{ "cells": [ { "cell_type": "markdown", "id": "9fe12a0c", "metadata": {}, "source": [ "# Aeropendulum (theory)\n", "\n", "Analysis of a **damped driven pendulum** with a **step torque input** starting at $t = 0$, including:\n", "\n", "1. **Equations of motion**\n", "2. **Equilibrium point shift**\n", "3. **Natural frequency change**\n", "4. **Time-dependent solution** with initial conditions:\n", "\n", " $$\n", " \\theta(0) = 0, \\quad \\dot{\\theta}(0) = 0\n", " $$\n", "\n", "**System Description**\n", "\n", "* Mass: $m$\n", "* Length: $L$\n", "* Damping coefficient: $b$ (viscous damping torque $= b \\dot{\\theta}$)\n", "* Gravitational torque: $\\tau_g = -mgL \\sin\\theta$\n", "* Driving torque: $\\tau_d(t) = \\tau_0 \\cdot u(t)$ (step function)\n", "\n", "Using Newton's 2nd law for rotation:\n", "\n", "$$\n", "I \\ddot{\\theta} = -b \\dot{\\theta} - mgL \\sin\\theta + \\tau_0 u(t)\n", "$$\n", "\n", "where $I = mL^2$ is the moment of inertia.\n", "\n", "**Linearized Equation of Motion**\n", "\n", "For **small angles** ($\\theta \\ll 1$), we linearize $\\sin\\theta \\approx \\theta$. Then:\n", "\n", "$$\n", "mL^2 \\ddot{\\theta} + b \\dot{\\theta} + mgL \\theta = \\tau_0 u(t)\n", "$$\n", "\n", "Dividing through by $mL^2$:\n", "\n", "$$\n", "\\ddot{\\theta} + \\underbrace{\\frac{b}{mL^2}}_{2\\zeta\\omega_0} \\dot{\\theta} + \\underbrace{\\frac{g}{L}}_{\\omega_0^2} \\theta = \\frac{\\tau_0}{mL^2} u(t)\n", "$$\n", "\n", "Define:\n", "\n", "* $\\omega_0 = \\sqrt{\\frac{g}{L}}$ (undamped natural frequency)\n", "* $\\zeta = \\frac{b}{2mL^2 \\omega_0}$ (damping ratio)\n", "* $A = \\frac{\\tau_0}{mL^2}$ (normalized torque)\n", "\n", "So the linear equation is:\n", "\n", "$$\n", "\\boxed{\\ddot{\\theta} + 2\\zeta\\omega_0 \\dot{\\theta} + \\omega_0^2 \\theta = A u(t)}\n", "$$\n", "\n", "**Equilibrium Position Shift**\n", "\n", "Set derivatives to zero in the steady-state:\n", "\n", "$$\n", "0 + 0 + \\omega_0^2 \\theta_{\\text{eq}} = A\n", "$$\n", "\n", "$$\n", "\\boxed{ \\theta_{\\text{eq}} = \\frac{A}{\\omega_0^2} = \\frac{\\tau_0}{mgL} }\n", "$$\n", "\n", "This is the new equilibrium position under constant torque.\n", "\n", "\n" ] }, { "cell_type": "code", "execution_count": 1, "id": "4cbef945", "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "C:\\Users\\Utilizador\\AppData\\Local\\Temp\\ipykernel_15004\\749860262.py:12: RuntimeWarning: invalid value encountered in arcsin\n", " tetaeq=np.asin(I*km/(m*g*L))*180/3.14\n" ] }, { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "# Physical constants\n", "g = 9.81 # gravity (m/s^2)\n", "L = 0.1 # length of pendulum (m)\n", "m = 0.1 # mass (kg)\n", "\n", "km=0.02\n", "I=np.linspace(0,6, 100)\n", "\n", "tetaeq=np.asin(I*km/(m*g*L))*180/3.14\n", "plt.plot(I, tetaeq, label='eq', color='blue')\n", "plt.xlabel(\"Current (A)\")\n", "plt.ylabel(\"Equilibrium Angle (deg)\")\n", "plt.grid(True)\n" ] }, { "cell_type": "markdown", "id": "c7e91779", "metadata": {}, "source": [ "**Time-Dependent Solution**\n", "\n", "This is a standard second-order linear ODE with step forcing. The general solution is:\n", "\n", "$$\n", "\\theta(t) = \\theta_{\\text{hom}}(t) + \\theta_{\\text{part}}(t)\n", "$$\n", "\n", "Particular solution:\n", "\n", "$$\n", "\\theta_{\\text{part}} = \\theta_{\\text{eq}} = \\frac{A}{\\omega_0^2}\n", "$$\n", "\n", " Homogeneous solution:\n", "\n", "Solve:\n", "\n", "$$\n", "\\ddot{\\theta} + 2\\zeta\\omega_0 \\dot{\\theta} + \\omega_0^2 \\theta = 0\n", "$$\n", "\n", "The solution depends on damping regime.\n", "\n", "Case: **Underdamped** ($\\zeta < 1$)\n", "\n", "Let:\n", "\n", "$$\n", "\\omega_d = \\omega_0 \\sqrt{1 - \\zeta^2}\n", "$$\n", "\n", "Then:\n", "\n", "$$\\boxed{\\theta(t) = \\theta_{\\text{eq}} \\left( 1 - e^{-\\zeta \\omega_0 t} \\left( \\cos(\\omega_d t) + \\frac{\\zeta}{\\sqrt{1 - \\zeta^2}} \\sin(\\omega_d t) \\right) \\right)}$$\n", "\n", "This satisfies:\n", "\n", "$$\n", "\\theta(0) = 0, \\quad \\dot{\\theta}(0) = 0\n", "$$" ] }, { "cell_type": "code", "execution_count": 15, "id": "fba23419", "metadata": {}, "outputs": [ { "data": { "image/png": 