMHd+aMhyaiKuUMhvArb6/bwUwK9wTEqJ6OjFERIqMMVY7+ngAr7hoR69UXwYBuBnAZmPMRt9tvxWRDyvxnBRFqbrcDeANnxPncwA/rOTzUSoZEVltjHkPwHqwgtYGaKv5CxJjzFsAhgC4yBizB8AfADwB4B1jzO3gRu2asMfR9vKKoiiKoiiKEhuqk+xEURRFURRFUao0anwriqIoiqIoSoxQ41tRFEVRFEVRYoQa34qiKIqiKIoSI9T4VhRFURRFUZQYoca3oijKeYoxppEx5qd+/7fylUSLxmtdZYx5KMT93Y0xr0XjtRVFUaoTWmpQURTlPMUY0wHAXBHpFoPXWgFggogcDPGYhQB+JCJfRvt8FEVRzlfU860oinL+8gSAJGPMRmPMk8aYDsaYPAAwxtxmjPmvMWaOMeYLY8w0Y8x9xpgNxphVxpgmvsclGWM+MsbkGGM+NcakBr6IMSYFwGnL8DbGXGOMyTPG5Bpjlvo9dA7YMVRRFEUJghrfiqIo5y8PANghIj1F5Jc293cDcAOAfgAeBXBCRDIBrARwi+8xMwDcLSK9AfwCwPM2xxkEdvezeAjAKBHJADDB7/Z1AAZH8H4URVGqPdWmvbyiKIpSjmwROQrgqDHme9AzDQCbAfQwxtQDMBDAu8YY6zk1bY7TEsA3fv8vB/CaMeYdAB/43X4AQKsKPH9FUZRqhxrfiqIo1ZfTfn8X+/1fDM7/cQAOi0jPMMc5CaCh9Y+I3GWM6Q9gLICNxpieInIIQC3fYxVFUZQgqOxEURTl/OUogPpenywiRwB8YYy5BgAMybB5aCGAZOsfY0ySiKwWkYcAHATQ1ndXCoA8r+ejKIpyIaDGt6IoynmKz9u83Jf8+KTHw9wI4HZjTC6AfAATbR6zFECmKdWmPGmM2exL7lwKINd3+1AAWR7PQ1EU5YJASw0qiqIoYTHGPA1gjogsDHJ/TQCfALhURIpienKKoijnEer5VhRFUZzwGIA6Ie5vB+ABNbwVRVFCo55vRVEURVEURYkR6vlWFEVRFEVRlBihxreiKIqiKIqixAg1vhVFURRFURQlRqjxrSiKoiiKoigxQo1vRVEURVEURYkR/x9J7UBUXVZ16AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 864x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "T  = 10\n",
    "N  = 200\n",
    "t  = np.linspace(0, T, N)                   # time domain\n",
    "\n",
    "uk = np.array([u0k[0]*np.cos(wk[0]*t),\n",
    "               u0k[1]*np.cos(wk[1]*t)])     # modal responses\n",
    "\n",
    "u  = np.dot(Phi, uk)*100                    # total responses (cm)\n",
    "\n",
    "plt.figure(2, figsize=(12, 4), clear=True)\n",
    "plt.plot(t, u[0,:], 'b', t, u[1,:], 'r')\n",
    "\n",
    "plt.xlim( 0, T);  plt.xlabel('time (s)') \n",
    "plt.ylim(-2, 2);  plt.ylabel('u(t) (cm)') \n",
    "plt.legend(('upper','lower'))\n",
    "\n",
    "plt.grid(True) \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The accelerations are obtained from the twofold derivative of the cosine sum:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ak = np.array([-u0k[0]*wk[0]*wk[0]*np.cos(wk[0]*t),\n",
    "               -u0k[1]*wk[1]*wk[1]*np.cos(wk[1]*t)])    # modal accelerations\n",
    "\n",
    "a  = np.dot(Phi, ak)/9.81                               # nodal accelerations (G)\n",
    "\n",
    "plt.figure(3, figsize=(12, 4), clear=True)\n",
    "plt.plot(t, a[0,:], 'b', t, a[1,:], 'r')\n",
    "\n",
    "plt.xlim(   0,   T);  plt.xlabel('time (s)') \n",
    "plt.ylim(-0.2, 0.2);  plt.ylabel('a(t) (G)') \n",
    "plt.legend(('upper','lower'))\n",
    "\n",
    "plt.grid(True) \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finnaly, answering the question, the peak displacement and acceleration amplitudes in\n",
    "the upper floor are:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Peak upper displacement: 0.876cm\n",
      "Peak upper acceleration: 0.064G \n"
     ]
    }
   ],
   "source": [
    "print('Peak upper displacement: {0:5.3f}cm'.format(u[0,:].max()))\n",
    "print('Peak upper acceleration: {0:5.3f}G '.format(a[0,:].max()))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It can be seen that, as expected, the second mode dominate the structural response.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 3 <a name=\"P2_2019_3\"></a> \n",
    "\n",
