diff --git a/CompMech05-BVPs_project.ipynb b/CompMech05-BVPs_project.ipynb new file mode 100644 index 0000000..82f605e --- /dev/null +++ b/CompMech05-BVPs_project.ipynb @@ -0,0 +1,691 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Computational Mechanics Boundary Values - Project 05\n", + "\n", + "![6-string guitar diagram](../images/guitar.png)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In this final project, we will consider all six strings of a guitar and the deflection of the neck of the guitar. Here are the inputs for each of the strings, all L=0.64 m:\n", + "\n", + "|string|density (g/m)|tension (kg)|\n", + "|---|---|---|\n", + "|E|0.401|7.28|\n", + "|B|0.708|7.22|\n", + "|G|1.140|7.32|\n", + "|D|2.333|8.41|\n", + "|A|4.466|9.03|\n", + "|E|6.790|7.71|" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "1. The neck of the guitar can be considered a cantilever beam with an applied moment, shown above. At the tip we have a moment equal to the sum of the (tensions in the strings) $\\times$ (bridge height). Here we will consider it as $h=4~mm$. \n", + "\n", + "a. Use a finite difference approximation to determine the deflection of the guitar's bridge if the Young's modulus is E=10 GPa and it is a rectangular cross-section $2\\times4~cm^2$ and $I=\\frac{4\\cdot2^3}{12}~cm^4.$\n", + "\n", + "b. Demonstrate that your finite difference solution has converged. _e.g. decrease the step size $h$ and show the solution converges to a final value._" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from IPython.display import Audio\n", + "\n", + "E=10e9\n", + "t=0.04 \n", + "I = (0.04*0.02**3)/12\n", + "L = 0.64\n", + "g = 9.81\n", + "h = 0.004\n", + "\n", + "rho_e = 0.401e-3\n", + "T_e = 7.28*9.81\n", + "rho_b = 0.708e-3\n", + "T_b = 7.22*9.81\n", + "rho_g = 1.140e-3\n", + "T_g = 7.32*9.81\n", + "rho_d = 2.333e-3\n", + "T_d = 8.41*9.81\n", + "rho_a = 4.466e-3\n", + "T_a = 9.03*9.81\n", + "rho_e2 = 6.790e-3\n", + "T_e2 = 7.71*9.81" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "M = (T_e+T_b+T_g+T_d+T_a+T_e2)*h\n", + "x = np.linspace(0,L,100)\n", + "w = (M*L**2/(2*E*I))*(x/L)**2\n", + "\n", + "N = 9\n", + "h1 = L/10\n", + "xnum = np.arange(0,L+h1/2,h1)\n", + "\n", + "A1 = np.diag(np.ones(N)*6)\\\n", + "+np.diag(np.ones(N-1)*-4,-1)\\\n", + "+np.diag(np.ones(N-1)*-4,1)\\\n", + "+np.diag(np.ones(N-2),-2)\\\n", + "+np.diag(np.ones(N-2),2)\n", + "A1[0,0]+=1\n", + "A1[-1,-1]+=1\n", + "A1[7]= [0,0,0,0,0,1,-4,5,-2]\n", + "A1[8] = [0,0,0,0,0,0,2,-4,2]\n", + "b1=-np.zeros(N)\n", + "b1[7]=-h1**2*M/(E*I)\n", + "b1[8]=2*h1**2*M/(E*I)\n", + "w1 = np.linalg.solve(A1,b1)" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "plt.figure(figsize=(8,8))\n", + "plt.plot(x,w,label='Analytical Sol')\n", + "plt.plot(xnum[:-1],np.block([0,w1]),'ro',label='Numerical Sol')\n", + "plt.ylabel('Delfection [m]')\n", + "plt.xlabel('Length of Beam [m]')\n", + "plt.title('Deflection along Beam')\n", + "plt.legend();" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "h = np.array([L/4,L/8,L/12,L/16,L/24])\n", + "w2 = []\n", + "xnum2 = []\n", + "for i in range(len(h)):\n", + " shape = int(L/h[i])\n", + " A2=np.diag(np.ones(shape)*6)+np.diag(np.ones(shape-1)*-4,-1)\\\n", + " +np.diag(np.ones(shape-1)*-4,1)+np.diag(np.ones(shape-2),-2)+np.diag(np.ones(shape-2),2)\n", + " A2[0,0]+=1\n", + " A2[-1,-1]+=1\n", + " A2[-2]= np.zeros(shape)\n", + " A2[-2,-4:]=[1,-4,5,-2]\n", + " A2[-1] = np.zeros(shape)\n", + " A2[-1,-3:] = [2,-4,2]\n", + " b2=-np.zeros(shape)\n", + " b2[-2]=-h[i]**2*M/(E*I)\n", + " b2[-1]=2*h[i]**2*M/(E*I)\n", + " w2.append(np.linalg.solve(A2,b2))\n", + " xnum2.append(np.arange(0,L+h[i]/2,h[i]))\n", + "plt.figure(figsize=(7,7))\n", + "plt.plot(x,w,label='Analytical Sol')\n", + "plt.plot(xnum2[0],np.block([0,w2[0]]),label='4-Step')\n", + "plt.plot(xnum2[1],np.block([0,w2[1]]),label='8-Step')\n", + "plt.plot(xnum2[2],np.block([0,w2[2]]),label='12-Step')\n", + "plt.plot(xnum2[3],np.block([0,w2[3]]),label='16-Step')\n", + "plt.plot(xnum2[4],np.block([0,w2[4]]),label='24-Step')\n", + "plt.ylabel('Delfection [m]')\n", + "plt.xlabel('Length of Beam [m]')\n", + "plt.title('Convergence of Deflection along Beam')\n", + "plt.legend();" + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "ana = np.max(w)\n", + "error = []\n", + "for i in range(len(h)):\n", + " num = np.max(w2[i])\n", + " error.append(np.abs(ana-num))\n", + "step = np.array([24,16,12,8,4])\n", + "plt.figure(figsize=(7,7))\n", + "plt.plot(step,error)\n", + "plt.ylabel('Absolute Error from Max Delfection')\n", + "plt.xlabel('Number of Steps of h')\n", + "plt.title('Convergence of Deflection along Beam');" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "2. Here we will record the first three frequencies of the 6-string guitar. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "a. Consider the G-string on the guitar, L=0.64 m, $\\mu=1.14~g/m,$ and T=71.81 N [1]. \n", + "\n", + "__Guitar string equation:__ $\\mu\\frac{\\partial^2 y}{\\partial t^2}=T\\frac{\\partial ^2 y}{\\partial x^2}$\n", + "\n", + "a. Calculate the first, second, and third natural frequencies using 6, 30, 45, and 60 nodes. Plot the mode shapes and determine the number of nodes needed to converge for the first three modes. " + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Natural frequencies of 6-element string (Hz)\n", + "[194.43708661 379.1242858 544.80061022]\n", + "Natural frequencies of 30-element string (Hz)\n", + "[195.99464595 391.48617646 585.97276757]\n", + "Natural frequencies of 45-element string (Hz)\n", + "[196.04043699 391.85229959 587.20727994]\n", + "Natural frequencies of 60-element string (Hz)\n", + "[196.05687232 391.9837462 587.65070938]\n" + ] + }, + { + "data": { + "image/png": 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gth5IIfJrRAopEoFEAkIhMPN/JhL+8uG46aabiEQizJgxg5aWFq699tpBb3vRRRcxY8YMTjnlFM466yxuvPFGjjrqKC666CKmTZtGfX09y5YtY86cOYCf+H/PPffwzW9+k1NOOYWZM2f25IsVig10uW7YOzcLAz91zp2cY93PgO86557MPG4GvuGcey7Hc6P4tyypq6s7NZVKFazNIlIYgYB/VacvM9i3L/c2nucRi8X4i1SKP59hXHOeIz1+//pgdZDE/ASR+gjhxjCpzv59Q2g7JBthH1B1PZCjqLU52HcDBK73A6t+HHAD7AU+cpUfgPV7nZoQyauSfrszAWG6a38k9YEu+FTzoTz6mzR1dXXE43Eiec5QB/NZiYy03/3ud2OuJESh5frMzOw559zsXM8v5pWwduC4rMe1wBu5nuicSzjnZjvnZh9++OGj0jgRGVlDzRnxPI9oNMonUykSwPfO6h2AAaS70sSa/ctDbZ25M4DbajI/ATrztKGz989+Ovfvo3t//V4n6/VjzbFeARjAn6rhd3++078tmkoRjUbx8txfLER+jYiUnmIGYQ8CX8iMkvxzoNM5V9iCHCJSNAfKGel7+275D5eTTqf5J/zbiQcKfupqckcodZ2wE7gWoBno6tOGLog3Z9rY7D/ONt7GU/2Ef4/yWqA2XyCX9fr5AsL2msxtUSD90TQLWxbmvF1ZiPwaESk9hSxRsRZ4GviYmbWb2RfNbKmZLc085SFgC7AZuA0Y2cxWESkpA+WM9ORzZSXEd3yyA+qhO7TJd5WqO/iJN8QJVveOXIK74R9+fSjfmjyZu8wIvRdi2bHLCNWEMIxQTYjEscuIvOc3KvKe/zh7/e0X3c4dX72DUCjEXWac8OtD+UDfQK46SLxhf4Q0UEBYB1APzIe9E/fuHyDQFO0JxAqVXyMipaWgOWGFMHv2bLdhw4ZiN0NEhsvz/EzztjbCXw+Qmri3/3O2w2uNEMZPsI/OJ29OGPjBXKw5RltnG3U1dcQb4j3rRrTpB3gdr9Uj+uPLSI/b378Gd0OiCf6iFaZeBeTKK9tRRfJ7+/z7jvG4oi4pKuWEDd1Qc8LGjUqrRESydddgyAwBbDs0RwAGUOPfArwNiLT6i2IN/q3JuurJxOev7BX8ROojBQm6+jrQ60TqI/DkU8ReuZW2wxx1nf6tzgtbYQlAvlurh+71BwF016QABWIiY5imLRKRUdEr56tlId5H9yeu57vVOLl6Mr8KhYgC7VVVfL4VkveF2DdtDcnYO6MScB2syLJbSJ56J/t+FCK50rjgjck9t0Wrdlbl3KbX55BO4/1w+UGXuRCR0qcgTEQKrl/O18S9PTW8IJMQv7v3NsHqICvPX0kymcRzjto9ezDnIJksn6tDkYjf3n37mPjOO/zgnXfYt28fq7+wOmf+WvcAAcjcfv1kR+/Csll5YyJj3a5duzj99NM55ZRTOOmkk7j++ut71r377rvMmzePadOmMW/ePP7whz+M+OuvW7eO8847b8T3m01BmIgURPZUPwv/fWG/kg3p8fuLpEZa/Xyp0I6q/QnzWbleY02kPkJifmL/AIAdVSSa9t9yhUzh2BwlORbcvoBwOJy3vIVIsWQf8yPxHT3kkEP45S9/yQsvvEBLSwsPP/wwv/71rwFYsWIFDQ0NvPLKKzQ0NOSdU7LUKQgTkRHXXeMrlUrhnGNvnpyv7LITkVeDJGeuZt/1+0helRyzAVi3SH2E5FVJ//3OXE3k1d5XxvKV5KCGA9YZExltfY/5kfiOmhkTJ04E/CmEurq6euZ8fOCBB1i4cCEACxcu5P777++3/bp165g7dy4XX3wx06dPJxKJ9Mwn2dzczKxZs6ivr2fx4sW8//77ADz88MNMnz6dM888k3vvvbdnXzt37mTx4sWcdtppzJo1iwceeOCg31c2BWEiMnIys05fumABL6bTXNq9PF95iZ1VqsEAOWtS1FVPzv3czGd5QTrNnIULNcO3lIRYLEa6z1xb6XSa2DDn2tq7dy8zZ87kiCOOYN68eT2Teb/11lscfbQ/3fTRRx/N22+/nXP7jRs30tjYyEsvvcSWLVt46qmn2LVrF4sWLeLuu++mtbWVPXv2sGrVKnbt2sWSJUtoamriiSee4M033+zZTzwe56yzzuLZZ5/lscce4+qrr2bnzp3Dem+gIExERkDfibPX1vtlJW4DPxBrBnLkfMW/sNqfh6ec8rwKJSt/jGSS+Pkr++WNsRto9j/T24DHT9xb0InERQarrS3PjBV5lg9WVVUVLS0ttLe388wzz/Db3/52SNuffvrp1NbWEggEmDlzJslkkk2bNjF16lSOP/54wL+Stn79el5++WWmTp3KtGnTMDMWLFjQs59HHnmEFStWMHPmTObOncuuXbuG/d5AQZiIDNOgJs5uBZqgqkJyvkZCdt4YDtgONAGtQ5tIXGQ01OWZUyvf8qGaNGkSc+fO5eGHHwbgyCOPZOtWf5KdrVu3csQRR+Tc7pBDDun5vaqqij179jBQfdTu2519Oef4yU9+QktLCy0tLbS1tY1IDTUFYSIyLLnmScxOuu/ugoOvBlldQTlfI6E7b2zNtDUEE0E/mMX/THMm7o9zPXNpioymeDxOsM9cW8FgkPgw5tratm0b27dvB+BPf/oTv/jFL5g+fToA559/PqtXrwZg9erVXHDBBYPe7/Tp00kmk2zevBmAO++8kzlz5jB9+nRee+01Xn31VQDWrl3bs83ZZ5/NTTfd1BPAbdy48aDfVzYFYSIyLIOZODsUCpFIJIhU+i3HgxSJREgkEoRCIcyMN6qqBjWRuMho6fsdHYljfuvWrXzmM59hxowZnHbaacybN6+nZMQ111zDo48+yrRp03j00Ue55pprBr3fCRMmcMcdd3DJJZdQX19PIBBg6dKlTJgwgUQiwbnnnsuZZ55JKBTq2ea6666jq6uLGTNmcPLJJ3Pdddcd9PvKpmmLRGRYwo1hUp2pfstD2yGZCFZ2wn2heB7h5y4jVdO//w7VhEhelRz9NsmYo2mLhm6o0xbpSpiIDEu+ibPjLZMVgBVKJEJ82lKCe3rnr/SdSFxESpuCMBEZUK/phnJMndOv8GhNiMTn1hB57B0FYAUUWXYLib+7s/fnnmOww4H+fiJSPLodKSJ5eauuJPrGraTH7e8ngtVBjWwsE90jV7MHTgT3GIljlhJZdksRWyblQLcjh063I0Vk2DzP4ytTpvCt/1rVKwADf+ocjcArDzlHro5zfOu/VvGVKVNUcV+kyMYVuwEiUlq6px95MZ3mZo3AK2v5/k7tNfC1jg5OikYBNGpVpEh0JUxEeumefqQOqMs33VDNyBRglMLK93eq6/RrjY3EtDIicvAUhIlIL91TcbQB8WZ/pGO24B7TCLwyEW+I9x9Budv/u3ZfIxuJqVdECmnv3r3MmjWrp0YYwLe//W2OPfZYZs6cycyZM3nooYdG/HXXrVvX6zULQUGYSIXrO3ruw3M+DMC1wIWtkGjya36Zg1BnJqlbSfllIVIfIXHMUkKd5v/9tvt/zwtb/b8v+NPKaASljAjP8yeTH+FJ5VeuXJlzgMBXv/rVnmmEPvvZz47Ia402