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cmkqgoNvfZwq6/8K+byPpNYrMNOtzQAFqZIfwotLjLOg19X+/q7/3xxat96c/fUEHMAr73ABAaWBONwBEmeYUqlOvvuQqCIjM3OXNWqorc6yoM3HknHJlBvWFXl2PRfNI9QVeSx2pFD4vvzRYY9fBA3UPV2dwn0rBlWGMFlUCaL6lSqTzUpm45qwWRVHGrQBGZft5l7pSJ+7C0POY92/VkbykmfKSUqdwzXn2qwd69+7tnhcd/Mm7L+pnf5qCyp7V6T2Sgm8FyHkrH7744otcpcAqFVd2XN3FdV/adF6XRZYzq/u1AmXNCdbzXxQ6iJV3/H7W1x+f9ic9/3ov5qXHljeAy69yQPJe77jjjnO3qy7okVTmrkDZf3/lR2X2er7y488F98vwC7r/wr5vI/nLn/nUrV52NtZd0fOgDvWRj0fvUe33CuZ3NS9bc/r1mulAQ+T0GnWMzzsfXAG69qNdvT9V2aCpBjrQlN8Br/yeGwAoDWS6AaAE9EVZmR19iVcQoYBbXxr1JVHrwvpzMxVUaEmmBx54wAVBmlOokl0tYRQrWo5IAY2W4NHYFOCrtNJfwkwB1LPPPuu+eGs+ra6ncSnwVZMpjdmfg60gTWtga26vlujS49UXd/2d1h+P1jJrWtrnsssuc/evDKWCGz2/ulyBsrKBRVGUcSs7quXZdKr70Rd8ZWwLQ3+jcWuesUqFlbHXePPOvy9tCnwUHOl11hxXBWvKUg8YMMAFwJoHrWyt9kM1tFMTLGXFtR9riS6twayMuJ43VW8o8NFjjKR5uzq4EblkmP/c+3Sfel9of9TrUKZMGRcYKUDWe6KoFKjpfrT8kx6TDi7pYI32WT1eUYm2lv5SSbwadinwVwZUmVU1WXv00Uddc8GC+M3h9Lj0+PTYtU655l0fddRRrgGansP27du797IOKvTr1y9Xc7/8gm4ts6XsvKaVKMOvoFoHhrTUl14PNVgT3Y4axqmyQK+RgnDNodbBvMK+b316fTUnX/epIFnrtGv+u8ZeXFqO67XXXnPj0HOkzxu9LrovLc2lz5dd0Wtz/PHHu/1C0wPU98J/f0YeUFAfAe2L+p0ObOi5URO9/Obl6wCDbk8HifRZp+y3Pv/0uDXtR+9NACh1gfRMB4A45y9t5G9aDqlevXpuuR4tyRW5VJfvjz/+cMsoVa9e3S0nduqpp7plk/IuN+Uv86Oln/JbdisvLeOj5XzyLq+j5cgGDBiQXadOHbesmJaOilw6y6floXr37p1dq1at7HLlyrmlevr27Zs9fvz4XNebNGlS9v777+8eq5b/0VJe/lh3paCx57V169bs+++/3z0ejaVGjRruPu+6667stWvX5lxP96klnXa1RFdRxq3ljC688EL32mgpKz0Hy5cvL9SSYVoe6eabb86uXbt2dsWKFd1yTlo6qqAlw/IudVXU13xX+0CkiRMn7vAY3nnnnezDDjvM3bY2LTem53P+/Pnu9z///HP2BRdckL3HHnu45b9q1qyZfdRRR2V/8sknuW7bfx1eeeUVtxyTXjMtk5d3aSeZNWuWe14qV67sniPd3ueff57rOgU9P3mXjNJtnXHGGdmNGzd296l9/IQTTsi1dJnvmWeeca+/3gN6XbWc1E033eTeezujJduuvvrq7N12280tBRa5v2hpLC1j1qBBg+z09HT32B988MFcS3jlJyMjI3vYsGFuCTXtGxq7ngs9Z/r7LVu25Lr+e++9l92mTRu3nFvepbEK87719ystk6clzfT49Z666qqrsv/+++/sosi7ZJhoCTjdrj7TtJ907Ngx+4MPPsj3tSto+Tnti61bt3aPQY915MiRbr+PXDJM9N7o06ePe770GC699NLsb7/9Nt9l1TSuf/3rX+4zWa+Plk/U/vH2228X6TEDQLSk6J/SD/UBALEyceJEl4lTNm9nmTygpJR11HJUeUutEQ533nmnqzhQWXXQVRdFoQZ++hwrSXd1AAgT5nQDAAAAABAjBN0AAAAAAMQIQTcAAAAAADHCnG4AAAAAAGKETDcAAAAAADFC0A0AAAAAQIyUsQSXlZVlixcvtipVqrilTQAAAAAAKCnN1F6/fr01aNDAUlNTkzfoVsDdqFGjoIcBAAAAAEhAv//+uzVs2DB5g25luEVPRNWqVS2sMjIybOzYsda9e3dLT08PejhALuyfCDP2T4QV+ybCjP0TYZYRJ/vnunXrXILXjzmTNuj2S8oVcIc96K5YsaIbY5h3LCQn9k+EGfsnwop9E2HG/okwy4iz/XNX05hppAYAAAAAQIwQdAMAAAAAECME3QAAAAAAxEjCz+kGAAAAEF8yMzPdvF4kp4yMDCtTpoxt3rzZ7QtB0XzytLS0Et8OQTcAAACA0Kx7vHTpUluzZk3QQ0HA+0G9evXcClS7alIWa9WrV3djKck4CLoBAAAAhIIfcNepU8d1rw464EIwsrKybMOGDVa5cmVLTU0NLPDftGmTLV++3P1cv379Yt8WQTcAAACAwKmM2A+4a9WqFfRwEHDQvXXrVitfvnxgQbdUqFDBnSrw1n5Z3FJzGqkBAAAACJw/h1sZbiAs/P2xJD0GCLoBAAAAhAYl5Ui0/THQoPvOO+90DyJy22uvvXJ+r251V155pSsvUT1/nz59bNmyZUEOGQAAAACA+Ml0t23b1pYsWZKzTZkyJed31113nb3//vv21ltv2aRJk2zx4sXWu3fvQMcLAAAAAHk9/fTT1rhxY6tUqZKLWVasWLHDdW6//Xa75JJLdno78+bNs4YNG9rGjRsLncB85JFHLBbOO+8869WrV66f/fscNWpUTO4zEQUedGv9NbVg97fatWu7y9euXWvPPfecPfTQQ3b00Ufb/vvvb8OHD7fPP//cpk2bFvSwAQAAAMAZOXKk3Xjjjfb444/bzJkzbf369XbKKafs0Jn90UcftVtvvTXnsiOPPNL69euX63pt2rSxgw8+2MVBhU1g7iqQjxaNX/eHOOtevmDBAmvQoIHrTNepUycbPHiwO0L01VdfucnqXbt2zbmuSs/1uy+++MLtiPnZsmWL23zr1q1zp7qtkkx+jzV/bGEeI5IX+yfCjP0TYcW+iTAL4/6psWiZJnWu1hZP7rvvPjcttmfPnu5nJQubNGlikydPtsMOO8xdNmzYMBfvNGrUKNfj8x9zpHPPPdcuvfRSu/nmm12SMj/6O/1OXbWloOdMz2t6enqxHpfuI3J8VapUcZt/f7F6nbKzs3NOg94XdP8ah57HvN3LC/v+CTToPuigg+yFF16wVq1auSMmd911lx1++OH27bffuiNBZcuWdYuRR6pbt677XUEUtOt28ho7dmxcdEIcN25c0EMACsT+iTBj/0RYsW8izMK0f/oVsFqfWctFxQstczZr1iy75557chJ+6kfVunVrGz16tLVr185dNmLECLvgggtyrnPFFVe4KbTaHnvsMXfZnDlzXJJRcdKqVavso48+siOOOCLf+1WiUcus+bfnq1Gjhv3nP/+xTz75xAX9V199tcvCK6Oun7X8lcrXL7zwQrvsssty/k63dccdd9grr7zigsuzzz7bvQ7btm3b4T7k77//zvfyaFq/fr0FTc+BHqueOz0XkbSOd+iD7mOPPTbnvHZG7Vw6IvTmm2/mrIlWVAMGDLD+/fvn/KwdQUeTunfvblWrVrWw0lESfeh169at2Eei4oV64c2cmWI1aujAS7YVc7k7lKJk2j8Rf9g/EVbsmwizMO6faqL8+++/u4BVVbCihGch45qoUq6usE2rVbkr7du3zxVvKLH4559/ussUQM+fP99lvf3rPPnkk/brr7+6EnE/abjbbrvlZFP33XdfF8z72fO8ypUr566bX4zzwAMP2KBBg1y5uw5m6Dlt1qyZC8DVpFpTdhVwN23a1Pr27ev+5sEHH7TXXnvNTfHVAQOVt3/44Yd21FFH5XsfitdiFV9lZ2e7gFtZ9aC72Wu/1GPt3Llzzn7pK+xBh8DLyyMpq73nnnvaTz/95D4AdFRBR44is93qXq4jYAXRzqctL32YhOUDZWfiZZzFkZlpdtttekN756VlS7NXXjHr2DHo0SHZ90/EP/ZPhBX7JsIsTPunMq0KsFJTU90m6iUWRN5swwazSpUKH5T5QXbeTPRJJ53kHssff/zhAkllmP3Hpoy0KnvVeE3TbfPSZYsWLcq5fl5+MJrf788880yXyY50991355zfY4897Msvv7S3337bTj/99Jz52kpg+nPR/+///s9VC/uvSV6Rr1O0Zf1TUl7QfZcm3b/Gkd97pbDvncAbqUVSKcnChQutfv36rnGaHsT48eNzfq+jQ9rxNBcC8UVHKc8/32zIEC/gbt3arFo1HRk0U8XM558HPUIAAACg6PwprBMnTrTZs2fnbKq09X+n8mTJmyndGWVXC1u+nNcBBxyww2XKrCvGUjZdme9nnnnGxVZ+E2tN91XlsU8Z8vxuB0UXaKb7hhtucOUSKinXcmADBw50JRJnnHGGVatWzR2dUal4zZo1XemCyiEUcBfURA3h9eSTZi+/rDev2UsvmZ1xht7cZqpmGTvW7OSTzb79ViU1QY8UAAAAYaGYVVnnIO63sJo3b+5OFa+0aNEiVwbc/52/QtPq1atd0FsYKklXRro4lD2P9Prrr7vY67///a+Lp1S2rXJyZbuR4EG3yiwUYP/1119u59McBy0H5u+IDz/8sEvn9+nTx5Vn9OjRw5566qkgh4xi+OMPs5tv9s7/979ewC3KdI8caaZjKAq4r7rK7I03Ah0qAAAAQkQV1IUt8w6KysSVQf7ss8/cVFm/glcrLvkl3QqeFZRrDW7/OqLycpXV50fNpfMuO1ZcU6dOtUMOOcQ1b/OpwtinhKeqjRWEa+6yqGmYVpTq0KFDVMaQzAINunXEZWdUfqEyCG2IX1qKUJUxhx5qdvXVuX+nD1FlvlW58uab3u//WVUBAAAAiAtae/vyyy93DZzVsOy2225zpdqH6gvwP/OCtRTylClTrFevXjl/p0ZmCnTVUE0l36rw1XX1s5qwRS6fXBItW7a0l156ycaMGePG9/LLL9uMGTPced+1115rQ4YMcdfVUs1qpKb+Wii5UM3pRuL5+WevUZo89FD+XSD328/M7/Nwxx2lOz4AAACgpE4++WS788473fRYdTFXd3ityBTpoosucknHyHWnVfKt6bVt2rRx1b7+HGt1EdeccE3DjQat+d27d2877bTT3MEAVRpHZr3l+uuvt3POOcetEe6XoOtxoeRC1b0cieeRR9R90KxHj513KL/9drPhw80mTDCbNs0rOQcAAADihZbgilz3Oq9jjjnGdSR/44033BRbUam5ytAjaQWnoUOHunW9i0Nd0vPS6k7Dhw93W6TBgwfnapz2yCOPuA3RRaYbMbNli9mrr3rn+/Xb+XUbNTI7+2zv/NNPx35sAAAAQGnSslPqGK650jujbPctt9ySU5q+M3PnznVl6aXV90oHFXR/KBoy3YiZ//1PXRfNGjY069Zt19e/9FKzF14we+sts8ce8xqtAQAAAIli3333ddvOqAN6ZBf0glxzzTV29j9Zq8J2RC8pNYZTSbyo8RoKh6AbMeNXr/zrX2Zpabu+vpYFbNvW7LvvNI9FR9JiPkQAAAAgLqnpmrbSVKdOHbehaCgvR0wsXmw2Zox3/rzzCvc3arJ20UXe+Wefjd3YAAAAAKC0EHQjJt57z2ug1qmTligo/N+pQiY93eyrr8y+/z6WIwQAAACA2CPoRkx88IF3etJJRfu72rXN/OUI3303+uMCAAAAgNJE0I2o27jRbPx47/wJJxT97/3lAEeNiu64AAAAAKC0EXQj6hRwa7mwpk3N2rQp+t+feKI3v3vGDLM//ojFCAEAAACgdBB0I2al5cpyK3guqrp1zQ45xDtPthsAAABAPCPoRlRlZ5t9+KF3vmfP4t+OX2KuhmwAAAAAzF544QWrXr16zs933nlnrnW/zzvvPOvVq1fOz0ceeaT169evyPfz66+/WkpKis2ePduC8M4779iBBx5olSpVsi5dutjChQstnhF0I6oWLPCWCytXzqxz5+LfzvHHe6effWb2999RGx4AAAAQdQp2FaTm3Y455pio3s9pp51mP/74Y4G/f/TRR11gXlKNGjWyJUuW2N57722lbfr06Xb22Wfbtddea3PmzLH69evbsccea1u3brV4RdCNqJo0yTs96CCz8uWLfzutWpk1bOjNDVfgDQAAAISZAmwFqpHba6+9FtX7qFChgtWpU6fA31erVi1XJrw4FNympaVZvXr1rEyZMlba7r//fpetV+DdokULe/bZZ23lypX25ptvWrwi6EZMgu4jjijZ7WgueLdu3vlPPin5uAAAAIBYKleunAtUI7caNWrk/H7BggXWuXNnK1++vLVp08bGjRvnsuGj/mliNHHiRPfzmjVrcv5G5d26TOXe+ZWX55W3vFy2bdtmV111lQvIa9eubbfffrtla07oP5o2bWr33HOP/etf/7KqVavaJZdcskN5eX73q3HrOnlL3Z9//nlr3LixVa5c2a644grLzMy0Bx54wD0fOmBw33337fR5HD9+vB133HE5P+v5Ovzww+2TQgYFv//+u/Xt29eNt2bNmnbSSSflPH+i8fTv39/9vlatWnbTTTfZueeeu8PzFk0E3YgavXejFXSLv173uHElvy0AAADEL2VfC9oUVBb2uhkZGbu8bixkZWVZ7969rWzZsvbll1/a0KFD7eabb7bS8OKLL7qMtcq2VX7+0EMPuexxpP/85z/Wvn17+/rrr11QXlyae/3RRx/Zxx9/7LL8zz33nB1//PH2xx9/2KRJk1wW+7bbbnPPQX7++usvW7t2rctwR2rZsqX9/PPPu7x/vb49evSwKlWq2GeffWZTp051wb+qEPzX9r///a87iKCDA1OmTLFVq1bZu+++a7FU+vUCSFi//OIt8ZWebtapU/SCbh1gW77cbCeVNAAAAEhggwcPLvB3CsjOPPPMXAFk3uDa16RJE5cN9ikI3bRpU67rDBw4sFhj/OCDD1yAF+mWW25xm7K0P/zwg40ZM8YaNGjgfjdo0CA3VznWND/74YcfdlnpVq1a2dy5c93PF198cc51jj76aLv++utzfo7MDBf14IKCWQW9yuYfddRRNn/+fBs9erSlpqa6+1fgPWHCBDtI81Hz8F+L7t2758qi6/XcZ599dnn/b7zxhhuDDir4fz98+HCX1VYlgW73kUcesQEDBriDIKIDIHpdYomgG1HjZ7kPPNCsYsWS356C7PbtzebM8db+PuOMkt8mAAAAEAsKMJ9++ulcl6m8Wb7//nsX/PoBt3SKRpaqEA4++OBcAazuV9lelVlr7rYccMABUbkvlaor4PbVrVvX3YcC7sjLliujlo+K/wQRr7zyijtAooMY+tsHH3zQPYe7osZrP/30U64xyObNm10WXll0zbWPDPhVBaDHH1lyH20E3Ygav+FZNErLfUcf7QXdkycTdAMAACQrZSYLEhnQyQ033FDgdSODT1GH7GjR8lZ5y6KLwn8ckcFfQRn7aNPYdzW2vEFpfmNLV8lrnuc7PZ/LlI3Oj+ZYa+65GsY1b97czTHXfWsKgX7elQ0bNtj+++9vr7766g6/22233SwozOlG1Eyf7p1G86Dd4Yd7p1OmRO82AQAAEF80F7qgLW+H7Z1dN28AmN91YqF169auwZeyrL5p06blGxRGXica62TnnT+t+1VJvp/lLgyNbf369bZx48aoji0/Xbt2dXOtfQr2VZ7fze+yvBMdOnRwDevUsE0HQCI3BfPatARZ5HOigP6rr76yWCLoRlSsX282b9728vJoOfRQ7/Tbb81Wr47e7QIAAADRtGXLFlu6dGmuTUtd+YHknnvu6bpkqwRaTb5uvfXWXH+vwFAl6OoCrsDxww8/dGXgJbVo0SLXrVtzq9Xc7PHHHy9yhl/l2Cr91vx0lWmPGDEiKuuB50cN5jRGdUf/5ZdfrF+/fq6D+amnnmq7ctZZZ7kO7epYrudYf6+53Ndcc41r5iZ67EOGDHG3r3n26rAe2TE+Fgi6ERU6OKSKk8aNzerVi97tal631uyWqVOjd7sAAABANKljt7Kokdthhx3mfqcSaXXI/vvvv61jx4520UUX7bB0lrLwCooVCLZr1841HLv33ntLPC4tBebf75VXXumCTi0LVhSam6551mqIpoZmGqcODsTCgQce6Lqe67G3bdvWNX5TR/TCVCHowMDkyZPdkmVqlKYKgwsvvNDN6Vapuqhh3DnnnOMOgGh+u+Z/n3zyyRZLKdmxnDEeAuvWrXNlBJo07z/RYaQ5EdqJtSZd3rKXePDAAzoqZXbKKWZvvRXd277oIrPnnvNuf8iQ6N42kmP/RGJj/0RYsW8izMK4fyowUmayWbNmLrOZDDS/WcF4LNeIjkdZWVkujvPndMeaOtor2+2vmV7Y/bKwsSaZbkTFjBnRLy33/XOAkHndAAAAAOIOQTei2kStY8fYBd0K7Ddvjv7tAwAAAAi/QYMGuWXE8ttKY83z4mLJMJTY0qVq0KDyGLP994/+7e+xhze3W8v5zZpldsgh0b8PAAAAoLQl+EzfqLvsssusb9+++f5Oy4wVV6yawvkIulFiM2d6p61bm+VZhz4qFMxr/fr33/ey3QTdAAAAQPKpWbOm2+IN5eUoMX+Jvg4dYncfftm6X8YOAAAAAPGAoBslNmeOd9q+fezug6AbAAAgeTpXA4m0P1JejrgIug84wDv96SezVatUWhK7+wIAAEDp0zrMWh5q8eLFtttuu7mftaQWkjPQ3bp1q1uuqzSWDCtovr3GsGLFCjeGwqwTXhCCbpTIhg1eIBzroFtBdsuWZgsWePO6e/SI3X0BAACg9Cmw0VrIS5YscYE3kld2drb9/fffrjla0AdeKlasaI0bNy5R8E/QjRKZO1dvCrP69b0O47GkEnMF3SoxJ+gGAABIPMomKsDZtm2bZWZmBj0cBCQjI8MmT55snTt3tvT09MDGkZaWZmXKlClx4E/QjdCXlkcG3a++avbll7G/LwAAAARDAY4CrSCDLQQrLS3NHXgpX758QuwHNFJDVDqXl0bQfeCB3ulXX8X+vgAAAAAgGgi6ETeZ7nbtvDW7ly71NgAAAAAIO4JuFJu652tOd2kF3ZUqme25Z+4MOwAAAACEGUE3im3RIrONG800zcIPhmNtv/28U4JuAAAAAPGAoBvFNm+ed9qqlVmZMqUbdH/9dencHwAAAACUBEE3iu2777zTtm1L7z733dc7JegGAAAAEA8IulHiTHebNqUfdP/0k9n69aV3vwAAAABQHATdiKugu04dswYNzLKzzb75pvTuFwAAAACKg6AbxaKg1w+6S7O8XGimBgAAACBeEHSjWP74w2zDBq+BWosWpXvfNFMDAAAAEC8IulGiJmpaKkxLhpUmmqkBAAAAiBcE3SiWoErLIzPd335rlpFR+vcPAAAAAIVF0I24aaLma9rUrGpVs61bzb7/vvTvHwAAAAAKi6AbxTJ/vne6116lf9+pqdtLzGmmBgAAACDMCLpRLAsWbJ/THYR99sk9txwAAAAAwoigG0W2dq3ZsmXe+ZYtgxnD3ntvn9cNAAAAAGFF0I1iZ7nr1TOrUiWYMRB0AwAAAIgHBN2Iu9LyyK7pixZ5mXcAAAAACCOCbhTZjz8GW1ouNWqY7b67d5553QAAAADCiqAbxQ66g8x0CyXmAAAAAMKOoBtxWV4e2cGcoBsAAABAWBF0o0iys8NRXi5kugEAAACEHUE3imTFCq9xWUqK2R57BDsWgm4AAAAAYUfQjSLxs9xNmpiVLx/sWFq39oJ/HQhYvjzYsQAAAABAfgi6Uaz53EGXlkvFituz7WS7AQAAAIQRQTfisnO5jxJzAAAAAGFG0I0iCUsTNR9BNwAAAIAwI+hGXC4Xljfonjs36JEAAAAAwI4IulFoWVnhC7rbtvVO583zljMDAAAAgDAh6Eah/fGH2ebNZmXKeN3Lw0Bl7qmpZuvWmS1ZEvRoAAAAACCkQfeQIUMsJSXF+vXrl3PZ5s2b7corr7RatWpZ5cqVrU+fPrZs2bJAx5nMFi70Tps18wLvMChXbnsH8x9+CHo0AAAAABDCoHvGjBn2f//3f9auXbtcl1933XX2/vvv21tvvWWTJk2yxYsXW+/evQMbZ7L75RfvtHlzC5W99vJOv/8+6JEAAAAAQMiC7g0bNthZZ51lw4YNsxo1auRcvnbtWnvuuefsoYcesqOPPtr2339/Gz58uH3++ec2bdq0QMecrH7+OZxBd+vW3ilBNwAAAICwCbxIWOXjxx9/vHXt2tXuvffenMu/+uory8jIcJf79tprL2vcuLF98cUXdvDBB+d7e1u2bHGbb50m+5q529IWVv7YwjzGn35Kc8dpmjTJtIyMLAuLPfdMcbvyvHlZlpGRGfRwElI87J9IXuyfCCv2TYQZ+yfCLCNO9s/Cji/QoPv111+3WbNmufLyvJYuXWply5a16tWr57q8bt267ncFGTx4sN111107XD527FirWLGihd24ceMsrGbPPtzMatqqVV/Z6NHh6Vq2apUqJDrb7NlbbfToMUEPJ6GFef8E2D8RVuybCDP2T4TZuJDvn5s2bQp30P3777/btdde657I8uXLR+12BwwYYP3798+V6W7UqJF1797dqlatamE+SqLnolu3bpaenm5hdMkl3u7Sp89+tt9++1lYHHqo2U03ma1eXd4OPfQ4q1Yt6BElnnjYP5G82D8RVuybCDP2T4RZRpzsn35VdWiDbpWPL1++3Dp06JBzWWZmpk2ePNmeeOIJGzNmjG3dutXWrFmTK9ut7uX16tUr8HbLlSvntrz0YoX5BQv7ODdsMFu+3DvfqpXGGPSItqtd26x+fW/JsIUL0+2gg4IeUeIK6/4JCPsnwop9E2HG/okwSw/5/lnYsQXWSK1Lly42d+5cmz17ds52wAEHuKZq/nk9iPHjx+f8zfz5823RokXWqVOnoIZtyd65vGZNC2UmmWZqAAAAAMIosEx3lSpVbO+99851WaVKldya3P7lF154oSsVr1mzpisNv/rqq13AXVATNcQ+6NYa3WGkZcM+/ZSgOwzUx3DVKjMVqFSoEPRoAAAAgCRfMmxnHn74YTvhhBOsT58+1rlzZ1dWPnLkyKCHlZTCulxY3kz3Dz8EPZLk9fHHZt266YCaWYMGOohmpqKUl14yywpPs3sAAAAguZYMizRx4sRcP6vB2pNPPuk2BCtegm4y3cHM97/kErPXXtt+WUqKWXa22bRp3jZsmNnbb2v1gSBHCgAAAJS+UGe6ER5hLy/3g+6FC73yZpSOjRvNjjnGC7jLlDG77jqzH39UU0SzP/4wGzTIy3xPmWKuwd28eUGPGAAAAChdBN1IiEy3upcruFMZ84IFQY8mOejgRq9eZlOnevO3Vajy0ENmLVt6me7dd9cSfmYzZ3qX/fabV36uYBwAAABIFgTd2CWVCfuZ7rAG3QrymNddum691eyTT7y52x995K2Xnp899zT74guztm3NFi82O+EEs/XrS3u0AAAAQDAIurFLy5aZ/f23WWqqWePGFlrM6y4948aZ/fe/3vkRI8x2taBArVpmH3xgVqeO2Zw5Zv37l8owAQAAgMARdKPQpeWNGmkBeAstLRsmBN2xb5x2/vne+csvNzvxxML9XdOmZm++6VUlPPus2YcfxnSYAAAAQCgQdCPu53P7yHSXjvvvN/vzT29/+M9/iva3RxzhNVuTiy4yW7cuJkMEAAAAQoOgG3HfuTxv0D1/PutCx4qaofmBtk4rViz6bdx3nzfPe+lS7zwAAACQyAi6kTCZbo1P5e+af75oUdCjSUx33plmmzebHXmk17m8OMqX97qcyyOPeMu8AQAAAImKoBsJk+nWOtFamkroYB59S5dWtNdfT3HnH3zQm5tdXMcdZ9ajh9nWrWY33xy9MQIAAABhQ9CNQpUUS5MmFnoqW5Yffwx6JIln1KgWlpmZ4oLlAw4o2W0pYFd5uk7fecfsm2+iNUoAAAAgXAi6sVOZmWZ//BE/QXerVt4pQXd0LVmiNbm99eJuuSU6t7n33mZ9+3rn7747OrcJAAAAhA1BN3YZbG3b5pVu169vcZPpVjM1RM8zz6Tatm1p1qlTlh1+ePRu9/bbt2e7586N3u0CAAAAYUHQjUKVljdsaJaWZqFHpjv6MjLMnn/e+6i48sqsEs3lzqttW7NTTvHOF3X5MQAAACAeEHQjYeZzR2a61b1cXcxRcv/7nyoeUqx69c3Wq1d21G//xhu909deM1u8OOo3DwAAAASKoBsJFXTXrm1Wvbp3fsGCoEeTGJ56yjvt1u03K1s2+rd/4IFmhx3mZdSffDL6tw8AAAAEiaAbCRV0q/SZEvPoLhf36ad6XrOte/d/doYY6N/fOx061GzTppjdDQAAAFDqCLqxUyrTjqegW