    "The cantilever beam shown in the figure has a constant flexural stiffness $EI = 1000{\\rm kNm^2}$ and mass per unit length $\\mu = 200{\\rm kg/m}$. \n",
    "\n",
    "1. Propose a function that resembles the first vibration mode. Calculate the associated potential elastic energy $V$ and the reference kinetic energy, $T_{\\rm ref}$. With these energies, estimate the natural vibration frequency for the first mode using the Rayleigh quocient (2 pts).  \n",
    "2. Calculate the modal mass and the modal stiffness and then use these parameters to estimate the static displacement at the cantilever tips, caused by a point load $W = 10{\\rm kN}$ placed at this same position (1 pts). \n",
    "\n",
    "<img src=\"resources/tests/P2_2019_figs3e4.png\" alt=\"P2_2019_figs3e4\" width=\"360px\"/>\n",
    "\n",
    "**Answer:** We will try and compare two different solutions: a parabolic and a sinusoidal functions.\n",
    "They are:\n",
    "\n",
    "$$ \\varphi_1(x) = \\frac{1}{27} \\;(x - 3)(x - 9) $$\n",
    "\n",
    "and\n",
    "\n",
    "$$ \\varphi_2(x) = 1 - \\sqrt{2}  \\, \\sin \\left( \\frac{\\pi x}{12} \\right) $$\n",
    "\n",
    "Both tentative solutions respect the kinetic condition of zero displacement at supports\n",
    "(located ate coordinates $x = 3$m and $x = 9$m.\n",
    "The script below shows a comparison plot:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "EI  =  1000000.                             # flexural stiffness\n",
    "mu  =  200.                                 # mass per unit length\n",
    "L   =  12.                                  # total length\n",
    "N   =  200                                  # number of segments\n",
    "X   =  np.linspace(0, L, N)                 # length discretization\n",
    "\n",
    "ph1 =  lambda x: (x - 3)*(x - 9)/27                    # first solution\n",
    "ph2 =  lambda x: 1 - np.sqrt(2)*np.sin(np.pi*(x/12))   # second solution\n",
    "\n",
    "plt.figure(4, figsize=(12, 4), clear=True)\n",
    "plt.plot(X, ph1(X), 'b', X, ph2(X), 'r')\n",
    "plt.plot([3, 9], [0 , 0], 'bo')\n",
    "\n",
    "plt.xlim( 0, L);  plt.xlabel('x   (m)') \n",
    "plt.ylim(-2, 2);  plt.ylabel('phi (nondim)') \n",
    "plt.legend(('phi_1','phi_2'))\n",
    "\n",
    "plt.grid(True) \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The sine function has an important feature, which is zero curvature at cantilever tips where\n",
    "bending moments must be zero. \n",
    "The parabolic function is the simplest, but presents constant curvature along all beam length.\n",
    "\n",
    "The rotations are calculated as:\n",
    "\n",
    "$$ \\phi_1^{\\prime}(x) = \\frac{1}{27} (2x - 12) $$\n",
    "\n",
    "and:\n",
    "\n",
    "$$ \\phi_2^{\\prime}(x) = -\\frac{\\pi \\sqrt{2}}{12} \\; \\cos \\left( \\frac{\\pi x}{12} \\right) $$\n",
    "\n",
    "while the curvatures are given by:\n",
    "\n",
    "$$ \\phi_1^{\\prime\\prime}(x) = \\frac{2}{27} $$\n",
    "\n",
    "and:\n",
    "\n",
    "$$ \\phi_2^{\\prime\\prime}(x) = \\frac{\\pi^2 \\sqrt{2}}{144} \\; \\sin \\left( \\frac{\\pi x}{12} \\right) $$\n",
    "\n",
    "The script below compares the curvatures for each solution:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ph1xx = lambda x: (2/27)*x**0                                       # first solution\n",