BWEiFaxn3sfOFA5HqjPFHz/zR6pPrWYtsAT4i1bY0gjv3TyZ5Kl3alRdmYksu4XkqXfy3s2T2dLo/z2XAGvxp5X57Dc/2+87EG2KKhCTofE8iEYhlQLn/J/R6LADsfb2dn72s59xxRVXDHnbdevWMXfuXC6++GKmT59OJBLpmXaoubmZWbNmUV9fz+LFi3n//fcBePjhh5k+fTpnnnkm9957b8++du7cyeLFiznttNOYNWsWDzzwwLDeVzcFYSIVLNfoud1uN4dddBihUIi7zJgbCrF2zRomvqO6X2UrEmHiO++wds0a5mb+rt3Tyjz0/kP9R1BqBKwMVSwG6d7fI9Jpf/kwXHXVVdx4440EAv3DlZtvvpkZM2awePFi/vCHP+TcfuPGjTQ2NvLSSy+xZcsWnnrqKXbt2sWiRYu4++67aW1tZc+ePaxatYpdu3axZMkSmpqaeOKJJ3jzzTd79hOPxznrrLN49tlneeyxx7j66qvZuXPnsN4bKAgTqWj5Rs+9u+ddkskk+/btI5lMavTcGBGJRPr9XfPO/akRsDIU+XILh5Fz+NOf/pQjjjiCU089td+6ZcuW8eqrr9LS0sLRRx/N17/+9Zz7OP3006mtrSUQCDBz5kySySSbNm1i6tSpHH/88QAsXLiQ9evX8/LLLzN16lSmTZuGmbFgwYKe/TzyyCOsWLGCmTNnMnfuXHbt2jUi+ZQKwkQqTVbeRt2O3F2ARj9WjrwjKHcERjy3R8awujx9Rr7lg/DUU0/x4IMPEg6H+dznPscvf/nLnsDoyCOPpKqqikAgwJIlS3jmmWdy7uOQQw7p+b2qqoo9e/YwUJF6M8u53DnHT37yk54ctLa2thEpZKsgTKSS9MnbiP98L8Gu3k/R/IOVJefcn10Q//neEc3tkTEuHodg7+8RwaC//CB997vfpb29nWQyyV133cVZZ53FmjVrANi6dWvP8+677z5OPvnkQe93+vTpJJNJNm/eDMCdd97JnDlzmD59Oq+99hqvvvoqAGvXru3Z5uyzz+amm27qCeA2btx40O8rm4IwkUrSJ28j0gqJByG0o2rA+Qdl7Oo39+eOKhIP+t+NHiOQ2yNjXCQCiQSEQmDm/0wkCpZH+o1vfIP6+npmzJjBY489xve///1BbzthwgTuuOMOLrnkEurr6wkEAixdupQJEyaQSCQ499xzOfPMMwmFQj3bXHfddXR1dTFjxgxOPvlkrrvuuhF5H5o7UqSSBAL+1Y2+zGDfvtFvj5QefUckQ3NHDp3mjhSRHv3qP2VqgPUzjLwNGWNyfBe8egh/PaA6YiIjTEGYyBiVqwZY9DN/xDu1uvcTh5m3IWNMn9werx6i50Nq4l7VERMZYQrCRMYYz/MIh8MsuH1B//pPbjexiw4btbwNKUN9cntiZ1eR7hO3p7vSLLh9AeFwGE8J+yIHbVyxGyAiI8fzPKLRKOl0GmpyP6dtz7uQfGd0GyblJRLpCczbbsjzf/UaSKVSRKPRzCYK5EWGSlfCRMaQWCzmB2AAnbmfoxpgMhR5vy+Z71c6nSamkZMiB0VBmMgY0quCczOwu/d61QCTocpVR4zd+N+vjJGoHC5SiRSEiZS7rAr4bYEAl3YvbwWagO2AQzXA5KBk1xHD4X+fmvC/X8ClQFtA1fWlMLZv394zAfcJJ5zA008/DcC7777LvHnzmDZtGvPmzcs7d+RwrFu3jvPOO2/E95tNQZhIOetTAb92715ug16BWDARZM20NSSvSioAk4MSqY+QvCrJmmlrCCaCvQKw24DavaquLzlK4ozACNrly5dzzjnn8PLLL/PCCy/01OBasWIFDQ0NvPLKKzQ0NLBixYphv1YxKAgTKWPeD5cTjqYJXA/hq/xyAocCN1ZVYWaEQiESiYSSpmVERCIREokEoVAIM+PGqioOzazz6v3vYODqNOGWhSphUWFylsQZZimT9957j/Xr1/PFL34RgPHjxzNp0iQAHnjgARYuXAj4E3Dff//9/bZft24dc+fO7bmSFolEeqYdam5uZtasWdTX17N48WLef/99AB5++GGmT5/OmWeeyb333tuzr507d7J48WJOO+00Zs2axQMPPHDQ7yubgjCRMuW1ekQ/2UFqEjiD1CSIzvdPhrX79rFv3z6SyaQCMBlRkUiEZDLJvn37qM1U0Pfq/e9ez3dx4l7VEqswseZY/5I4XWlizQc/aGPLli0cfvjhXH755cyaNYsrrriCnTt3AvDWW29x9NFHA3D00Ufz9ttv59zHxo0baWxs5KWXXmLLli089dRT7Nq1i0WLFnH33XfT2trKnj17WLVqFbt27WLJkiU0NTXxxBNP8Oabb/bsJx6Pc9ZZZ/Hss8/y2GOPcfXVV/e0ZTgUhImUqVhzjPT43svS4yHWgCrgy+jIfM9iDfT/Lg7zBCzlpa0z9+CMfMsHY8+ePTz//PMsW7aMjRs3cuihhw75tuPpp59ObW0tgUCAmTNnkkwm2bRpE1OnTuX4448H/Ctp62OK2DEAACAASURBVNev5+WXX2bq1KlMmzYNM2PBggU9+3nkkUdYsWIFM2fOZO7cuezatWtEBqQoCBMpU3k7vRpUAV9GR6a6flu+mnTDOAFLeclXymQ4JXFqa2upra3ljDPOAODiiy/m+eefB+DII49k69atAGzdupUjjjgi5z4OOeSQnt+rqqrYs2cPA82ZbWY5lzvn+MlPfkJLSwstLS20tbWNyLyaCsJEylTeTq96sirgy+jIVNev21mVc7Vq0lWOXKVMhlsS56ijjuK4445j06ZNgJ/HdeKJJwJw/vnns3r1agBWr17NBRdcMOj9Tp8+nWQyyebNmwG48847mTNnDtOnT+e1117j1VdfBWDt2rU925x99tncdNNNPQHcxo0bD/p9ZVMQJlKm8nZ6568sUoukIkUixL+wesRPwFJeskuZGDZiJXFuuukmIpEIM2bMoKWlhWuvvRaAa665hkcffZRp06bx6KOPcs011wx6nxMmTOCOO+7gkksuob6+nkAgwNKlS5kwYQKJRIJzzz2XM888k1Ao1LPNddddR1dXFzNmzODkk0/muuuuG9b76mYDXZYrRbNnz3YbNmwodjNESoLX6hFrjtHW2UZdTR3xhrjKUEhR6Ls49vzud78bkVtulSTXZ2ZmzznnZud6vuaOFCljkfqITnRSEvRdFBk63Y4UERERKQIFYSIiIpJTuaUsFdPBfFYKwkRKVCGmABEpFn2fy8+ECRPo6OhQIDYIzjk6OjqYMGHCkLZTTphICfJWXUn0jVtJj/M7v+4pQADl3UjZ6Z7SpruieqozRfTHl8GTTxFZdkuRWyf51NbW0t7ezrZt24rdlLIwYcIEamtrh7SNRkeKlBDP8/jN8uXcf1kHr0/qvz5UEyJ5VXLU2yUyHOHGMKnOVL/lx22HC++czBkrV2p6LRmzNDpSpAx4nkc0GuXFdJqbVYFcxpB839v2GvhaRwcnRTNXeRWISYVRTphIiYjFYqTTaeqAus7cz1EFcilHeWd36IQ6IJ1OE4tpnkmpPArCREpE92SwbUC8GYK7e68P7jFVIJeyFG+IE9zTe06+4G7/e959jWwkJkMWKTcKwkRKRF2df7XgWuDCVkg0QWg7mINQp5E4ZqmS8qUsReojJI5ZSqjT/O/zdv/7fWGr/32H/d9/kUqiIEykRMTjcYLBIGuBJcBftMKWRnjv5skkT71To8ikrEWW3ULy1Dt57+bJbGn0v99LgLVAMBgkHtdVXqk8CsJESkQkEiGRSBAKhbjLjLmhEGvXrGHiO++AEpZlLIhEmPjOO6xds4a5me95KBQikUgoKV8qkkpUiIiIiBTIQCUqdCVMpAhUPVykPx0XUmlUJ0xklOWsHq5q+FLhdFxIJdKVMJFRFmuO9ZxouqW70sSaVSdJKpeOC6lECsJERlm+6uGqhi+VTMeFVCIFYSKjxfMgHKZue+7BMKqGL5Usb1X9HQEIBCAc9o8hkTFEQZjIaPA8iEYhlcpdDb86qGr4UtHiDXGC1cFey4JdEP/5XnAOUin/GFIgJmOIgjCR0RCLQdrPd4n0rYZfEyIxP6HkY6lokfoIifkJQjUhDCO0o4rEg/7x0iOd9o8lkTFCdcJERkMg4P9vvi8z2Ldv9NsjUup0zMgYoTphIsWWb148zZcnkpuOGakACsJERkM8DsHe+S4Eg/5yEelPx4xUgIIGYWZ2jpltMrPNZnZNjvU1ZtZkZi+Y2Ytmdnkh2yNSCJlBj70GcPWr/D0DSCQgFPJvp4RC/mPNlyeSWySS85jxZtCvqn6uY1CkHBQsJ8zMqoD/AuYB7cCzwKXOuZeynnMtUOOc+6aZHQ5sAo5yzu3OtU9QTpiUlu5Bj+msGpPVp3rY+VF2u/0Lg9VBJd+LDFPfqvoA4y2IezBB13P7j61gUP/HkdJRrJyw04HNzrktmaDqLuCCPs9xwAfNzICJwLvAngK2SWREZQ167NH1qVivAAxU+VtkJOSqqr/bpen6VO9jS4MopVwUMgg7Fng963F7Zlm2m4ETgDeAVmC5c07DXqRstOUq5l2jyt8ihZD3GMpxzOU8NkVKTCGDMMuxrO+9z7OBFuAYYCZws5kd1m9HZlEz22BmG7Zt2zbyLRU5SL0HanlAGDpVEV+kEPIeQ50B/OMv67k63KQMFDIIaweOy3pci3/FK9vlwL3Otxl4