1g2LHpeecU7PfrobNttt9jV6594olnTpmarVpm9/XbM7gYAAAAodQTdSKhMt/iZbjqYl0x2ttnLL3vnzzwzK6b3pSZ9F1/snX/mmZjeFQAAAFCqCLqx06ArHoNuMt3RMX26Ny++YkWzk0+OfgO1vM4/3wu+p041mzcv5ncHAAAAlAqCbhRIpb4bN3rnGzWyuBG5VrcOHKB4/Cz3ySebVa4c+/vTOvA9e3rnhw2L/f0BAAAApYGgGwXys9x165qVL29xo2VL73T1arO//gp6NPEpM3P73Oozzyy9+73kEu/0pZfMNm8uvfsFAAAAYoWgGwWKx9JyUTm0n5mnxLx4Pv/cbNkys2rVzLp2Lb377d7drHFjr8pi5MjSu18AAAAgVgi6kVCdy300UysZP8t90kkWk7W5C6I53Rde6J1//vnSu18AAAAgVgi6kXCZbqGZWvFlZW3PMp9ySunf/znneKdaH3zx4tK/fwAAACCaCLpRIILu5O1a/scfZlWqmHXrVvr336yZ2aGHek3wXn+99O8fAAAAiCaCbiRk0E15efH973/e6fHHB9dA76yzvNNXXgnm/gEAAIBoIehGQme6f/rJ68SNwvvwQ+/0hBOCG0PfvmZlyph9/TVrdgMAACC+EXQjX1qfe+VK77y6SccbHShQA7AtW8x+/z3o0cQPPVfffGOWmmp2zDHBjaNWLbNjj/XOv/pqcOMAAAAASoqgG/nyA9WqVc2qV7e4oy7YLVp45ykxL7zRo73Tgw/2At8gnX22dzpihNfcDQAAAIhHBN3Y6XJh8Zjl9tFMrfil5ZrPHbSePb1mbr/+ajZtWtCjAQAAAIqHoBs7zXQ3amRxy2+mRtBdOJs3m40fH56gu0IFsxNP9M6/807QowEAAACKh6Ab+dKSUfEedPuZbsrLC2fiRLNNm8waNjRr185CwV8n/O23vSXEAAAAgHhD0I2dZroVgMUrMt3FKy0/7jizlBQLhR49zCpV8qY7zJwZ9GgAAACAoiPoRsKWl/uZbgVsf/8d9Gjip4laGErLI0vM/aXLlO0GAAAA4g1BNxK2vLx2ba/zusqStV43Cvbzz96mtbGPOspCxS8xf+stSswBAAAQfwi6kbDl5SqRpsS8cPwGaloqTB3Dw0Tl7hUrmv3yi9nXXwc9muSlAx6LF5tNmWL26admX35ptnZt0KMCAAAIP4Ju7EBfpNevj/+gW2imVrSgu0sXCx0F3Aq8hRLz0jdvntnVV5s1aWK2++5mhx/u7Sc6QKNKkr33Nrvnnu0H6gAAAJAbQTcKLC2vUcNrYhXPyHTvWlaWl7kMa9AtlJgH8znQt68XVD/xhBdUp6WZNW1q1ratWYMG3vW++87sjjvM9tjD7KqrzJYvD3rkAAAA4ULQjYRsopY3003QXbC5c81WrPAyygcdZKGkTHf58t7cfI0XsfXSS15g7R/kOPlks/ff9ypgVOb/7bdmf/7pBdi67hFHmGVkmD355Pa/AwAAgIegG0kRdFNevuvScgVOZctaKGme+THHeOcpMY+dzEyz/v3Nzj3XbN06s44dzebMMRs50usir27ykXbbzeycc7w13lUtsc8+ZitXehlyZb0ViAMAACQ7gm4UWF4e7/O5pWVL73TVKrO//gp6NOEU5vnckZRtlffeC3okiWnbNrMzzzR7+GHvZ5WMf/65Wbt2hft7db3XWuq33OL9rKy31llX8A4AAJDMCLqR0JlulUz7j4Ns9462bjWbNCk+gm6tH645xd9845U4I7oB91lnmb35pll6utkbb5jddZf3fBeFKiXuu887MKLqhAkTzLp25YAXAABIbgTdSMjlwiLRTK1g06ebbdzorWle2IxmUGrVMjvsMO/8//4X9GgSh+Zs9+vnBdwKmlVKrvLwkjjxRK/kXK/ZjBlm3buT8QYAAMmLoBsFlpcnQqZbaKZWMGUi/dLg1Dj4NDjpJO+UEvPoefRRrxRc69qPGOHN3Y6GDh3MJk/25n3PmuVND9iyJTq3DQAAEE/i4Gs2SjvrlUjl5ULQXTAFRXLkkRYX/KBb49Y8fZTM1KlmN9zgnX/wQbM+faJ7+23amH30kVnlyl6jtbPP9pq1AQAAJBOCbuSyZo1Xbiy7724JgQ7m+VNnaTXKks6dLS40b+6tG63AbfTooEcT33TQ4owzvOdSDdTUtTwW9t/fq0xQ6bo6zw8cGJv7AQAACCuCbuRbWq65mGpClkhzuhcsMMvKCno04aGS302bzGrW9DKS8YIS8+hUtFx4oVfV0qKF2dChXnl5rBx9tNnzz3vn1WhN88YBAACSBUE3ErqJmjRp4nVk1nxS//Fhe2n54YfHx3zuvEH3xx8zR7i4nnvObNQoL/usTuXqNB5r6o5+3XXeea0DPm9e7O8TAAAgDOLoqzZKQ6LN5xYte6RsnlBivmPQHS+l5ZHlyg0amG3Y4M0TRtEsWbJ9HvegQV7Ds9LywANe0z69dpo/7k9lAQAASGQE3UjozuU+lg3LTfN4P/ssPoNuZeW1JJVQYl50Wh5s7VqzAw7wzpemMmW8zHr9+mY//LA98w0AAJDICLqR8OXlQgfz3L791gu81FV6330t7vgl5lqvm3n6hffBB9563Kr+eOYZ77S0aQmxl1/25pAPG+Y1VwMAAEhkBN1I+PJyIejOv7T80EO97GO8UYmy5iGrVHrmzKBHEx/+/tvsyiu98+pUvt9+wY2lSxezm27yzl988fYKGwAAgERE0I1c/C+/iZbp9svLmdOdO+g+4giLS+XKmR1zjHdeDcGwaw8/bLZokXdALQzLdt1zj9mBB3rLFCrwVkd1AACARETQjRz60pvome7ffjPbvNks2V/nyM7l8SqyxBw7t3Sp2eDB3nmdVqoU9Ii8FQVefNE7gKJO9C+8EPSIAAAAYiMOC0sRK6tWeSWoiZjp1jzSatW8ecwLF5q1bWtJ69dfzZYv94IeNdOKV8cd581J/u477zXdY4+gRxRed9zhdQxXZvmMMyw0Wrc2u/tus5tv9pq6deuWeJ89YZaRYTZ9utk333hVQP5BSS3FpwMz9ep5r4c63B90kFmdOkGPGACA+ETQjR1Ky2vXNitf3hKKmjYp2z1jhvflMpmD7mnTvFM1UIvn17lGDa/z+oQJZu+/X/qduOPF3Lneutzy0EPhW5P9+uvNRo40+/JLs0suMfvwQ+/9itjYutXr+v/662affGK2bl3RDpJoqbdTTjFr147XCQCAwiLoRo5ELS2PnNetoDvZm6kpuJGDD7a4p6XDFHSrxJygO3+33eZ1eFewdNhhFjqqVhg+3Gvs9tFHXsn5eecFParEs3q1N69fXeuXLdt+uQ6yKou9115mzZt7GW6V/K9f7zUq/Pln73Pz+++97d57vU1VMtdcY9a3r3d9AABQMIJuJPxyYT46mOfOdOuLdiIE3VrrWXPUFVQo+43tVDqsAxLKbt93n4VWZJm5Oqtr6gClzNGxaZPZ44+bDRniNa0TlY1fcIFZr15e6Xhhlo7T+2v0aLN33vFOtWrAv/5l9u9/ewd2LrzQrGzZmD8cAADiUsgKDRGG8vJEzXT7QXcydzDXXM2vv06cTLcyc3vvbZaZ6WVJseNcbjn77O0d/MNKwbamPCi4u+GGoEeTGD791GyffbzAWAG33itvvOF1sddBGM3xL+xa7TqgddZZ3lQAHaDV3+++u9nixWZXXOFlyhWQ04UeAIAdEXQj4ZcL8/lBRzJnuufM8eZ0qqRUAWsiULZbNE8V202ZYjZmjLcOux98h5nGqdJnzRN++WVvvjGKR03zND9e66GrPFyf6S+9ZDZ7tlcOriaKJW1MecstXgNDZdGVOf/lF2+ut6oUfvopWo8EAIDEQNCNHH/+6Z0qe5GIWrTwTleu9Dq1J3NpeceOidMEyQ+6lenWAQV4/ED7/PPjp7O7Mq9XXeWdv/zy7aspoPDUzV/P47Bh3s/KQuuyc84pfFa7sDSXW6+Xgu/bb/fKy7X8mzLqyoRv2xbd+wMAIF4RdCNpgu7Klbc/tmTNdidSEzWfAoy6db3GT5MmBT2acPj8c6/BnDKamm8bT9Skq0EDL1s6aFDQo4kvI0Z4B9R++MF7DrUPPPmkWdWqsb3fihW9OfnqlN+1qzeNRfvd4YebLVgQ2/sGACAeEHQjaYJuSfYS80RqouZTk7CePb3zahoGs8GDvVM1umrc2OKKAkSVLMv995vNmxf0iMJP86jvvNObc63GaQp81bvhyCNLv2/G2LHe9IBq1bzPG83T/7//Y643ACC5BRp0P/3009auXTurWrWq2zp16mQfRXRD2rx5s1155ZVWq1Ytq1y5svXp08eWRa51gqhRllBbogfdydzBfMUKb36nKBuWSPwScwXdyf7lXtnGDz7wpg/cdJPFpZNP9g6kZGSYXXqpt+QZ8qcpFTq4ctdd3s96zVXiHVT3d+13atyn/fDoo72DAJddZnbaaUVbExzFp/eNphSo6Z0qHW691eyii7zX5fTTvbn3Z5zhTT3Q3PyHHjJ7913vNdu4MejRA0BiCnTJsIYNG9qQIUOsZcuWlp2dbS+++KKddNJJ9vXXX1vbtm3tuuuusw8//NDeeustq1atml111VXWu3dvmzp1apDDTkjqQCtVqnhbokrmDuZ+abm6DFevbglFmb0KFbyuzN98Y9a+vSUtLQ0l+mLt7+/xRoHbE0943bfVEO6558wuvjjoUYWP5rzrddYSXpqv/fTT4XmetArGuHFmjz7qLQX31lte9v3tt5P7/RkLy5d77xVtWspNAXdJ+luoB4Sm7Wjr1Mk7VaNDAEDxBfox2tOvCf3Hfffd57Lf06ZNcwH5c889ZyNGjLCjdbjczIYPH26tW7d2vz84kSalhqi0XPMAE1kyl5cn4nxunwLu7t29DubKdifrl3pVMrz+und+wACLayqLv+cebykxZW9VzaC5+9jeofykk7xAS/u/MpU9eljopn5cd53ZIYd4XdM1T19TWzR9QJnXRGnmGITvvzd7800vm60Djfn1MGnTxutcX7++12Fec+8VPGtTUK7l+bSUnAoI9dmh10eXqTGeNv+zRFMF9DWsWzdvH0uUlS8AoDSF5thlZmamy2hv3LjRlZl/9dVXlpGRYV2VwvrHXnvtZY0bN7YvvviiwKB7y5YtbvOt+6eeTbelLaz8sQU1xt9+07efMtagQZZlZGRaomrWTP+m24IF2bZlyzb3pTBZTJum1sWpdsABmZaRkRVX+2dhHH98ir33XhkbNSrL/v3vxN2Hd+b++1MtKyvNunfPsr331utscU1lyS+9VMZmz06x667LshdfzIzb/TOa1q7VQYg0++KLVKtcOdveey/TDj88O7Svd4cO3kG/Cy9Ms9GjU91yZpMnZ9kTT2S6QDCRRXPf1BShl15KtREjUm3u3NxHLPbZJ9u6dMmyTp2yrX37bGva1DvoUVRa3ePrr1Ns5kxvmzo1xVatSnEHdbRJu3bZ1qdPlvXunZVzIBvxKdk+OxFfMuJk/yzs+FKyVde9CzVr1izSnaekpNisWbOsSZMmu7zu3LlzXZCt+duat63M9nHHHedOzz///FwBtHTs2NGOOuoou18ddvJx55132l3+5LYIur2Kif6/ewm8805Le/nlNnbkkb9bv36zLFFlZqZY374nWGZmqg0bNtZ22y051iTSnNizzz7ONm1Kt4cemmjNm6+1RLNmTTk7//welp2dYs8+O8Zq195syWT16nJ2ySXdLCMjze67b4q1bfuXJYKffqpuN93U2bKyUuzOOz+3ffddYclsy5Y0Gziwk/3wQy2rVGmrDRz4he255xqLl8+hUaNa2CuvtHGvZ9Oma+2mm2ZYgwZMJC6IvqF9/31N+/jjpvb55w1s2zZv3be0tCzbd9/lduihi61Dh2VWvXps1kvMzFQWvLrNnl3HZs/ezY0lK2t7NN+48Tr3veGII/6wWrWS6zM36PfS5s1lbOPGdNu4sYzbL3SZ3lf6PzAlJdvKlct0W/ny2rZZhQrbqC4BYmDTpk125pln2tq1a12PshIF3ampqfbII4+4edW7opu74oor7Ntvv7XmhahB2rp1qy1atMgN9O2337Znn33WJk2aZLNnzy5W0J1fprtRo0a2cuXKnT4RYThKMm7cOOvWrZula52fUtavX6o99VSa3Xhjpt13X2J3Ldp77zL2448p9tFH26xLl+ykKUVs3z7dKlTItr/+2lbk+XlB75+F1blzmk2bluoyaJdcktj7cV4DB6ba4MFpdvDBWTZpUmZCfbnq31+vaZrtsUe2zZqlL4/xuX+WlNa9PuUUL1tcvXq2jR27zXUHjzeTJqXYWWel2fLlKVa1arY991ymnXRSYn4WF3ffVAA1enSKPfBAqvtM83XokGUXXaQsc7YVMR8SFX/9Zfb++yn2zjupNn58im3b5n3QpKZmW9eu2Xb22Vnutcz7HkXRbN7sTYObPz/FFixIsT/+MPvjD52muB48mgag4LooypXLdg0W69TxTjVdp3HjbbZ+/Td24ol7W8uWabbbbkz7QHhkxMn/7Yo1a9euvcugu9BfvU8//XSrU8h2qFdffXVhb9bKli1rLVq0cOf3339/mzFjhj366KN22mmnuYB8zZo1Vj2i65O6l9fT5KQClCtXzm156cUK8wsW9DiXLPFOGzdOs/R070h6olIjMf1n9vPPZeyYYywpfPWVd3rAASlWoULx96+wv4969fKWKfrwwzS78srE3o/zNtQaNsw7f8MNqVa2bGLNm9B63SptXbgwxe6/P93uuy8+98+S0OFxdXJX07Ty5dWhPsUOPDA+H6u/pJnmeat8+dRTy7h5+3pdE7VhV2H3TR1Y0VxqNURUQzQpW9brPH755foM13s7uPe3vn6pWZ82BX7vvKOSd7PPPkuxsWO1pbpl/9St/rzzvEZsBHE7p+dR/0erCd6MGWazZ5v98kvhVuLQLqWvyDrIoekE2tRUUfuR/l/Q6gHqSK+KhS1bUuz3381+/z3yBSmrbwb28MPbewG0bm3Wtu32be+9vd4AvI6lT6+j+i/4+4JO/fN6nRXu6DTRpYf8//bCjq1Q/71lFXG9lvX+2lPFoPtSploBuB7E+PHj3VJhMn/+fJcVVzk6oisZ1uhO5g7midxELZKabf3732bjx3uNpvQFIhmMGOHNxdSMHjXXSjRaUUHdzLWU2AMPmJ15pvdlMJlo2afhw70vWGqgdeihFtfUtHPCBO/9qiWr9LpOn2722mteYJds9EVaS/3p+fDXptd+r2W9rr3Wa4YWNjVqeA3xtKnxmoJvbb/+6h0E1Kb/bxV8a1m7ZPh+UdjvW2qAqP+ntDqDnrv8KJhWAKx58/psV+CrVQH0PNau7TW40wG4wgTDCsDV5V5N8/xTZcwXLsyymTNX2bp1tVwWXf9vKvDXFkkHUvSZu88+mtPvbTqfaCuhxOq9rQoRvS900EO9GVat8i7zT7WpqaEOkvgHS7QVZqqw4j3tB/6mAzD67NBro/do3lNtfqWDTmvVKl7/BxRdoMeUBwwYYMcee6xrjqZAXfOuJ06caGPGjHGl7BdeeKH179/fzSlXul4ZdAXcxelcrqy5tvxK58tEHFrP7zqRc9Ujj2YU5boqkSiokl/XjbSr6xb2dv1KgsJcVx/COuLp/6e4bdu2nR5sibzdXV1X4/UfY6yuq0Z82gpz3ZYtMy09PdN1as3vJdT+oP2iMLdbGtfVc6DnoiBpaWlu29l19R+odpuOHXW9tGLdrsarfT6/fSjyuvr9zppKFOW6ke/Pwlx3r73KmApnfvop2z7+OCNn/e6d3e6u3sth+Ywo6Lo6UTdo/fqqq/ySw+h/RuS9bml/Rhx3nBd0v/tuul16aYpNnqz90ruuxp3f/lncz4gwvO8jr/vUU2b/+Y/3Guu8OkhnZUX3M6I0rxv5Xh482OtorqZ5Wg20Y0ezV15Jc1NF8l43Gp8RxbluST4j8u6b+X1G6KDo7bebff65d5m+CF9zjSobUqx27ZJ/RpTkuoX9jFAwqANDAweWde/NF19UZ/Vt9ssvWTZwoLeOvDqgn3OOml56wUE8f48oynt5xYpMmzhRm7ltwYLc101JKWPNm6vJqSo+M23ffTNdsJ1fqXfez4idNb6NvG65cllWv/62HQ7e6PUcM2ay66WUklLedbH/7rss++67be7gj6am6buSAkFl4rVlZmr+uPf+bNw4y9q33+ay4f6m/4P9t0M0PiNKet1Yf0Zo91q6VAcwtrqg+rffvOBaS5j65zduTLVt27Z/RqSnF/x5ov/Dt21LL/R1MzLSXXCufGd6usZb0HvZu+722