    "ph2xx = lambda x: (np.pi*np.pi*np.sqrt(2)/144)*np.sin(np.pi*x/12)   # second solution\n",
    "\n",
    "plt.figure(5, figsize=(12, 4), clear=True)\n",
    "plt.plot(X, ph1xx(X), 'b', X, ph2xx(X), 'r')\n",
    "\n",
    "plt.xlim(    0,    L);  plt.xlabel('x (m)') \n",
    "plt.ylim(-0.05, 0.15);  plt.ylabel('phi_xx (1/m^2)') \n",
    "plt.legend(('phi_1','phi_2'))\n",
    "\n",
    "plt.grid(True) \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The curvatures are quite close at the center, but it is overestimated by the parabolic function\n",
    "at the cantilever tips.\n",
    "\n",
    "The potential elastic and the reference kinetic energy finally are:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Potential elastic energy for solution 1: 32757.2J\n",
      "Potential elastic energy for solution 2: 28044.6J\n",
      "\n",
      "Reference kinetic energy for solution 1: 203.5J\n",
      "Reference kinetic energy for solution 2: 238.1J\n"
     ]
    }
   ],
   "source": [
    "dx = L/N\n",
    "\n",
    "V1 = EI*np.trapz(ph1xx(X)*ph1xx(X), dx=dx)/2\n",
    "V2 = EI*np.trapz(ph2xx(X)*ph2xx(X), dx=dx)/2\n",
    "\n",
    "T1 = mu*np.trapz(  ph1(X)*ph1(X),   dx=dx)/2\n",
    "T2 = mu*np.trapz(  ph2(X)*ph2(X),   dx=dx)/2\n",
    "\n",
    "print('Potential elastic energy for solution 1: {0:5.1f}J'.format(V1))\n",
    "print('Potential elastic energy for solution 2: {0:5.1f}J\\n'.format(V2))\n",
    "\n",
    "print('Reference kinetic energy for solution 1: {0:5.1f}J'.format(T1))\n",
    "print('Reference kinetic energy for solution 2: {0:5.1f}J'.format(T2))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And the natural vibration frequencies estimated with Rayleigh quotient are:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Natural vibration frequency for solution 1:  2.02Hz\n",
      "Natural vibration frequency for solution 2:  1.73Hz\n"
     ]
    }
   ],
   "source": [
    "wn1 = np.sqrt(V1/T1)\n",
    "wn2 = np.sqrt(V2/T2)\n",
    "\n",
    "fn1 = wn1/2/np.pi\n",
    "fn2 = wn2/2/np.pi\n",
    "\n",
    "print('Natural vibration frequency for solution 1: {0:5.2f}Hz'.format(fn1))\n",
    "print('Natural vibration frequency for solution 2: {0:5.2f}Hz'.format(fn2))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If one recalls that the true vibration mode minimizes the Rayleigh quotient, the lowest value\n",
    "obtained with the sinusoidal function is likely to be closer to the exact solution.\n",
    "The relative error between both tentative functions is approximately 17% and the\n",
    "correct natural frequency must be a little below 1.73Hz.\n",
    "\n",
    "Now, we will proceed with the calculation of modal mass and modal stiffness:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "Mk1 = mu*np.trapz(ph1(X)*ph1(X), dx=dx)   # modal mass and ...\n",
    "Kk1 = Mk1*wn1**2                          # ... stiffness for solution 1\n",
    "\n",
    "Mk2 = mu*np.trapz(ph2(X)*ph2(X), dx=dx)   # modal mass and ...\n",
    "Kk2 = Mk2*wn2**2                          # ... stiffness for solution 2\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For static analysis, the modal displacement is obtained from modal \n",
    "force divided by modal stiffness:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Static displacement of cantilever tip for solution 1:  5.09cm\n",