DZjed0fOuYRzbrZzbvbhhx9esAaLDNX+AVweEAVS0AyoIr7IiMtVVZ/dQPNe/OPPD8Q0iFLKRSGDsGeBaWY21czGA58DHuzznDagAcDMjgQ+BmwpYJtERlT3AK6qqhiQyVVpBZqA7YAq4ouMmO6q+lU7qvz7Ktvxj7VW8I+/mAYeS1kpaMV8M/ss0AhUAbc75+JmthTAOXermR0D/Ag4Gv/25Qrn3JqB9qnRkVKKAoEAuY4lM2OfqnuLjCgdb1JOBhodOa6QL+ycewh4qM+yW7N+fwP4q0K2QWQ01NXVkUqlci4XkZGl403GClXMFzlYWRUif7tjB4uqq3utDgaDxJWYIjLi4vE4wT7V9BdVV/PbHTtUsVXKioIwkYPRXaU1lQLnmNjRwW1mfHnyZMyMUChEIpEgosQUkREXiURIJBKEQiEsc9zdZsbEjg5/0u9Uyj8+FYhJiStoTlghKCdMSkI47Hf0fYVCkEyOdmtEKpuORylhxaqYLzImea0e4YtSBK6H8FXg1WetVIVIkdGXddx59f5xGbgewhel8Fp1NUxKl4IwkSHonrsuNQmcQWoSROdnBWJKDBYZfZnjzqv3j8dex2dTVIGYlCwFYSJDkGvuuvR4iDWgCpEixZKpmhxr8I/HbJq3VUpZQUtUiIw1+eaua6tBFSJFiiVz3LW9siDnas3bKqVKV8JEhiDf3HV1k0IKwESKKRLxj8McNG+rlCoFYSJDkGvuOs0LKVIadHxKuVEQJjIE3XPXhWpCGKZ5IUVKiI5PKTeqEyYiIiJSIKoTJiIiIlJiFISJiIiIFIGCMBEREZEiUBAmIiIiUgQKwkRERESKQEGYSB9eq0e4MUzghgDhxrDmnRMZY3SMS6nQtEUiWbxVVxJ941bS4/zSLanOFNGmKIBqDYmMAV6rR7Qp2jMHbKozRfTHl8GTTxFZdkuRWyeVRlfCRLp5HrFX9gdg3TQBsMjYEWuO9QRg3dLjHLFXbgVPV8RkdCkIEwE8z6N94ULaDstdvFgTAIuMDfmO5bbDHDuWLx/l1kilUxAmFc/zPKLRKMfs3UtdZ+7naAJgkbEh37Fc1wnBjg48XQ2TUaQgTCpeLBYjnU7TBsSbIbi79/rgHtMEwCJjRLwhTnCP9VoW3O0f+234/YHIaFEQJhWvrc2/PXEtcGErJJogtB3MQajTSByzVEn5ImNEpD5C4pilHNd9jG/3j/kLW/0+oLs/EBkNGh0pFa+uro5UKsXazON/aoUtrfBGVRW1q1dDRAGYyFgSWXYLv5nyY77W0UEd/hWwJcBaIFSn1AMZPboSJhUvHo8TDAYBvxOeCnwwGORxBWAiY9YZK1dyUjBIFf4xvxYIBoPE40o9kNGjIEwqXiQSIZFIEAqFMDNCoRCJRIKIAjCRMUvHvZQCcy73kPxSNXv2bLdhw4ZiN0NERETkgMzsOefc7FzrdCVMREREpAgUhImIiIgUgYIwqSiauFdEDkT9hIwWlaiQipFz4l5Nzi0iWdRPyGjSlTCpGDkn7tXk3CKSRf2EjCYFYVIx8k7cq8m5RSRD/YSMJgVhUjHyTtyryblFJEP9hIwmBWEytnkehMMQCBB/YAdBG99rdbA6qMm5RaRHvCFOsDrYa1nQxhN/YAcEAn5/4ilRX0aGgjAZuzwPolFIpcA5Ius6SDzoCI2bjGGEakIk5ieUbCsiPSL1ERLzE4RqQn4/MW4yiQf9/gPn/P4kGlUgJiNCFfNl7AqH/Q6zr1AIksnRbo2IlCP1IzJMqpgvlaktTyJtvuUiIn2pH5ECUhAmY1ddnkTafMtFRPpSPyIFpCBMxq54HIK9E2wJBv3lIiKDoX5ECkhBmIxdkQgkEn7uhpn/M5Hwl4uIDIb6ESkgJeaLiIiIFIgS86UiaNJdERkt6m9kJGgCbxkTNOmuiIwW9TcyUnQlTMYETborIqNF/Y2MFAVhUvY8zyO1PUcxRTTproiMvHz9Smp7Ck+V9GUIFIRJWfM8j2g0Cp2512vSXREZaXn7lU6IRqMKxGTQFIRJWYvFYqTTaWgGdvdep8m5RaQQck3yzW6gGdLpNLGYbkvK4CgxX8paW/fUIa2ZBQ1ADdAJicWanFtERl53v7Lg9gU9/Q3N9PRDbZrSSAZJdcKkrIXDYVI5JtcNhUIkNbmuiBSQ+h8ZDNUJkzErHo8T7DOlSDAYJK4pRUSkwNT/yHApCJOyFolESCQShEIhzIxQKEQikSCiKUVEpMDU/8hw6XakiIiISIHodqSMLZ4H4TAEAv5PDQcXkVKh/kmGQKMjpbx4HkSjkM5Uq06l/McAugUgIsWk/kmGSLcjpbyEw37H1lcoBBqNJCLFpP5JctDtSBk78tXfUV0eESk29U8yRArCpGx4rR7hrwcIXA/hq8Crz1pZp+mJRKTI+vRDXr3fVwX+wRFuDOO1Kj9MelMQJmXBa/WINkVJTdyLM0hNguj8TCAWDILq8ohIscXjfn+E3zdF5/t9lTNIdaaINkUViEkvCsKkLMSaY6S70r2WpcdD7OwqSCSU9CoixReJ+P1RKESswe+jsqW70sSaNa+k7KcgTMpCW2funIq2ifsUgIlI6YhEIJmkbZLlXJ2vL5PKpCBMykJdTe6cr3zLRUSKSX2WDIaCMCkL8YY4weo+c7RVB4k3KBdMREqP+iwZDAVhUhYi9RES8xOEakIYRqgmRGJ+gki9bkWKSOlRnyWDUdBirWZ2DrASqAJ+6JxbkeM5c4FGoBp4xzk3Z6B9qliriIiIlIuBirUWbNoiM6sC/gWYB7QDz5rZg865l7KeMwm4BTjHOddmZkcUqj0iIiIipaSQtyNPBzY757Y453YDdwEX9HnO54F7nXNtAM65twvYHhEREZGSUcgg7Fjg9azH7Zll2Y4HPmRm68zsOTP7Qq4dmVnUzDaY2YZt27YVqLkiIiIio6eQQViuIil9E9DGAacC5wJnA9eZ2fH9NnIu4Zyb7Zybffjhh498S0VERERGWcFywvCvfB2X9bgWeCPHc95xzu0EdprZeuAU4L8K2C4RERGRoivklbBngWlmNtXMxgOfAx7s85wHgE+Z2TgzCwJnAL8rYJtERERESkLBroQ55/aY2ZeAn+OXqLjdOfeimS3NrL/VOfc7M3sY+E9gH34Zi98Wqk0iIiIipeKAV8LMLGhm15nZbZnH08zsvMHs3Dn3kHPueOfcR51z8cyyW51zt2Y953875050zp3snGs82Dci5c1r9Qg3hgncECDcGMZr9YrdJBGRglK/J4O5HXkH8D7wiczjduA7BWuRVByv1SN632JSnSkcjlRniuh9i9UhiciY5bV6RJuivfu9H1+Gt+rKYjdNRtFggrCPOuduBLoAnHN/IvfIR5GDEntwOWm3u9eytNtN7MHlRWqRiEhhxZpjpLvSvZalxzlir9wKnv4DWikGE4TtNrMPkCkvYWYfxb8yJjJsnufR1tWRc12+5SIi5a6tsy338sMcO5brP6CVYjBB2PXAw8BxZuYBzcA3CtoqqQie5xGNRjmuM/f6ujzLRUTKXV1NXe7lnRDs6MDT1bCKcMAgzDn3KPA3wCJgLTDbObeusM2SShCLxUin03y1GYK970YS3A3xlsnFaZiISIHFG+IE9/TO7AnuhngztOH3jzL25Q3CzOzj3f+AELAVv9hqXWaZyLC0tfmX459phZuaILQdzPk/Ez+vJnLFyiK3UESkMCL1ERLHLOW47H6vCS5shWvZ3z/K2DZQnbDvZX5OAGYDL+An5M8AfgOcWdimyVhXV1dHKpViLUArrGuFOuCNqipqV98BkUiRWygiUjiRZbfwmyk/5msdHdThXwFbgn/LKVSX+3aljC15r4Q55z7jnPsMkAI+npm78VRgFrB5tBooY1c8HicYDAJ+pzMV+GAwyOOrVysAE5GKcMbKlZwUDFKF3weuBYLBIPF4vMgtk9EwmMT86c651u4HmYr2MwvXJKkUkUiERCJBKBTCzAiFQiQSCSIKwESkQqgfrGzmnBv4CWZrgZ3AGvwyFQuAic65SwvfvP5mz57tNmzYUIyXFhERERkSM3vOOTc717rBzB15ObAM6C5csh5YNUJtExEREalIBwzCnHO7gO9n/omIiIjICDhgEGZmr5Gplp/NOfeRgrRIREREpAIM5nZk9n3MCcAlwIcL0xwRERGRyjCYivkdWf9+75xrBM4ahbbJGOK1eoQbwwRuCBBuDOO1akoOEZEDUd85tg3mdmR2dfwA/pWxDxasRTLmeK0e0aYo6a40AKnOFNGmKOBXjRYRkf7Ud459gylR8VjWwz3Aa8D3nHObCtmwfFSiovyEG8OkOlP9lodqQiSvSo5+g0REyoD6zrFhuCUqvuic29Jnh1NHpGVSEdo6c8+Blm+5iIio76wEg6mYf88gl4nkVFeTew60fMtFRER9ZyXIG4SZ2XQz+1ugxsz+JuvfIvxRkiKDEm+IE6wO9loWrA4Sb9DcaCIi+ajvHPsGuh35MeA8YBIwP2v5H/EnehcZlO4E0lhzjLbONupq6og3xJVYKiIyAPWdY99gEvM/4Zx7epTac0BKzC8TngexGLS1QV0dxOOgCWlFRA6e+tWydFCJ+Wb2DefcjcDnzazfZN3Oua+MYBtlLPE8iEYh7Q+rJpXyH4M6DBGRg6F+dUzKeyXMzOY755rMbGGu9c651QVtWR66ElYGwmG/g+grFIJkcrRbIyJS/tSvlq2DuhLmnGvK/CxKsCVlrC3P8Ol8y0VEZGDqV8ekgW5HNpFj4u5uzrnzC9IiKX91dbn/x1anYdUiIgdF/eqYNNDoyH8etVbI2BKP985dAAgG/eUiIjJ06lfHpIFuRz7e/buZjQem418Z2+Sc2z0KbZNy1Z0kqlE8IiIjQ/3qmDSYEhXnArcCrwIGTAX+m3Pu/xW+ef0pMV9ERETKxXDnjvwe8Bnn3ObMzj4K/AwoShAmIiIiMhYMZu7It7sDsIwtwNsFao+IiIhIRRhMEPaimT1kZosyNcOagGe755IscPukDHitHuHGMIEbAoQbw3itXrGbJCJSEdT/lrfB3I6cALwFzMk83gZ8GH8+SQfcW5imSTnwWj2iTVHSXf6InVRnimiTX8VZ85uJiBSO+t/yd8DE/FKjxPzSMiU+hY49Hf2Wh2pCJK9Kjn6DREQqRLgxTKqzf+2wqh1VrJ65mohGTpaEYSXmm9lU4MtAOPv5KtYqnufR0dXhj5nto61TVZxFRAopXz+799C9RDPzSioQK22DyQm7H0gCN+GPlOz+JxUuFotBZ+51dTWq4iwiUkh5+9lOSKfTfh8tJW0wQdgu59wPnHOPOece7/5X8JZJyWtra4NmoG/p3t0Qb1AVZxGRQoo3xAlWB3sv3I3fL5Ppo6WkDSYIW2lm15vZJ8zs493/Ct4yKXl1dXXQij9edjv+MI3tMPlXk5UUKiJSYJH6CIn5Cap2VPX0vzTh98tk+mgpaYMZHVkPXAacBezLLHOZx1LB4vE40WiUdGu656APBoOsTKwsbsNERCpEpD4CM/H74qx5JYPBIHHNK1nyBhOEXQR8RPNFSl/dCZ+xWIy2tjbq6uqIx+NKBBURGUXqi8vXYOaOvBv4snOuJKrkq0SFiIiIlIvhzh15JPCymT0LvJ9Z5pxzF4xUA0VEREQqzWAS86/HvyX5T8D/AZ4B/qyQjZIS5XkQDkMg4P/0ND2GiEjJUp9d8g54Jcw597iZzQQ+D/wd8Bpwa6EbJiXG8yAahe7Ez1TKfwygvAMRkdKiPrss5M0JM7Pjgc8BlwIdwN3A/3DOhUavef0pJ6xIwmH/IO4rFIJkcrRbIyIiA1GfXTIONifsZeAJYL5zbnNmR18tQPukHOQr+qdigCIipUd9dlkYKCfsb4E3gcfM7DYzayDnLIFSEfIV/VMxQBGR0qM+uyzkDcKcc/c55/4emA6sA74KHGlmq8zsr0apfVJkXqtHuDFM4PIU4a8aXn3WymAQVAxQRKT0xON+Hw149RC+CgLXQ3jJDrxWJeiXigOOjnTO7XTOec6584BaoAW4puAtk6LyPJjyGY8Fd0VJdaZwQKrGEb0gE4iFQpBIKMFTRKQURSKQSODNnUx0PqQmgTNI7engsh9HsRmeBkyWgAMWay01SswvvJ5BNdEwTOqf2BmqCZG8Kjnq7RIRkaEJN4ZJdeZI0N8egsYkwaD+P11oAyXmD6ZOmFSYWCwzqrkmdwJnW6cSO0VEykHe/jrTv6fTfp8vxaEgTPrpGTzTmTuBs65GiZ0iIuUgb3+d1b9rwGTxKAiTfnoGzzTHYXew17pgdZB4g5LxRUTKQbwhTrC6dz/O7qDfv2dowGTxKAiTfnoG1bRGoCnh5w44Y/K4EIn5CSL1Sh4QESkHkfoIifkJQjUhwLDOkN+vt/r9uAa5F5cS8yUnz/PzBNra/P8lxeNK3BQRKXfq20ffQIn5CsJERERECkSjI0VERERKjIIwERERkSJQECYiIiJSBArCRERERIpAQZiIiIhIESgIExERESkCBWEiIiIiRaAgTERERKQIChqEmdk5ZrbJzDab2TUDPO80M9trZhcXsj3Sn9fqEW4ME7ghQLgxjNfqFbtJIiIyynQuKI5xhdqxmVUB/wLMA9qBZ83sQefcSzme9/8BPy9UWyQ3b9WVRN+4lfQ4f9aEVGeKaFMUQPNDiohUCK/VI3rfYtJuN5A5F9y3GNC5oNAKeSXsdGCzc26Lc243cBdwQY7nfRn4CfB2AdsifXke176yqicA65buShNrjhWpUSIiMtpiDy7vCcC6pd1uvv4fS4vUospRyCDsWOD1rMftmWU9zOxY4CLg1oF2ZGZRM9tgZhu2bds24g2tRDuWL+f1w3Kva+tsG93GiIhI0bR1deRc/nZgB56n25KFVMggzHIs6ztbeCPwTefc3oF25JxLOOdmO+dmH3744SPWwEoW7OigrjP3urqautFtjIiIFE2+c8FxnRCL6c5IIRUyCGsHjst6XAu80ec5s4G7zCwJXAzcYmYXFrBNktEGxJsh2PsKNB/YDfGGeFHaJCIioy/eMrnfuSC4G77aDG1tujNSSIUMwp4FppnZVDMbD3wOeDD7Cc65qc65sHMuDNwDXOmcu7+AbZKM/zN5Mhe2QqIJQtvBHBy3HT71y0OViCkiUkEiV6zk5qxzQWg73NQEz7RCXZ3ujBRSwUZHOuf2mNmX8Ec9VgG3O+deNLOlmfUD5oFJYZ2xciVfuvxyrm/tYkurf2Xshupq/vKOfy1200REZDRFIkx76inWr7yVWudoA64FHggGScR1Z6SQzLm+aVqlbfbs2W7Dhg3FbsaY4HkesViMtrY26urqiMfjRCK6CiYiUol0TigMM3vOOTc75zoFYSIiIiKFMVAQpmmLRERERIpAQZiIiIhIESgIExERESkCBWEiIiIiRaAgTERERKQIFISJiIiIFIGCMBEREZEiUBAmIiIiUgQKwsYwr9Uj3BgmcEOAcGMYr9UrdpNERKQM6XxSGAWbO1KKy2v1iDZFSXelAUh1pog2RQE0QbeIiAyazieFoythY1SsOdZzwHRLd6WJNceK1CIRESlHOp8UjoKwMaqts21Iy0VERHLR+aRwFISNUXU1dUNaLiIikovOJ4WjIGyMijfECVYHey0LVgeJN8SL1CIRESlHOp8UjoKwscbzIBwmcsplJB79AKFxkzGMUE2IxPyEkihFRGRIIvUREvMThGpC/vlk3GQSj36AyCmXQTjsn3fkoJhzrthtGJLZs2e7DRs2FLsZpcnzIBqFdFYCZTAIiQREFHyJiMgw6TwzZGb2nHNuds51CsLGkHAYUqn+y0MhSCZHuzUiIjLW6DwzZAMFYbodOZa05Rmpkm+5iIjIUOg8M6IUhI0ldXlGquRbLiIiMhQ6z4woBWFjSTzu35vPFgz6y0VERIZL55kRpSBsLIlE/OTIUAjM/J9KlhQRkZGi88yIUmK+iIiISIEoMV9ERESkxCgIExERESkCBWEiIiIiRaAgrIx5rR7hxjCBGwKEG8N4rZo6QkREikPnpKEbV+wGyMHxWj2iTVHSXf7UEanOFNGmKIDmhxQRkVGlc9LB0ZWwMrX8weU9X/Zu6a40seZYkVokIiKVKtYcy3lOWv7g8iK1qDwoCCtDnufR0dWRc11bp6aOEBGR0ZXv3NPR1YHn6bZkPgrCylAsFoPO3OvqajR1hIiIjK68557OzDlLclIQVoba2tqgGdjdZ8VuiDdo6ggRERld8YZ4znMSzZlzluSkIKwM1dXVQSvQBGwHnP9z8q8mKwFSRERGXaQ+wuRfTe51TqIJaM2csyQnjY4sQ/F4nGg0Sro17QdjQDAYZGViZXEbJiIiFWvlFSv9c1N6f4J+MBgkrsm989KVsDIUiURIJBKEQiHMjFAoRCKRIKIJVEVEpEh0bho6TeAtIiIiUiCawFtERESkxCgIExERESkCBWEiIiIiRaAgrBx4HoTDEAj4P1V9WEREyoXOYXmpREWp8zyIRqF7yG8q5T8G0IgTEREpZTqHDUijI0tdOOx/afsKhSCZHO3WiIiIDJ7OYRodWdbyTfegaSBERKTU6Rw2IAVhpS7fdA+aBkJEREqdzmEDUhBWorxWj3BjmMDlKcJfNbz6rJXBIGgaCBERKXXxuH/OArx6CF8FgeshvGQHXqsS9JWYX4K8Vo9oU5R0l5/ImKpxRC8wwBF5L+R/qZXQKCIipS5zrvJ+uJzoJztIj/cXp/Z0EG3yE/Qj9ZV7PlNifgkKN4ZJdfZPCY87rwAAD3ZJREFUZAzVhEhelRz9BomIiAxDJZ/XlJhfZto6cycs5lsuIiJSynRey01BWAmqq8mdsJhvuYiISCnTeS03BWElKN4QJ1gd7LUsWB0k3qBkfBERKT86r+WmIKwEReojJOYnCNWEMIxQTYjE/ERFJy+KiEj50nktNyXmi4iIiBSIEvNFRERESoyCMBEREZEiUBAmIiIiUgQKwkRERESKQEGYiIiISBEoCBMREREpAgVhIiIiIkWgIExERESkCBSEiYiIiBSBgjARERGRIlAQJiIiIlIECsKKwFt1JeGrxxH4thG+ehzeqiuL3SQREZGiq7TzY0GDMDM7x8w2mdlmM7smx/qImf1n5t+vzOyUQranFHirriTavorUxL04g9TEvUTbV435L5qIiMhAvFVXEv197/PjkvZV/MPX/rLYTSuYggVhZlYF/Avw18CJwKVmdmKfp70GzHHOzQD+EUgUqj2l4lub/5X0+N7L0uP95SIiIpUqtiVBurr3sj+Nh9tdM57nFadRBVbIK2GnA5udc1ucc7uBu/7/9u49Rq7yvOP497HXbllITWSDaEx2F1KqKIhLy3JJClWQq6hBRS4NEqgjUIF2E7s0gQorlbZKgWhVRVVaJ2pt2FgQkU6bVClOcQhNVfdCWi7BELChoQgivCG0DXGCqXEEvjz9Y8Z0dz1rn41n5pyZ/X6k0c68c3b24fFh39/OnHNeYPX0DTLzocz8UfPhI8CpHaynEl5628F5jUuStBBMHX+g5fjLy2B8fLzL1XRHJ0PYSuC70x6/1Bybyw3AA62eiIixiNgWEdteeeWVNpbYfe/YPb9xSZIWgqHXF7ccf8dumJqa6nI13dHJEBYtxrLlhhGX0ghhH2/1fGZOZuZoZo6edNJJbSyx+8585HiOe3Pm2HFvNsYlSVqoJk4fazk/Dm2FoaGhcorqsE6GsJeAd057fCrw8uyNIuJsYBOwOjN3dbCeSrj2xjsZvX8RK1+FSFj5Kozev4hrb/SYMEnSwlVbs4Fbfrxqxvx47hZ46oVBJiYmyi6vIzoZwh4DzoiI0yJiKXA1cN/0DSJiCLgXuCYzn+tgLZVRq9X48Lp7GNg8DLcHA5uH+fC6e6jVamWXJklSqW7/03/kU6N/ydDnh3n5M8HLrw0zOTnZt3NkZLb8hLA9Lx5xGbAeWAzclZkTEfERgMy8IyI2AR8Cdja/ZX9mjh7pNUdHR3Pbtm0dq1mSJKldIuLxubJNR0NYJxjCJElSrzhSCPOK+ZIkSSUwhEmSJJXAECZJklQCQ5gkSVIJDGGSJEklMIRJkiSVwBAmSZJUAkOYJElSCQxhHVDfUWdk/QiLblvEyPoR6jvqZZckSVJf6Kc5dqDsAvpNfUedsS1j7N23F4Cdu3cytmUMgNpZ/bn2lSRJ3dBvc6zvhLXZ+Nbxt3aOQ/bu28v41vGSKpIkqT/02xxrCGuzqd1T8xqXJEnF9Nscawhrs6FlQ/MalyRJxfTbHGsIa7OJVRMMLhmcMTa4ZJCJVRMlVSRJUn/otznWENZmtbNqTF4+yfCyYYJgeNkwk5dP9uQBg5IkVUm/zbGRmWXXMC+jo6O5bdu2sss4XL0O4+MwNQVDQzAxAbXe3CkkSeoJPTD3RsTjmTna6jkvUdEO9TqMjcHe5hkbO3c2HkPldgZJkvpCH8y9vhPWDiMjjX/82YaH4cUXu12NJEn9r0fm3iO9E+YxYe0wNcepsXONS5KkY9MHc68hrB2G5jg1dq5xSZJ0bPpg7jWEtcPEBAzOPGWWwcHGuCRJar8+mHsNYe1Qq8HkZONz6IjG18nJnjkwUJKkntMHc68H5kuSJHWIB+ZLkiRVjCFMkiSpBIYwSZKkEhjCJEmSSmAIm6e1G9cysG6AuDUYWDfA2o1ryy5JkiQ11XfUWTGxgrg1iJuDFZeuoF6vl11WS4aweVi7cS0bv7eRAyccgIADJxxg4/c2GsQkSaqA+o4612++nl37d0EAJ8Ku9+3iuj+7rpJBzEtUzMPAuoFGAJtl8Z7F7P+T/SVUJEmSDhlZP8LO3S3Wk3wVhjcP82IJa0p6iYo2OXD84QHsSOOSJKl7pnbPsW7kMpiq4JqShrB5WPz64nmNS5Kk7hlaNse6kbthqIJrShrC5mHs9DHYN2twX3NckiSVamLVBEtj6czBN2HJN5YwUcE1JQ1h87BhzQbWrFzD4j2LIRvHgq1ZuYYNazaUXZokSQte7awad11xF8sHlkMCr8Lyh5Zz9813U6vgmpIemC9JktQhHpgvSZJUMYYwSZKkEhjCJEmSSmAIkyRJKoEhbC71OoyMwKJFja8VXO5AkiQVVMF5faDsAiqpXoexMdi7t/F4587GY4AKnuIqSZKOoKLzupeoaGVkpPEPNNvwMJSw7pQkSToGJc7rXqJivuZaX6qC605JkqSjqOi8bghrZa71pSq47pQkSTqKis7rhrBWJiZgcHDm2OBgY1ySJPWWis7rhrBWajWYnGx8VhzR+Do56UH5kiT1oorO64awWeo76oysH2HR89cwchPUn/pC46A9A5gkSb2rVmvM5wcPUt8ywcgr4yy6bREj60eo7yjnchVeomKa+o46Y1vG2LuvcQrrzt07GdvSOIW1dpYhTJKkXlelud53wqYZ3zr+1j/KIXv37WV863hJFUmSpHaq0lxvCJtmanfrU1XnGpckSb2lSnO9IWyaoWWtT1Wda1ySJPWWKs31hrBpJlZNMLhk5imsg0sGmVjlpSkkSeoHVZrrDWHT1M6qMXn5JMPLhgmC4WXDTF4+6UH5kiT1iSrN9a4dKUmS1CGuHSlJklQxhjBJkqQSGMIkSZJKYAiTJEkqgSFMkiSpBIYwSZKkEhjCJEmSSmAIkyRJKoEhTJIkqQSGMEmSpBIYwiRJkkpgCJMkSSpBzy3gHRGvADuP8WVWAD9oQzkLnX1sD/vYHvaxPexje9jH9uiHPg5n5kmtnui5ENYOEbFtrhXNVZx9bA/72B72sT3sY3vYx/bo9z76caQkSVIJDGGSJEklWKghbLLsAvqEfWwP+9ge9rE97GN72Mf26Os+LshjwiRJksq2UN8JkyRJKpUhTJIkqQR9HcIi4lcj4j8j4vmI+IMWz0dEfLb5/PaI+MUy6qy6An18d0Q8HBFvRMQtZdRYdQV6WGvug9sj4qGIOKeMOquuQB9XN3v4ZERsi4iLy6iz6o7Wx2nbnR8RByLiym7W1ysK7I/vj4jdzf3xyYj4RBl1Vl2R/bHZyycj4pmI+Ndu19gxmdmXN2Ax8AJwOrAUeAp4z6xtLgMeAAK4CHi07LqrdivYx5OB84EJ4Jaya67arWAP3we8vXn/g+6LP3EfT+D/j3U9G3i27LqrdivSx2nb/RPwNeDKsuuu2q3g/vh+4Ktl11rlW8E+ngj8BzDUfHxy2XW369bP74RdADyfmd/JzDeBLwKrZ22zGrgnGx4BToyIn+12oRV31D5m5vcz8zFgXxkF9oAiPXwoM3/UfPgIcGqXa+wFRfq4J5u/pYHjAc88OlyR340Avwf8LfD9bhbXQ4r2UUdWpI+/CdybmVPQmHO6XGPH9HMIWwl8d9rjl5pj891mobNHx26+PbyBxju0mqlQHyPiioh4FrgfuL5LtfWSo/YxIlYCVwB3dLGuXlP0/+v3RsRTEfFARJzZndJ6SpE+/jzw9oj4l4h4PCKu7Vp1HTZQdgEdFC3GZv9VXGSbhc4eHbvCPYyIS2mEMI9lOlyhPmbmZmBzRPwy8EngVzpdWI8p0sf1wMcz80BEq81FsT4+QWPdwD0RcRnwFeCMjlfWW4r0cQA4D1gFHAc8HBGPZOZznS6u0/o5hL0EvHPa41OBl3+CbRY6e3TsCvUwIs4GNgEfzMxdXaqtl8xrX8zMByPiXRGxIjN7fQHgdirSx1Hgi80AtgK4LCL2Z+ZXulNiTzhqHzPztWn3vxYRG9wfD1N0rv5BZr4OvB4RDwLnAD0fwvr548jHgDMi4rSIWApcDdw3a5v7gGubZ0leBOzOzP/qdqEVV6SPOrKj9jAihoB7gWv64a+7DinSx5+LZnJonu28FDDQznTUPmbmaZk5kpkjwJeBtQawwxTZH0+Ztj9eQGPOdX+cqcgc83fAJRExEBGDwIXAt7tcZ0f07Tthmbk/Im4Evk7j7Iu7MvOZiPhI8/k7aJz1cxnwPLAXuK6sequqSB8j4hRgG/AzwMGIuInG2S2vzfnCC0jBffETwHJgQ/N39v7MHC2r5ioq2McP0fjDah/wY+CqaQfqi8J91FEU7OOVwJqI2E9jf7za/XGmIn3MzG9HxN8D24GDwKbMfLq8qtvHZYskSZJK0M8fR0qSJFWWIUySJKkEhjBJkqQSGMIkSZJKYAiTJEkqgSFMUiVFREbEp6c9viUibp3na+yZ5/YnRMSdEfFCRDwTEQ9GxIXzeY0CP+Pc5tXTJS1whjBJVfUG8BsRsaKLP3MT8EPgjMw8E/gtGleMb6dzaVyfUNICZwiTVFX7gUng5tlPRMRwRGyNiO3Nr0PN8dMi4uGIeCwiPjnre9Y1x7dHxG0tXvNdNK7E/YeZeRAgM7+Tmfc3n//9iHi6ebupOTYSEU9Pe4233q1rLjb8qYj4ZkQ8FxGXNK8IfjtwVUQ8GRFXtaVTknqSIUxSlf0FUIuIZbPG/xy4JzPPBurAZ5vjnwE2Zub5wH8f2jgiPkBj4eQLaLwTdV5zge/pzgSezMwDs4uIiPNorKhxIXAR8DsR8QsF6h/IzAuAm4A/ysw3aayO8KXMPDczv1TgNST1KUOYpMpqLn11D/DRWU+9F/ir5v0vABc37/8S8NfTxg/5QPP2LeAJ4N00QllRFwObM/P1zNxDY53PSwp8373Nr48DI/P4eZIWgL5dO1JS31hPIzjdfYRtco77hwTwx5l55xFe4xngnIhYdOjjyFnf38p+Zv4x+9Oznn+j+fUA/r6VNIvvhEmqtMz8IfA3wA3Thh8Crm7erwH/1rz/77PGD/k6cH1EnAAQESsj4uRZP+cFGgvR3xbNVdQj4oyIWA08CPx6RAxGxPHAFcA3gP8BTo6I5RHxU8CvFfhP+l/gbQW2k9TnDGGSesGnmXmW4keB6yJiO3AN8LHm+MeA342Ix4C3jiPLzH+g8fHlwxGxA/gyrYPQbwOnAM83t/sc8HJmPgF8Hvgm8CiwKTO/lZn7aBxo/yjwVeDZAv8t/wy8xwPzJUVmq3fuJUmS1Em+EyZJklQCQ5gkSVIJDGGSJEklMIRJkiSVwBAmSZJUAkOYJElSCQxhkiRJJfg/FUDWGOdObHYAAAAASUVORK5CYII=\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "L = 0.64\n", + "mu = 1.14e-3\n", + "T = 71.81\n", + "\n", + "N6=6 # 6-node guitar string\n", + "dx6=L/(N6+1)\n", + "k6 = T/dx6**2/mu\n", + "A6 = k6*(np.diag(np.ones(N6)*2)\\\n", + " -np.diag(np.ones(N6-1),-1)\\\n", + " -np.diag(np.ones(N6-1),1))\n", + "e6,v6=np.linalg.eig(A6)\n", + "isort = np.argsort(e6.real)\n", + "e6=e6[isort]\n", + "v6=v6[:,isort]\n", + "\n", + "N30=30 # 30-node guitar string\n", + "dx30=L/(N30+1)\n", + "k30 = T/dx30**2/mu\n", + "A30 = k30*(np.diag(np.ones(N30)*2)\\\n", + " -np.diag(np.ones(N30-1),-1)\\\n", + " -np.diag(np.ones(N30-1),1))\n", + "e30,v30=np.linalg.eig(A30)\n", + "isort = np.argsort(e30.real)\n", + "e30=e30[isort]\n", + "v30=v30[:,isort]\n", + "\n", + "N45=45 # 45-node guitar string\n", + "dx45=L/(N45+1)\n", + "k45 = T/dx45**2/mu\n", + "A45 = k45*(np.diag(np.ones(N45)*2)\\\n", + " -np.diag(np.ones(N45-1),-1)\\\n", + " -np.diag(np.ones(N45-1),1))\n", + "e45,v45=np.linalg.eig(A45)\n", + "isort = np.argsort(e45.real)\n", + "e45=e45[isort]\n", + "v45=v45[:,isort]\n", + "\n", + "N60=60 # 60-node guitar string\n", + "dx60=L/(N60+1)\n", + "k60 = T/dx60**2/mu\n", + "A60 = k60*(np.diag(np.ones(N60)*2)\\\n", + " -np.diag(np.ones(N60-1),-1)\\\n", + " -np.diag(np.ones(N60-1),1))\n", + "e60,v60=np.linalg.eig(A60)\n", + "isort = np.argsort(e60.real)\n", + "e60=e60[isort]\n", + "v60=v60[:,isort]\n", + "\n", + "print('Natural frequencies of {}-element string (Hz)'.format(N6))\n", + "print(e6[:3].real**0.5/2/np.pi)\n", + "\n", + "print('Natural frequencies of {}-element string (Hz)'.format(N30))\n", + "print(e30[:3].real**0.5/2/np.pi)\n", + "\n", + "print('Natural frequencies of {}-element string (Hz)'.format(N45))\n", + "print(e45[:3].real**0.5/2/np.pi)\n", + "\n", + "print('Natural frequencies of {}-element string (Hz)'.format(N60))\n", + "print(e60[:3].real**0.5/2/np.pi)\n", + "\n", + "x6 = np.linspace(0,L,N6+2)\n", + "x30 = np.linspace(0,L,N30+2)\n", + "x45 = np.linspace(0,L,N45+2)\n", + "x60 = np.linspace(0,L,N60+2)\n", + "\n", + "plt.figure(figsize=(10,7))\n", + "plt.plot(x6[1:-1],-1*v6[:,0]/np.max(-v6[:,0]),'bo',label='6 node')\n", + "plt.plot(x30[1:-1],-1*v30[:,0]/np.max(-v30[:,0]),'ko',label='30 node')\n", + "plt.plot(x45[1:-1],1*v45[:,0]/np.max(v45[:,0]),'ro',label='45 node')\n", + "plt.plot(x60[1:-1],-1*v60[:,0]/np.max(-v60[:,0]),'go',label='60 node')\n", + "plt.title('First Eigen with Increasing Node Count')\n", + "plt.xlabel('Node Count')\n", + "plt.ylabel('Amplitude')\n", + "plt.legend();" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", 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" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "plt.figure(figsize=(10,7))\n", + "plt.plot(x6[1:-1],1*v6[:,1]/np.max(-v6[:,1]),'bo',label='6 node')\n", + "plt.plot(x30[1:-1],-1*v30[:,1]/np.max(-v30[:,1]),'ko',label='30 node')\n", + "plt.plot(x45[1:-1],1*v45[:,1]/np.max(-v45[:,1]),'ro',label='45 node')\n", + "plt.plot(x60[1:-1],-1*v60[:,1]/np.max(-v60[:,1]),'go',label='60 node')\n", + "plt.title('Second Eigen with Increasing Node Count')\n", + "plt.xlabel('Node Count')\n", + "plt.ylabel('Amplitude')\n", + "plt.legend();" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "scrolled": false + }, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "plt.figure(figsize=(10,7))\n", + "plt.plot(x6[1:-1],-1*v6[:,2]/np.max(-v6[:,2]),'bo',label='6 node')\n", + "plt.plot(x30[1:-1],-1*v30[:,2]/np.max(-v30[:,2]),'ko',label='30 node')\n", + "plt.plot(x45[1:-1],1*v45[:,2]/np.max(-v45[:,2]),'ro',label='45 node')\n", + "plt.plot(x60[1:-1],-1*v60[:,2]/np.max(-v60[:,2]),'go',label='60 node')\n", + "plt.title('Third Eigen with Increasing Node Count')\n", + "plt.xlabel('Node Count')\n", + "plt.ylabel('Amplitude')\n", + "plt.legend();" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Converges at 45 nodes\n" + ] + } + ], + "source": [ + "print('Converges at 45 nodes')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "b. Use the number of nodes necessary for convergence to calculate the first 3 modes of vibration for the other 5 strings on the guitar. Display the first three natural frequencies for all six strings. " + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [], + "source": [ + "rho_e = 0.401e-3\n", + "T_e = 7.28*9.81\n", + "rho_b = 0.708e-3\n", + "T_b = 7.22*9.81\n", + "rho_g = 1.140e-3\n", + "T_g = 7.32*9.81\n", + "rho_d = 2.333e-3\n", + "T_d = 8.41*9.81\n", + "rho_a = 4.466e-3\n", + "T_a = 9.03*9.81\n", + "rho_e2 = 6.790e-3\n", + "T_e2 = 7.71*9.81" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Natural frequencies of E-string (Hz)\n", + "[329.63523775 658.88613556 987.36880161]\n" + ] + } + ], + "source": [ + "N_e=45\n", + "dx_e=L/(N_e+1)\n", + "k_e = T_e/dx_e**2/rho_e\n", + "A_e = k_e*(np.diag(np.ones(N_e)*2)\\\n", + " -np.diag(np.ones(N_e-1),-1)\\\n", + " -np.diag(np.ones(N_e-1),1))\n", + "e_e,v_e=np.linalg.eig(A_e)\n", + "isort = np.argsort(e_e.real)\n", + "e_e=e_e[isort]\n", + "v_e=v_e[:,isort]\n", + "\n", + "print('Natural frequencies of E-string (Hz)')\n", + "print(e_e[:3].real**0.5/2/np.pi)" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Natural frequencies of B-string (Hz)\n", + "[247.05411832 493.82018258 740.01047461]\n" + ] + } + ], + "source": [ + "N_b=45\n", + "dx_b=L/(N_b+1)\n", + "k_b = T_b/dx_b**2/rho_b\n", + "A_b = k_b*(np.diag(np.ones(N_b)*2)\\\n", + " -np.diag(np.ones(N_b-1),-1)\\\n", + " -np.diag(np.ones(N_b-1),1))\n", + "e_b,v_b=np.linalg.eig(A_b)\n", + "isort = np.argsort(e_b.real)\n", + "e_b=e_b[isort]\n", + "v_b=v_b[:,isort]\n", + "\n", + "print('Natural frequencies of B-string (Hz)')\n", + "print(e_b[:3].real**0.5/2/np.pi)" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Natural frequencies of G-string (Hz)\n", + "[196.03934499 391.85011687 587.20400904]\n" + ] + } + ], + "source": [ + "N_g=45 # 45-node guitar string\n", + "dx_g=L/(N_g+1)\n", + "k_g = T_g/dx_g**2/rho_g\n", + "A_g = k_g*(np.diag(np.ones(N_g)*2)\\\n", + " -np.diag(np.ones(N_g-1),-1)\\\n", + " -np.diag(np.ones(N_g-1),1))\n", + "e_g,v_g=np.linalg.eig(A_g)\n", + "isort = np.argsort(e_g.real)\n", + "e_g=e_g[isort]\n", + "v_g=v_g[:,isort]\n", + "\n", + "print('Natural frequencies of G-string (Hz)')\n", + "print(e_g[:3].real**0.5/2/np.pi)" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Natural frequencies of E-string (Hz)\n", + "[146.88611909 293.60097554 439.97350639]\n" + ] + } + ], + "source": [ + "N_d=45\n", + "dx_d=L/(N_d+1)\n", + "k_d = T_d/dx_d**2/rho_d\n", + "A_d = k_d*(np.diag(np.ones(N_d)*2)\\\n", + " -np.diag(np.ones(N_d-1),-1)\\\n", + " -np.diag(np.ones(N_d-1),1))\n", + "e_d,v_d=np.linalg.eig(A_d)\n", + "isort = np.argsort(e_d.real)\n", + "e_d=e_d[isort]\n", + "v_d=v_d[:,isort]\n", + "\n", + "print('Natural frequencies of E-string (Hz)')\n", + "print(e_d[:3].real**0.5/2/np.pi)" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Natural frequencies of E-string (Hz)\n", + "[110.00809489 219.88792527 329.51137616]\n" + ] + } + ], + "source": [ + "N_a=45\n", + "dx_a=L/(N_a+1)\n", + "k_a = T_a/dx_a**2/rho_a\n", + "A_a = k_a*(np.diag(np.ones(N_a)*2)\\\n", + " -np.diag(np.ones(N_a-1),-1)\\\n", + " -np.diag(np.ones(N_a-1),1))\n", + "e_a,v_a=np.linalg.eig(A_a)\n", + "isort = np.argsort(e_a.real)\n", + "e_a=e_a[isort]\n", + "v_a=v_a[:,isort]\n", + "\n", + "print('Natural frequencies of E-string (Hz)')\n", + "print(e_a[:3].real**0.5/2/np.pi)" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Natural frequencies of E-string (Hz)\n", + "[ 82.43894585 164.78177157 246.93246912]\n" + ] + } + ], + "source": [ + "N_e2=45\n", + "dx_e2=L/(N_e2+1)\n", + "k_e2 = T_e2/dx_e2**2/rho_e2\n", + "A_e2 = k_e2*(np.diag(np.ones(N_e2)*2)\\\n", + " -np.diag(np.ones(N_e2-1),-1)\\\n", + " -np.diag(np.ones(N_e2-1),1))\n", + "e_e2,v_e2=np.linalg.eig(A_e2)\n", + "isort = np.argsort(e_e2.real)\n", + "e_e2=e_e2[isort]\n", + "v_e2=v_e2[:,isort]\n", + "\n", + "print('Natural frequencies of E-string (Hz)')\n", + "print(e_e2[:3].real**0.5/2/np.pi)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "c. Create an audio signal that has the 18 frequencies (6 strings $\\times$ 3 modes) in an array and display it using the `from IPython.display import Audio` library. \n", + "\n", + "_Hint: you don't need to solve the differential equations here. You can use the calculated frequencies to add sine-waves together:_ $\\sin(f_12\\pi t)+\\sin(f_22\\pi t)+...$" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": { + "scrolled": true + }, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + " \n", + " " + ], + "text/plain": [ + "" + ] + }, + "execution_count": 16, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "samplerate = 20000\n", + "t = np.arange(0,3,1/samplerate)\n", + "data = 100*(np.sin(t*e_e[0].real**0.5)+np.sin(t*e_b[0].real**0.5)+np.sin(t*e_g[0].real**0.5)+\n", + " np.sin(t*e_d[0].real**0.5)+np.sin(t*e_a[0].real**0.5)+np.sin(t*e_e2[0].real**0.5)+\n", + " np.sin(t*e_e[1].real**0.5)+np.sin(t*e_b[1].real**0.5)+np.sin(t*e_g[1].real**0.5)+\n", + " np.sin(t*e_d[1].real**0.5)+np.sin(t*e_a[1].real**0.5)+np.sin(t*e_e2[1].real**0.5)+\n", + " np.sin(t*e_e[2].real**0.5)+np.sin(t*e_b[2].real**0.5)+np.sin(t*e_g[2].real**0.5)+\n", + " np.sin(t*e_d[2].real**0.5)+np.sin(t*e_a[2].real**0.5)+np.sin(t*e_e2[2].real**0.5))\n", + "Audio(data=data,rate=samplerate)" + ] + } + ], + "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": "3.7.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +}