{ "metadata": { "name": "" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Simple Harmonic Motion" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We will start by reviewing simple harmonic motion (SHM) as it contains\n", "many of the important concepts that we will meet in waves. The most\n", "general form of the equation for a simple harmonic oscillator (SHO)\n", "including damping and driving forces can be written: \n", "\n", "$$\n", "m\\frac{\\partial^2 \\psi}{\\partial t^2} = -s\\psi -b \\frac{\\partial \\psi}{\\partial t} + F_0 \\cos \\omega t,\n", "$$\n", "\n", "where $\\psi$ is the displacement, $m$ is the mass of the oscillator,\n", "$s$ is the constant giving the restoring force (e.g. a spring\n", "constant) sometimes known as the *stiffness*, $b$ is the damping\n", "coefficient or *resistance* and $F_0$ gives the magnitude of the\n", "driving force (which has *angular* frequency $\\omega$). " ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "SHM and Circular Motion" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If we set the damping and driving coefficients to zero, we recover the original, SHM equation:\n", "\n", "$$m\\frac{\\partial^2 \\psi}{\\partial t^2} = -s \\psi,$$\n", "\n", "which can be shown to be solved using sinusoidal motion:\n", "$$\n", " \\psi = A \\sin \\omega_0 t + B \\cos \\omega_0 t = C \\cos \\left(\\omega t + \\phi\\right),\n", "$$\n", "where $\\omega_0 = \\sqrt{s/m}$, $\\phi$ is a constant phase and $A = -C\\sin \\phi$ and $B = C \\cos \\phi$. As you have seen in both PHAS1245 (Mathematical Methods I) and PHAS1247 (Classical Mechanics), we can use De Moivre's theorem to write the sinusoidal terms as a complex exponential ($e^{i\\theta} = \\cos \\theta + i \\sin \\theta$ where $i = \\sqrt{-1}$), giving: \n", "$$\n", " \\psi = Re\\left[A e^{i(\\omega_0 t + \\phi)}\\right] = Re\\left[D e^{i\\omega_0 t}\\right] \n", "$$\n", "where $D = Ae^{i\\phi}$.\n", "\n", "We often call the argument of a trigonometric function the\n", "_phase_, and $\\phi$ is often called the phase difference when\n", "there is more than one oscillation; it represents the offset in the\n", "phase at $t=0$. Now that we have the representation of the\n", "oscillation in terms of the complex exponential (or the sum of a\n", "$\\cos$ and $\\sin$) we can see that there is an immediate link with\n", "circular motion: the amplitude of the oscillation is just the\n", "projection of circular motion onto the $x$-axis (or any other axis\n", "that is chosen). (Note that from now on I will not write the need to\n", "take the real part of $\\psi$ explicitly, but will assume it.) All\n", "these ideas are illustrated in the figure below.\n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from IPython.display import Image\n", "Image(filename='SHMCircularMotion.png',width=300) #Size is in pixels" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": { "png": { "width": 300 } }, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAf4AAAHwCAIAAADIB5oCAAAACXBIWXMAAAxNAAAMTQHSzq1OAAAA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\n", "prompt_number": 4, "text": [ "" ] } ], "prompt_number": 4 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can also illustrate this more dynamically, with the python code below. The blue line shows the phasor at $t=0$ (with the angle relative to the x-axis being $\\phi$. The red line shows the phasor at a later time $t$, with the real part shown as a dashed red line along the x-axis." ] }, { "cell_type": "code", "collapsed": false, "input": [ "phi = 30.0*pi/180. # Phase in degrees\n", "rad = 1.0\n", "x = (0,rad*cos(phi))\n", "y = (0,rad*sin(phi))\n", "axis('scaled')\n", "axis([-1.1,1.1,-1.1,1.1])\n", "plot(x,y)\n", "# Add arc to indicate phi\n", "theta = linspace(0,phi,1000)\n", "plot(0.5*rad*cos(theta),0.5*rad*sin(theta),'b-')\n", "# Label\n", "text(0.5, 0.18, r\"$\\phi$\", fontsize=20, color=\"blue\")\n", "# Now draw full circle\n", "theta = linspace(0,2.0*pi,1000)\n", "plot(rad*cos(theta),rad*sin(theta),'g-')\n", "# Plot the phasor\n", "omega = 2.0\n", "t = 2.5\n", "psix = (0,rad*cos(omega*t+phi))\n", "psiy = (0,rad*sin(omega*t+phi))\n", "plot(psix,psiy,'r-')\n", "# Projection on the x-axis\n", "plot(psix,(0.,0.),'r--')\n", "# Label\n", "text(0.4,-0.2, r\"$\\psi$\", fontsize=20, color=\"red\")\n", "# Arc showing omega t\n", "theta = linspace(phi,omega*t+phi,1000)\n", "plot(0.6*rad*cos(theta),0.6*rad*sin(theta),'r:')\n", "# Label\n", "text(-0.5, 0.6, r\"$\\omega t$\", fontsize=20, color=\"red\")" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 16, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAAQgAAAD7CAYAAACWhwr8AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAIABJREFUeJztnXdYFFfbxu+lCAgoGumgKFIEFSmKXVSwoVgSY0mMUYM1\n0VT90gxJ1FeNmkSJwZJErLHGisRoBBVpomABBFSULlVAOnu+P04UUVgWtszMcn7XtZc7u7Pn3LPr\n3Jz6PCJCCAGDwWA0gBrXAhgMBn9hBsFgMBqFGQSDwWgUZhAMBqNRmEEwGIxGYQbBYDAah/CEYcOG\nEQDswR7soeTHsGHDGr0vedOCCA0NBSGkycc333wj1XlCeKjKtajKdbTWawkNDW30vuSNQTAYDP7B\nDILBYDSK4AzCw8ODawlyQ1WuRVWuA2DX8jIiQgiRpYC5c+fizJkzMDIywq1btxo8Z+nSpTh79iza\ntm2LXbt2wdnZ+VUhIhFklMJgMFqApHtP5hbEnDlzEBwc3Oj7QUFBSElJQXJyMrZv345FixbJWiWD\nwVASMhvEkCFD0KFDh0bfP3nyJGbPng0AcHd3R1FREXJycmStlsFgKAGFj0FkZGTA0tLy+bGFhQXS\n09MVXS2DwZADGsqo5OX+jUgkUka1jBZSUlmCh08eIrMkE3llech9movcslwUlBegrLoMZdVlKK8p\nR3l1OcREDDWR2vOHhpoG9NrooZ1WO+i30Uc7rXbo1LYTTPVNYapnChM9E5jpm0FLQ4vry2RIgcIN\nwtzcHGlpac+P09PTYW5u3uC5fn5+z597eHio1Igy36iurUZSfhLu5N7B7ce3EZ8bjwdFD5BalIry\n6nJYGVjBTN8MhrqG6KTTCYa6hnAwdICupi7aaraFjqYOdDR0oK6mDjERP39U11ajtKoUxZXFKKkq\nQXFlMW4/vo3zD84jqyQLWaVZyCnNgZGuEbp37P780cuoF/qY9IGJngn7A6JgQkJCEBISItW5Ms9i\nAEBqaiomTJjQ4CxGUFAQ/P39ERQUhIiICHz44YeIiIh4VQibxVAYYiJGUn4SItMjEZlBH/G58ejc\nvjMcDR3haOgIB0MHWHe0hpWBFQzbGir0Jq0R1yDtSRpSClKQUpCCpPwk3Hp8C7HZsVBXU4eTsRPc\nzNwwyHIQBloORAedxse4GLIj6d6T2SBmzJiB0NBQ5OXlwdjYGN9++y2qq6sBAAsWLAAAvP/++wgO\nDoauri7++OMPuLi4NEsko3kQQpBckIwL9y/gwoMLCEkNQTutdnC3cIe7OX30MekDHU0drqXWgxCC\nzJJMxGbHIiojCmFpYYjKiELn9p0xuPNgjLIehZFdR6K9dnuupaoUCjUIecEMQjaqaqtw8cFFnLh7\nAqeTToOAYGTXkRjZdSRGdB0B83YNd+v4To24BnHZcQh9GIpz984hLC0MfUz6YLT1aEyynwRHQ0fW\nJZERZhAqSmVNJc4kn8GhO4fw972/0aNTD0y0mwgfOx/Yd7JXyRunvLoclx5ewtmUs/gr8S/oaOjg\n9R6v4w2HN9DHpI9KXrOiYQahQoiJGJcfXsa+W/twNOEonIydML3ndPjY+cBEz4RreUqFEIJrmddw\nJP4IjiQcgaaaJt7t8y5m9Z4l2BYTFzCDUAFyn+bi9xu/Y1vMNui20cWs3rMwo+cMWLa3bPrDrQBC\nCMLTw7ErdheOxB9BP/N+eM/lPUy0mwhNdU2u5fEaZhACJjwtHP7R/jiTdAaTe0zGIrdF6GvWlzWl\nJVBWXYa/Ev7CtphteFD0AIvcFsHXxReGuoZcS+MlzCAEhpiIcTrpNNaHrUdmSSY+6PcBZveZjY46\nHbmWJjhis2OxJXILjiUewxT7KVgxeAVsX7PlWhavYAYhEGrENdh3cx/Wha2DjqYOVgxagSk9pkBD\nTSkLXlWavLI8bI3eii1RWzCy60h8MeQL9DbuzbUsXsAMgueIiRgHbx+EX6gfTPRMsHLoSozoOoJ1\nIxRASWUJtsVsw8bwjXA3d8eqEavQ06gn17I4hRkETyGE4OTdk/jq4lfQ1dTFqhGrMLLrSGYMSqC8\nuhwB1wKwNmwtxnYfi289vkUXgy5cy+IEZhA85GbOTXz090fIKc3BWs+18LbxZsbAAU8qnmDD1Q3Y\nem0r5vSZg5XDVqKdVjuuZSkVhQaMYTSP3Ke5WHh6ITx3e2KK/RTELozFeNvxzBw4or12e3w/4nvc\nWXwHheWFsPe3x564Pa3qj5UkWAtCSRBCEBgXiBXnV2C643R84/ENm5XgIRHpEVgStAQ6GjrY6r21\nVQxksi4Gx6QUpGDB6QUoqijCjgk74GL66mY1Bn+oFddi5/Wd+OriV1jktghfDvlSpeNXsC4GR4iJ\nGJvCN6H/zv7wtvFG5HuRzBwEgLqaOha4LUDcwjjEZsfCdbsrojKiuJbFCawFoSAePXmE2cdno0Zc\ng8BJgejWoRvXkhgtgBCCP2//iY/+/gi+Lr5YOWylyi3dZi0IJbP35l64bnfFaOvRCJkdwsxBwIhE\nIszoNQOxC2MRnRmNobuG4kHhA65lKQ3WgpAjZdVleD/ofYSnh2P/lP1wNn01/wdDuIiJGD9H/Iz/\nXfkffh7zM2b0msG1JLnABimVQHJ+Mt44/AZ6GvXEtvHboNdGj2tJDAVxI+sGph2ZhtHWo7Fx9Ea0\nUW/DtSSZYF0MBXM88TgG/j4QC10XYu/kvcwcVBxnU2dE+Ubh4ZOHGLl7JLJKsriWpDCYQcgAIQRr\nLq/BB2c/QNDMICzqu4gteGolGGgb4Pj04xjVbRT67uiLiPRXAzGrAqyL0UIqairw3sn3cDf/Lk5M\nPwEzfTOuJTE44nTSacw9MRe/jPsFUx2nci2n2bAuhpzJL8vHiMARqKqtQui7ocwcWjnjbcfj3Kxz\n+Pjcx1gftl5Qf+iaghlEM3n05BEG/zEYQ7sMxZ9v/Im2mm25lsTgAX1M+iB8Xjj23dqHxWcWo1Zc\ny7UkucC6GM0gITcBo/eOxof9P8THAz7mWg6Dh5RUlmDinxNhpGuEPZP3CGJRFZvmlAPXMq9h/P7x\n+MHrB8xymsW1HAaPqaipwNTDU0EIweGph3mXoOhl2BiEjFzLvAbv/d7YNn4bv8zh3j3AzAx4Ifcp\ng3u0NbRx7M1j0NfSx7j94/C06inXkloMM4gmiM6Ihvd+b+yYsAMT7SdyLac+p04BhYWAsTHXShgv\noamuib2T98LKwAo+f/qgvLqca0ktghmEBJ61HHZM2AEfOx+u5bzK5ctA//5AG2Gv5FNV1NXUsXPC\nTpjomWDywcmorKnkWlKzYQbRCIl5iRi/fzy2T9jOT3MAgCtXgKFDuVbBkIC6mjoCJwVCr40eph6e\niuraaq4lNQtmEA2QXpyO0XtHY63nWkyyn6T4CisrgY8+knzOokVAz57AoUPA2LG05ZCbC/z7Lz3e\nulXxOhktQkNNA/tf3w8xEcP3lC+vB+NfgfAEvkjJe5pHHH5xID+E/aC8StesIeTWLcnnzJlDiEhE\nSG4uPQ4IIERLi5CKCsXrY8iF0spS0m9HP/LlhS+5llIPSfcea0G8QGVNJSYfnIxx3cfh04GfKqfS\n8nIgK4u2Dl6koKD+8ZYtQIcOgIEBPb54EejXD9BS3VBoqoZuG12cnnEaB+8cRMC1AK7lSAUziP8g\nhGDhmYV4re1rWOe1TnkVBwUB48fXf+3ECWDWS9OpurqAhweg8V+WrZAQYNgwZShkyBFDXUMEvxWM\nb0O/RXBKMNdymoQZxH/8cPUHxGbHYs/kPVATKfFruXgRcHev/9rJk4CbW/3XUlMBR0f6/M4d4PFj\nZhACxbqjNQ5PPYx3/noHSflJXMuRCDMI0N14myM349SMU8qP5fDo0avTlHfuAG+9Vf+1b76pa1Vc\nvEhbEgMH0uMnT4D0dMVrZciNwZ0HY9WIVZj450Q8qXjCtZxGafUG8aDwAeaemItDUw/Bop2F8gUQ\nAvz9d93xli1AdHTd2IJYDKxcSQ3Bxoa+dvky4OwMtP1vo9jPP9d1PRiCYb7rfAy3Go63/3obYiLm\nWk6DtGqDqKipwBuH38AXQ77AQMuB3IhwdQVmzwbmzAFGjgTWrAEGDwZGjQIWLKCDlydPAj/+WPcZ\nsRjo8l8eyehoahQmJtzov3eP6nnGkyfU9BhS8dOYn5Bflo8NVzdwLaVhlDeZIhkupCw8tZC8fvB1\nIhaLlV73cwoKCBk5khBdXfpvUhIhcXGEdO9OSKdOhMydS0h+fv3PxMURMmgQIZ9+SsgPCp6O/eYb\nQsrL644NDQkpKqo7dnIipKSk7tjMrP7xxImElJXVHaelEcLl981DUgtTidEPRuTqo6uc1C/p3mu1\nBnE0/iix/tmaPKl4otR6eY+XFyEPH9Ydb95MSHFx3XFZmfQ3uFhMyNmzdedXVFDjq6mhxzU1hISG\nyke3wDmecJx0+bELKSgrUHrdku69VrndO6skC87bnHF8+nH0t+ivlDp5y8yZtCvzbEYkJYV2XzSV\nEMcgOxtYtgw4eJAel5UBxcXcdZc4ZtnZZcgszcShNw4pNbYpiwfxAoQQjNs/Dv3M+uHb4d8qvD7e\nsWcP0K4dMPG/nakZGfSGVFfnVhcAXL1KB2kPHOBaCSdU1FTAZZsLVg5biek9pyutXmYQL7A1eit2\nxe5C2NwwQUT7kZnqarqG4tkMyI0bgJ5e3TGf2bKFDsDOm8e1EqURnRGN8QfGI3ZBLEz1TZVSJzOI\n/0h7kgbnbc64MvcK7DvZK7Qu3nD1KrBzJ/D771wraT6PH1ODMzenx3fvAra2gIqnFvjq368QlxOH\nk9NPKqWrodCIUsHBwbC3t4eNjQ3WrXt1iXJISAjat28PZ2dnODs7Y9WqVbJW2SIIIVgStARL3Zeq\ntjmIxcDGjUBFBT0eOFCY5gAARkZ15kAIsHQpbQ2pOCuHrURqUSoO3jnItRTZpg5qamqItbU1efDg\nAamqqiJOTk4kPj6+3jkXL14kEyZMaLIsGaU0yaHbh4jDLw6ksqZSofXwgg0b6nZ9qip5eYRcucK1\nCoVx5eEVYrbRjBSVFzV9soxIuvdkakFERUWhe/fusLKygqamJqZPn44TJ040ZEKyVCMzxZXF+PDv\nD7Fjwg7B51FskOjo+gN7n3wCdOrEnR5lkJoKnDvHtQqFMajzIIyxHoOvL37NqQ6ZDCIjIwOWlpbP\njy0sLJCRkVHvHJFIhKtXr8LJyQnjxo1DfHy8LFW2iNWXVmO09WjuVksqGj09QF+faxXKxdUV+PaF\nWaiYmPorOlWAdV7rcPDOQVxKvs6ZBpkW8EszgOLi4oK0tDS0bdsWZ8+exaRJk5CU1PAONj8/v+fP\nPTw84OHhIYs8AEBKQQp+u/Ebbi++LXNZvOL0aWDIEKB9e6BHD/porYjF1Cy2bQNMlTPyrwxyH3ZC\np5vfY2reR8j+X4jcBixDQkIQEhIi3cmy9F3Cw8PJ6NGjnx+vWbOGrF27VuJnrKysSP7LS4eJ4sYg\nJh6YSNZelqxJkHz3HSF37nCtgp9UCnucqaCAkGXL6Er7DZuqiYO/IzmecFxh9Um692S6K6urq0m3\nbt3IgwcPSGVlZYODlNnZ2c/3OkRGRpIuXbo0W2RLufjgIun2czdSUa0iYdlKS7lWIAwWLybk0CGu\nVTSb6mpCtm4lxMiIkAULCHn8mL4elBREbLfYkqqaKoXUqzCDIISQoKAgYmtrS6ytrcmaNWsIIYQE\nBASQgIAAQggh/v7+xNHRkTg5OZEBAwaQ8PDwZotsCWKxmAzYOYDsjdsr13I5o6KCEAcHQgoLuVbC\nf548IaRKMTeTorhwgZBevQgZNoyQ2Nj674nFYuK525P4R/orpG5J957KLpQKSg7C8n+WI25hHNTV\neLCMWB48fUpDzzGkJyqKLiN3deVaSYPcvw989hlw/TqwYQMwZUrD68CuZ13HhAMTcG/pPWhraMtV\nQ6tLvScmYnz171f4bvh3wjaHmhrgjz/qRueZOTSfnBwaFJhnlJYCX34J9O0LuLgA8fHA6683vkjU\nxdQFbmZu2BGzQ6k6VdIgTt6lS1Qn20/mWopsVFcDCQk0bwajZUyY8GpQYA4Ri4HduwE7O5pS9eZN\nahQ6UuT3XTl0JdaGrUVFTYXihf6HynUxCCEY8NsAfDbwM7zu8LoclDFUhh9/BCwtgTfe4KT6iAi6\nux2gUQL7tyDSgM8BH4yyHoX3+70vN12tqotx+dFlFJQXKCcjliKoraVZtHJyuFaienh707UjSiYj\ng8Ybfv11YMkSIDy8ZeYAAF8O+RIbwzeiVlwrX5GNoHIGsS5sHT4d+Klwxx7U1Wkqvdde41qJ6mFr\nW5cJXQkN5/JyYPVqoHdvoHNnuhn1nXcANRnuOncLd5jpm+F44nH5CZWAShlEfG48YjJj8I7TO1xL\nkQ0fHxalWpEQQgPmKGjZPyHA0aOAgwOdnYiOpkahJ6eMCh/3/xibIjbJp7AmUCmD+DX6V/i6+Mp9\nGkgpbNpEl08zFI9IRL9vBSxPj4sDhg+nK79/+40aRbdu8q1jkv0kZJZkIjI9Ur4FN4DKGERpVSn2\n3dqH+a7zuZbSMkaPpvNdDOXQvbtcA8/k5gILF9JsBdOm0ZbDiBFyK74e6mrqWNJ3CX699qtiKngB\nlTGI/bf2Y2iXobBsb9n0yXzE0REwM+NaResjLo7mJGkhVVV0csTBAdDWBhIT6RizonuIs51m43ji\ncRRVFCm0HpUxiG0x27DIbRHXMprH06d0Ery6mmslrRcHB+Cjj1r00bNn6QDk338DoaHATz/RBOzK\nwFDXEKOsR+HALcUG+FUJg7j9+DYeP30ML2svrqU0H2trNiDJJZqa9C5vBnfv0hnTZctodL+zZ6nP\nKJv3XN7DjuuKXVmpEgax7+Y+zOw5U7lZueWBri4wd67KB2EVBGIx8N13QElJo6cUFdFgXYMG0fGF\n27epUXD183l280ReWR5u5dxSWB0Cu6NeRUzE2HdrH97u/TbXUqSnoIBm8GbwBzU1Gmym9tUFSLW1\nwPbtgL09zetz5w41ipeTsisbNZEapjlOU2hwW8EbxOWHl9FRpyN6GffiWor0xMW12uQwvMbXFzAw\nqPdSaCjdCLp3L+1K7NhRt9aKD0zvOR0H7xxUWNxXwXd+jyYcxZuOb3Ito3kMH04fDH7y8CFSq8yw\n/EtNREYCP/wATJ3Kz56giymdGr+edR2uZvLf0i7oFgQhBMcTjwtn3wU/9sUxJPD0KXDT+3MscItB\nz550M+2bb9Y3hxZOeigEkUiENx3exOH4wwopX9AGcSP7BrQ1tNGjk0ACtu7fD6xcybUKRgMQAuzb\nR8cZ1vXeh523+2PlSpr570Xi4+kaKz7hY+eDM8lnFFK2oLsYz1oPysyELBOvv06X3DF4RXQ0nbKs\nqgL+/BMYNKjx/0+XL/Ovd9jXvC9ySnPwsOghuhh0kWvZgm5BBCUHYbwtf4KBNIm2No1HwOAFWVnA\nu+/SfVu+vjQ63aBBkj+TkMDNmgdJqInUMNZmrEJaEYI1iILyAiTlJ6G/RQs31iuTigqaRJfBCyoq\ngLVrgV69ABMTujx6zhzptmHztbHqbePNDOJFLj64iMGdBwsjld6DB3SejMEphNDl0KamNLpTRAQ1\ninbtGv9MdTWd2pw3jwZ9SU6mrY4tW/iVyGtE1xG48ugKasQ1ci1XsAZx/v55jOw6kmsZ0tGjB7B1\nK9cqWjW3bgGenkBAAG0FfPdd04ONDx8CQ4fSsYmdO2n343//A3btosutZ89WinSp6NS2Ezq374wb\nWTfkWq5gDeJi6kWM6Kqg/bQMlSE/n4Z5GzkSmDyZLo/28wM+/7zpz3l40FbDkiXUVG7dot0SAPji\nCzrrcfOmoq9AeoZ1GYbQh6FyLVOQBlFQXoDMkkz0Nm7eJhtOWLWKJj9gKJXqamDzZtp4U1Ojg4vv\nv0/3xS1YQKcrw8Ia//yKFXRR5eLF9PjlJSzPdm0mJipGf0tgBvEfURlRcDVzFUbcSRsbwNCQaxWt\ninPnACcn4NQp4N9/6XjBiyE+tbRospr16xv+fHU1XbLyzguRC2/fpiE7npGQQP/l06TUoM6DEJ4W\nLtdl14I0iIj0CPQ3F8DsBUDDC+nrc62iVZCcTMN5Ll5MBx/PnQN69mz43HffpYOUDbUACgroTIed\nXd1rISG0m/KMffvo+wMGyPMKZMNM3wya6pp49OSR3MoUpEFEZkTC3cKdaxkMnvDkCW0RDBgADB5M\nd1v6+EiekmzblhrJTz+9+p6REd2QVVxc91pCQp1h3LxJByr/+EOulyEX3MzccC3zmtzKE6RBxGXH\nPd+kwlsePar/J4chd2praWBYe3s6qHj7NrB8Oe1CSMN77wGHDgFlZfVfF4loC+Tnn2lLQiyuWyNx\n8SJtfRw71vLcForE1dQVMVkxcitPcEut88vyUVpVCst2POr8NYS5ObBtG9cqVJYrV+jyaG1tOtbg\n5tb8MszN6U1+9CidrXiR2bNpz3D2bKB9e+DePTqbYWZGuxuS1k5wiYupC7ZGy3FKXSH5xFuAtFJC\nU0PJgJ0DFKyGwVcePiRk2jRCLC0J2b+fELFYtvIOHSJkxAjJ52zeTEhysmz1KIvk/GTS5ccuzfqM\npHtPcF2MO4/voKdRIyNPfKG2tsHIRIyWU1ZG1y84O9MuRUICMGOG7Eufvb3pZq38/MbPSUri3w7O\nxrAysELO0xyUVZc1fbIUCM4gEvIS+L+9+9o1miCBITOE0B2W9vZ0xuH6dWoUurryKb9tWxpf8uxZ\nyRqEgoaaBrp16Ibk/GS5lCc4g0gtSkXXDl25liEZd3fgjGL257cmYmJort316+m04p9/Al3ku5sZ\nAJ3xOHGi4ffS0oTTeniGfSd7JObJZwWX4AziQdEDWBlYcS2jabQFmP6PJ2Rn081R48fTGYPoaMUm\n5R49ms5ONLT5ytIS+PBDxdWtCLoadMXDJw/lUpagDIIQgtSiVP4bRF4e1woESWUljf/YsyfQsSPt\nUrz3Hk14rkjMzemMxd27iq1HWVi0s0B6cbpcyhKUQRSUF0BdpA4DbYOmT+aKwkLaxRBSx5VjCAFO\nnqRLmS9doqEzfviBTi8qiyFD6NSpKmDZzhJpxWlyKUtQBpHzNAcmeiZcy5BMhw5ASgp/I4vwjDt3\naBN/xQrA35+uabC1Vb6O/v1pRClVoNW2IHKf5sJQVwAbn5g5NElBAbB0Kd1SPX48Xb48Zgx3ehwd\nVSeXkZm+GbJKsuRSlrAMoiwXhm15bhA5OUB5OdcqeEtNDfDLL3Qbdk0NXc+wdClNkckljo50C7gq\n9Aw76HRAYUWhXMoSlkE8FYBBrFlD0z0zXuHCBaBPH7q0+Z9/aJCtTp24VkXp1ImaVHY210pkR1dT\nF9W11aiqrZK5LEHtxSiqKEIHHSXlV28pP//MtQLece8e8OmnNOPgxo3ApEn87IVZWACZmTRmpZAR\niUQw0DZAYXkhjPVkyxMoqBZEWXUZ2mq2bfpEBi8oKQH+7/+Afv3oIz6ehn3jozkAdCNWZibXKuSD\nvLoZMhtEcHAw7O3tYWNjg3Xr1jV4ztKlS2FjYwMnJyfcuNHyoJqCMIh79/gV7pgDxGIaL8HOjuae\nuHWLxoDk+9oxU1OqVxXQ0dBBRU2FzOXI1MWora3F+++/j/Pnz8Pc3Bx9+/aFj48PevSo2ysRFBSE\nlJQUJCcnIzIyEosWLUJERESL6iuvKee3QRBCd//cvMl9bniOuHqVbsNWVweOH6ctB6HQvn39IDFC\nRlNdUy5jEDK1IKKiotC9e3dYWVlBU1MT06dPx4mXFrWfPHkSs/+LD+7u7o6ioiLk5OS0qL7ymnJo\na/D4z5BIRJf/tUJzSE8H3nqLJrpdtowahZDMAQB0dGiAGFWgjXobVNdWy1yOTAaRkZEByxeidlpY\nWCAjI6PJc9LTW76IQwSedmBbKeXF1QicdwlOTkC3btQf335buixVfENbW3VmqK9lXkNGSUbTJzaB\nTD+jtElzyUuTy419zs/P7/kjJCTk1foggpjwuH8vFtMxiFZE/o1HeDPQG/kFIny/SgQ9fRFtSYlE\ndF92Q/j51Z0j4s/5X30twuo1CtTT2PsKoKq2qtHgtSEhIfXuNYnIEr0mPDycjB49+vnxmjVryNq1\na+uds2DBAnLgwIHnx3Z2diQ7O7tZUW2eMe/EPLIjZocMihVMWRkhjo5cq1A+CQk0xJO/P9dKGP8x\n9I+hJORBiFTnSrr3ZGpBuLm5ITk5GampqaiqqsLBgwfh4+NT7xwfHx/s3r0bABAREQEDAwMYG7ds\nbpb3LQgdHRo5tbVhbw+EhgKbNtForwzOqRHXQENN9mVOMpWgoaEBf39/jB49GrW1tZg3bx569OiB\nbf8Fa12wYAHGjRuHoKAgdO/eHbq6uvhDhljhbdTboLKmUhbJDEXRtSvdiunlRRdArFrF3wUPrYCq\n2ipoqsu+fl30XxODc0QiUZMZgf7v/P+hvVZ7fD6kicSKXJKaSlfctMKZDABAbi7dnjlkCPDjj8Ic\nrVQB7P3t8de0v9DDsOnwjJLuPUH9evpt9FFSVcK1DMksWkRzYrRWDA1pvrvoaMDXlwXv5YjiymK0\n05I9Nr+wDEJLHyWVPDeIs2eFF8RQ3hgY0Lx3Dx/SxRHVss/HM5pHSVUJ9LVkT/koKINop9UOTyqf\ncC2DIQ16esDp0zRe/ZQpwlmB9NlnXCuQGTERo6y6DHpt9GQuS1AGYaxrjMdPH3MtQzJFRTQUMoOu\nPDp6lMao9/YGSku5ViSZggKVSLScV5aHDtodoCaS/fYWlEGY6Jkgu5TnG/ZDQ4Fff+VaBX/Q1KQx\n67t2pblCioq4VtQ4ly4Bw4dzrUJmskqyYKovnz3rzCDkzcSJNGgMow51dWD7dro5Y/hwOtPBRyIj\naYpwgZNVmgUzfTO5lCUogzDUNUR+eT5qxDVcS2E0FzU1Ou05fjwwdCiQIfs+AblTXQ1oCCqGUoNk\nlmTCVK8VtiA01DRgomeCjGIe/ud6kaQk1YhdJm9EIuD772k2nKFDgQcPlK8hNZUGwZwwATh0qO71\nvDzAyIjuS7fkAAAbnUlEQVQ+r6ykyTlWr1a+PjmQXpwOc31zuZQlKIMAAOsO1kgpSOFahmQOHQJi\nY7lWwV9WrAA++YSaRKJ8UsRJzYYNdEm4l1d9A7h0iYbYfkZVFRAerlxtciKlIAXdO8pnql1wBtG9\nY3f+G8RXX3Ebw10ILF5Ml2MPH648M302xqChAZw/T/enP+P6dcDNjT7X0qKxRZ+1KARGckGy3AxC\ncB0u6w7WuFfYurZUqyyzZ9Mp0NGjafbc/v0VW1+XLoCTE+3+BQcDe/fWvScW118WbmEB9O2rWD0K\nIqUgBTav2cilLMG1IGxes8HdfAEkUbx8Gbh/n2sV/OeNN2gASx8fukRbkZiY1K3N0NCgdQJ0VsXk\npYxtUVF0gZfAKCwvREVNBYx1ZYtm/QzBGURv496Iy47jWkbT3LmjOhFQFc3YsXTcZvp04MwZxdd3\n5QrNn/osim5oaP3xh4oKup+mhWEJuOTW41voadRT6mBOTSE4g7DuYI28sjwUVfB4wQ0ALFwIDBrE\ntQrh4OFBE3POnVt/dkERPH1af3zh9m2gd2/6XCym61g+/lixGhREbHYs+hj3kVt5gjMIdTV19DLu\nhZs5N7mWwpA37u405daHHwIyxA1pkmnT6IBlWVn913Ny6AzLO+8oN7W4HInNjkUfE/kZhOAGKQGg\nj3Ef3Mi6gaFdhnItRTJXr9J595kzuVYiHHr3Bi5epNOQpaXABx/Iv4633qIthffeozMXkZF0ylNb\nm8aN1NWVf51KIjY7FgtcF8itPEEaRD/zfjh3/xyWYRnXUiRjYCDYqTJOsbOj6xI8PalJfK6AAEGz\nZtGHvz/NB+jgIP86lExZdRnu5t9FL+NecitTcF0MABjUeRCupl3lWkbTODjQ/+SM5mNlRU1i717g\niy8Ul3b79m2VMAcAiMqIQi+jXnJNLiVIg7DpaIPy6nKkF7c8v4ZS4UdUP+FhZkZnGP7+m2bjkXdK\nQxX7XcIehWFw58FyLVOQBiESiTDQciDCHoVxLaVpxGK6AKiF2cRaPZ060fUR168D8+bJN4RdfDzQ\nR34DelxzJe0KM4hnDO0yFBdTL3Ito2nU1ICDB9lYhCy0b09bEenpwIwZdJ+EPHB0pNPRKkB1bTXC\n08IxyFK+U+uCNYhR1qPw972/m4yEzQusrFgIeFnR1aXrJCor6QpHVcmRJyci0iNg3dEahrqGci1X\nsAbhaOiI6tpqJBckcy1FOqqqgGvXuFYhbLS1gSNHgHbtaAi7Ep4HMFYiwSnBGGMt/w2CgjUIkUhE\nWxEpf3MtRTrS0+k2Y4ZsaGoCe/YA1tY0hF1hIdeKeEHwvWCM6c4Moh5juo9BUEoQ1zKko1s3YP9+\nrlWoBs9C2A0YQLeLP+Z5IGMFk1Oag3sF99DfQv67YQVtEGO7j0XYozD+78tgyB+RiC5wmjgRGDaM\nttBaKccTj2OczTi5pNp7GUEbhL6WPoZ3HY5Td09xLUV6bt+mcRAYsiMSAd9+Szd4DR3aarfXH0k4\ngjcc3lBI2YI2CAB4vcfrOJpwlGsZ0mNnJ9idgrzls8+A5ctpSyIhgWs1SiWvLA9RGVEKGX8AVMAg\nJthOwL8P/kVxZTHXUqRDU5NGNWLIl4UL6TbtESOAGze4VqM0jicex2jr0XJdXv0igjeIDjodMKLr\nCByJP8K1lOZRUQHs3s21CtVi1izgl19oPNCrAtirIwf23NyD6T2nK6x8wRsEALzb513sit3FtYzm\nExdHF/4w5MeUKUBgIDBpEnDhAtdqFMq9gntIyE3AeNvxCqtDJQxinM04JOYl8j/a9Ytoa9NReC0t\nrpWoHmPG0AVVM2bQ1Zcqyq7YXZjZaybaqLdRWB0qYRBt1NtgZq+ZCIwN5FpKy7h/n4ZBY8iPoUNp\nfEtfX7oXRsUQEzEC4wIxp88chdajEgYBAL4uvth5YyeqauW0kUeZbNkChAlgZ6rQ6NuXhrD7+GPg\n99+5ViNXziafhZGuEZxMFDvgLciIUg3haOSIHp164Ej8EczsJbAQbz/+yLUC1aVXr/oh7JYu5VqR\nXNgctRlL3RV/LSrTggCAD/p9gC1RW7iWIRvJAtl8JiRsbWl0qi1baOxJIewAlkBiXiLisuMwzXGa\nwutSKYOYYDcBmSWZiM6I5lpKyygooIlt5RXvgFFHly7UJPbvp3lBBWwS/lH+mO86H1oaih/gFhGe\nBFQQiURyie2wKXwTItIjcGiqgnMrKApCWOwIRZKbS/OVjBoFbN5cP92eAMgry4PtFlvcXnwbZvpm\ncilT0r0nrG9HCua7zkdIaggS85ScNVpePDOHsjI2cKkIDA2B6Gi6BmXuXKCmhmtFzeKniJ8w1WGq\n3MyhKVTOIPTa6OGDfh9gXdg6rqXIRkoKzSHJkA+LFtF0iAANYRccTFMjyjOEnYIpqihCwLUArBi8\nQml1qlwXA6AJTLtv6Y6Y+TGwMrCSS5kMgRMeDri41F+YVllJ84FWVlIz1tHhTp8UrL60Gnfz72L3\nZPku0W9VXQyA7s9Y7LYY34Z+y7UU+XDzJvD111yrEB6PH9cNRg4Y8OqqVS0tmge0Y0dg3Dheh7Ar\nqijCz5E/44shXyi13hYbREFBAby8vGBra4tRo0ahqKjhoC1WVlbo3bs3nJ2d0a9fvxYLbS6fDvwU\nQclBuPP4jtLqVBhdutDISYzm8c47wK1bks/R1KSb5mxt6VoJnoawWx+2Hj52PrDvZK/UelvcxVi+\nfDk6deqE5cuXY926dSgsLMTatWtfOa9r166IiYlBx44dJQuRYxfjGZvCN+HSw0s4Pv24XMvlFEKA\nxESgRw+ulfCf6mpqANJACI0r8c8/wLlzgLGxYrU1g8ySTPT6tRfiFsbBop2F3MuXeO+RFmJnZ0ey\ns7MJIYRkZWUROzu7Bs+zsrIieXl5TZYng5RGKa8uJ5abLEnYozC5l80Z6emEeHoSUlvLtRL+UVFB\nyJw5hBQXt+zzYjEhfn6E2NkR8uiRfLXJwPyT88nyc8sVVr6ke6/Fd6WBgcHz52KxuN7xi3Tt2pX0\n6dOHuLq6ku3bt7dIpCzsjt1NXLe5kpraGoWUzwlicd3zqirudPCREycIqZHxt96wgRArK0JSUuSj\nSQZis2KJ0Q9GpKCsQGF1SLr3JO7F8PLyQnZ29iuvr169+pUmiqiRxT1hYWEwNTVFbm4uvLy8YG9v\njyFDhjR4rp+f3/PnHh4e8PDwkCRPKt7u/Ta2xWzD7zd+h6+rr8zl8YIXv2tPT2DbNsBeuX1T3vDg\nAV3TMGkSPfbxkb3MTz4B9PVpCLtz5zhL7ksIwftn38f3w79HB50Ocis3JCQEISEhUotoEXZ2diQr\nK4sQQkhmZmajXYwX8fPzIxs2bGjwPRmkNMn1zOvE6Acjkl+Wr7A6OCP/hWuqra3fumgNJCUR4u+v\nmLL37iXExISQmBjFlN8Eymr9Srr3WjyL4ePjg8BAGn8hMDAQk545+AuUlZWh5L+po6dPn+LcuXPo\n1atXS6tsMc6mzphiPwVf/fuV0utWOC8O/h45QrNgqzoBAUB+Pn1uYwMsWaKYet56C9i6FRg7Vumr\nWp9UPMGK8yvwy7hfoK6mrtS669FS18nPzycjR44kNjY2xMvLixQWFhJCCMnIyCDjxo0jhBBy7949\n4uTkRJycnIijoyNZs2ZNi1xMHhSUFRCzjWbkUuolhdbDKbW1hDx+XHecmamaLYpffyUkNVV59f39\nNyGGhoT884/Sqpx/cj7xPemrlLok3XuKvSubgaINghBCjsUfIzabbUhZVZnC6+IFXl6E3LzJtQrZ\nCQ4mZNUqbjVcukRN4sQJhVd1/t55YrnJkhSVFym8LkIU1MUQIpN7TIazqTP8Qvy4lqIcgoNpwBSA\nbv76+mthbHOurgZiYuqOnZyANxSTGEZqhgwBgoKA+fOBAwcUVk1pVSl8T/kiYHwA2mu3V1g90tKq\nDAIAtozdgsC4QISnhXMtRfG8uJW5shKwsqqbAcnLA+7d40RWg7y4q7K0FPjiC0AspscmJjThENe4\nuQHnzwOffgrs3KmQKj4//zmGdBmCcTbjFFJ+s1FKG0YKlCnlr4S/SNefuiqtCcdLzp8nZMWKuuOc\nHEKKlPh9iMX1F3v16EFIWpry6peFpCRCunQh5Mcf5VrsqbunSJcfuyh0zUNDSLr3VHI3pzQsOr0I\nJVUl2Dtlr9Lq5DUBAfQv96ef0uPr1wEDA5qVXB7cv0/Lezbr4u0NrFhBo08DNKq3rq586lIGjx7R\nNSjvvAN8+aXMQX6ySrLgst0FR6YewaDOg+QkUjoUstRa3ihbytOqp6SHfw+yJ26PUusVDDt3EnLm\nTN3x998Tcvp03fHJk4TExtYdnz5NyK1bdccbNhDy7791x598Qlstzygpkb9mZZOVRUjPnoQsXy7T\nbFGtuJZ47vYkfhf95ChOeiTde622BQEAcdlx8NzjiZDZIXA0clRq3YIjJQXQ06PjAQBw+DDQvTvg\n7EyPDx6kx66u9PjGDcDICDA350avssjPp4l6+vYF/P1bFMLu+9Dv8c/9f/Dv7H+hoab8QPOS7r1W\nbRAAEBgbiNWXVyPaN5oXo8YMAVJcDIwfD3TtCvz2G6Ah/U0elByE+afmI9o3Gqb6pgoU2TjMIJpg\nyZklyCjJwLFpx6AmanUTOwx5UFZG84Lq6dHI2W2aToeXUpCCQb8PwrE3jyl93OFFWl1Eqeby45gf\n8fjpY3wX+h3XUhhCpW1b4MQJOjU7cSI1DAmUVpViysEp+GbYN5yaQ1MwgwDN7Xls2jHsit2FvTfZ\nrAajhTwLYWdoSPdvFBc3eFqNuAbTjkyDu7k7FrktUrLI5sEM4j9M9ExweuZpfPz3x7j08BLXchhC\nRUMD2LWLbhH39KTJkF6AEIIPgj5AjbgGW723NhomgS8wg3iBnkY9sW/KPrx5+E3czbvLtRyGUFFT\no7tAhw0DPDyAnJznb20M34iwtDAcnnoYmupShsPjEGYQL+Fl7YX/jfwfRu0dhUdPHnEthyFURCJg\n/Xpg6lS6jyMtDYGxgfg58mecmXkG7bTaca1QKlQmu7c8meM8B0UVRfDc7YnLcy7DWI8/AUwZAkIk\nohvk9PTwdIAbAmaJ8c8nl2HZ3pJrZVLDDKIRPhrwEZ5UPsGovaMQMjtEriG/GK2L4Ak9EBxejl1D\nfoGdksPWywpbByEBQgg+OfcJLj28hHOzzqGjjuTQ/QzGy/xz7x+8dewtnJh+AgMsB3Atp0HYOogW\nIhKJsHHURgy3Go4RgSOQ+zSXa0kMAXE66TTeOvYWjk07xltzaApmEE0gEomw3ms9xtuOx/DA4cgu\nfTXKN4PxMscSjmHeyXk4PfM0BncezLWcFsMMQgpEIhFWjViFaY7TMOSPIbhXwKNAKwzeERgbiMVn\nFiP4rWD0M1deuklFwAYpm8HXw75Gp7adMOSPITg54yTczNy4lsTgEYQQrL68Gr/d+A0h74YoPY+m\nImCDlC3geOJx+J7yxZ7JezCm+xiu5TB4QI24BkvOLEF0ZjTOzDzD2c7MlsB2cyqAq2lXMeXgFHw1\n9Css6buE90tmGYqjqKIIM4/ORC2pxZGpR6Cvpc+1pGbBZjEUwEDLgbg67yoCrgVg/qn5qKqt4loS\ngwMS8xLhvtMd3Tt2x+kZpwVnDk3BDEIGunXohvB54cgrz8OIwBFshqOVcTrpNIb+MRQrBq3A5rGb\nBbG3orkwg5ARfS19HH3zKDy7ecJ1uysu3L/AtSSGgqkR1+DLC19i4emFODH9BOY6z+VaksJgYxBy\n5Pz983jnr3fg6+KLlcNWcptTkaEQ0p6kYcbRGdBto4s9k/fASNeIa0kyw8YglIRnN0/EzI/BlbQr\nGLl7JB4WPeRaEkOOHE88DrcdbphgOwFn3zqrEubQFKwFoQBqxbVYH7YemyI2YZ3nOszpM4fNcgiY\nwvJCLAtehqtpV7Fn8h7BLptuDDbNyRE3c25i9vHZMNc3x44JOwQ1N86gnE0+C99TvphsPxlrPddC\nt42AkvtICeticERv496IfC8SLqYu6B3QG79G/4pacS3XshhSkFOag3f+egeLgxZj9+Td2DJui0qa\nQ1OwFoSSuP34NhadWYSq2ioEeAfA2dSZa0mMBqgV12J7zHZ8E/IN3u3zLlYOWwm9Nnpcy1IorIvB\nE8REjF2xu/D5hc8x1WEqvhn2DQx1DbmWxfiPq2lX8WHwh9DS0MKv3r+ip1FPriUpBdbF4AlqIjXM\ndZ6L+MXxUBOpwWGrA9aHrUdFTQXX0lo1KQUpeOPQG5h2ZBqW9F2C0HdDW405NAUzCA54re1r2Dx2\nM8LmhiE8PRz2/vbYHbcbNeIarqW1KnJKc7Ds7DL039kfLqYuuPv+XczuM5tlV3sB1sXgAZceXsLX\nF79GVkkWvh76NWb0msFJEtfWQnZpNtaHrceu2F14u/fb+GroV61iTUNjsDEIAUAIQUhqCPxC/ZBV\nkoUVg1bgrd5vQVtDm2tpKsOjJ4+wKXwTdsftxqzes7Bi8AqY6ZtxLYtzmEEIiGdG8cPVH3A96zoW\n912MRW6L2GCmDESmR2JTxCacv38e7zq9i08HfsrWpLwAMwiBEp8bj58ifsLh+MOYbD8Zvi6+6G/R\nn63KlILy6nIcTTiKrdFbkVWahWXuyzDXea5gEtYoE2YQAif3aS7+iP0DO67vgJa6FnxdfDHLaRYL\nw98AN7Ju4Lcbv+HA7QNwN3eHr4svfOx82MY5CTCDUBEIIQh9GIod13fQWARdhmK643RMtJ+o8ot5\nJJGcn4yDdw7i4J2DKK4sxjzneXi3z7vo3L4z19IEATMIFaSksgQn757EgdsHcPnRZXh184KPnQ/G\ndh+r8uMVhBDczLmJM8lncDThKDKKMzDVYSqm9ZyGgZYD2TRlM1GIQRw+fBh+fn5ITExEdHQ0XFxc\nGjwvODgYH374IWpra/Hee+9hxYoVzRbJkExBeQFOJJ7AmeQzOH//PHoY9oC3jTcNYmPqqhKRjgrL\nC3Hp4SUEJQchKCUIbdTbwNvGG5PsJ2FYl2GsCyEDCjGIxMREqKmpYcGCBdi4cWODBlFbWws7Ozuc\nP38e5ubm6Nu3Lw4cOIAePXo0SyRDeiprKnH50WWcSTqDi6kXcb/wPgZYDoBHFw8M6jwIzibOvI+b\nSAhBRkkGItMjcenhJYQ+DMW9wnvob9EfY6zHwNvWG3av2bHBWjkh6d5r8Woce/umY/5HRUWhe/fu\nsLKyAgBMnz4dJ06caNAgGPJBS0MLnt084dnNEwCQX5aPy48uIyQ1BMv/WY5bj2+hS/sucDNzg4up\nCxwMHWDfyR4W7Sw4aZpX1FQgOT8Zd/PvIi47DjFZMYjJigEhBH3N+2Jo56EIGB+gMi0hoaHQ5XoZ\nGRmwtKxLdW5hYYHIyEhFVsl4idfavoZJ9pMwyX4SAKC6thp3cu8gJjMG17Ou41TSKSTmJaKoogh2\nr9nBysAKlu0sYdneEpbtLGGmb4YOOh3QUacjOmh3gI6mjlT1iokYxZXFKKooQmF5IR4/fYyMkgyk\nF6cjvTgdj548QlJ+EjJLMtG1Q1fYd7JHT8OemO86H66mrrBoZ8FaCDxAokF4eXkhO/vVSM1r1qzB\nhAkTmiyc/cD8Q1NdE31M+qCPSR/Mw7znrxdXFuNu3l2kFqUirTgN6cXpiMyIRFZJFgorClFQXoDC\n8kIAgI6mDtqot3n+UBOpobq2GtXialTVVqGqtgqlVaXQa6MHA20DdNDugE5tO8GyvSUs9C3gYuoC\nHzsf2L5mi64GXVnLgMdINIh//vlHpsLNzc2Rlpb2/DgtLQ0WFhaNnu/n5/f8uYeHBzw8PGSqnyE9\n7bTaoa95X/Q17yvxvPLqclTUVDw3gsraSoiJGJpqmmij3gaa6vRfvTZ6bD8JTwkJCUFISIhU58o8\nzTl8+HBs2LABrq6ur7xXU1MDOzs7XLhwAWZmZujXrx8bpGQweIZC4kH89ddfsLS0REREBLy9vTF2\n7FgAQGZmJry9vQEAGhoa8Pf3x+jRo+Hg4IBp06axAUoGQ0CwhVIMRiuHRZRiMBgtghkEg8FoFMEZ\nhLSjr0JAVa5FVa4DYNfyMswgOERVrkVVrgNg1/IygjMIBoOhPJhBMBiMxiE8YdiwYQQAe7AHeyj5\nMWzYsEbvS96sg2AwGPyDdTEYDEajMINgMBiNwnuDOHz4MBwdHaGuro7r1683el5wcDDs7e1hY2OD\ndevWKVGhdBQUFMDLywu2trYYNWoUioqKGjzPysoKvXv3hrOzM/r166dklZKR5jteunQpbGxs4OTk\nhBs3bihZofQ0dS0hISFo3749nJ2d4ezsjFWrVnGgsmnmzp0LY2Nj9OrVq9FzZPpNlDUI2VISEhLI\n3bt3iYeHB4mJiWnwnJqaGmJtbU0ePHhAqqqqiJOTE4mPj1eyUsl89tlnZN26dYQQQtauXUtWrFjR\n4HlWVlYkPz9fmdKkQprv+MyZM2Ts2LGEEEIiIiKIu7s7F1KbRJpruXjxIpkwYQJHCqXn0qVL5Pr1\n66Rnz54Nvi/rb8L7FoS9vT1sbW0lnvNiaDtNTc3noe34xMmTJzF79mwAwOzZs3H8+PFGzyU8HDeW\n5jt+8Rrd3d1RVFSEnJwcLuRKRNr/L3z8HV5myJAh6NChQ6Pvy/qb8N4gpKGh0HYZGRkcKnqVnJwc\nGBsbAwCMjY0b/ZFEIhE8PT3h5uaGHTt2KFOiRKT5jhs6Jz09XWkapUWaaxGJRLh69SqcnJwwbtw4\nxMfHK1umXJD1N+FFyB9VCW3X2HWsXr263rFIJGpUc1hYGExNTZGbmwsvLy/Y29tjyJAhCtHbHKT9\njl/+q8uX3+ZFpNHk4uKCtLQ0tG3bFmfPnsWkSZOQlJSkBHXyR5bfhBcGoezQdopC0nUYGxsjOzsb\nJiYmyMrKgpFRw+nmTU1pUllDQ0NMnjwZUVFRvDAIab7jl89JT0+Hubm50jRKizTXoq9flxpg7Nix\nWLx4MQoKCtCxo7DSHcr6mwiqi9FYn9DNzQ3JyclITU1FVVUVDh48CB8fHyWrk4yPjw8CAwMBAIGB\ngZg0adIr55SVlaGkpAQA8PTpU5w7d07i6LQykeY79vHxwe7duwEAERERMDAweN6t4hPSXEtOTs7z\n/29RUVEghAjOHAA5/CYtHz9VDseOHSMWFhZEW1ubGBsbkzFjxhBCCMnIyCDjxo17fl5QUBCxtbUl\n1tbWZM2aNVzJbZT8/HwycuRIYmNjQ7y8vEhhYSEhpP513Lt3jzg5OREnJyfi6OjIu+to6DsOCAgg\nAQEBz89ZsmQJsba2Jr1792501okPNHUt/v7+xNHRkTg5OZEBAwaQ8PBwLuU2yvTp04mpqSnR1NQk\nFhYW5LfffpPrb8KWWjMYjEYRVBeDwWAoF2YQDAajUZhBMBiMRmEGwWAwGoUZBIPBaBRmEAwGo1GY\nQTAYjEZhBsFgMBrl/wEsuN0vtYwNuQAAAABJRU5ErkJggg==\n", "text": [ "" ] } ], "prompt_number": 16 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The representation of a simple harmonic oscillator at a single point in time in the complex plane (e.g. figure above) is often called a _phasor_ diagram, and the arrow representing the amplitude and phase of the oscillator relative to a fixed time or phase is called a _phasor_; note that for a phasor we just need\n", "the arrow, not the associated circle. We will see later that we can combine two oscillations using phasors (or using complex arithmetic - the two are completely equivalent). \n", "\n", "A simple way to think about phase differences is to consider the velocity and acceleration of the oscillator: \n", "\n", "\\begin{eqnarray}\n", " \\psi &=& A e^{i(\\omega_0 t + \\phi)}\\\\\n", " \\dot{\\psi} &=& \\frac{\\partial \\psi}{\\partial t} = i\\omega_{0} A e^{i(\\omega_0 t + \\phi)} = i\\omega_{0} \\psi\\\\\n", " \\ddot{\\psi} &=& \\frac{\\partial^2 \\psi}{\\partial t^2} = -\\omega_{0}^2 A e^{i(\\omega_0 t + \\phi)} = -\\omega_{0}^2 \\psi\n", "\\end{eqnarray}\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice that both of these quantities vary harmonically and with the same frequency as the oscillation, though with different amplitudes; more importantly, however, note that the velocity is a factor of $i$\n", "different to the displacement, while the acceleration is a factor of $-1$ different. These are easier to understand when we write them in complex exponential form: $i = e^{i\\pi/2}$ and $-1 = e^{i\\pi}$. So we\n", "can write: \n", "\n", "\\begin{eqnarray}\n", " \\psi &=& A e^{i(\\omega_0 t + \\phi)}\\\\\n", " \\dot{\\psi} &=& \\omega A e^{i(\\omega_0 t + \\phi + \\pi/2)} = \\omega_{0} \\psi e^{i\\pi/2}\\\\\n", " \\ddot{\\psi} &=& \\omega^2 A e^{i(\\omega_0 t + \\phi + \\pi)} = \\omega_{0}^2 \\psi e^{i\\pi}\n", "\\end{eqnarray}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We see that the velocity leads the displacement by a phase factor of $\\pi/2$ and the acceleration leads the velocity by a phase factor of $\\pi/2$. This phase relationship is shown in the next figure. " ] }, { "cell_type": "code", "collapsed": false, "input": [ "from IPython.display import Image\n", "Image(filename='SHMDispVelAcc.png',width=300) #Size is in pixels" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": { "png": { "width": 300 } }, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAf4AAAHwCAIAAADIB5oCAAAACXBIWXMAAAxNAAAMTQHSzq1OAAAA\nHXRFWHRTb2Z0d2FyZQBHUEwgR2hvc3RzY3JpcHQgOC42MoMtCM8AACAASURBVHic7d1PqCzpWcfx\nt+O9k5gLAzXRjBsRqolEcDV1XZgEZgjVi7hwV82gESYuuglkM26qtoJI1cb4hyBdm5CAmy40K5lF\n1yJINC76HRJwIYPnFRFEGHNeDMyNkxs9Lt5Mpef86VOnu/689b7fz2K459xz73nvma5fP/XUW0/N\nrq6uBADAJx8ZewEAgKER/XDEYrGYzWZjrwKYBqIfALxD9AOAd4h+APAO0Q8HZVk2m82KopjP57PZ\nbDabLZfLqqqWy6X58OnTp1LKsZcJjIboh7OKokjTdLfbJUlicl8IsdvtNpuNUsp8CPjp0dgLAPqy\n3W7jOBZCRFFUVVUcx9vt1vyWUqooilFXB4yJqh/OMrkvhAiCQAgRRdGoywEsQvQDgHeIfgDwDtEP\nAN4h+gHAO0Q/XKOUGnsJgO1mDG2GY9br9WazGXsVgNWIfjhFKTWfz3lVA8fR8IFT6rpu/gvgLkQ/\nnPLd735XEP3AfYh+OOWtt94SQux2u7EXAliNXj/cYRr95te8sIEjqPrhjrqunzx5IoR4/PgxPR/g\nCKIf7vjud7/73nvvCSGeP39O9ANHEP1wh2n0G7T7gSPo9cMRh41+g9c2cBeqfjiiafQbtPuBI4h+\nOKJp9Bu0+4EjiH444rDRb9DuB+5Crx8uMI3+F1988fHjxz/4wQ8+8YlPPH/+/Ic//CEvb+BWRD9c\noLWWUppfLxaLpt6Posg8mBfAIaIfrpnNeFUD96DXDwDeIfoBwDtEPwB4h+gHAO8Q/QDgHaIfALxD\n9AOAd4h+APAO0Q8A3iH6AcA7RD8AeIfoBwDvEP0A4B2iHwC8Q/QDgHeIfgDwDtEPAN4h+gHAO0Q/\nAHiH6AcA7xD9AOAdoh8AvEP0A4B3iH4A8A7RDwDeIfoBwDtEPwB4h+gHAO8Q/QDgHaIfALxD9AOA\nd4h+APAO0Q8A3iH6AcA7RD8AeIfoBwDvEP0A4B2iHwC8Q/QDgHeIfgDwDtEPAN4h+gHAO0Q/AHiH\n6AcA7xD9AOAdoh8AvEP0A4B3iH4A8A7RDwDeIfoBwDuPxl4AcCellFLqrg+11ubDKIoOPxRCLBYL\nIUQYhkEQCCGklIcfGlEUHfkQcNvs6upq7DUAQghR17XWWkqplArDUEppMl08MKZns2OvavMtbv1Q\nShmGoRAiCIIoisIwbBYAOIboxwhM/S6lNKW6yfEgCOI4NrF7zl9+PPrbMG8/ZnlmtebNJooiTg7g\nBqIfQ6jruq5rU86b/xp9JOn50X9Tczpi3hKiKDL/ijiO4zju9nsBAyD60Zcm7oUQURQlSWLaKX3r\nI/pvJaWs61pKGQSBeRugQYSpIPrRJZOGSimtdRRFo6ThYNF/qHkbEEKM9Q8H2iP6cS7bUm+U6D90\neLpjOkLDnO4A7RH9OIVSygScEMK2lvfo0X+oeRswV4mTJOEqMWxg0UEC+2mtq6qq6zoIgiRJ7In7\nQ1ZFf0Nr3bxZRlG0Wq3GXhG8ZuNBAguZxNdaJ0mSJMnYyznGzuhvmLdP0x+z9u0TzrP6IMHopJQm\np0ziT6JZYXn0N5RS5mcbhuFqteJ6AIY0jYMEA5t0Kk0l+hvm/VUpFcfxVN5fMXUTO0jQKzd6EZOL\n/saEumqYuqkeJOhWXddVVQknrkBON/qNw2vpaZpO65QLUzHtgwTnq+u6LMswDNM0daPVMPXobyil\niqIQUz4Dg7UcOUhwAvdC33Am+g2tdVEUSqnVasUbALri1EGClsqyrOva9HZcCn3Dseg3eANAtxw8\nSHBEWZZVVSVJMvWG/hFORr/RvAHEcezw/0EMwNmDBI2yLKMoMjsI3Q59w+Hob/jwFo5e8Wxex2mt\n/+mf/uk3fuM3nj17ttvtSAo3rFar3W4nhFgsFmVZjr0cTI/79ZG3DrvDi8XiF37hF9555x33Ovs3\n+VD1H+IMACeg6ndTWZbL5TKO4+12a4YG/9d//Zd5WDkcY84AgiBYLBbm5gzgXkS/a6SUy+VSa73b\n7ZqtIObW0O9///tf+tKXRl0d+pIkyW63k1Ku12vzqADgCL9Ojd1mOjxa6zzPrzV26rr+whe+8JOf\n/OTRo0df+cpXvvrVr461yAH41vC5xtwIFgRBnudjrwX28vogcUlVVVVVHdn0PZvNzC9eeOGFv/iL\nv3C4L+x59BtVVZVlmaYpNwHgVhwkk6eUyrIsDMPjVd58Pj/sA2y3W1cHhBH9RnOdf7PZ+HB5Hw/C\nQTJhzbGd5/m9Q77efPPNP/3TP20+/PjHP/53f/d3Tj46nOg/ZCqDKIrSNB17LbAIl3mnqq7rZg9P\nm+GOn/nMZ548edJ8+OzZs1dffZXrgc4Lw9C8QhaLhXk8JCCo+qfotOt4Sqn5fH7tk7/6q7/6j//4\nj451A6j6b3VkFwA8xEEyMVmWtezw3PTJT37y3XffPfzMZz/72T/7sz9zrO1D9B8hpSyKgv4PaPhM\nhtZ6uVw25+8n/A3NZo+PfOQjv/RLv7Tb7b7zne84lvs4Loqi7XYrhDA3f4y9HIyG6J8G09lP0/Sc\nTZmvvfaaEOKVV175xje+8Z//+Z88/slbaZqmabpcLs3DOOEhTo0nIMuyTlq0Sikz71cIsV6vhRCb\nzaabJdqEhk9LWuv1ek3zx08cJFYzB2cfw9nNVd/DYQ/OIPofpCgKKSV7/33DQWKvuq6LothsNj11\nZsywFzP71yVE/0NJKbMsy/OcCz/+4CCxVJZlQohex7C4WvgT/Seg+eMbLvNax+zkiaKo7/FbYRiu\nVquiKHr9LpiEIAjY+eMV6iO79N3kuUZrPZ/PzUz/Ab7dMKj6z0HzxxMcJBYZoMlz6zeVUrrU8Sf6\nz0Tzxwc0fKwwWJPnpjRNpZRMd0GD5o8PqI/GZ2Yrpmk61im2Y4U/VX9XTPNnu92y79M9HCQjs+Ho\nMh3/PM/deH4L0d8hM/PntJlRsBkHyZhsyH0jy7Kqqi4uLsZdRieI/m6ZbiQXfh1Dr380ZVlWVWVD\n7gsh0jTVWpdlOfZCYB3T+jddwbHXgs5QH42jLEtz9/zYC/kZZwp/qv4+mHH/5l6QsdeCDlD1j6B5\nYurYC/kQs7mIwh+3Mo8GklLyCnED0T+09Xr90AdsDSZNU27uxRGbzcY8JG7sheBcRP+gzJ0y1p4y\nm4VR1uEIMzzcDP3GdBH9AzHbJJIksTb3DVP4cyMPjlitVlEUkf6TRvQPweR+mqb2j8ox70yc0eO4\n1WqVJAm3+04X0d87pdR6vZ7Qtug0Tcuy5JDGcXEcm6c88lKZIrbB9UtrbeYg2rB5v735fJ4kiZ3X\nou/F5s4h2XNbIh6Eqr9Hps+zWq0md1TkeU7hjzbMzEH6/pNDfdSjxWIxoT7PNYvFYpRJouej6h9e\nXddVVdl2qwqO4CDpy3q9Nhshxl7Iieq6Xi6XFxcXkztlIfpHYeEN6jiChk8vzP796ea+ECKO4yiK\n2OqDlkyhwwtmKoj+7mVZZvN9W+2ZrT5KqRP+bJZl5tawLMtms1nzyebXi8WiqqqulgobrFYrhgBO\nBdHfMfO6dyD3xRmFv5n0cvyHkKbper3mSrJjmPMzFXRFu+Reu7Ou68VicXFx8aAndbScU9TTlWR6\n/aPLsixJkkk3PJ3HQdIZU+y4lPvGYrEIw7D9v0spNZ/P9/v9vUd+WZZZll1eXp69xg8h+kfH010m\n4Apd2O/3cRxfXl6OvZDu7fd7IcTFxUXLrzdvEs2HaZo2Hx7++urqyoT+ZrPpcLVXV1e8qm1weXkZ\nx3H7lw0GRq+/A27f0GguWbfv+Nd13XJUURAEURTx7CcnNc/24nKOnYj+c5mnF7ma+8aDtvoopdqf\n5hP9DguCgCE/1iL6z2VGcjqc+0II81i+loX/g6I8CAKi32GMebAW0X+WLMsmfctue6bwr+t67IVg\nYsy9jdzqZRui/3TmjqQkScZeyBAeVPgDh9I0lVJyemcVov9ESqmyLKc43exkaZrWdX1v4R/Hcfve\nrlLK/sfX4HybzYZLvlYh+k+0Xq/d28J/XMvCPwzD9rMfpJQ+tMtgbvGj6W8Pov8UpsX/oBtc3WBu\n0z9e+EdR1PKSgFLqQduBMGk0/a1C9D+YyTVPWvzXBEFwb+GfJEkQBM1otjzPrz64t/bw10KIqqrC\nMPTzJ+knmv724Jb3hzF3qO92u7EXMhqt9Xw+3263R3r05rz+3obYfD5P07TzUXcMcrCZOYLcvg9m\nEqj6H8bDFv81bQr/NE2rqjp+Ta8sS/NXdb1AWI2mvyWI/gcw8wg9bPFfY07bjwzmDcMwz/PjHX/H\nRpyiPZr+NuDUuC2zr9Gr3ZxHZFlWVdXFxcXYC7kFDZ9JMLfBc5F/LBwkrdDiv8Z0/PM8t7BjQ/RP\nAk3/cdHwaYUW/zVtOv7AETT9x0X0348W/61M74tH8eFkNP1HRPTfw9x2xN7zW6VpynGLc5gtAwx4\nGB7Rf48sy7i0exfT6KfwxznSNM2ybOxVeIfoP6aqqiiKaPUcYQp/qjacLIqiIAiYBz4w9kLcSWu9\nXq+32+3YC7HdfD5PksSecyN2+EwOO+iGR9V/p6IoLNy5aCHzFBcKf5yMDWPDI/pvZy49MUq+jdVq\nFQQBxy3OkSQJ13uHxKnx7ZbL5Waz4WaTlqqqWq/XFxcXNvzEaPhMlFIqyzJarMOg6r9FWZbm0tPY\nC5mMJEnYoI0zhWEYhiHXe4dBfXQdV5xOU9f1crm0ofCn6p8ujr7BUPVfVxSFPZtVJiSOYwp/nMlc\n72Wb/wCI/g8xzw9imuBpzFaf9g/mBW5KkkRrzauob5waf8hisWCU4DkWi0UYhuOOuqPhM3VKqaIo\nGJjYK6r+nymKwjxXduyFTBiFP84XhiH39/aN+uinuHe3K6MX/lT9blgsFlzv7Q8HyU9lWRbHMfdw\nnU9K+fTp04uLi7FmHxH9bjCPd+aO+p5wkAhByd818/yNsQp/ot8ZFP794SARgpK/a0qp+Xw+VuFP\n9DujLMt//ud//t3f/V023XWOg4SSvxcjFv5E/6QppaSUb7311t/8zd/893//96c+9al33nln7EU5\niIOEkr8XpvDf7XbD/2CJ/smRUtZ1/f3vf7+u63fffffx48fPnz8XQrz55puf/vSnxQcPBUKHOEjE\ncrmk5O/Der1WSg3fqyX6J6Gu67qupZT/8A//8OzZs49//OPPnj1rfvdjH/vYd77zHdPnoePfB98P\nkrIsgyDg0bt9GKvwJ/rtZ14bt/7W48ePX3rppb/9279t+vvmCaAU/t3y/SChoOjVKIU/0T8Jv/M7\nv/PXf/3XP/7xjw8/+ejRo0996lN///d/f+3OSk7NO+f13bxlWVLv9yrPc9PGHXshsM4f/dEf3cz9\nJElu5r4QgmHOnfO6PqLkH0CWZVLKIX/OVP32K8uyKIp333332bNn//u//yuEePTo0Re/+MWvf/3r\nt3492/A652/VT8k/jDRNKfzRKMtyPp8XRZGm6fe+9z2T+0KIr33ta3flvhAiCAIK/45d+SqO47GX\n4Is0TYf8afv8qrbZZrMxc9nSNG0++cYbb3zsYx/bbDb3/vHLy8skSfpcoF88PTVmz8CQtNbz+TzP\n82F+4DR8bFPXdVEUUsrVapWm6WEr34zmb3mzLrfgdMjThk9d1+T+YMyjl3iAl4fqul4sFsvlMoqi\ni4uLPM+vXcINgqD9kIY0Tauq6mGZPvIx+uu6HmuopLfSNNVam5Mt+ODe0D+B+Rt4GkQnfIz+qqrS\nNB17FX6h8PeHCX3z2IauQr+Rpimvok54F/1aa/FB+YAhmYfdU/g7TCm1Xq+b0N9sNp0faGEYmkMY\nZ/Iu+suy5DLRWCjZXGVC38xmMKHfX081jmM6/ufzLvqllGznH4u5tE7h75IhQ99IkoToP59f0a+U\n4gLvuEzhzzm7A5rQV0rt9/sBQt8IgiAIAl5CZ/Ir+rmDd3Sm8KftM2la6yzLnj59ambz7Xa7gZ+i\nRc/nfH5Ff/ubR9CfNE3LsqRqmyIT+vP5XEq53W5HeRSPECJJEoY6nMmj6K/rmty3wWq1CoKAwn9a\nLAn9RhiGbPA/h0fRX1UVd/BaIs9zCv+paEK/qqrNZjN66BtJkrBf4By+RD/b+a2SJEkURRT+9ivL\n8unTp1VV5Xl+cXFhz6WyKIqo+s/hS/RXVUW3xyp0/C13OF354uLCwjPmKIro+J/Ml+g3UwPHXgV+\nJo5jCn872R/6Bhv8z+FF9HNiaCdT+PN/xx4m9LMsS5LE5tA3uEfnHF5Ef1VV9vQo0aDwt4eZudaE\nvhm4ZL8oirjYexovol9KacOeBNxE4T+6PqYrDyZJEinl2KuYJPejX0rJiaG1zEOXnC/8sywzxWmW\nZbPZzHxysViM26qedOgbZsFsFjiB+9FPt8dyZo+/w4W/lLIsy5t98zRN1+v1KLHV60j9gTHU4TTu\nRz/DGywXRZHbT3E5zP08z5vnBo9yqWOAkfoDi+OYns8J3I9+2M/hjr9S6sjQwCFvSR1+uvIwmOJ5\nGsejn7k9kxCGoauFv7nnqHkRHvb6hRBJkgzwyGJXQ7/BPJ8TuB/9NPonwRT+7t2cWdf1kd1lQRBE\nUdRfv2KskfoDi+PYvVdO3xyPfq21k69197ha+N97qamn6B99pP6QiP4TuBz9dACnJU3Tuq4dO4bv\njfUgCLqNftumK8NOLkc/jf5pcbXwH4zPoR+GIft8HsTl6Ocm3snJ81xK6VjhPwA7R+oPiZ7PQ7kc\n/TyEfXKCIHCs8I/j+HjjUSl1ZkxbO1J/SOzufyhno19rPfV7VfyUpqlLhf+9+w6llCe3JacyXRkW\ncjb6afRPlGOF//HHiSilTrjb3EyGIPSv6XWbrIOuHGUOibFXgVNcXl4GQbDZbE7741a9qs2/Zbvd\n3vq7eZ6HYfigv3Cz2bz88ss/93M/l6ZpFwt0x36/N3My0IazVT87+qfLpcI/CIIkSe4q/MuyTNO0\n5V/VjNR/4403XnjhhW9961tsXz7E03ofxM3op9E/dWmaDjDhYBhpmlZVdTOmy7I0b3L3/g03pyt/\n/vOff+edd37913+dsDvEe2F7bkY/D2GfOpcK/zAM8zy/WfhLKTebzfE/e9dI/T//8z9/8cUX/+M/\n/uPVV1/99re/3dPKJ4cHtT/A2B2nXqRpenl5OfYqcK4wDE/o+Lvxqm725q9Wq1tfzJ/+9KfNIRwE\nwde//vXBF2ij/X7PJZCW3Kz6afi4IU1TNwr/B2k5Uv/3fu/3zC+01n/wB3/wx3/8x8Mu00a0+9tz\nNvrHXgI6YPrgbnT823jQdOUvf/nLv/iLvyiEePTo0S//8i+//vrrwy3UYmzuaMnN6Od/vzNM4e/8\ne/kJI/WDIPi1X/u1IAiePn36rW99i9e8wdb+lhyMfq7zuMQU/g63fc4Zqf+Hf/iH+/3+r/7qrxz+\n+TxUFEXOFwqdcDD6zx+KAquYp7i4dzyfP1L/tddeC8PQvFXQ4zYY4dmSm9HPya9LVqtVEAQuFbad\nT1f283r4rYj+loh+TECe524U/j2N1Kfwb9DwacnB6Gdbp3uSJImiaNKFbd8j9Ver1b1boXzIxM6f\neuYqB6Pfh9e3hybd8R9gpH7LLe3L5bLzb20bTvrbcC366fa4Ko7jKRb+Q05XvvdJVUEQTPS9E50j\n+jEZpvCfSkd7+JH6PKTQoOfThmvRX9c10e+qqRT+JvSzLEuSZMjnqBx5IpjWer1er9drKeVLL73k\n9g3SjHNow7XoF0Kwqd9hlhf+zUh9E/p5ng+8gDAMb7Z0tNZmItBms4miaLVardfrqqoGXttg2N/Z\nhmvRb20ooBNxHMdxbGHhf9d05YFFUXQz04uiUEo1z4Qxv3A4+tnf2YZr0c/OTueZPf72vMdbEvpG\nHMc3C95rj68IgiAMQ4ejH20Q/ZgY07KwofA3od9MVx439I1bF3DzbdL562H2VAbWcir62d7gidE7\n/i1H6o/i5v6WW3e8uJ3+9Hzu5VT0CyF4LqMPwjAcq/A/YbrywG5u3o/jWGt9mP5aa+d3Q3Cl9zjX\not+e4gu9MoX/kOd59oe+cbP6SdM0CIIsy8yHSqnDq77wk1PRT4PPH0MW/ueM1B/FtXfEKIp2u53W\n2vwTsizb7XaW/xPOxNn/vYh+TFWapnVd91r4nz9Sf3i3nvhGUbTf703ib7db+/8VZ+Ls/15ORT+8\n0mvh39N05QEciXW3K/1ruMx7nGvRP5XjE53I81xK2W3hP93Qb3CFU/BDuI9r0Q+vBEHQYeHf90j9\nwdxV3TPcEA2nop9TPA+ladpJ4T/ASP3B3NXpNmN8Bl7MKJy/mHG+R2MvoEtc5vVQU/ifXKGXZWnO\nG9I0HWzKZq+ogbjMey+nqn74yRT+JwwiHn6k/jCIfsEP4T5ORT9neX46oeM/1kj9AWitaegL13sA\ns9s8ffq0/VHgVMMH3jI395ZleW+I13VdFIWUcrVamdtch1nhYNjZ4okwDK9dkaqqKssyrXWbB0U4\nFf2c4nmrKfyPRP9h6G+3W/dCvzHRjUndcr4HEIbhtYhP09TsT2sT/U41fNw+xcNx5uV+a8ffqpH6\nfWN+rf16KlKDIGj/cEqnoh+eS9P0Wq/TwpH6A3C+4G3D5h6AlHI2m0VRlGVZ5zckHl7pUUotl8vm\nSsBhYeRU9POK99xht8fmkfq9Ukp58i89zvIewIsvvvj2228XRfFbv/VbnbwNaK1NP7OZyaqUevr0\naV3XaZqaT67X62aAq7hySJqmYy8BIzO3LJn3gNVqdXFxMfaKhrZarcZeghXiOB57CXfa7Xaf+MQn\nbsb348ePhRCvvPJKmqa73e7I33DXG0Ce583XrFarIAguLy+bz5jLwuagmF1dXWVZZsPj7gAAh8w+\ntJu7dWez2c0dPqafs9/vzde/9NJLcRwfngpLKc3I7jiOqfrhgsvLS/MSN5tbxl4OxjfFqv/JkydC\niJdffvmNN97YbDZHzlnNS/3mXysOCv+73k7M+YRTmzvhIdPiLMvSPJAkjuPZbDb2ooB7PH/+3Pzi\nyZMn77333ssvv/yFL3zhN3/zN+M4PvmOPFP3NNe3zY7nuzZ6OhX9ll/YQbcOQ3+73bKZHYcsv6X5\nhz/8YSdxf0SSJFVVHd63WBRFXdfmphanot/m7VzoEKGPe9m8zcncX9LTm1MTg6vVqqqq+XxueqFm\nwO3P3gn662cNL0mSsZeA3pm9+Wb+8K1f4NirGqdx+8qfuK3Xf3V1ZTa4b7db8+F+v28uBYdhePgz\ncarqt/wUD2dqpivnee7SwDU8iNa6TUXv9iyjqzuu4u73+8MPzWnxrV/p1C1dcFW305Xrus6yLIoi\nLghPUX+3wnrFqarf7fd5P5lKX2vd3JF4mrqu67re7XZvv/3248ePm/0VmCJzK+zbb7/9J3/yJ8+f\nP3/llVcWi0Ucx4dXfegBHOdU9MMl509Xvivuyf2pM3e9ig/+V976NmDzZV4bOBX9/M92w/nTldfr\n9bURnnfF/c9GmmAi/u3f/u3Zs2c3P3/4NmCuCZmTRcr/W83uulwwRVmWtRlUDWt18hyV2Wx2dXWl\nlKrr+tvf/nZd1+++++5HP/rR999//+YXu/T690Rd16+//voPfvCDa58/vDcqCIIf/ehHf/mXfznK\nCifBqaof02Wmi9R13dVzVMIwXK1W5oJwy7cBTMW9t8KaXt+oa7Sda9Ff1zU3+EyLUsrcn2UGbfZx\nen7kbaDz74UBtLkVlj7Pca5FPyZkgNC/6drbwADfEd1qeSss0X+cU/v6+Z89FeY5KvP5XGttnqMy\nyv87XjBTZO7lPv41zHS5F9GPQWmtsyybz+dKqd1ut91u+b+GznGLz71o+GAgN6crj70iwF9OVf1C\nCC7rW6ip9KWU2+2W3O/VYrGYzWbL5XLshYyMJ3Uf51T0Eyi2aUK/qipCfwBm/5IQoqoqny9iSym5\nwfM4p6JfsGfDJmVZPn36tKqqPM8vLi4I/QFUVSWEMMOOzK/9RMl/L9ein7d6G3Q7aBPtVVUVhqF5\npIHP0U8JeC+iH10i9EckpTQzMIQQSZJIKf1MQCkl28bu5Vr0C670jqSu6/l8nmVZkiT7/Z7QH54p\n881TmczP38/CX2tNw+derkV/HMfczTGwuq4Xi8VyuUyS5OLiwnQbxl6Uj6qqSpLEFLxRFIVheG18\nqSfquuYVeC/X9vUHQVDXdfM4SvTq/OnK6Epd10qpMAybMdRBEJgWkG8lsNaaPQX3ci36oygyo7rR\nKxP6dV2naUro28D0dm5OrKyqyrfoRxuuNXwE4xx6ppRaLpeLxSIMQ9o79jARf/VhYRh62O6n5duG\ng9HvwPiOLMtMlzbLsubR4Xf9erFYDHN4NzPXgiAYceYabqqqSmt989J6kiRKKQeOiPa01rws23Aw\n+qMomvSeNimlmWPc8uvTNF2v171WOk3oCyEIfQuZJs/NS1zmM15d7OU+3raunLPZbHa73dirOJ15\nNuGD/kgcxw/9Iy1dXl6aN6E4jvf7fR/fonNOvqqPuLy8FEIkSXLr74ZhGATBwEsa0dQP/8E4WPWH\nYTjdM1ylVFmWD92hlCRJ55XdtenKu92Oq4V2Msm+3W5v/d2Liwvz3uAJs81p7FVMgIPRH0XRdK/z\nmDP3JmTb9PqFEEmSaK27Sn8GbWK6iP6WXNvcKYQIgmC6vf7Tni0cBEEUReef6zQj9cMw3G63JD4m\nh9xvycGqX0x5ko9S6rS+yvnRXxSFma6c5/l+vyf3MUXTLfsG5mb0i8nu7T05vs2tm6f9WTNzrSxL\nM12Z8TuYKLo97bkZ/VEUMcStDQZtwiVVVXG22pKb0R/H8XQ3+QzDhH6WZavVitCHG6SURH9LDl7m\nFUKEYTjRlt/Jk0eVUi1f9Icz19I0ne51EQAnc7PqF0IEQTDFdv/Jb1ptBjQ205WjKGL8DhzDE1oe\nxNnon2i7/7RlK6WObw0i9OG80zZG+2vs24n7cnl509v0zAAADWVJREFU2dNsg15dXl4GQbDdbh/0\np/I8D8Pw1t9qnoduevpdrNF2Dr+qccRdoyxwK2er/ok2fIIgSJLkoYV/WZZpml77ZDNzzUxXZuYa\n3MaJ7IM4G/1isrf1pmlqZvC2/PqyLIMgONyiw6BN+IZG/0O5HP0TbfeHYZjnefuVSyk3m435dRP6\nWmtCH/6g0f9Qs6urq7HX0BczhqyJRbeZ8TtFUZgBzj4fBrOZy69q3Gq5XN41uxS3cnNfvzHRdv9D\nNTPXoihiyiaANlxu+Igp39vVBtOVASFEXdc8TOKhHI/+ibb72zChX1WVeSwRoQ9v0eg/gePR7+Qw\nHzN+x0xXvri4eOgjvQDHnDzq3Gcu9/qFc1t9y7IsikIIkaYpA9cAnMzx6Bcf7O6f+h5HQh+4Fd2e\n0zje8BFCJElSVdXYqzhdXddmunKSJPv9ntwHDjGj/zTuR/90r/Q2M9eSJGHmGnArrfXUz+lH4X70\niy6eWzswBm0CbVDyn8yL6F+tVlPp+ZjQXywWhD5wr6qq2OF2Gi+iPwxD+6t+M35nsViYQZuEPnCc\n1joIAg6T03gR/cLui70M2gROQLfnHB5Fv4UXewl94GR0e87hS/Sbs0J7prlprU3oK6X2+z2hDzwI\nd/CeyZfoF0LEcWxDz6eZuaaU2u12u92OVzDwUGVZUvKfw6PoH73dz6BNoCtU/Wdyf5DDoSiKRhnq\n0IzUD8Nwu92S+MA5mNJ8Po+qfiFEkiRlWQ78TQ8Hbe73e3IfOBMXeM/nXdWfZdlg366ZuZbnObN3\ngA6xLeJMfkW/+GCCf99niwzaBHpinkU69iomz6+Gj+i/52PaO1mWrVari4sLch/olpSSbs/5vKv6\nwzDsaXd/XddFUUgpV6tVmqbcXw50zhy8HFzn867qF0LEcdxt4c+gTWAYRVFQ8nfCx+jvcJAnoQ8M\nRmutlGKPXCd8jH7xwI6/1vrm/B+llJmuHIbhfr8n9IG+FUXBxbOueBr97Qt/rfVnP/vZL37xi81n\nmplrZroy43eAAVDyd8vT6BftCn8p5dOnT//lX/7l8vKyLEsGbQJjKcuSkr9Ds6urq7HXMJrFYrHb\n7e76XSnlq6+++t5775kPf/7nf/5HP/pRHMckvuVmM69f1a5aLpfb7XbsVbjD36pfCBHH8V1D/Kuq\n+tznPtfkvhDi/ffff/3113e7HbkPDKwsS1o93fK6PjJD82+WEmVZrtfrm1//5MmTf//3f+dyruWo\n+t1z/AQdJ/C66g+CIAzDa4X/m2+++ZWvfOXWr3/vvffMeAYAg2E0fx98r4+uFf5f+tKXvvnNb/7f\n//1f8wUf/ehH33///Zdffvnzn//8a6+9FscxDR/LUfU7hpK/D94NcrimKfyjKIqi6F//9V+FEC+8\n8MKPf/zjMAx/+7d/+zOf+UwcxzR5gFFQ8veE+khorX//93//rbfeev/993/lV37l9ddfj6KIuJ8u\nqn6XUPL3xPeqXwihtf6f//mfr371q1/+8pfHXguAn6nrmo09PfH6Mq8RhuHXvva1733ve2MvBMCH\ncBtXf4h+IT544o9SauyFAPipuq7DMKTv2hO6oj+llCqKYrPZjL0QnItevxsWi8V2uyX6e0LV/1Om\nvuhqmDOAc5i5/OR+f6iPPoRCwwFU/VN312326BBV/4ekacr9usC4sixL03TsVTiO6P+QOI6VUlzv\nBcZS13UQBFEUjb0Qx3FqfB0nm1NHw2fSlsvlZrOh6do3qv7rTMXB9V5geFmWcXV3GET/LdI0LctS\naz32QgCPmF4rE3uGQfTfjuu9wMCyLMvzfOxV+ILov10cx1prKeXYCwG8YO7dZSL6YLggdieu904U\nl3mniFtqBkbVfydzvbcsy7EXAjjObOQn94dE9B+Tpmld11zvBfqjlNJaM5x5YET/PbjeC/SKe3dH\nQfTfw9xVeO3R7QA6UZYlV3dHwQWx+2mtl8sl16Cmgsu8UyGlLIqCnRSj4CBphdfohBD9k0BFNS4a\nPq2YB7VnWTb2QgBHsKtnXER/W6vVSmtN0x84X1mWQRCwq2dEnBo/AKeok0DDx3JKqSzLaJ+Oi4Pk\nYWj624/otxw37tqAhs/D0PQHzkGL3xJE/4PR9AdOYx6DQYvfBpwan4Kmv81o+NiJFr9VOEhORNPf\nWkS/nWjxW4WGz4lo+gPt0eK3DdF/Opr+QBu0+C1E9J8lz3Oe4gscIaWsqoonL9qG6D9LEARpmi6X\nS9IfuElrXRTFZrMZeyG4jgtiHZBSZlm22+3GXgiE4DKvNcxGuM1mw0xmC1H1dyCKojzP1+v12AsB\nbGFyP89zct9ORH83oiiKooj0B4yiKPI8N086goWI/s6sVqsoitjuCazX6zAMyX2bEf1dWq1WQoiy\nLMdeCDCaLMuiKDLHAqxF9Hcsz3MpJekPP5lB/OS+/Yj+7m02GymllHLshQCDKstSSpmm6dgLwf2I\n/l5sNpssy0h/+KOuayklW/ingujvy3a7Jf3hCdPkJPcnhOjvSxAE2+2WMQ9wnrmlkdyfFqK/R+Z6\nF2Me4DAzqoFpzJND9PfL3OhL+sNJpt7P85zcnxymnQzBdELNPV9jr8V9zPAZhsl96v2J4iAZSDPS\nhPTvG9E/gLqu6fNMGgfJcEj/YRD9fTP797muO2kcJENbr9fc5t4ror9X5L4bOEhGQPr3iujvj5lO\nyCO3HMBBMg4Oof4Q/T2hZHEJmzvHYR5hwXx/TILWmtx3DPXRmGib9oGqv1tsT3ASB8nI2BzdOaK/\nQ+S+q2j4jIzbfWEtKeV6vSb3nUR9ZAUpZVEUq9UqjuOx1zJ5VP2dqKqqLEvOR13FQWILrXWWZUEQ\nsO3nTET/+diB5jwaPrYIgmCz2YRhSPMHI1JKLRYL04ccey3oEfWRdcyF3zRNaf6chqr/ZKbJY0qQ\nsdeCfnGQ2IjmzzmI/tPQ5PEKDR8b0fzBkEyTJ45jct8f1EdWo/lzAqr+B6mqqqqqzWbDTh6vcJDY\njubPQxH97dHk8RYNH9vR/EEfaPJ4jvpoMmj+tETVfy+aPOAgmRKtdVEUQog0TTlo70L0H6GUyrIs\nDEOKfc9xkEyPmfoQxzETdG9F9N+lKAoppRkYPvZaMDJ6/dMTRdF2uzUjFaWUYy8HE1DX9XK5DMNw\nu92S+xBU/ZPWbP6h/3OIqv8QLxLcioNk8uq6NlM/kyQZey1WIPobpsOTpilTl3ENDZ/Ji+N4t9sp\npZbLpVJq7OXAClLKxWIRBMF2uyX3cRP1kTvMA1TDMPT81N7zqp9tYGjD64PESfR/fI7+siyrquK5\nWrgXDR/XmP6PlJL+j1fMHh6t9W63I/dxL3/rI+cppcyJ/2q18ioLfKv667ouy5JGHx7Er4PEQ0qp\nsiyVUv48+Nef6Cf0cTJfDhLPmUt/nrwB+BD9ZggPoY+TuX+QoOHJG4Db0W8u5CZJwhgPnMPlgwS3\n0lqXZVnXtavx4Wr0E/roEDt8vGPu6d/tdkKIxWJRluXYK8I9yrJcLBZCiN1uR+6jE27WR2jPvVrS\nmaq/adAxpRWdc+QgwZnKsjRDQB3YCepA9FdVVde1ECJJEoevymBEkz9I0CGllAmdKIpWq9VEp/tO\nN/qllFVVKaXMz5+tO+jPVA8S9MpkkJQySZIkSaaVQZOLfnPhXUoZhuF033ExLRM7SDCwpvMQx/FU\nhgJNKPqbPhuNHQxsMgcJRqS1NvcQRVGUJInlFwPsj/66rquqEkKYn+e0TqrgBtsPEljFjIWQUpqT\nADtbE9ZG/2Er39qfHjxh6UECy5kU01oLIaIoiuPYniCzKvrNlXPT1QnD0P5zJnjCooMEU6S1ruta\nSqmUCoLAhreB0aNfKWV+JkKIIAjiOKaPD9sQ/ehM8zagtW7eBoZvZI8S/U3ca63DMBzr3w60RPSj\nF+bK8ChROFj0W/JWB5yA6EfvDhsg5p3ABGUYhn20hnqKfqWU+YeYXwdBEASBad8T95gcoh9D01qb\nawNKKSllFEVKKXNmYDrjZ/79nUR/XdfqA0EQmHcsg8Y9HED0wwqmbWJqaq21iVohxLWcbRO7LaPf\nfC/DvBvd/NY9nZcAoyP6Ya8mnU313XxeKdXsK20+NO0XIURRFGmaCiGaLZWmIXPtQ+Mw3Cnn4Q+i\nH64ZfXMnYD8e1QIA3iH6AcA7RD8AeIfoBwDvEP0A4B2iHwC8Q/QDgHeIfgDwDtEPAN4h+gHAO0Q/\nAHiH6AcA7xD9AOAdoh8AvEP0A4B3iH4A8A7RDwDeIfoBwDtEPwB4h+gHAO8Q/QDgHaIfALxD9AOA\nd4h+APAO0Q8A3iH6AcA7RD8AeIfoBwDvEP0A4B2iHwC8Q/QDgHeIfgDwDtEPAN4h+gHAO0Q/AHiH\n6AcA7xD9AOAdoh8AvEP0A4B3iH4A8A7RDwDeIfoBwDtEPwB4h+gHAO8Q/QDgHaIfALxD9AOAd4h+\nAPAO0Q8A3iH6AcA7RD8AeIfoBwDvEP0A4B2iHwC8Q/QDgHeIfgDwDtEPAN4h+gHAO0Q/AHiH6AcA\n7xD9AOAdoh8AvEP0A4B3iH4A8A7RDwDeIfrhmjRNx14CYLvZ1dXV2GsAAAzq/wHWiYIEWZrWuAAA\nAABJRU5ErkJggolQTkcNChoKAAAADUlIRFIAAAH+AAAB8AgCAAAAyAeaAgAAAAlwSFlzAAAMTQAA\nDE0B0s6tTgAAAB10RVh0U29mdHdhcmUAR1BMIEdob3N0c2NyaXB0IDguNjKDLQjPAAAG5UlEQVR4\nnO3UMQEAIAzAMMC/5+ECjiYKenXPzAKg5PwOAOA16wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6A\nHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc\n6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzr\nB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsH\nyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfI\nsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ix\nfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+\ngBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6A\nHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc\n6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzr\nB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsH\nyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfI\nsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ix\nfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+\ngBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6A\nHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc\n6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzr\nB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsH\nyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfI\nsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ix\nfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+\ngBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6A\nHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc\n6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzr\nB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsH\nyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfI\nsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ix\nfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+\ngBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6A\nHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc\n6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AHOsHyLF+gBzrB8ixfoAc6wfIsX6AnAtRZAbdmiMV\npQAAAABJRU5ErkJggg==\n", "prompt_number": 3, "text": [ "" ] } ], "prompt_number": 3 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The simple harmonic oscillator, as with all mechanical systems, has two forms of energy: kinetic and potential. If we have $W$ as the total energy and $T$ as kinetic and $V$ as potential, we can write: \n", "\n", "\\begin{eqnarray}\n", " T &=& \\frac{1}{2} m\\dot{\\psi}^2\\\\\n", " V &=& \\frac{1}{2} s\\psi^2\\\\\n", " W &=& T + V\n", "\\end{eqnarray}\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The form of the potential energy follows directly from the form of the force (and is why the motion is called harmonic). As there is no form of dissipation in the motion, we can assert that the total energy does not change with time, just exchanging between potential (at a maximum when the displacement is at a maximum and the velocity is at a minimum) and kinetic (at a maximum when the displacement is at a\n", "minimum). They are out of phase - as we expect from the phase differences between displacement and velocity. \n", "\n", "As the total energy is constant, we can write:\n", "\n", "\\begin{eqnarray}\n", " \\frac{d W}{d t} = \\frac{d T}{d t} + \\frac{d V}{dt} &=& 0\\\\\n", " \\Rightarrow m\\dot{\\psi}\\ddot{\\psi} + s\\psi\\dot{\\psi} &=&0\\\\\n", " \\Rightarrow m\\ddot{\\psi} &=& -s\\psi\n", "\\end{eqnarray}\n", "\n", "which is of course just the original equation for simple harmonic motion." ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Damping Oscillations" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Restoring the damping term changes the equation and its solution. We will use $\\gamma = b/2m$ in the equations below. These solutions were derived in PHAS1247 and are discussed in many textbooks, so I will not derive them here. We find: \n", "\n", "\\begin{eqnarray}\n", " m\\frac{\\partial^2 \\psi}{\\partial t^2} &=& -s \\psi -b \\frac{\\partial \\psi}{\\partial t}\\\\\n", " \\psi &=& \n", "\\begin{cases}\n", "A e^{-\\gamma t}e^{i\\omega t} &\\gamma < \\omega_0\\\\\n", "\\hskip 0.5cm\\omega = \\sqrt{\\omega^2_0 - \\gamma^2} &\\, \\\\\n", "A e^{-\\mu_+ t} + B e^{-\\mu_-t} & \\gamma > \\omega_0\\\\\n", "\\hskip 0.5cm\\mu_\\pm = \\gamma\\mp \\sqrt{\\gamma^2 - \\omega_0^2} &\\, \\\\\n", "A(1+\\omega_0 t)e^{-\\omega_0 t} & \\gamma = \\omega_0\n", "\\end{cases}\n", "\\end{eqnarray}\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The cases for damped harmonic motion correspond to light (or\n", "underdamped, $\\gamma < \\omega_0$), heavy (or overdamped, $\\gamma >\n", "\\omega_0$) and critical ($\\gamma=\\omega_0$) damping respectively. The\n", "total energy is \\emph{not} conserved in this case (contrast the\n", "undamped oscillator) as the damping force opposes the motion at all\n", "times. We write the energy as before, and find the change with time: \n", "\n", "\\begin{eqnarray}\n", " W &=& \\frac{1}{2}m\\dot{\\psi}^2 + \\frac{1}{2}s\\psi^2\\\\\n", " \\frac{dW}{dt} &=& \\frac{d W}{d \\dot{\\psi}}\\frac{d \\dot{\\psi}}{d t} + \\frac{d W}{d \\psi}\\frac{d \\psi}{d t} = (m\\ddot{\\psi} + s\\psi)\\dot{\\psi}\\\\\n", " &=& -b\\dot{\\psi}^2\n", "\\end{eqnarray}\n", "\n", "where the last line comes from using the SHM equation for damped motion. Notice that\n", "the change in energy is always less than zero (unless $\\dot{\\psi}=0$)\n", "and so the total energy of the system decreases with time (as we would\n", "expect). \n", "\n", "We illustrate this behaviour in the interactive figure below: test the effect of changing `b` and `k` (and re-run the cell to render the figure)." ] }, { "cell_type": "code", "collapsed": false, "input": [ "t = linspace(0,10.0,1000)\n", "m = 1.0\n", "k = 2.0\n", "b = 5.0\n", "omega0 = sqrt(k/m)\n", "gamma = b/(2.0*m)\n", "print \"omega0 and gamma are \",omega0,gamma\n", "A = 1.0\n", "if gammaomega0 : \n", " print \"Heavy damping\"\n", " mum = gamma + sqrt(gamma*gamma - omega0*omega0)\n", " mup = gamma - sqrt(gamma*gamma - omega0*omega0)\n", " B = 1.0\n", " plot(t,A*exp(-mum*t) + B*exp(-mup*t),label='sum')\n", " plot(t,A*exp(-mum*t),'r--',label=r\"$\\mu_{-}$\")\n", " plot(t,B*exp(-mup*t),'g--',label=r\"$\\mu_{+}$\")\n", " legend()" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "omega0 and gamma are 1.41421356237 2.5\n", "Heavy damping\n" ] }, { "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAAXUAAAEACAYAAABMEua6AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nQmCC\n", "text": [ "" ] } ], "prompt_number": 27 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Driving Oscillations" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we will introduce a driving term to the system. As we have a\n", "harmonic oscillator, it will make sense to use a harmonic driving term\n", "(though we could, for instance, use impulses at regular or irregular\n", "intervals). Note that the driving term will have an angular frequency\n", "$\\omega$ which is \\emph{different} to the natural frequency of the\n", "system, $\\omega_0 = \\sqrt{s/m}$. Also remember that $\\omega = 2\\pi\n", "\\nu$ where $\\nu$ is the frequency. We will retain the damping term to\n", "give a damped, driven oscillator (again derived in PHAS1247 and \n", "textbooks). \n", "\n", "\\begin{eqnarray}\n", " m\\frac{\\partial^2 \\psi}{\\partial t^2} &=& -s \\psi -b \\frac{\\partial \\psi}{\\partial t} + F_0 \\cos \\omega t\\\\\n", " \\psi &=& A\\cos (\\omega t + \\phi)\\\\\n", " A &=& \\frac{F_0}{m} \\left( \\frac{1}{(\\omega_0^2 - \\omega^2)^2 + 4\\gamma^2 \\omega^2}\\right)^{\\frac{1}{2}}\\\\\n", " \\tan \\phi &=& \\frac{-2\\gamma\\omega}{\\omega_0^2 - \\omega^2}\n", "\\end{eqnarray}\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The response of the system (including both the phase and the\n", "amplitude) depends strongly on the frequency; we can consider three\n", "regimes (though there isn't really time to do this in depth). \n", "\n", "* When $\\omega$ is small (i.e. $\\omega \\ll \\omega_0$), the amplitude can be shown to be $A \\simeq F_0/m\\omega^2_0 = F_0/s$ and the motion is dominated by the spring constant (_stiffness controlled_).\n", "* When $\\omega$ is large (i.e. $\\omega \\gg \\omega_0$), then the amplitude is $A \\simeq F_0/m\\omega^2$ and the motion is dominated by the mass (_mass controlled_).\n", "* When $\\omega \\sim \\omega_0$, we are at _resonance_, and the response is controlled by the drag term $\\gamma$ (also known _resistance limited_).\n", "\n", "The behaviour of $A$ and $\\phi$ are plotted below; you should try changing the values of `b` (damping) and `s` (stiffness) to see the effects. In particular, you may need to increase or decrease `F0` to see the graph on a sensible scale. (Remember to run the cell by pressing the play button in the toolbar above.)" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# Set up parameters of oscillator\n", "m = 1.0\n", "k = 2.0\n", "b = 0.2\n", "gamma = b/(2.0*m)\n", "# Find resonant frequency\n", "omega0 = sqrt(k/m)\n", "print \"omega0 and gamma are \",omega0,gamma\n", "F0 = 1.0\n", "# Create array of values of omega from omega0-1 to omega0+1\n", "omega = linspace(omega0-1.0,omega0+1.0,200)\n", "# Calculate phi\n", "phi = arctan(-2.0*gamma*omega/(omega0*omega0 - omega*omega))\n", "# arctan in NumPy is defined to lie between -pi/2 and pi/2 so we correct\n", "# by -pi to give physical reasonableness\n", "phi[100:200] -=pi\n", "# Calculate amplitude\n", "fac = (omega0*omega0 - omega*omega)**2 + 4*gamma*gamma*omega*omega\n", "A = F0/(sqrt(fac)*m)\n", "plot(omega,phi,label='phi')\n", "plot(omega,A,label='A')\n", "legend()" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "omega0 and gamma are 1.41421356237 0.1\n" ] }, { "metadata": {}, "output_type": "pyout", "prompt_number": 30, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAAXUAAAEACAYAAABMEua6AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAIABJREFUeJzt3XlcFfX+P/DXYVX2TUCBBJHVBXAjS+O4oKmBWt7cc00z\nrezeltu3jUpcbt/uTbOszCup1/JmX4PMzCWP/rQIF1BUwFRQdkJkDVkO8/tjFERQgTOcOcvr+XjM\nYw6Hw8zbeQwvPn7mM59RCIIggIiIDIKJ3AUQEZF0GOpERAaEoU5EZEAY6kREBoShTkRkQBjqREQG\nRJJQV6vVCAsLQ1RUlBSbIyKiDpIk1NeuXYvg4GAoFAopNkdERB2kcajn5ORgz549WLhwIXgfExGR\nvDQO9RdffBHvv/8+TEzYPU9EJDeNknj37t1wdXVFWFgYW+lERLpA0MBrr70meHp6Ct7e3oK7u7tg\nZWUlzJ49u9lnfH19BQBcuHDhwqUdi6+vb4dyWaNQv51KpRIee+yxljuAZLvQe2+//bbcJegMHosm\nPBZNeCyadDQ7Je0I5+gXIiJ5mUm1oYiICEREREi1OSIi6gAOWdEipVIpdwk6g8eiCY9FEx4LzSlu\n9t103g4UCo6MISJqp45mp2TdL0REUnBycsL169flLkNrHB0dUVJSItn22FInIp1ibJlxt39vR48D\n+9SJiAwIQ52IyIAw1ImIDAhDnUhCRVVFuFF/Q+4ySMtUKhW8vLzu+v0lS5ZgxYoVWqmFo1+IJJBe\nnI4n/vsEcspzYGNhgzWj12BW/1lyl0U6YsOGDVrbF0OdSENpf6RhxJcjsGrUKswNnYuT+Scx6etJ\nsLe0R1QAnwZG2sXuFyIN1DfUY853cxCjjMG8sHlQKBQY1GMQvn3yWyxIWICrZVflLpEk5O3tjdWr\nV6NPnz5wcnLC/PnzUVNT0/j9f/7zn3Bzc0OPHj0QFxfX+P7cuXPx5ptvaqVGhjqRBtb9tg42FjZY\nPHBxs/fDPcOxeOBivHlIO7/IpD3bt2/Hvn37cOnSJVy4cAErVqyAQqFAQUEBysvLkZeXh02bNmHp\n0qUoKysDII4519aEhwx1og6qqKnA6qOr8fH4j1v9hX354Zfx08WfcKbwjAzVGTaFQpql/ftVYNmy\nZfDw8ICjoyNef/11fPXVVwAAc3NzvPXWWzA1NcW4ceNgY2ODjIyMxp/V1g1VDHWiDlqftB6je41G\nULegVr9vZ2mHVx5+BWuOrdFyZYZPEKRZOuL2US4PPPAA8vLyAADOzs7NHutpZWWFyspKjf6dHcFQ\nJ+qA6rpqfPjbh3jzkXt3rywIW4Aff/8ReRV5WqqMOtvVq1ebve7Ro4eM1bTEUCfqgO2p2zHEY8hd\nW+m32Hexx/S+0/HZic+0VBl1JkEQ8MknnyA3NxclJSWIjY3FtGnT2vRz2sJQJ2onQRCw/vh6LBu8\nrE2fXzpkKTae2gh1g7qTK6POplAoMGPGDIwZMwa+vr7w8/PDG2+8AUEQ7nkhVJsXSjlLI1E7Hbt6\nDPMT5iNtaRpMFG1rFw3eOBgrR65EpG9kJ1en/3Q5M3x8fLBp0yaMHDlSsm1ylkYimW1K3oSnBzzd\n5kAHgFn9ZmFb6rZOrIpIpFGo37hxA+Hh4QgNDUVwcDBee+01qeoi0kmVtZXYlb6r3VMATOs7DQkZ\nCaiqreqkyohEGk0T0KVLFxw6dAhWVlaor6/HsGHDcPToUQwbNkyq+oh0ys7zOzH8geFwt3Fv18+5\n2bhhqOdQxGfEY0a/GZ1UHXW2zMxMuUu4L427X6ysrAAAtbW1UKvVcHJy0rgoIl21OWUz5oXO69DP\nzuo/C9vOsAuGOpfGod7Q0IDQ0FC4ublhxIgRCA4OlqIuIp1zqeQS0v5IwwT/CR36+YkBE/Frzq8o\nrCyUuDKiJhqHuomJCVJSUpCTk4MjR45ApVJJUBaR7olLicPMfjNhYWrRoZ+3trBGdEA0vj77tcSV\nETWRbOpde3t7TJgwASdOnIBSqWz2vZiYmMbXSqWyxfeJdJ26QY0vT3+J3TN2a7Sd6X2n453D7+CF\nB1+QqDIyFCqVSpJGsUbj1IuLi2FmZgYHBwdUV1dj7NixePvttzFq1KimHejwmFOittp/aT9ePfAq\nTi0+pdF26tR16PHPHjjx9An0dOgpUXWGxdgyQ6fGqefn52PkyJEIDQ1FeHg4oqKimgU6kaHQ5ALp\n7cxNzTE5cDL+e+6/ElRF1BLvKCW6j9IbpfD+0BuXnr8EZytnjbd38PJBvHrgVZxYdEKC6gyPvmSG\nUqnEmTNnUFBQAAuLjl1nAXSspU5kDL4++zUifSMlCXQAiPCOQE55Di6WXJRke6R9WVlZSEpKgqur\nKxISEuQupxmGOtF9bErehAVhCyTbnpmJGZ4IeoJdMHpsy5YtGD16NGbPno0vv/xS7nKaYagT3cOZ\nwjMorCxEZC9pJ+Ka1ncadpzbIek2SXu2bNmCqVOn4sknn8RPP/2EoqIiuUtqJNmQRiJDtOnUJswN\nnQtTE1NJt/vwAw/j2p/XkPZH2n3nZKeWFO9IM42t8Hb7+6yPHj2K3NxcREdHw9bWFsHBwdi+fTuW\nL18uSU2a4oVSoruoqa+B5788kbQwCT6OPpJv/8W9L8K+iz1ilDGSb1uf6XpmPP300ygqKkJ8fDwA\nIDY2Fjt37kRycnKHtif1hVKGOtFd7Di7AxtPbcSBpw50yvYTcxIxL34ezj97XmsPUNAHupwZ1dXV\ncHd3R0NDA2xsbAAANTU1KC0tRUpKCvr379/ubXL0C5GWSH2B9E7hHuGorqtGalFqp+2DpPXdd9/B\nzMwMaWlpOH36NE6fPo20tDQMHz4cW7Zskbs8AAx1olZdvn4Zp/JPYXLQ5E7bh0KhwNQ+UzkXjB7Z\nsmUL5s+fD09PT7i6usLV1RVubm5YtmwZtm/fjoaGBrlLZPcLUWte2vcSFFDg/THvd+p+UgtTMWH7\nBGQtz2rXk5QMmbFlBrtfiDpZVW0V4lLi8OzgZzt9X/3c+sHZyhmHMg91+r7IODDUie7wn9T/4OEH\nHu6UES+tmRMyB1vO6EZ/LOk/hjrRbQRBwEdJH+G5Ic9pbZ8z+s1AfHo8KmsrtbZPMlwMdaLbHLly\nBPUN9Rjlo73ZRl2tXTG853B8e/5bre2TDBdDneg2a39bi2WDl2l93Di7YEgqDHWim9L+SMOx7GOY\nGzpX6/t+zP8xpBSk4ErpFa3vmwwLQ53oplVHV+GF8BdgbWGt9X13MeuCmf1mYuOpjVrft65xdHSE\nQqEwmsXR0VHS48dx6kQQbzYasnEILj1/CfZd7GWpIb04Hco4Ja4svwJLM0tZaiDdwXHqRBpYc3QN\nnhn0jGyBDgCBLoHo69oX36bxgil1HEOdjF5ueS6+Of8Nlj8o/9SpSwcvxcfHP5a7DNJjGod6dnY2\nRowYgT59+qBv375Yt26dFHURac2KIyswP2w+XKxc5C4FUQFRyC7LRkpBitylkJ7SuE+9oKAABQUF\nCA0NRWVlJQYOHIjvvvsOQUHixP/sUyddlvZHGiLiIpC+LB1OXZ3kLgcAEHskFlfKruDzqM/lLoVk\nJFufuru7O0JDQwEANjY2CAoKQl5enqabJdKKVw+8ir8P+7vOBDoALBywEDvP70RRle48Io30h6R9\n6llZWUhOTkZ4eLiUmyXqFIcyD+Fs0VksHbxU7lKacbNxw7S+0/Bh4odyl0J6SLJnlFZWVmLKlClY\nu3Zt4xNBbomJiWl8rVQqoVQqpdotUYc0CA14af9LWDVqlU4OH3z5oZcxaOMgvPLwK3Do4iB3OaQF\nKpUKKpVK4+1IMk69rq4Ojz32GMaNG9fi4avsUydd9OmJT7H1zFYcnXdUZx8lN+e7OfB38sfrj7wu\ndykkA9meUSoIAubMmQNnZ2f861//kqwwos6SW56L0M9CoZqjQh/XPnKXc1fpxemIiIvA5ecvy3KX\nK8lLtgulx44dw7Zt23Do0CGEhYUhLCwMe/fu1XSzRJ1CEAQs3bMUzw56VqcDHRBvRnqk5yP47ORn\ncpdCeoTTBJBR+fb8t3jj0BtIWZyik33pd0otTMXoraORsSyDfetGhtMEEN1HQWUBnvvxOWyM2qgX\ngQ6Ij7uL8o/Cqv+3Su5SSE+wpU5GoUFowNhtYzHUcyjeHfGu3OW0S35FPvpu6IuTi07C28Fb7nJI\nS9hSJ7qHNUfXoKa+Bm9FvCV3Ke3W3bY7nh/yPP7n4P/IXQrpAYY6GbxjV49h7W9rsf2J7TAzkezW\nDK166aGXcPjKYfya/avcpZCOY6iTQcsqzcJfvvkLNk/cDE87T7nL6TBrC2t8MOYDLNq9CLXqWrnL\nIR3GUCeDVV5TjqivovDqw69inN84ucvR2NQ+U+Ht4I3VR1fLXQrpMF4oJYNU31CP6K+i0dO+Jz6Z\n8InO3jXaXjnlOQj7LAyH5x5GcLdgucuhTsQLpUQ3qRvUmPvdXAgQsG7cOoMJdADwtPPEeyPew4KE\nBahvqJe7HNJBDHUyKA1CAxbvXoy8ijx8++S3MDc1l7skyS0auAg2FjZ4R/WO3KWQDmKok8FoEBqw\nbM8ypBenI2F6AqzMreQuqVOYKEywbfI2/Dvl3zhw+YDc5ZCOYaiTQahV12LW/81CalEq9szcAxsL\nm/v/kB5zs3HD1slb8dSup5BfkS93OaRDGOqk9ypqKjBh+wRU11dj36x9sLO0k7skrRjpMxKLBi7C\n1J1TUVNfI3c5pCMY6qTXMq9nYtjmYejl0Avf/OUbdDXvKndJWvVWxFtwsXLB098/zVFmBIChTnps\n/6X9GLppKBaGLcSnj32qt3eLasJEYYJtj29DenE6VhxZIXc5pAOM77eA9F59Qz1WHFmBz09+jh1T\ndiDCO0LukmRlZW6FhOkJePCLB+Fp54l5YfPkLolkxFAnvXKp5BJm7ZoFO0s7nFh0Aj1se8hdkk5w\nt3HHT7N+wsgtI2Fuao5Z/WfJXRLJhN0vpBfUDWqsTVyL8C/CMbXPVPw480cG+h0CXAKwf/Z+vLL/\nFXx99mu5yyGZsKVOOi85PxmLdy+GlbkVjs0/hgCXALlL0lnB3YKxb/Y+jNk6BpW1lVg4YKHcJZGW\nce4X0lk55Tl44+c3sPfiXsSOjMX8sPkGdct/Z7pw7QLG/Wcc5oTMwZuPvMnjpodkm/tl/vz5cHNz\nQ79+/TTdFBEAcXbF1w++jpBPQ+Bh64ELz13AggELGEzt4O/sj1/m/4L4jHg8/f3THMduRDQO9Xnz\n5mHv3r1S1EJGrvjPYrx96G34rvNFbkUuTj9zGrGjYo3mZiKpudm4QTVHhZLqEkTERSCnPEfukkgL\nNA714cOHw9HRUYpayEhll2Vj+d7l8P/IH3kVefhl/i+ImxSn1w+10BW2lrbY+eROTAyYiCEbh+Dn\nzJ/lLok6GS+UkiwahAYcuHwAG05sgCpLhfmh85G6JBUedh5yl2ZwTBQmeG34axjUYxBm75qN6X2n\nI3ZkLCzNLOUujTqBVkI9Jiam8bVSqYRSqdTGbkkHZZdlY3vqdmw8tRG2lrZYMmgJtkzaAltLW7lL\nM3iRvpE4/cxpLPp+EQZvHIzNEzdjYI+BcpdFN6lUKqhUKo23I8nol6ysLERFRSE1NbXlDjj6xehd\nr76Oned34j+p/0FqUSoeD3wcCwYsQLhHOC9+ykAQBGw7sw0v738Z0/pOw3sj3uMfVR3EJx+RTsm8\nnol1v63D6C2j0fPDnth3eR+WP7gceX/Nw8bojXjQ80EGukwUCgVmh8zG2WfPorymHEEfB2Fz8mao\nG9Ryl0YS0LilPn36dBw+fBjXrl2Dq6sr3n33Xcyb1zT3BFvqxqGmvga/5f6GfZf2ISEjAYVVhXjM\n7zFEB0RjdK/RsLawlrtEuovEnES8vP9llN4oxT9G/wOP9n6Uf3B1QEezkzcfUYeoG9Q4lX8KP2f+\njJ+zfsav2b8i0CUQo3xGITogGkM8hsDUxFTuMqmNBEHA9xe+x6sHXoW7jTveGP4GRvqMZLjLiKFO\nnaqgsgC/5fyGxJxEJOYm4mTeSTxg/wBG+YzCSJ+RiPCOgEMXB7nLJA3VN9Rj25ltWHNsDWwtbPH3\nYX/HpMBJMFGwp1bbGOokCUEQUFBZgDOFZ3C68DRO5Z9CYk4iymvKEe4Zjgc9HkS4ZziGeAyBU1cn\nuculTtIgNCAhIwGrjq5C6Y1SLBm0BHNC5sCxK+9J0RaGOrVbSXUJLly7gPTidJwpPNMY5IIgIMQ9\nBP1d+yOsexjCPcLh5+zH1poREgQBx7KPYcOJDfjhwg+YHDQZiwcu5sglLWCoU6uqaquQWZqJC9cu\n4MK1C8i4liGuizNQq65FgEsA/J390d+1vxjkbv3R3aY7f2GphT+q/kBcShw+P/U5AGBmv5mY2W8m\n/Jz9ZK7MMDHUjZAgCCipLsGVsiu4Unql+frm6z/r/kRPh57wd/ZHgLMY4Ldeu1q7Mryp3QRBwPG8\n49h2Zht2nNuBB+wfwOOBjyM6IBrB3YJ5TkmEoW5A6hvqce3Pa8ivzEdBZQHyK/KRX5nftL75uqCy\nAJZmluhp3xM9HXqK69tfO/REN6tu/CWjTlPfUI9DmYcQnxGP+Ix4WJpaYmLAREwMnIiHvB4yyufG\nSoWhroPqG+pRXlOOshtlKKspQ9mNMpRUl6D4z+LG5Vr1tRavK2oq4NjVEd1tuqO7bXe427iLr29+\nffua479JVwiCgOSCZCRkJCA+Ix6Z1zPxSM9HMMpnFEb3Gs1WfDsx1GXwv7/8Ly6WXETpjdLG0L59\nXV1XDVtLW9hb2sO+iz3sLe3hbOUM567OcLFygYuVS7PXLlYucLZyhkMXB16UJL1XVFWEnzN/xsHL\nB3Ew8yCq66uh9FZiqOdQPOT1EELcQmBuai53mTqLoS6DuJQ43Ki/0Sy0b60dujjAxsKGLROimzKv\nZ0KVpcKvOb/i15xfkXk9EwO6D8BQz6EY6jUUg3sMRg/bHvyduYmhTkR6pexGGY7nHcev2WLIn8g7\nAQAIdQ9FiFsIQt1DEeoeigCXAKPsm2eoE5Feu3XjW0pBClIKUpBckIyUghTkVuTCz8kPgS6BCHIJ\nQqBLIAJdAuHv7I+u5l3lLrvTMNSJyCBV1lYiozgDacVpSC9OR3pxOtKK03D5+mV0t+mOoG5B6O3Y\nGz6OPujl2As+Dj7wcfSBjYWN3KVrhKFOREalvqEel69fRnpxOi6VXMLl65eRWZqJy9cvI6s0CzYW\nNmLIO/rAx8EH3g7e8LTzhIetBzztPOHU1Umn++8Z6kREN93qyrkV8pnXM5FVmoXcilzkVuQipzwH\nN+pvwMPWAx52Ho1hfyvwu9t2h6u1K1ytXWFrYStL+DPUiYjaoaq2Sgz5cjHkb4V9bkUuCioLUFhZ\niKKqIqgFdWPAu1q7ws3ardnXrtau6GbVDc5WznDq6gRrc2tJ/ggw1ImIOkFVbRWKqoqaLYVVhS3e\nK6kuQUl1CdSCGs5dxYC/dV+KU1cnOHd1Ruyo2DaP5OlodhrfOCEionawtrCGj4V48bUtquuqUVJd\ngmvV18T1n9caA99U0fkPjmFLnYhIB/HB00REpHmo7927F4GBgfDz88OaNWukqImIiDpIo+4XtVqN\ngIAAHDhwAB4eHhg8eDC++uorBAUFNe2A3S9ERO0mS/dLUlISevfuDW9vb5ibm2PatGmIj4/XZJNE\nRKQBjUI9NzcXXl5ejV97enoiNzdX46KIiKhjNBrS2NYB9jExMY2vlUollEqlJrvVGdXVgIUFYNr5\no5SIyMCpVCqoVCqNt6NRqHt4eCA7O7vx6+zsbHh6erb43O2hbkiGDQOSk8VQ79JFusXSsvnXXbsC\n1tYtFzPeZUBkMO5s8L7zzjsd2o5GF0rr6+sREBCAgwcPokePHhgyZIjRXSgVBKC+Hrhxo/OW6mqg\nqqrlYmbWetjfa7G1Beztmy92duLa1hYw4SBXIp0gyx2lZmZmWL9+PcaOHQu1Wo0FCxY0C3RjoFAA\n5ubiYmurvf0KAlBT03rY320pLAQuXgTKy4GyspZLVRVgY9My9B0cAGdnwMWlaX374uws/o+CiOTH\nO0qpkVoNVFS0DPvSUqCkBCgubn25dk28tuDsDHTrBvToAXTvLq5vLbe+7taN/xsgagtO6EWyEQSg\nslIM+KIiID8fyMtrvtx6r7wc8PICvL0BH5+mtY8P4Osrhr4OT3FNpDUMddILN24AV64AmZlAVlbz\n9e+/ixedg4KA4GBxufXaw4NhT8aFoU56TxDEfv/z54G0tKb1uXNi19CQIcDgwU2Lm5vcFRN1HoY6\nGbTcXOD4cSApSVwfPw44OQEjR4rLiBFivz2RoWCok1FpaBBb8ocOAT//DBw+LHbRTJoETJ4MhIWx\nu4b0G0OdjJpaLbbid+0Sl7o64IkngLlzgX795K6OqP0Y6kQ3CQJw9iywYwfw5ZeAuzuweDEwaxbH\n05P+YKgTtUKtBvbvB9avF/vhlywBXngBcHSUuzKie+OTj4haYWoKPPoosHu32O+ekwP4+wMrV4p3\n0BIZGoY6GY3AQOCLL4Bjx4AzZ8Rw37lT7K4hMhTsfiGjdfQosGiReCfrJ5+Id7oS6Qp2vxC1062p\nkwcPBgYNAvbskbsiIs2xpU4EsdU+fTrw1FPAu+/ywSckP45+IdJQUREwbZo41fD27Rz+SPJi9wuR\nhlxdgR9/FB8+MmGCOA0xkb5hqBPdxtIS+OoroFcvYPRoBjvpH3a/ELVCEMS7UK9eBb7/XnyyFZE2\nsU+dSGL19cDEieIUv5s2cYIw0i72qRNJzMxMnD8mNRWIjZW7GqK26XCof/PNN+jTpw9MTU1x6tQp\nKWsi0hk2NkBCAvDxx8CRI3JXQ3R/HQ71fv36YdeuXXjkkUekrIdI53TvLna/zJolPoCbSJd1ONQD\nAwPh7+8vZS1EOmv8eGDKFGDhQs4VQ7qNfepEbbRqlfiQ7Lg4uSshujuze30zMjISBQUFLd5fuXIl\noqKi2ryTmJiYxtdKpRJKpbLNP0ukKywtxVkex48HoqMBZ2e5KyJDolKpoFKpNN6OxkMaR4wYgQ8+\n+AADBgxofQcc0kgG5rnngNpa4LPP5K6EDJmsQxoZ2mRM3ntPvCHpt9/kroSopQ6H+q5du+Dl5YXE\nxERMmDAB48aNk7IuIp3l4AC8/774aDy1Wu5qiJrjHaVEHSAI4nzszzwDzJ4tdzVkiDhNAJGWHTkC\nzJ0LpKcDFhZyV0OGhtMEEGnZI48AAQHAxo1yV0LUhC11Ig0kJ4tzr//+O2BtLXc1ZEjYUieSQViY\n2GJfu1buSohEbKkTaejCBeDhh4FLlwA7O7mrIUPBljqRTPz9gUcfBT76SO5KiNhSJ5JERgYwfLjY\nWre1lbsaMgRsqRPJKCAAiIwE1q+XuxIydmypE0kkLQ2IiGBrnaTBljqRzIKCgFGjxKckEcmFLXUi\nCZ0/D4wYIbbWbWzkrob0GVvqRDogOFgM9U8+kbsSMlZsqRNJ7Nw5sRvm0iXeZUodx5Y6kY7o00e8\ny3TDBrkrIWPEljpRJ0hNFYc4srVOHcWWOpEO6ddPnG/900/lroSMDVvqRJ3kzBlg7FixtW5lJXc1\npG/YUifSMf37Aw89xAdUk3axpU7UiU6fBsaNE1vrXbvKXQ3pE1la6i+//DKCgoIQEhKCxx9/HGVl\nZZpsjsjghIQA4eHA55/LXQkZC41CfcyYMTh37hxOnz4Nf39/rFq1Sqq6iAzG228Dq1cDpaVyV0LG\nQKNQj4yMhImJuInw8HDk5ORIUhSRIQkNBaKigJgYuSshYyDZhdJ///vfGD9+vFSbIzIoK1cC27eL\n49eJOpPZ/T4QGRmJgoKCFu+vXLkSUVFRAIDY2FhYWFhgxowZrW4j5rYmilKphFKp7Fi1RHrKxUVs\nqT/3HHDoEKBQyF0R6RqVSgWVSqXxdjQe/RIXF4eNGzfi4MGD6NKlS8sdcPQLEQBArQYGDQJeegmY\nOVPuakjXdTQ7NQr1vXv34m9/+xsOHz4MFxcXSQsjMkTHjwMTJgCnTgGennJXQ7pMllD38/NDbW0t\nnJycAABDhw7FJ3fMOcpQJ2puxQqxC2b/fsCEt//RXcgS6m3aAUOdqJn6evGxd48/Dvztb3JXQ7qK\noU6kRzIzgSFDgB9+ENdEd+LcL0R6xMcH2LQJmDQJyMqSuxoyJPcd0khEnSM6Grh8Wbxw+ssvgL29\n3BWRIWD3C5GMBEEcu56eDnz/PSf9oibsUyfSU/X1wFNPAQUFQEICYGMjd0WkC9inTqSnzMyArVuB\nXr2AMWM48RdphqFOpANMTcXpeYcMER+skZEhd0WkrxjqRDrCxAT48EPgxRfF55vu2iV3RaSP2KdO\npIOOHwemTAHGjwfWrAHs7OSuiLSNfepEBmTwYPFRePX1QN++4k1KRG3BljqRjjt4EFiyBPD2Flvt\nYWFyV0TawJY6kYEaNQo4d068+3T8eGDqVODkSbmrIl3FUCfSA+bmwLPPAhcuiA+ynjwZGDkS+L//\nA2pr5a6OdAm7X4j0UF0d8N//Al98AZw9C0ybBsydCwwYwKcqGQreUUpkpDIzgS1bgC+/FIdFRkWJ\ny/DhYguf9BNDncjICQKQkiLOIbN7N3DxIqBUinO3R0QA/fvzoRz6hKFORM3k54tPWDp8WFyKisT+\n+IEDmxYvL3bX6CqGOhHdU0EBkJQEnDghjp45eVJ8GPaAAUBQUPPFxYVhLzeth/qbb76JhIQEKBQK\nODs7Iy4uDl5eXpIVRkSdSxCAvDyxyyYtTVzS08W1QgH4+ooP8/DxEcfI31r37Al06SJ39YZP66Fe\nUVEBW1tbAMBHH32E06dP44svvpCsMCKShyCIXTWXL4sXYbOyxPWtJTtbnLagRw+ge3dxffvr7t3F\nlr6Li/jgD/bjd0xHs7PDTz66FegAUFlZCRcXl45uioh0iEIBuLmJy9ChLb/f0AAUF4ut/Pz8pvX5\n88CBA+Lr4mLg2jWgqgpwdBQD3tm55drJSQz+W4udXdPrrl3ZBdQRGvWpv/7669i6dSusrKyQmJgI\nBweHljup3NMJAAAF/0lEQVRgS53IaNXVASUlTSF/57qkBCgrE5fy8qbXZWXivDd2ds2D3s4OsLYG\nrKzE9e3Lne/d7TMWFvrxx6JTul8iIyNRUFDQ4v2VK1ciKiqq8evVq1cjIyMDmzdvlqwwIjJutbVN\nQX/7uqpKXP78s+n13d5r7TNqNWBpKV4XuLXc+XVr793tM5aW4h+K25fW3rOwAPz92/4HpVO6X/bv\n39+mjcyYMQPjx4+/6/djYmIaXyuVSiiVyjZtl4iMl4VFU9+8lNRqoKYGuHGjaX37cud7rX2mslL8\n38aNG+Ifn9pa8XO3Xre21NSIF6FNTVuvS6VSQaVSafzv63D3y++//w4/Pz8A4oXSpKQkbN26teUO\n2FInImo3rY9+mTJlCjIyMmBqagpfX19s2LABrq6ukhVGRGTMePMREZEB4XzqRETEUCciMiQMdSIi\nA8JQJyIyIAx1IiIDwlAnIjIgDHUiIgPCUCciMiAMdSIiA8JQJyIyIAx1IiIDwlAnIjIgDHUiIgPC\nUCciMiAMdSIiA8JQJyIyIAx1IiIDwlAnIjIgDHUiIgOicah/8MEHMDExQUlJiRT1EBGRBjQK9ezs\nbOzfvx89e/aUqh6DplKp5C5BZ/BYNOGxaMJjoTmNQv2vf/0r/vGPf0hVi8HjCduEx6IJj0UTHgvN\ndTjU4+Pj4enpif79+0tZDxERacDsXt+MjIxEQUFBi/djY2OxatUq7Nu3r/E9QRCkr46IiNpFIXQg\njc+ePYtRo0bBysoKAJCTkwMPDw8kJSXB1dW12Wd79+6NS5cuSVMtEZGR8PX1xcWLF9v9cx0K9Tv5\n+Pjg5MmTcHJy0nRTRESkAUnGqSsUCik2Q0REGpKkpU5ERLpBsjtK9+7di8DAQPj5+WHNmjWtfub5\n55+Hn58fQkJCkJycLNWudc79joVKpYK9vT3CwsIQFhaGFStWyFBl55s/fz7c3NzQr1+/u37GWM6J\n+x0LYzknAPH+lhEjRqBPnz7o27cv1q1b1+rnjOHcaMuxaPe5IUigvr5e8PX1FTIzM4Xa2lohJCRE\nOH/+fLPP/PDDD8K4ceMEQRCExMREITw8XIpd65y2HItDhw4JUVFRMlWoPUeOHBFOnTol9O3bt9Xv\nG8s5IQj3PxbGck4IgiDk5+cLycnJgiAIQkVFheDv72+0edGWY9Hec0OSlnpSUhJ69+4Nb29vmJub\nY9q0aYiPj2/2mYSEBMyZMwcAEB4ejtLSUhQWFkqxe53SlmMBGMcQ0OHDh8PR0fGu3zeWcwK4/7EA\njOOcAAB3d3eEhoYCAGxsbBAUFIS8vLxmnzGWc6MtxwJo37khSajn5ubCy8ur8WtPT0/k5ube9zM5\nOTlS7F6ntOVYKBQK/PLLLwgJCcH48eNx/vx5bZepE4zlnGgLYz0nsrKykJycjPDw8GbvG+O5cbdj\n0d5z4543H7VVW0e/3PnXxhBHzbTl3zRgwABkZ2fDysoKP/74IyZNmoQLFy5ooTrdYwznRFsY4zlR\nWVmJKVOmYO3atbCxsWnxfWM6N+51LNp7bkjSUvfw8EB2dnbj19nZ2fD09LznZ27dsGRo2nIsbG1t\nG2/cGjduHOrq6oxylktjOSfawtjOibq6OjzxxBOYNWsWJk2a1OL7xnRu3O9YtPfckCTUBw0ahN9/\n/x1ZWVmora3Fjh07EB0d3ewz0dHR2LJlCwAgMTERDg4OcHNzk2L3OqUtx6KwsLCxFZKUlARBEIzy\nxi1jOSfawpjOCUEQsGDBAgQHB2P58uWtfsZYzo22HIv2nhuSdL+YmZlh/fr1GDt2LNRqNRYsWICg\noCB89tlnAIDFixdj/Pjx2LNnD3r37g1ra2ts3rxZil3rnLYci507d2LDhg0wMzODlZUVvv76a5mr\n7hzTp0/H4cOHUVxcDC8vL7zzzjuoq6sDYFznBHD/Y2Es5wQAHDt2DNu2bUP//v0RFhYGAFi5ciWu\nXr0KwLjOjbYci/aeG7z5iIjIgPBxdkREBoShTkRkQBjqREQGhKFORGRAGOpERAaEoU5EZEAY6kRE\nBoShTkRkQP4/4pOnFHCFMpIAAAAASUVORK5CYII=\n", "text": [ "" ] } ], "prompt_number": 30 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Power absorption" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's think a little about the power _absorbed_ by the oscillator; to do that, we must consider the\n", "work done _against_ the drag. If the displacement changes from\n", "$\\psi$ to $\\psi + \\Delta \\psi$ then the work done is $-F_d\n", "\\Delta\\psi$, where $F_d = -b\\dot{\\psi}$ is the work done against the\n", "drag. If that takes a time $\\Delta t$ then the rate of work is $-F_d\n", "(\\Delta \\psi/\\Delta t)$ which tends to $-F_d \\dot{\\psi}$ as $\\Delta t\n", "\\rightarrow 0$. So the instantaneous power adsorption becomes: \n", "\\begin{eqnarray}\n", " P &=& -F_d \\dot{\\psi} = b\\dot{\\psi}^2\n", "\\end{eqnarray}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We are not actually interested in the instantaneous power, but the\n", "time averaged power (often written $\\langle P\\rangle$), which for a\n", "harmonic force can be shown to be: \n", "\\begin{eqnarray}\n", "\\langle P \\rangle &=& b\\langle \\dot{\\psi}^2 \\rangle = \\frac{1}{2}b\\omega^2A^2\\\\\n", "&=& \\frac{F_0^2}{m^2}\\frac{m\\gamma\\omega^2}{(\\omega_0^2 - \\omega^2)^2 + 4\\gamma^2\\omega^2}\n", "\\end{eqnarray}\n", "where we have substituted $b = 2m\\gamma$. This has a maximum value of\n", "$F^2_0/(4m\\gamma)$ when $\\omega = \\omega_0$. Note that this is\n", "inversely proportional to the resistive force. \n", "\n", "Finally we introduce the idea of _impedance_ which is a measure\n", "of the resistance to the motion of the oscillator. It is defined as\n", "the amplitude of the driving force divided by the complex amplitude of\n", "the velocity, $Z(\\omega) = F_0/\\dot{\\psi} = b + i(m\\omega -\n", "s/\\omega)$. At resonance, $Z(\\omega) = b$; in the resonance region,\n", "it only departs very slightly from this value. Away from resonance,\n", "it includes a phase lag or lead, which reflects the relation of\n", "velocity to the driving force. We will encounter impedance again with\n", "waves.\n", "\n", "We plot below the power absorbed, imaginary part of the impedance and the magnitude\n", "of the impedance, using the system given above. Note how the impedance has a minimum\n", "at the point that the maximum power is absorbed (and is purely real at that point).\n", "Away from that, the power is mainly dissipated (by the impedance)." ] }, { "cell_type": "code", "collapsed": false, "input": [ "# We use the values of b, m, k, omega and A defined above\n", "power = 0.5*b*omega*omega*A*A\n", "impi = m*omega - k/omega\n", "Z = sqrt(b*b+impi*impi)\n", "plot(omega,power,label='Power')\n", "plot(omega,impi,label='Imaginary part of impedance')\n", "plot(omega,Z,label=r'$\\vert Z\\vert$')\n", "legend(loc=4)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 35, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAAXUAAAD7CAYAAACVMATUAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nCNEhlNQJIUSHUFInhBAd8n/MsqXGOu9vfgAAAABJRU5ErkJggg==\n", "text": [ "" ] } ], "prompt_number": 35 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Transients" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that so far we have discussed a _steady state_\n", "solution: there is a transient behaviour at the beginning of the\n", "oscillation which takes the form of a damped oscillator\n", "added to a solution of the homogenous equation. So a real system will\n", "respond at its natural frequency, but damping will cause that solution\n", "to die off, while the driven oscillation persists into the steady\n", "state: \n", "\n", "\\begin{equation}\n", " \\psi(t) = A_s \\cos(\\omega t +\\phi) + A_t e^{-\\gamma t} \\cos(\\omega_0 t + \\phi_t)\n", "\\end{equation}\n", "where the subscripts s and t stand for _steady state_ and\n", "_transient_ respectively. For lightly damped system near\n", "resonance we can approximate and find that $A_t \\simeq A_s$ and\n", "$\\phi_t \\simeq \\phi_s + \\pi$, giving: \n", "\\begin{equation}\n", " \\psi(t) = A_s \\left(\\cos(\\omega t +\\phi) - e^{-\\gamma t} \\cos(\\omega_0 t + \\phi_s)\\right)\n", "\\end{equation}\n", "\n", "We plot an example of this below, with the total response of the system over time in blue, and the transient in green: notice how the initial response of the system is at its natural frequency, $\\omega_0$, and how this dies down and leads to the steady-state, driven response at $\\omega$. You should try changing the value of the driving frequency `omegaD` (though you will also have to change the amplitude of the driving force, `F0`, and the initial response, `F1`)." ] }, { "cell_type": "code", "collapsed": false, "input": [ "t = linspace(0,50.0,1000)\n", "# Driving frequency\n", "omegaD = 7.0\n", "# Amplitude of driving (F0) and response (F1)\n", "F0 = 10.0\n", "F1 = 0.10\n", "fac = (omega0*omega0 - omegaD*omegaD)**2 + 4*gamma*gamma*omegaD*omegaD\n", "A = F0/(sqrt(fac)*m)\n", "# Define transient\n", "transient = 2*(F1/m)*exp(-gamma*t)*cos(omega0*t+pi/2)\n", "steady = A*cos(omegaD*t+pi/2)\n", "plot(t,transient+steady)\n", "plot(t,transient)\n", "#plot(t,A*cos(omegaD*t+pi/2)+2*(F1/m)*exp(-gamma*t)*cos(omega0*t+pi/2))\n", "#plot(t,2*(F1/m)*exp(-gamma*t)*cos(omega0*t+pi/2))" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 44, "text": [ "[]" ] }, { "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAAX0AAAEACAYAAABfxaZOAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nECz6DMMwKQSLPsMwTArxP6t1Bk6MKhdoAAAAAElFTkSuQmCC\n", "text": [ "" ] } ], "prompt_number": 44 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Combining Oscillations" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Instead of combining a single driving force with a damped response,\n", "what happens if we have an oscillator with _two_ driving forces ?\n", "The full equation looks like this: \n", "\n", "$$\n", "\\begin{equation}\n", " m\\frac{\\partial^2 \\psi}{\\partial t^2} = -s \\psi -b \\frac{\\partial \\psi}{\\partial t} + F_1 \\cos (\\omega_1 t + \\phi_1) + F_2 \\cos (\\omega_2 t + \\phi_2),\n", "\\end{equation}\n", "$$\n", "\n", "We know that the SHO equation is _linear_, so we can try solving\n", "this problem by summing the steady state solutions arising from each\n", "driving force independently. We will consider three specific cases\n", "(where it is easier to understand the behaviour) with the general case\n", "more complex: \n", "\n", "* Same amplitude and frequency ($F_1 = F_2 = F$, $\\omega_1 = \\omega_2 = \\omega$), different phase ($\\phi_1 \\ne \\phi_2$)\n", "* Same frequency ($\\omega_1 = \\omega_2$), different phase and amplitude ($F_1 \\ne F_2, \\phi_1 \\ne \\phi_2$)\n", "* Same amplitude and phase ($F_1 = F_2=F, \\phi_1 = \\phi_2$), different frequency ($\\omega_1 \\ne \\omega_2$)\n", "\n", "In all cases, we assume that we can write the displacement as a\n", "superposition of the responses to the individual oscillations: \n", "$$\n", "\\begin{equation}\n", " \\psi = A_1 \\cos(\\omega_1 t + \\phi_1) + A_2 \\cos(\\omega_2 t + \\phi_2)\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Same amplitude and frequency" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For simplicity, we will set\n", "$\\phi_1 = 0$ and set $\\phi_2 = \\phi$. Then we can perform the\n", "following manipulations, using the complex exponential form for\n", "simplicity: \n", "$$\n", "\\begin{eqnarray}\n", " \\psi &=& Ae^{i\\omega t} + Ae^{i(\\omega t + \\phi)}\\\\\n", " &=& Ae^{i\\omega t}\\left(1 + e^{i\\phi}\\right)\\\\\n", " &=& Ae^{i\\omega t}\\left(e^{i\\phi/2}e^{-i\\phi/2} + e^{i\\phi/2}e^{i\\phi/2}\\right)\\\\\n", " &=& Ae^{i\\omega t}e^{i\\phi/2}\\left(e^{-i\\phi/2} + e^{i\\phi/2}\\right)\\\\\n", " &=& 2Ae^{i(\\omega t+\\phi/2)}\\cos(\\phi/2) = 2A\\cos(\\phi/2)e^{i(\\omega t+\\phi/2)}\n", "\\end{eqnarray}\n", "$$\n", "where the identification of $\\cos (\\phi/2)$ follows from De Moivre's\n", "theorem. So the _resultant_ oscillation has magnitude\n", "$2A\\cos(\\phi/2)$ and phase $\\phi/2$. The same result can be found\n", "using trigonometry and a phasor diagram" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from IPython.display import Image\n", "Image(filename='PhasorSameFreqAmp.png',width=200)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": { "png": { "width": 200 } }, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAXMAAAFyCAIAAACIhK1gAAAACXBIWXMAAAxNAAAMTQHSzq1OAAAA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\n5yxAz1mAnrMAPWcBes4C9JwF6DkL0LtVfAXhCmQJfgAAAABJRU5ErkJggg==\n", "prompt_number": 28, "text": [ "" ] } ], "prompt_number": 28 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Same frequency" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " We can write a solution as before, by using\n", "superposition: \n", "\n", "$$\n", "\\begin{equation}\n", " \\psi = A_1 \\cos(\\omega t + \\phi_1) + A_2 \\cos(\\omega t + \\phi_2) = A \\cos(\\omega t + \\theta)\n", "\\end{equation}\n", "$$\n", "This is illustrated in a phasor diagram below." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from IPython.display import Image\n", "Image(filename='Phasors.png',width=200)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": { "png": { "width": 200 } }, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAXMAAAFyCAIAAACIhK1gAAAACXBIWXMAAAxNAAAMTQHSzq1OAAAA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\nAD1nAXrOAvScBeg5C9C7VXwF4QpkCX4AAAAASUVORK5CYII=\n", "prompt_number": 27, "text": [ "" ] } ], "prompt_number": 27 }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is just an oscillation with frequency $\\omega$ but a total\n", "amplitude and phase which depend on the amplitudes and phases of the\n", "two driving forces. If, for instance, the phase difference is $\\phi_1\n", "- \\phi_2 = \\pi$ then the two driving forces are out-of-phase and we\n", "will get destructive interference. The use of phasors allows a simple\n", "visualisation of the resultant. Using trigonometry from the phasor\n", "diagram (or the cosine rule), we can write the amplitude of the\n", "resultant as: \n", "$$\n", "\\begin{equation}\n", " A^2 = A_1^2 + A^2_2 + 2A_1A_2\\cos(\\phi_2 - \\phi_1)\n", "\\end{equation}\n", "$$\n", "Now we can see that the amplitude will vary between $A_1+A_2$ if\n", "$\\phi_1=\\phi_2$ and $|A_1 - A_2|$ if $|\\phi_1 - \\phi_2| = \\pi$. If\n", "the driving forces are _in phase_ then we have the maximum\n", "amplitude, while if they are in _antiphase_ we have the minimum\n", "amplitude. \n", "\n", "The phase is also found using trigonometry; by projecting onto thereal and imaginary axes, we can write: \n", "$$\n", "\\begin{equation}\n", " \\tan \\theta = \\frac{A_1\\sin \\phi_1 + A_2\\sin \\phi_2}{A_1 \\cos \\phi_1 + A_2 \\cos \\phi_2}\n", "\\end{equation}\n", "$$\n", "We can get the same result using complex notation. Start by finding\n", "that, for two general complex numbers $z_1$ and $z_2$: \n", "$$\n", "\\begin{eqnarray}\n", " \\left| z_1 + z_2\\right|^2 &=& (z_1+z_2)(z_1 + z_2)^\\star = (z_1+z_2)(z^\\star_1+z^\\star_2)\\\\\n", " &=& |z_1|^2 + |z_2|^2 + (z_1z^\\star_2 + z^\\star_1z_2)\\\\\n", " &=& |z_1|^2 + |z_2|^2 +2\\mathrm{Re}(z_1z^\\star_2)\n", "\\end{eqnarray}\n", "$$\n", "Now we have $z_1 = A_1 e^{i(\\omega t +\\phi_1)}$ and $z_2 = A_2\n", "e^{i(\\omega t +\\phi_2)}$ with $A_1$ and $A_2$ real. This gives: \n", "$$\n", " \\begin{eqnarray}\n", " \\left| z_1 + z_2\\right|^2 &=& |z_1|^2 + |z_2|^2 +2\\Re(z_1z^\\star_2)\\\\\n", " &=& A_1^2 + A_2^2 + 2\\mathrm{Re}(A_1 e^{i(\\omega t +\\phi_1)}A_2 e^{-i(\\omega t +\\phi_2)})\\\\\n", " &=& A_1^2 + A_2^2 + 2\\mathrm{Re}(A_1A_2 e^{i(\\phi_1 - \\phi_2)})\\\\\n", " &=& A_1^2 + A_2^2 + 2A_1A_2 \\cos(\\phi_2 - \\phi_1)\n", " \\end{eqnarray}\n", " $$\n", "while we can find the phase from:\n", "$$\n", "\\begin{eqnarray}\n", " \\mathrm{arg}(z_1+z_2) &=& \\tan^{-1} \\left[ \\frac{\\mathrm{Im}(z_1+z_2)}{\\mathrm{Re}(z_1+z_2)} \\right]\\\\\n", " \\Rightarrow \\theta &=& \\tan^{-1} \\left[\\frac{A_1\\sin \\phi_1 + A_2\\sin \\phi_2}{A_1 \\cos \\phi_1 + A_2 \\cos \\phi_2}\\right]\n", "\\end{eqnarray}\n", "$$" ] }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Same amplitude and phase: beats" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For this case, we use the\n", "principle of superposition to write: \n", "$$\n", "\\begin{equation}\n", " \\psi = A \\cos \\omega_1 t + A \\cos \\omega_2 t \n", "\\end{equation}\n", "$$\n", "Note that we have assumed that the common phase can be set to zero for\n", "simplicity. But we can rearrange this using a standard\n", "trigonometrical formula for the sum of two cosines: \n", "$$\n", "\\begin{eqnarray}\n", " \\psi &=& A \\left(\\cos \\omega_1 t + \\cos \\omega_2 t\\right)\\\\\n", " &=& 2A \\cos\\left(\\frac{\\omega_1 + \\omega_2}{2} t\\right) \\cos\\left(\\frac{\\omega_1 - \\omega_2}{2} t\\right)\n", "\\end{eqnarray}\n", "$$\n", "\n", "Now let's define two \\emph{new} angular frequencies, and rewrite the solution:\n", "$$\n", "\\begin{eqnarray}\n", " \\omega &=& \\frac{1}{2}\\left(\\omega_1 + \\omega_2\\right)\\\\\n", " \\Delta \\omega &=& \\frac{1}{2}\\left(\\omega_1 - \\omega_2\\right)\\\\\n", " \\psi &=& 2A\\cos (\\omega t) \\cos (\\Delta\\omega t)\n", "\\end{eqnarray}\n", "$$\n", "If the two angular frequencies are relatively close, then the form of\n", "the resulting oscillation is rather simple. There is an oscillation\n", "at the average frequency (the term $\\cos \\omega t$) whose amplitude is\n", "\\emph{modulated} by a slow oscillation at the difference frequency\n", "(the term $\\cos \\Delta\\omega t$ - also known as the envelope). This\n", "is a phenomenon known as \\emph{beats}, and is illustrated in the figure below. It is important to understand\n", "that, while the angular frequency of the modulation is $\\Delta \\omega\n", "= \\frac{1}{2}|\\omega_1 - \\omega_2|$, the angular frequency at which\n", "\\emph{peaks} of activity occur is $|\\omega_1 - \\omega_2|$ (or\n", "equivalently zeroes of activity). The number of minima per second is\n", "$\\Delta \\omega/\\pi$. The perceived effect (say for sound) will be of\n", "a sound at the average angular frequency with its amplitude varying according\n", "to the envelope." ] }, { "cell_type": "code", "collapsed": false, "input": [ "fig = figure(figsize=[12,3])\n", "# Add subplots: number in y, x, index number\n", "ax = fig.add_subplot(121,autoscale_on=False,xlim=(0,65),ylim=(-2,2))\n", "ax2 = fig.add_subplot(122,autoscale_on=False,xlim=(0,65),ylim=(-2,2))\n", "t = linspace(0.0,65.0,1000)\n", "o1 = 1.3\n", "o2 = 1.0\n", "omega = 0.5*(o1+o2)\n", "domega = 0.5*(o1-o2)\n", "# Plot the resulting pattern\n", "ax.plot(t,2.0*cos(omega*t)*cos(domega*t))\n", "# Show the envelope in a dashed red line - we need two lines\n", "ax.plot(t,2.0*cos(domega*t),'r--')\n", "ax.plot(t,2.0*cos(domega*t+pi),'r--')\n", "# On a separate plot, show the two individual oscillations\n", "ax2.plot(t,cos(o1*t))\n", "ax2.plot(t,cos(o2*t))" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 48, "text": [ "[]" ] }, { "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAAsYAAADICAYAAADr/SfsAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n/w9yjJ1PrwUMWgAAAABJRU5ErkJggg==\n", "text": [ "" ] } ], "prompt_number": 48 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The result of two driving forces on one SHO with different frequencies but the same amplitude. Left: two forces shown together. Right: the resultant motion with the envelope superimposed in dashed lines. The original frequencies are $\\omega = 1.0$ and $\\omega=1.2 s^{-1}$; on ce how the peaks coincide at regular intervals, and this leads to the maxima in the envelope function." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The figure above shows an illustration of exactly this behaviour for the frequencies $\\omega_1 = 1.2 s^{-1}$ and\n", "$\\omega_2 = 1.0 s^{-1}$ which gives a resulting motion at frequency\n", "$\\omega = 1.1 s^{-1}$ modulated by an envelope with frequency $\\Delta\n", "\\omega = 0.1 s^{-1}$. \n", "\n", "The resulting motion will show a true periodicity if the ratio of the\n", "frequencies can be written as a ratio of integers\n", "(i.e. $\\omega_1/\\omega_2 = n_1/n_2$). The motion from two frequencies\n", "which are almost non-periodic is shown in\n", "the figure below (I used 25/3 and 7.1 here but\n", "we could have made it properly non-periodic if we'd tried harder). \n", "\n", "Note that if there is a phase difference between the two oscillations\n", "then beats still result, but the phase difference will appear in both\n", "sum and difference phases. We can write $\\cos(\\omega_1 t + \\phi_1) +\n", "\\cos(\\omega_2 t + \\phi_2) = 2\\cos(\\omega t + \\phi)\\cos(\\Delta \\omega\n", "t + \\Delta \\phi)$ where $\\omega$ and $\\Delta \\omega$ have the same\n", "meaning as before and $\\phi = (\\phi_1 + \\phi_2)/2$ and $\\Delta \\phi =\n", "(\\phi_1 - \\phi_2)/2$. " ] }, { "cell_type": "code", "collapsed": false, "input": [ "fig = figure(figsize=[12,3])\n", "# Add subplots: number in y, x, index number\n", "xmax = 15.0\n", "ax = fig.add_subplot(121,autoscale_on=False,xlim=(0,xmax),ylim=(-2,2))\n", "ax2 = fig.add_subplot(122,autoscale_on=False,xlim=(0,xmax),ylim=(-2,2))\n", "t = linspace(0.0,xmax,1000)\n", "o1 = 25./3\n", "o2 = 7.1\n", "omega = 0.5*(o1+o2)\n", "domega = 0.5*(o1-o2)\n", "# Plot the resulting pattern\n", "ax.plot(t,2.0*cos(omega*t)*cos(domega*t))\n", "# Show the envelope in a dashed red line - we need two lines\n", "ax.plot(t,2.0*cos(domega*t),'r--')\n", "ax.plot(t,2.0*cos(domega*t+pi),'r--')\n", "# On a separate plot, show the two individual oscillations\n", "ax2.plot(t,cos(o1*t))\n", "ax2.plot(t,cos(o2*t))" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 24, "text": [ "[]" ] }, { "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAAsYAAADICAYAAADr/SfsAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nrkJggg==\n", "text": [ "" ] } ], "prompt_number": 24 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "General case" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If the amplitudes and frequencies all differ, then the phasor diagram\n", "must be dynamic as illustrated \n", "below." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from IPython.display import Image\n", "Image(filename='PhasorsDiffFreq2.png',width=200) #Size is in pixels" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": { "png": { "width": 200 } }, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAXMAAAFyCAIAAACIhK1gAAAACXBIWXMAAAxNAAAMTQHSzq1OAAAA\nHXRFWHRTb2Z0d2FyZQBHUEwgR2hvc3RzY3JpcHQgOC42MoMtCM8AACAASURBVHic7d1PqOxm/T/w\nZ37cS9Haluco/kGFmtNbQRG0OVDd9FrI4M5SS2YpVjSzk1aQBMSFuDARpJuiJKALEYSJmxbExQSF\nitDFpC3SRf/NQ7uQVqQTbS+Vy714vouPN780MyeTyTzJ8yR5vxbae+6cmdxzMu95ns/zb3J+fs4A\nAKT6f6ovAAAGCMkCAPIhWQBAPq2TJcuyyWQymUziON7+W8/zJpPJ2dlZ9xcGANW0ThbOuW3bjLEk\nSbb/Nooixhg9AAC0onWyMMYsy2KMbbdZ4jjOsowhWQC0pHuyUHBkWVYKF2rF2LZtGIaaKwOAiylO\nFs/zqh+ws0OUBw21aABANyqTJUmSIAj2Pmy7Q0RdoTx0AEA3KpPlu9/9bp2HbXeI8q4Q57y9ywOA\nxlQmy7vvvssuGPcp4pxTs4Ueia4QgP5UJsvbb7/NGPvVr36195HFDhH9r2EY6AoBaEtZsiRJcvny\nZcbYK6+8svfBxQ4RprEA6E9lspycnDDGnnvuub0PNgzDNE3GWBRFaZoyJAuA3molSxAE8/m8+jGe\n5+0dQi5K0/TOO+9kjN1xxx17Sy3sVpTQI03TpKABAE2d77Nerznn6/W6+mGbzYYxtvdhOcbY/fff\nzxi7evWq67p1LiO/Zt/3a74KACixv80SBEGdqa6cc8dx6sxPYYwlScI5v+OOOxhjV65coQ5OtbxD\nxNAVAtDenmShomnN8V3LsqIoouU81ZIkyZ/zvvvuq9MbYrdGiDCjH0B/e5KFZruW2gie552enk4m\nk9PT02JthR62c8eDkjRN8wbIlStXOOd1woVGhTCNBUB/e5IlTdPSO9nzvCAIDMNwXdeyrCAIiuFi\nWVadrk2xzULftTdZMKMfoEcuVf91sXHBGBNCBEHguq7v+/QVwzCKoWAYxt5kSdOUc158WtM0L/ou\nIQRN4acKDmb0A/RDdYGXMVYcuKFAqRgAcl1373PmqNmyXC4rHrNcLvNLrTNEBQA6OGymHFVnuyyg\ncs6pkWLb9nK5RO0WoBf29IZK6E1O9Q76SpZl1GNqqZNimibNlAGAHtnTZrEsSwhR/CO7VfIgQRBM\np9N8pFkIgbEbANjTZjFNs1igNU2TpsNROyXLsiiKihNMhBCYdw8A+5MlCIJi9ycMQ855HMc0j9Z1\nXarasls9o/yPADBae3pD1B4pTX7zfZ/GaDabje/7eejEcYz5JgDA6qx1tm27zrRaxliSJI7jHH1J\nANB7+5PFdV0hxN45svQYdIUAgNVJFs55GIZ1Zt9TCUbShQFAj9Waz2JZ1t6xZLRWACCn+xmJANBH\nSBYAkA/JAgDyIVkAQD4kCwDIh2QBAPmQLAAgH5IFAORDsgCAfEgWAJAPyQIA8iFZAEA+JAsAyIdk\nAQD5kCwAIB+SBQDkQ7IAgHxIFgCQD8kCAPIhWQBAPiQLAMiHZAEA+ZAsACAfkgUA5EOyAIB8SBYA\nkA/JAgDyIVkAQD4kCwDIh2QBAPmQLAAgH5IFAORDsgCAfEgWAJAPyQIA8iFZAEA+JAsAyIdkAQD5\nkCwAIB+SBQDkQ7IAgHxIFgCQD8kCAPIhWQBAPiQLAMiHZAEA+ZAsACAfkgUA5EOyAIB8SBYAkA/J\nAgDyIVkAQD4kCwDIh2QBAPmQLAAgH5IFAORDsgCAfEgWAJAPyQIA8iFZAEA+JAsAyIdkAQD5kCwA\nIB+SBQDkQ7IAgHxIFgCQD8kCAPIhWQBAPiQLAMiHZAEA+ZAsACAfkgUA5EOyAIB8SBYAkA/JAgDy\nIVkAQD4kCwDIh2QBAPkudfliSZIU/7jZbBhjzz//PP3RNE3OeZfXAwAtaZ4saZpmWVYMCyFElmWM\nMdM088cwxgzDKEWGZVn5f7/77rv0JPS/WZYJIfInKf1x+xmQRwAaqpUsSZJkWUZRwhjLsoxzzjk3\nTdOyLPqPBq99cnLCGHvggQeKQVNNCEFBQ9I0TZIkTVPTNIUQhmFQ0NR/QgBow/uSheKD3r15iBiG\nwRizLMu27WYJIpFhGHQ9pJQgeTMqjmN26+Ip+ErfCACtusQYS9M0jmP65Kf3IVF9bQejay7GTZ6V\naNoAdOkSY6ynOVLHzvigRlmSJNTLK3brlFwkwPB0OjakCeoZ5YmTZVkcx57nUe8JKQNwvDEmSwnn\n3HEc+m9qy+QpY1kWekwADSBZ3scwjFLKzGYz+jpSBqC+yfn5uarXnk6nSZIsl0v937H58DZjjLpL\nQ61MAUiBNkstxSI3pUwURVmWWZaVt3EAIIdkOVieMlT6nc/njDHbtvVveQF0BsnSXF76FULEcRwE\ngWmajuNgSh4A1jpLYBiG67rL5dK27SiKptMp9ZVUXxeAMmizyJR3lPIJMrZt27at+rrgQrQMDc1M\n6dBmaYVt22EYhmGYZdlsNvM8j8aVQDnqunqeZ5rm7bfffnp6GkWR6osaILRZWkSFGMdxhBBRFHme\nRws78QnZJSFEmqZpmi6Xy5deeum999677bbbrl+/Tn976dIlzLduA5KlC4Zh+L7PGEvTNAgCDFe3\nSgjxt7/97amnnvrTn/705ptv3rhx4/Llyzdu3MgfkMcKY+yuu+7C1KQ2YKacGlEUxXFs2zbypSZa\nR0obZdTZHuz69euvv/76K6+88tJLLzHG/vvf/+582suXL99///2f+9znaPEqdtuQBW0WNaiXRANJ\n1H5Bm7woSRJxC+eclnHl60UbPNsf/vCH3/3ud//4xz9K7ZcbN2586lOfYoy9+OKLTz/99F133fXe\ne+99+tOfvnLlysc//nEqydMF4Bd0ELRZ1KMZvTR0Pdrblza1KI7UtNGCoPLtr3/961deeSWvthTv\nQNrQJ03Tt95665lnnrl8+fJkMplMJu+8886HPvShZ599VuLFDBuSRRcjzBdaJ0EbGHa8GovmTz/1\n1FN/+ctf3njjjYofOC1MDYLgzjvv/NrXvoaFqTUhWfSS58tQ5/L2dG3n6elpGIaGYdDuiwzL3/dB\nsugoTVOaZOG67jDyhf5FQghKk379xoUQp6enm82m2LQpRiTmQ25DBVdHpmmGYSiECIKA9TlfaCJP\nmqaWZfX3X5EkyfbhM6WFqbPZzDAMHXah1wTaLLrrY77Qmy1JkmG82ebzOeecZiRVKMYo5kOizaI7\nwzCo/UK9Cdd1dX6jUqDQgqnFYqH6cuRIkmRvrLAL5kPatj2SenwJ2ix9kmVZ3n7R6n6lC0vTlCoO\nWl3bkXYWWWqK4ziOY855j9qb0pyrQ4GyXC4VXkMfURaHYaj6Qs7Pz883m43rurZtD/X3GIahaZrH\nPMN6vaaJkev1WtZV6Q9rnfvHsqzlckmrqItn0XZMCOF53nw+tyxrsVgMteEphDjyn0b9Wdd1gyCY\nz+cKf2VdQp2lr1zXzbLM8zxqbHfZAcmLyo7j6Fz0kSJJEilru0r1MsdxhprF/6OwvYTekBTUOVos\nFh281tga9pvNhjEm/R87+C7k+fk5kmUg6E5t7w2/2WxGlSlksVgYhtHSk+f5slqtWnoJhdAbGgjf\n96nwkY99SkTjPpoPeLeBJqe09OQ0R0ZVl7ZtqOAOh2EYVEml4Xwpz5mm6Ww245wvFouxxQq7Nfu2\n1ZfgnIdhaFnWfD6P47jV1+oSkmVoaOSIzo09ZhiCPkvjOA7DcLTbU7XaZimiSpkQQu14n0wKe2Ko\ns7RqvV7btu26boPvpU/RQfb/61sul5zzjl80/61tNpuOX1outFkGizpHpmlOp9P6JwdQ9yfLsuVy\nOcLuT1GSJN0PDOdd2tlsJqtLqwSSZeBo/U4QBDQDpZrneUEQ0LSuDq5Nc2maqsrWYpe2p0fiIVmG\nj+qvjLGK25Rm9NIH5pBGKI6hpM1S5Pu+67qz2ayXh1Up7ImhztKx1Wq1c/YEzbUbeVWlREmRZafN\nZmPbtu/7qi/kMGizjAhtKBVFUbFnRANA4xxUrqC8wZKr0+TUEJJlXGj2BOd8Pp+//vrr1AOir6i+\nNL0IIbTa98B13X71jJAsY+Q4zhe/+MUvf/nLDz300GjnqlTTp82SM02zfjFeOSTLGHme98ILLzz7\n7LPPPPNML27TjtFJjLolC+tVzwjJMjrz+Zx6QHfffXfeM9L8Nu2Yhg2WorxnpPNvDckyIlmWTadT\nWrKcf5H+OJ/P+9KB74DCmSw1Uc9I57ILkmUsaMaK7/vb75mdY0ZjpnmbhVDPyPM8PcMFyTIKaZp6\nnrczVkhxzEjnNrbneZNbWrpObYss2yhcoijScJE0kmX49sZKTv+eUfEt1NLbqbP1zVLQRwId2qv6\nWt4HyTJwFCv15+znPSPd7lTGWBzHtMMARWRLC/b0L7JsC8MwP7FXE0iWIaOAoInq9b+LPgZ1u1PZ\nrSixLIsq0HnQSH+V3iULYywMQ8bYfD5XfSH/g2QZLDoJlG64Bmijec/z5F5VY3SiK3v/8ezSO0RC\niOOPAVGFjlLQJFyQLMN0ZKwQ3/cNw9DkTo3jmEq2dAYjhYv0ZNl5OHyP6BMuSJYBiuP4+Fgh+typ\n1BVyHIfe9tSsSNNUbrG5X+XbnehXpryxiWQZmjRNkySREiuEBoym06nC0WghBDVP8rd9HjFymy19\nLN9uozqU4jKZwh0csD+LdKvVyrKsNrZQbe+Z66BzTkobptD7R+J5QC2dW6aK4zgKD/9Gm2U4aLf9\nljaFM03T931VMz7p47e0LJs+mYQQsoafkyQxDEOrzROOQQN8yqYmqYq0c7RZpNpsNh20KehVOt59\nbrVa0b26/bqUoY7jSHkh13VlPZU+VO0WiDbLEORrgtoe1FCyViWvpJydnU3ej0o/skotPZ3JUk3Z\n2qLuwyyHNossjuN0/LnUZR++TlwuFosjX2VgRZYiOpO74xoZ2iy953meZVkdf9h2Nkk3n8ZyUXTK\nGiEaWJGliHPuOM5sNuvyRZEs/ZZPS+3+pbuZpEvVWcMwLorOfMrckYPiwxhvvohpmq7rdjnJBcnS\nYzTLg0ZklWh7km6WZdQsqojOfIbLzmaL53l5deb09LTirTXsZGG3flDd7bfQZderBHWWIymcYFJE\nR8q39Mx0l1ZXkahDZJpm6et0gxmGQds7UnBsP4zsfZVhsCyrm1oSkqWvXNfV50cXhmEbR23l0VD9\nsHyeS/E9Q6lUCl965HbtWZ9zy9q2Xq8ty+rghdAb6qXSVHflHMfJey6y5FPg9laRdnaIaJw1XwRA\nqOe43SPoxfaUUhiG4ThOBwUXJEv/CCGiKFJYXtnJ9325o0V5U2Xvv9S2bXpk8aB7zrllWaW8uGgA\ne/BFlqKWlomXddAuugh6Q8101lVuwHVdnasVFFLbHTc2jiJLUdt3EZKlZ1zXPX5WWHuUTP+viWLF\nNM1S2ZtWD6i6KlXaLrigN9QnNesOCuXT/9vYR7KxJEloyNk0ze29O8dTZClqu+CCZOmTIAiKpQQ9\n5eGiw+kiQojpdDqdThljYRiuVqvtUsuoiixFtm3T7pytPHt7zaG90Bs6iFbDzHup3c+F0AlKnPPq\nEXHOeY9+sHKt1+u8/i0X2iz9IIToy/FahPZzUbjNJa08MAxjtVpVNPR6dG5ZG2jZRBvjREiWfvA8\nT/9+UIlpmnQumpJXD4IgyzJaf1DxsHEWWYpc1z1+1dW2S3KfDtoQx3FPl+FaliWEmM/nEvflrYk+\nh4Mg2D6smtpT9N9CiHEWWYocxwmCQPIMqTa6WDWhzlIHjeOqvoqjtDT3v1rFPV/8eRqGofMofmek\nb/GD3pDuejEeVK2Nuf97Vdz0y+WSHtPrc8vk8n1/u3F3DCSL1gZz60uf+y9F388tk4hzbpqmxF8Q\nkkVrnud1X6FoieKt5HcZwLllEskt5SJZ9EX7Jw7pE5Wa3DrMoCOD3FL7GK7ryuoTIVn0FUVR3yss\nJZzzjvdMrDCYnqZENJYnJfqRLJoaXoOFmKZpGIYOBRea1D+8n/CRaAT6+OdBsmhqeA2WnOu6OhRc\nRrtcqJqsZguSRUdDbbDkdCi4oMhyESnNFiSLjgbcYCHKCy5ZlmFg6CJSmi1IFu2kaTrsBgtRW3AZ\n8LllUjiOc+QyRSSLdgbfYMkpLLigwVLNsiwky6DQNjyDb7DkVBVcUL7dy7btY1qUSBa9DGCV0EFU\nFVxQvt3ryA4RkkUj1GAZW+efJpV0WXBJkoSWyXT2ij11TLMFyaKRKIry4/5Gxff9JEk6K7hgt6ea\nHMdp/EtBsmhkzJ3/MAw725R7zD/nBpptwY1k0cXIRyu6LLigzVKfbdvNqi1IFl1EUaTzQUIdsCyr\ng4ILiiwHsSyLTrk6FJJFF1mWja12u40KLq2egoau0KFM02xQbUGyaCGOY7TPCRVc2nt+JMuhmg0/\nI1m0EMfxyLtCOenbJpagyHIowzDQZumlLMvoKD/VF6KLlk7AYaM/t6yxBnVcJIt66Aptk7htYhEa\nLM3Ytn1oHRfJol6SJOgKlViWRRsdyH1aFFma4ZxnWXZQKxLJopgQAkNCO0k/AYehzXKEQztESBbF\ncK9fhHNuWZbEUi7tZoSfdjO2bR/UhESyKIZkqeA4TpIkskq5WN/cJSQLaE3ilP+Rr584Hue8frMF\nyaISbUyp+iq0RnssNJtgXoI2y5EOmumPZFEJXaE6pIxA49yy41mWhTZLP6B9Xgfn3LbtBx988JiC\nCw6H7xiSBXrAcZx33333s5/9bONwQYNFivoz/ZEsymDW1kF+/vOf//Of//zKV77S7NvpGBC5lzRC\n9UstSBZlUGQ5yNWrVz/wgQ+8/PLLjzzyyKHfi3PLZKFDzuo8EsmijBACbZaDfOtb32KM/fGPf3z0\n0UcP+kacW9Y9JIsyak817qMHH3zw0qVL//nPf377298eFC5osEhUs9SCZFEDM1kasG375s2bjLGb\nN28eFC6YySKRaZp1Si1IFjXQFWrm/vvvp/+gcHn88cf3fguKLHKZplmnuY1kUQNtlmYefvjhy5cv\n03/fvHnzySef3LtkMU1Tzjl+2rLUnOOPZFEjyzK0WRqwLOvGjRv5H2/evPm9732vOlwwBiddnZhG\nskCfmKZ5++23F79y/fr1+XxesXUI5g0pgWRRAwNDjX39618vfeWxxx6raJWgzSJdnQ4RkkUBHC10\njK9+9au33XYb/fdnPvOZ9Xr9xBNPXLQgCOeWtcE0zb3z5ZAsClBNUfVV9JVt29evX7/33nt/9rOf\n5RFzETRY2lBnSguSRQEMOR+Dc75YLF5++eUf/OAHhmFUb7CAIksb6gw8I1kUwK7aR8qPOnjooYde\neumlipY5kkUVJIsCmMwii+M4165du2jUGeeWtQd1Fh3hU1Si+Xx+0WRzFFnas7dDhGRRQPoBXWPm\nOM577723cz4LukKtqr6NkSzQe47jPPnkk9tfR5tFISSLAvggleuxxx6jDbSLX0SRpVV7C4VIliHz\nPG9yy7Bn/T788MM//OEPi1/B+uZWIVm0U3OzPymK1YeDDuXtnR//+Md//etfi19BkaVtqODqpbNk\nieOYXoveYFIOA9MW5/zzn//897///fwr2O2pbajgjhRFiWVZjuOwQtAM1S9/+cvf//739N84t0w5\nJIsCHXyWZllG3R/btvMZq8PuEN19992f/OQnn376aYZzy9q39x5GsijQwR0fxzF1g23bpjMG2dCT\nhTHm+/5Pf/pThvJt+/bew0iWrnXTJaGukOM4dAfQ2yxN02FP0rt69erf//53hiKLBpAsXesgWYQQ\n1DzJP7fziBl8s+Xq1auPPvooiiwdQAV3dCg+8k4QGUmH6JFHHlksFlevXkWRRS0kywDR2l8aEsrR\nZ7gQYpDDz1mWBUFwenr67W9/+/Lly1/4whdUX9HYIVmGJk1T6nAVGyzsVimXDa7ZkiTJfD4/OTmJ\n49h13fV6/eCDD37iE59QfV1jh2TpWtuVxTw4zs7OJu9Ho0XDSJa8kTKbzRhjq9VqtVpROemjH/3o\nn//8Z9UXOHzVdzKSpWtt9//3nuyVT3Xpqe1GShiGpbu8zhbQ0CokiwLtLQ7Mp7GsVqvzXfrbIapo\npBQfRuPNtm3vTVg4BvaU01F7k0qoOmsYxkUt1XyEqEdLn+s0UooPNk3TNM1hz9xRDskyIlmW0Qd1\nqXZblM/yqGi2xHE8mUykX96hajZSSvJzESzLQrgohGTpWnsV3DwsKpIlHyGq6Cwo7ysd1Ei5CDpE\nbauei4hk6Vp7FVxKhIquEKHcyQenc2maBkEwnU5VJUuzRkpRcVI/TkdQC8kyEPkUuIoGC7moQ+R5\nnud5SubRSWmksK2Nb03TRLOlJXvrLJe6uQ4oaqP/bxjG+fl5nUfatr3zkcvlkv5jOp12ky9UGIqi\nKMsy27ZXq9WRXcXS4ZO2bXued/Rlwg57T+NDsoACSZLEcRxFkWmaruvm1Z8jlZ6E/phlGdYQdQ+9\nIQVGWwI4vpJSYefJk5ZloUPUkuoKLtosCvRoLoksLTVSSi+x3ZmybZtSDOTCfBYdGYYxkrnnrTZS\nSi7aR248P+0uYU85HY3hXpc13HM8TGxRAsmigGEYg9wkhXXbSCnaWWQhWJ0oXZIkaLPoaJAbKapt\npOAI5y7VOdYWyaLGYD5FVTVSSqo36zdNc6iNRCXSNEWbRVMDmGGhTyWF7ft52raNZJGoNCNxJ4w6\nQ1k+GXcn6RNnjyeEqE4WwzBGONLfnjoTstBmUYNz3rtPUa0aKaULq1NkQbjIUmd5CpJFDcuy+nKj\na1JJqZCm6d6AQ6lFljpdIYZkUYVzrv++RNo2UkrqrAzCRlCy7F2LSJAsaug8yUL/RkpRzRt9DLMT\nu1HzZFtUcOH/62B1j3T1j3DmnGPd8/HQG9KdPj3/fjVSSqpnshTp8wMfAySLMpZlKb/R+1JJqVYz\nAVFqOV79FiKSRRmFpZZeN1KKDurdYFbL8ZIk2bsdKkGdRaXu38l9rKRUqP8RSmqWWijxR7tBV4Us\ny2r+WNBmUamznv9gGikldWayFJmmWfNkgul02vSiBuugFh+SRaUOSi3DqKRcpOaQc65mqQVD1Dsd\n1EJEb0il9u5gDVf3SNdgCJl6Qy1dz+Claeo4Ts0HI1kUkz7JYmCVlAqHFlkIpTlqKA0c9HNDb0gx\nWaWWoVZSKtSfyVK09wceRdF8PmeMzWYzavQ1v8QBOfTzD8mi2PGllmFXUirUH6coqii1pGl6enoa\nxzG9hYQQ8/n89PQUs2AYY3EcH3RTIVkUa9wsH2EjRYqLfj5pmk6nU8uylsul7/uMMfp5Zlk2nU4R\nLgcVWRiSRQeHHj882kZKUbMiC9nZwaGApkzJ+b5PhbCRn+LaYJwByaKebdt1PhLRSCk6Jlm2h+Si\nKBJCWJa1fX4rfVAnSTLmceg4jmtOvc0hWdTLjx++6AFopGyrueJ2p+1koelzO3umeZFY+SIvhRoU\ny5EsWth5/DAaKXsFQdBg7MYwjFIjsSI18vwabZul4iynCkgWLZQ6RDUbKUKIcd7ueVfINM35fH5o\nEcQ0zfp5hByP4/ig2i1BsujCMIwXXnjh0EbKOJe35FtqW5a1WCwsy5pOp/Wr4JzznYlcXe0a7eQ6\ntFl6LEmSV1999Utf+tJBlZTRLm8pFVloqJgxNp1Oa1ZDSmFNOVWdLOM8g7FB7ZYgWVQqVlI+8pGP\nWJaFSkodO38+juMsl8skSWaz2aHVVnrz0GKr0l/RU1mWVf257Xne2dnZZDKZTCanp6eDGaWuvyFL\n2bk69CGwXC4VXoMqy+WS+q6maYZhuNlszs/Pfd8/9Keh9jeoxGq18n2/4gGbzcZ1Xdd1Kx7jum7p\nR02NIMMw6HeR/2Dp17RarSqeje5kwzDodfMaUN1/kq42m41t282+F8nSqc1m4/u+YRg0UaJ0v242\nG8dxaj5VGIZ009u2nWfTGNTM3zAMwzC86G+Xy+VisSh+Zb1eU6vEtu31ek3JEoYhY6ziefLHWJZV\n/BXQr6b6G/XX4KMuh2TpyM5GyjbHcdbrdfVTrVYrwzAsy3Jdl90aFuWcV3+uDkb9T1Hf9y96b69W\nq+1GDTV28q6WaZrUP61+Ffq1lnJqs9lQ3NS8VD0dc/1IlnZVN1K2rdfr6mbLarWip6I/0kcr3dwj\nCZeD2ueO41wULhXPQ/XgvRFPXNctNVhI35OlutG3F5KlLTsbKXV+VdXNFgqp/D6mZNlsNvRJ2+tb\nuY6dbY1qF4VL9fMcWcCi9UfV9SDNHXkvYWxIsr0TZ2nvjwqu6wZBsPOvsLyl5uHwRWEYpmm6PejT\n3vLlIAg8zzNNs8EEM01EUdRwSOgWJIs0dSbOOo5D00YrnofqiDsDAstbmu32tDNc2jiVJUkSGnI2\nTXO5XPZ39kCzebdFSJZjHbq6p064XNRswfKWxu/VMAxLDbqKeYYNfoZCiOl0SlOiwzCkclizS1Xu\n+AYLQ7Ico/ES5L3hQsWUg9ba9fc+rk8Iccw/0/f9Yl5vr0ssvtBBLaMois7OztI09X2favCNL1IH\nSZIc/09AshxMyhJkCpeL6imMMcuyLvrb0S5vaVBkKSp1MyvWJebLBeoQQnieZxgGVZcbX54mjtn4\n5n1kVZIb6N3YUM05KfXRpKyKvy29BP3EOOf5V4q/QfqF1hwr7SPHcY78mZcG9RtPMC2iW6JHt3G1\n7buuGbRZ9mtvnxTf9ysWmDiOU2q2HL+8pdeOPz6l1GyR0oWksnoQBNMtvVs9lCQJ9cQlPNfx4dSY\n/m0W6Y2Ubb7vl6ZvFm03ao5Z3tJr6/X60JksFz1P3myR0mapeHP1bnrRzil/zSBZdjh04uyRKtqf\n6/W6dPc3Xt7Sd2EYVkTwQfK5iFKiajC2V2keA8nyPh00UrZR3+qiv91u1DRb3tJ3xxdZcnmzBcmS\n27us5FCoszCmesfZ6lK867pxHBdHMTjnvu9vNhsav1gsFsvlcgwbbsv6dRSrLcOeWFif53lyB7bG\nniyabItffeTQdimX5NuCHPpycRxPJpNDv0shuUdfTXHBhwAACH1JREFUM8Zs2z7ojKdhi+PYMAy5\ntf+RJotu2+JXHzlkWVaWZRLXudBwRo9Im2RxSz5+L/E5e4qGGktHuB1vdMmiSSOlZO+RQ6X5o82k\naUqDo71LljRNpf+OsixT/nvXQRAEbUzwG0uy6NZI2WZZVsUbnnO+3WM6dHmL53me5/WxsiCEkD5P\np8FBRcNDB8u0sXn48JNFz0bKNtu2q9/z26XcQ++JfBiuX9vQSy+yEMMwrl271seclcjzPJqyIN1g\nk0X/RkoDpTm7By1v6S/pRRbCOX/11VelP22PRFFkmmZL74gBJktfGinb9v6OTdM0DGNsgxrN9mTZ\nyzTNf//739Kfti/SNE3TtL0llMNJlgE0Ui46u6/IdV26J7q5JB1kWdbGYijTNG/evCn9aXuB3izS\nx4OKhpAs/W2klNRJFnZrnAgFyCNxzkcV0EU0L67VD90eJ8sAGiklNU8y55y7rtu7dbTNtFRkIRWb\nPw1YFEWGYbT90dvLZBlMI6Wk/qfoeAourSZLfz+EGmu7vJLrU7IMr5FSctA/ZCQFl9Lh8NKffMDb\n2WzroLyS60eyDLWRUnLojY6Cy5E454P5WKqjg/JKTutkGXwjpSRN04OShbaPGXDBpdWuUJqmQ72R\nduqmvJLTNFlG0kgpabCShU41O7TgQpNxD/oWJY7cUrsaDWYPvjtJOiuv5PRKlrE1UkqanXrh+/7O\nMwAHAEUWKdI09Tyvfnnl7OzM87zSUpJD6ZIs42yklDS+0ekMwOF9/Lb6iULJMvhwoU/rxWJR84eZ\nZZnjOFmWeZ53cnLSOGUu0f+ladp9IXCz2TDGnnjiiW9+85vXrl174IEHfvGLX1y5coW1eeauzpbL\nZePGv23bjuN85zvfoR/gALz66qur1aq9FYMf/vCHhRDvvPPOgBclXrt27Sc/+cnjjz9+0BuKAte2\n7bfeeuu555578cUXf/Ob37z55pv33nvvfffd941vfGP7ZPEd+rj4FQCUoy1TL9oH939tFiX9juef\nf/7tt9++7777Tk5Oun/1Qbp58+Zrr712zz33XLp0SfW19AP9uFRfRStee+21j33sY3fccccxT3Lz\n5s3NZvPOO+/Q/37wgx88OTn50Y9+tP9kK4m7dR9Kw737B2C1Wkk8NWbwpBw5pCHHcRqfErPZbBaL\nheu6+eFW9GwHHb+JT7ahMU3T9/3ZbDaGrVuON8h5hp7nmabZ7NR3IcTp6alhGJZlOY7T+NRNJMsA\nUbjM5/OWtgsbkuFNaIiiiKZQNvt2wzDy0/KOocuoM8hlmqZpmvP5fJCfyRINbNTZ8zwhxJEz4qT8\nTJAsg+U4juM4s9kM4TIS8/ncMIxuFhzuhWQZsrzmgnC5yKHnH2hrPp83rq0cZDabTSaTs7Oz0teF\nECcnJ5PJ5H/TwZtVj6XA2FA31uv1GA5+bsayLNWXcKzNZtPl73ez2VBxqnQqNr2d87E2JMsodHzz\n9Ujfk0XJb3axWFA7JX9d6oJxzvPpDugNjQLnfLFYBEHQu9MR29brtWn5UsOO/xW2bdu2zRibz+eM\nMSEEHeAZhmE+1oZR57GgcKHRog5649A2ipX6Sw3lCsMwSRK6Blp16LouxQ1Bm2VcaFX0gDeLOkh/\nF77GcawwVhhjnHOaLRUEQZIkhmGUhrqRLKMThqFhGBgwYr2dgEvNhOVyqXaan23beZoU+0EEyTJG\njuO4rjubzQa8gcAgCSGm0ylNJlB9LYwxlt8/2zcSkmWkTNNcLBbUqFZ9LSr1aP+QOI5pxUaxnKEQ\nNZ2oqRIEQalriWQZL+oqo2fUC3kPSJPlCGma0njQYrEojhPlkCxjN+aeUS8m4FIPyLIsTXpAjLEs\nyyhHXNe1LIuKLKWRASQLjLdnpH+y5GNAWvXaqO+TjwcVx4nyPhGSBRi7dXOYpjmbzfo7Fjsk1C5I\n01Th0PJOSZJsz4srzZ1jDOuG4P02mw31j8awK11p5Ys+fN+3bVvD1RibzYYKPds/uvV6XVxPhDm4\n8D7UeKFDWvIPIugMVUYty8rX5miFc75er3f+lWEYdBoHQW8IdqA7Wwgxm830L0Y0ptU/jY74ieM4\nDMMBLL9AmwUu5LquEMLzPKrVadXbl0Kff1EURXSGn1aV2mOgzQJVDMOggYlxDkt3IE1Tmk90zDl2\nGkKbBfazLMuyLM/zoiii/dxVX5EE+fxRVXZuPjAYSBaoy/d9OiR4GPkihFD1T8gzxXVdTebUSodk\ngQNwzvN8ieO4m41XW5KmafcjX0mS0OZbA84UgmSBg1G+MMaiKJpOp3RYveqLOliWZV1uxZYkSRRF\ntLf+8Po+21DBheYcx6GTGKfT6f92bIctSZJQ/TsMw5HECkObBY5HBxtFUTSbzQzDsG27F5vLtr28\nWwgRxzGtrxlkjbYakgXkoHwRQkRRFAQBlWC0fTtlWdZSmSPLMgoU9v5d18YGyQIy5Wf0JUlCK6f1\nrPK2MeScV2dN08SJ2kgWaAVNgaEPcFr/atu2PgPVQghZXTZqptETjqeMsheSBVrEOacGCxUdoiji\nnJumaVmW2jHXIyezZFlGZ2IIIXpUWuoSkgW6kO8SRO9J+pCnlLFtu/vPeUqEQ7+LCij5lTuOM+w5\nKcdAskCnOOf55gyUMkEQZFmWt2W6SZn6iZAkSZIkNJCENKkPyQLKFFNGCJEXfWnghrLGMIw23sk7\n983LsoyaJEKINE1N06TqCTo7DSBZQAuGYRSHkPI3OZUz6E1uGIZpmpxzCp3GryWEuOeee2jpdt4e\nKcaZ8jLQACBZQEec8+0KK7UmKBFofFcIQT0pCoLiH7Mso42dKIOKf3zjjTf+9a9/JUmC9kh7VCYL\n/UYxSgc1Uc/o+KFrmsinzxD4IKlMFn3OT4FR6Xgt4jhhRSKMDtVTVF/FwP0fLb7p48luvjYAAAAA\nSUVORK5CYIKJUE5HDQoaCgAAAA1JSERSAAABcwAAAXIIAgAAAIiErWAAAAAJcEhZcwAADE0AAAxN\nAdLOrU4AAAAddEVYdFNvZnR3YXJlAEdQTCBHaG9zdHNjcmlwdCA4LjYygy0IzwAABKJJREFUeJzt\n1MEJACAQwDB1/53PJQqCJBP01T0zCyB1XgcAH3IWoOcsQM9ZgJ6zAD1nAXrOAvScBeg5C9BzFqDn\nLEDPWYCeswA9ZwF6zgL0nAXoOQvQcxag5yxAz1mAnrMAPWcBes4C9JwF6DkL0HMWoOcsQM9ZgJ6z\nAD1nAXrOAvScBeg5C9BzFqDnLEDPWYCeswA9ZwF6zgL0nAXoOQvQcxag5yxAz1mAnrMAPWcBes4C\n9JwF6DkL0HMWoOcsQM9ZgJ6zAD1nAXrOAvScBeg5C9BzFqDnLEDPWYCeswA9ZwF6zgL0nAXoOQvQ\ncxag5yxAz1mAnrMAPWcBes4C9JwF6DkL0HMWoOcsQM9ZgJ6zAD1nAXrOAvScBeg5C9BzFqDnLEDP\nWYCeswA9ZwF6zgL0nAXoOQvQcxag5yxAz1mAnrMAPWcBes4C9JwF6DkL0HMWoOcsQM9ZgJ6zAD1n\nAXrOAvScBeg5C9BzFqDnLEDPWYCeswA9ZwF6zgL0nAXoOQvQcxag5yxAz1mAnrMAPWcBes4C9JwF\n6DkL0HMWoOcsQM9ZgJ6zAD1nAXrOAvScBeg5C9BzFqDnLEDPWYCeswA9ZwF6zgL0nAXoOQvQcxag\n5yxAz1mAnrMAPWcBes4C9JwF6DkL0HMWoOcsQM9ZgJ6zAD1nAXrOAvScBeg5C9BzFqDnLEDPWYCe\nswA9ZwF6zgL0nAXoOQvQcxag5yxAz1mAnrMAPWcBes4C9JwF6DkL0HMWoOcsQM9ZgJ6zAD1nAXrO\nAvScBeg5C9BzFqDnLEDPWYCeswA9ZwF6zgL0nAXoOQvQcxag5yxAz1mAnrMAPWcBes4C9JwF6DkL\n0HMWoOcsQM9ZgJ6zAD1nAXrOAvScBeg5C9BzFqDnLEDPWYCeswA9ZwF6zgL0nAXoOQvQcxag5yxA\nz1mAnrMAPWcBes4C9JwF6DkL0HMWoOcsQM9ZgJ6zAD1nAXrOAvScBeg5C9BzFqDnLEDPWYCeswA9\nZwF6zgL0nAXoOQvQcxag5yxAz1mAnrMAPWcBes4C9JwF6DkL0HMWoOcsQM9ZgJ6zAD1nAXrOAvSc\nBeg5C9BzFqDnLEDPWYCeswA9ZwF6zgL0nAXoOQvQcxag5yxAz1mAnrMAPWcBes4C9JwF6DkL0HMW\noOcsQM9ZgJ6zAD1nAXrOAvScBeg5C9BzFqDnLEDPWYCeswA9ZwF6zgL0nAXoOQvQcxag5yxAz1mA\nnrMAPWcBes4C9JwF6DkL0HMWoOcsQM9ZgJ6zAD1nAXrOAvScBeg5C9BzFqDnLEDPWYCeswA9ZwF6\nzgL0nAXoOQvQcxag5yxAz1mAnrMAPWcBes4C9JwF6DkL0HMWoOcsQM9ZgJ6zAD1nAXrOAvScBeg5\nC9BzFqDnLEDPWYCeswA9ZwF6zgL0nAXoOQvQcxag5yxAz1mAnrMAPWcBes4C9JwF6DkL0HMWoOcs\nQM9ZgJ6zAD1nAXrOAvScBeg5C9BzFqDnLEDPWYCeswA9ZwF6zgL0nAXoOQvQcxag5yxAz1mAnrMA\nPWcBes4C9JwF6DkL0HMWoOcsQM9ZgJ6zAD1nAXrOAvScBeg5C9C7VXwF4QpkCX4AAAAASUVORK5C\nYII=\n", "prompt_number": 21, "text": [ "" ] } ], "prompt_number": 21 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The\n", "tip of the resultant will trace out a shape in time called a\n", "_cycloid_. For instance, the path followed by the tip of the\n", "resultant vector in the complex plane for different amplitudes and\n", "frequencies differing by a factor of two is plotted below - you should try playing with the parameters, and seeing what effects they have." ] }, { "cell_type": "code", "collapsed": false, "input": [ "# Define figure size to get square\n", "fig = figure(figsize=[5,5])\n", "# The range of t may need to change with angular frequencies\n", "t = linspace(0,10*pi,200)\n", "# Original frequencies: 1.2 and 0.6\n", "o1 = 1.2\n", "o2 = 0.6\n", "# Original amplitudes: 1.5 and 1.3\n", "A1 = 1.5\n", "A2 = 1.3\n", "# Calculate x and y projections of the two oscillations when added\n", "x = A1*cos(o1*t) + A2*cos(o2*t)\n", "y = A1*sin(o1*t) + A2*sin(o2*t)\n", "# Plot path of the tip of the vector\n", "plot(x,y)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 15, "text": [ "[]" ] }, { "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAATgAAAE4CAYAAADPf+9qAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nOCKyLSY4IrItJjgisq3/BzPg+smmwvcUAAAAAElFTkSuQmCC\n", "text": [ "" ] } ], "prompt_number": 15 }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [] } ], "metadata": {} } ] }