      "Static displacement of cantilever tip for solution 2:  7.38cm\n"
     ]
    }
   ],
   "source": [
    "W   = -10000.        # point load (downwards)\n",
    "\n",
    "Fk1 =  W*ph1(6)      # modal (static) force\n",
    "Fk2 =  W*ph2(6)\n",
    "\n",
    "uk1 =  Fk1/Kk1       # modal displacement\n",
    "uk2 =  Fk2/Kk2\n",
    "\n",
    "u1  =  uk1*ph1(12)   # displacement at cantilever tip\n",
    "u2  =  uk2*ph2(12)\n",
    "\n",
    "print('Static displacement of cantilever tip for solution 1: {0:5.2f}cm'.format(u1*100))\n",
    "print('Static displacement of cantilever tip for solution 2: {0:5.2f}cm'.format(u2*100))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The error in the displacement at cantilever tip for the two solutions is quite high, \n",
    "over 40%, for the two tentative functions diverge noticeably in that position \n",
    "(we recommend this result to checked with the Ftool software).\n",
    "\n",
    "A comparison of displacement solutions for the whole beam is shown below:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(6, figsize=(12, 4), clear=True)\n",
    "plt.plot(X, 100*uk1*ph1(X), 'b', X, 100*uk2*ph2(X), 'r')\n",
    "plt.plot([3, 9], [0 , 0], 'bo')\n",
    "\n",
    "plt.xlim( 0,  L);  plt.xlabel('x (m)') \n",
    "plt.ylim(-6, 12);  plt.ylabel('u (cm)') \n",
    "plt.legend(('phi_1','phi_2'))\n",
    "\n",
    "plt.grid(True) \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 4 <a name=\"P2_2019_4\"></a> \n",
    "\n",
    "The same point load from previous question is now applied suddenly from zero to its final magnitude, what causes a dynamic amplification on the beam displacements.  Estimate the peak displacement and the peak acceleration at the cantilever tip (2 pts).\n",
    "\n",
    "**Answer:** The solution for some impulsive loading is well known to be the static solution\n",
    "multiplied by a dynamic amplification factor. In the case of a step load (Heaviside's\n",
    "function) this amplification factor is 2. Hence:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Dynamic displacement of cantilever tip for solution 1: 10.18cm\n",
      "Dynamic displacement of cantilever tip for solution 2: 14.77cm\n"
     ]
    }
   ],
   "source": [
    "print('Dynamic displacement of cantilever tip for solution 1: {0:5.2f}cm'.format(2*u1*100))\n",
    "print('Dynamic displacement of cantilever tip for solution 2: {0:5.2f}cm'.format(2*u2*100))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The peak accelerations are:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Acceleration at cantilever tip for solution 1: 0.835G\n",
      "Acceleration at cantilever tip for solution 2: 0.887G\n"
     ]
    }
   ],
   "source": [
    "ak1 = uk1*wn1**2\n",
    "ak2 = uk2*wn2**2\n",
    "\n",
    "a1  = ak1*ph1(12)\n",
    "a2  = ak2*ph2(12)\n",
    "\n",
    "print('Acceleration at cantilever tip for solution 1: {0:5.3f}G'.format(a1/9.81))\n",
    "print('Acceleration at cantilever tip for solution 2: {0:5.3f}G'.format(a2/9.81))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is observed that the error in the acceleration response at cantilever tip, \n",
    "approximately 6%, is not as high as for the displacement response. \n"
   ]
  }
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