/Wuq11EB3H0uaN9zjtNsbp10/+ZjqDfZ1jt2tnuevlNHYnV94iUiDdAmD8jdvb/Q5HndEf6n9bjye+GUlKsfPnyrlS8mdciepcUVCuTvWTJEhdkt2vXzm6++WZXuy9qrnb99dfba6+95rLfPXr0sKeeemqn5eX51dnrtv/973+78eWlNcI1+d03aNCgAt9kagx3ng7Z/uPBBx90k+fz06BBA7s4YsFUzYlXrX9+dtttN3fd0aNHuw++YcOG2QodCsuHHku/fv1yftZ1F/uLbOehxnE33nhjzs8vvPCC/aZPgHxs3Zpugwbd4uYNKfDWAZAFef9niDBQ/4v+Q13n5/mH5gs4uOK/yUaNGmVz5swp8Lo33HCDVapUyZ3XGu0z9QlfgGuvvTZn6sHYsWNdV/uCXH755TnTI158caL9+uukAq970UUX2e7/HH3QmvCffPJJgdc999xzranaxJqyNNPto48+KvC6Z5xxhu35T5pdPQve0/pWBTjllFPcWvXy3XffuX4HBdHa9vv+M7Hzxx9/dO+XghxyyLHWrVtHd/7XX3+1F/XNqABaOeDQf9Jp2m+0/xTkiCOOsCOPPNKdX758uVv6ryA6cNZd63u55mdr3HSSghxwwAF2vL6hmf7T2mj/UeRRgPbt21uvXr3s+usVhG61W28dXOB127RpY6eeemrOz/k1XwzbZ4R6Zfj0OVjanxH6j+eWW27J+Tmoz4gnn7zBVqyoZM88o+cwNp8ROvir3iLJ+BmhA+HqnVLUz4g///zT9WSJxmfE5593smOO6W7ajdeujf5nhP8labAi/gA/I1Rt9corD1qZMon5GfHSSyPsl18K/ow47riBLtDU99l4+x6xq8+INm0usunTd3fZ7IoVp1q3bgV/Rpx88rnWrl2wnxEnnHCCqzAtzGfEb78da+PHd3RBZdOmv9p55xX8GVG2bFfbf/9DXWZ827Y/7bnnno2L7xGF+YxYsaKNjRlzqnseFHPdeWfBnxE//tjSJkw40y2FqV1ov/0GWWpq/p8RtWo1se7dz3MrOyjAfe65B+3vv/P/jKhTp4H17Hmx6wGgbfToR+zvv/P/jNi6VY0Qr3BZdX00dO36lNWqlf9nxJo11eyRR7Z/Rlx88TDbfff8PyMyMytadvaNOQH6kiUv2Nq10fkeccstt+TERnrfh/UzQvHqkCFDojen26edUQF23ljdv0ynhx12mHvANVTDsBNah3tnFCQ/+eSTbkPs6WhXMqyDq1Ialfkko0L0Qox7ym4r84vEpPXH/bW7/XnsSCz6eqHXWJlflZ4nGvVQUQyvYxR6nIk6j31Xj0vHdhQT6hhEIVsGxQ0d0/CPY2id+p0JcY/ffKkq5fnnvXJoVTWoP0NBNGXCj7H0WkccxwolHQhT1Ymfof7nGECBFaKqApBdzatW/6BXX83dp6SgJLqmxfnTIGVn0wf0HtO0A5+O/6oqIT86ThzZD0VVUwUcl3P7pKaEaBqCrhORnN6Bgv0HH9z+s97P/xxn3sHff3vTHPUYNR9dPZZ2tv/femuq/fxza+vc2RJCkTPdykzfeuutdt999+UcDdeRudtvv91uu+02dwT10ksvtYMOOmiXQXVp8DPdOuKb39GHsJSOin80R0qzLGzWLHV9VvlI2Zz/JBK5vHzbtkyrUyfTzV3SwS296RO5vFwNUvSfiOb6vv568Uq9VGmio4GqNsmvYUSYysL0kOrWzbb16zNs7Nj8v/AkSnm55gK2b+9dpoO6e+wRuykoea8b1GdEamq6HXxwims8dOaZ22z4cK+8XNOS8u6f8VxePn16ph1zjPc5pbYmms8dOe8u2lNQSvO6O3sva/cbNizN+vdPc19KW7XKtjfeyHDltvFaXu7vm5s3p9sjj6TYQw+lu/mactJJW10Ant/ji1XJeJCfEfpRCeKXX1YTtrLuC7uUKbPNDjlEnc/Nevb05jCH9XuEdpsZMzLdKhEKOnVwKG+g07BhGTv66FRXUn/EEZm2227h/B7h75/67ulXg5bkfa+nWpnfb7/1tu++S7M5c9JcSX12dpZ7nXe8PS8gbNQozW1eObNKmTNcubO69CtI8zO/SlBWrJhmZcqkuc+LrCzvvaynQvuTGsf5c6J1/q+/Um3VqjKu78nKlVrBJcMFzKrs1HdefcZuH3+qZWb67+Xsf0qwvaBTwa0y1do3NZ2iceNUa968jPtZ48/Kis/vEYW9rp7ndeu8IDxy03O5YkVZd6pt5coMW7tW38G81yCvjIztnxHaH1JSCn5/+tf9888Mq1MnJbSxhmJNVRtFPdOtNPszzzxjh0R8k+3SpYt7s15yySWuhEUlUBdccIGFif4jiPzPYGfXK8ptRquzXeR//kXp0BeN6+pIlu4+sslJ5IfDroThupH/Cez6drX0UJoLuDV/SWVP0bjdWF1XH9aF3dfyu66aE+n11TGyyLss6u1qvLr+rvY5fTgV9nZjcV3tNscfn2Ivv1zWPvxw50eqY/2+j+ZnRH7X1ZFqvbYnnJD/l/bi3m7Y3/f/93/e/jxiRBnX1blr15Rd7p9heC8X9rr6gnrCCWm2enWa6/StmR35LMoRtc+I0r7urt7L6k1w4IEqj/WWTDrkkLJuOsFZZ5Xsdot7XSnudXU/mn8/bFh5GzQoLSezpGyPmscdfnhsPyOCvG5B72X1JNCmIsY33vAOKE2bVsYFsNo0RUizIRSAa//Xez0y21fa3yMU8yib68/JVvOzDRu290cRBYoKsLt08bbcsyxzX7c4YyjpdQt6f2r/zHs7JX3fK0urrXfv7Zcp+Jo3L9Xmzi1r33zjdeZXNllVhwqUdQA5d0M5veCFfW+U/LqqgFQgHbk1bZpiTZqUdUG1ZrXuutlYfH6PKMp1dRBE2y6unXNO308UqGtWzNp/Nv2sAx1a3XnLljL/nObeFAPreICSZL/88qtVqNA41LFGYV+jIgfdCxcuzDeK12U/K4L5Z36T1sVGfEimzuU+/YegoFtLhyU6LaElalaULFRirkyKWlBElj0lEpX26cuqREzPTAqaeqiOzqri0JTXnZU3xmPpsQISBWd6nCNH7jzgTlT6vFIVlgLtceO8JbM0fVUHmtTMJx7oi+Nrr6XYTTd1sWXL0nL+79E0UTUFTPYlmDTd6ZJLvE2lvKNGefu7Am8tm6VNVQDKbB5+uNlRR3kHY/bbL3Zds5XE0kEvNR/VAWud6vMlb7JR2dcjjvA2BdkqnU7213NnlKXW/H1ted8jSvwo+I7s7u1lT71Nc5D9rLW2XdXnKv7RPuNv+rzIuylw9LvB61TjQ/QpVtcBDW3FkZGRZaNHf2uVKze2RFDkoFuNFtRU46WXXnKpdFHp9k033WQH6tPQHaVfYI20JyMuJGvQnQzLhvnlUzow90+PlKSgoEX/8eqgyg8/7DiFIBFo9o6+gKhbrDIsyebee735gvqCfMMNabmyKvFKWYBjj/XW6FUHYK3JraVfkpW+YqinlOYh3nOPmm15pcmqdPin11EoKSjQuDWXdc4cfc0q46a83Hlnil14ofdFFLnpK+PVV3ubcjbvv2/28cfe0lr6Wee1+Zo394Jv/V+u89q0DJ2+3CsgLigBrKBayzTp/0VtOsilYE9duvV/hf7PyK+aV7epaXiqnNKm5bJYZqnk9Byqo7q2wqwGrPeWMqH+tATxD3boVMFzovZIQPwr8q6pedrqhNqwYcOcwPr333+35s2b53RR3LBhg5vfjfjgz+NOpqBb615Kome6/Sy3viD806gxKShQUSCqL2nKnEQ0y0wImkLnN4tTljsZMyz6cqVMvzJgL72Uao0a1XXLisUrfYlUIKm5+WpoOWZM4jWXKg4FT3fc4U2hUGMflaUqS6znSpUOBTXsCYqWPVOzv88+836uWjXbevb83p54oqVVr060XRjKRJ5/vrcpE6q5wQq+9Zwq66yDUiqs/Ke4ssD/A3RwQwGY9iG9v5QtVcC2K5o33KGDl1VXabs2BfXJ+DkbNnoN/PWogYQPulu1auVatqttupYV8C/TMl9+wwa/3T7iQzJnuhM96Nb6r1KMpe3jnuaDKuh+553EC7p1fFPz4fTlNGKloqSj1arU/fm//1XZ8b4uSxaPKzAo+6byac0XVbCg/VZf8rGdgiA1z7vzTm/KiEqR9Tzpva0VrYL8Eq7sm0rgVTau11A0Hu2P/ftvsy+/XGCVKrUMboBxTF8r1XdFmz+NRplqv/zcD761aa1kf0U1NXHaGR3QUnZV2XHlj1QN5W9qllXIqdIAUGjFKsJQcH3MMce49fTKlSuXa/FyxJ9kDLpb/vP9R/OGNDc2VvPDwhJ0J9N8bp+a8Fx6qTcvVF/IEimIeeSR7Uu3KCuTzFR2/P772fbjj+Xtmmuy7PXX4ysjpYBNjcN0cEhTInRA5Z8ltZGH5rYrsNUBCj1nCnCVBVeTNRXXKTNahJ5DUTlYouBfY9IBAVFmVePQuDRXdCcN0lFMKvVWJVN+02r0fK9e7QXfqgjyNx0E8bpee3PJKfEHUNqKPCNFbdXvuece23333a1y5cr2i+p8TEsS3R6KJcJQdMkYdCubpCPciTyvW18+1CwuWTPdygL7ncsV0CQKvabqnqsvjZdfHvRogqeDDs8/n2mpqVn25pupOc3l4umgwdCh3oECreOqZlHYOTWtUrmxDrAoS6neFToApWlDOiCl7rix/j/z7ru90nZV1Cjg1n6o5n46wKeDAJFr56L06HNRWWwdWNeKDppapbnfOq8Mtv5fIOAGEBdB97333msvvPCCPfDAA7lapO+999727LPPRnt8iDE1YvLLsfwgNFn4zbXUQCURaR6c1g7VUX2/nD7Z6AuxvP22JYxHH/VOTzst+d6zBenYMdvOPPMHd14Z0Hh5Tz/xhNnAgd55LZ/k76/YNR2k0HtAU4Qee8ybVqCGWNdd5x1AVpWLumDvZPnVIlEXZX3FUaM7LSGk103Bvhp33XqrN91DAT89ZAEAUQm61bVc63SfddZZudYqa9++vf2g1o+Iyyy3yq52sp57QvLXNY6XL+jFbaKmJjDJ2mVV7SX05VxLvyxaZAnR9FDr2ibjMmG70rv3AuvSJcsdaOrb1zvgFGavvOLN+RUFcFQtFI8/d1oZZlUM6HNda8Aq26zlnJTdVMfw117zgvJdLTfk0zJFn3zilYmra7XW6dXydJpHrrJyXabKBAXe6qS/67VrAQDJrMhzuv/8809robVM8ik7z2DyUlyXlsfTPMhoSPSgO5mbqPn0RVndrZXxUhfzeA9Un37amzZw2GHJtQRcYejA0vDhmXbAAamuykMZTwVhYaT14887zzt/zTXbs90oPs3VVXZbaz5PmOAd1NC0Ev0f9/zz3uZPLdJnv5poKTvu90TQMlHKZmvNYGXPdZpfM7c+fbyKhGStHgIAlFLQ3aZNG/vss8+sieqrIrz99tu2nybOIK4k43JheYPuRC3Q8DPdydhELZK+ICvoVol5PAfdytz6QWQ8P45YH2RRsNW9u7eWs6o8LrjAQkUBoTLxypaee6637FWyHfCMJT2XfpMtlezrva+M9fjx3pQbdbVW5UthaM62OuQrY961q1mzZrEePQAgURU56L7jjjvs3HPPdRlvZbdHjhxp8+fPd2XnH3zwQWxGiZhJxiZqeed0L1zord2pzriJQt1b/QZxyR509+7tZRM//9w7yBSv86BHjDBbudKbT6rO7Mhft27eslLa/OZaCpzC4IsvzE480fu80dQHzRFO1qkfpUFZ7B49vE1UJaLPRWWytXKFNn/dZnUdV4m4mnAp2NZBWWXFAQAIJOg+6aST7P3337e7777bKlWq5ILwDh06uMu0VjfiSzIH3Sov1Dx2dbpdsEDNAC1hzJjhne6xh9etNZlp3+7UyQt4VGKuRlvxRvNQ/WXCNH4FCCjY7bebzZ3rlRfroIsym3mKs0qdMq7HHec1r1QWVnOMeR1Ll7pW63M+kT7rAQDxoVjH2A8//HAbN26cLV++3DZt2mRTpkyx7qrnQ9xJ5qBbZYiJWmLul5Yn83zuSH5X6DfftLik8liVxqrh4UUXBT2a8FP2+MUXvfWulc085hhvvm5QVNqsMSjgVpny++97DcAAAEByoLAtySVz0J3IzdT8JmrJXlru0xxaHWT57LP47GKueb+i+cnVqwc9mvigAxRqWKb1knVQTUGvvzxiafrwQ7MTTvDm5Gu5KQXcavoFAACSR6GC7ho1aljNmjULtSG+JHvQnYhrdasUmaA7NwVeaoYkKuuNJ9o3P/rIO2hw7bVBjya+aM3kceO8ubqzZpn17Ok10iotanyn+febN3tzud99lww3AADJqFAzyh7xJxO6tSv/snvvvdd69OhhnTRR0jWH+cLGjBljt2siHeJGVpbZkiXe+XhtLlVSiZjpVmM4rTGrxnAqr4XnzDPNJk70GpLdfLPFDf/jV8Gb5uij6AfWxowxO/JIr9JB86lHj47tusrqTD5ggNmDD3o/a3kwdVMvWzZ29wkAAOI86Fa3cl+fPn1cE7WrIroRXXPNNfbEE0/YJ598YtdpcVTEBc1x3LbNy6BpqZ1kDrrV0VYHIRKhk7A/n1sr+PElfzutr3vllWbffOPNj46HZkrqVv7SS955PlqLT+8FzYtXI7OZM71u5grEY7EE1NKlZmedZfbpp97P99xjduutLAsGAEAyK3KIoYz2MZocl4cuU9CN+Cstr1vX6+qajPSlW4Gp5lvG41zf/Pil5TRRy02zXzSnNp5KzJUdVWlyhw5qYBn0aOLbgQeaTZnidTHXagUHHGD28cfRvQ8F8grwFXBrTrn2s9tuI+AGACDZFTnorlWrlr333ns7XK7L9DvEj2Sfzy1asqdly8QqMWc+985LzEUl5pr7HmZbt5o9+eT2LDeBW8lpze6pU70AfNUqL/OtefLqKl4Sy5Z52W0dj1amu21bL6N++unRGjkAAIhnRV4l9K677rKLLrrIJk6caAf9863+yy+/tI8//tiGDRsWizEiRgi6t5eYf/ed1+HYz4TGK2VFZ8/2zpPp3pEaaVWubPbrr9783s6dLbTeeMPruaD15NV9HdGhzzu99jqQ8fTTZo895jU4u/tus3POMUtLK/xtqRv6o4+aPfSQd17TUzTzavBgOpQDAIASZLrPO+88mzp1qlWtWtVGjhzpNp3XWt36HeIHQXfiNVP7+muzjAyzOnW8MlrkpkDID2CHD7fQUhZegZxcfTVz86NNTQafesorB2/c2Oz3383OP99rVKeA+ccfd16BoHW3L7nE++wcONALuDUFQFUmCsIJuAEAQIky3aIM96uvvlqcP0WIEHQn3rJhfhM1FaFQjpy/Cy80e/55szff9LKcVapY6Iwd61UsKHhTcIfY6N7dq3B54gmzIUPMfvvN7JZbvE3LzKnZnppMKkhXYP3zz2Zz53o9IHxt2pjdcYfZqacmRiNGAAAQUNC9bt06l80urPXr11uVMH6TRS4E3YmX6aaJ2q5ppUPN7VXHepVwX3SRhY6yraKAm1YZsVWhgtmNN3pl4dofdDx50iSzP/7wtvyokuT447WyhzdFgQNcAACgxEF3jRo1bMmSJVZH3zQKYffdd7fZs2db8+bNC3V9BIOg26MATF+atba1lmiqXdsSItON/Om1vuACb61uZbzDFnR/8YUX9GlFgeuvD3o0yRV8a4aUtvXrvaXl5s3zPhfUK6FaNe+zsl07sz33JKsNAACiHHRnZ2fbs88+a5XVgagQMjSpFKG3eLF32qCBJTWV8Gr+s5prKdsdr0szqYOyymMVVKo7Mwr2r395JcQKcPWa+9UOYcpyq6mXSpxR+lSopbW8tQEAAJRK0N24ceMidSavV6+epSfrws9xQnMSV6/2zid7ptuf1x3vQbdfWq45pkWYDZKUNE9X5cH/+5/Zc8+Z/ec/Fgrffmv2/vvegZObbgp6NAAAACi1oPtXRSNIyNJyZXlVNpnslOn8+GOvnDTeS8uZz104F1/sBd0qMb/rLrNKlYIekdfMS/r08aY9AAAAIP4xKy1JRc7npgmQ16VYtF53vPIz3cznLhytya62E6r4CMNiDOqM/dpr3vkBA4IeDQAAAKKFoDtJ0UQt/6Bb5b3xKDPTbPp07zxBd+GkpXkdq0VLh2lt7CDdc49ZVpa3jJXWfAYAAEBiIOhOUgTduWketCxd6nUrjjeai75hg1ci3bZt0KOJH+pirudMFQ4TJgQ3Di1f9tJL24NvAAAAJA6C7iRF0J2bGvM3bRq/Jeb+fG51LVcGF4WjfgZaIkoefTS4cdx5p5flPvFEs44dgxsHAAAAoo+gO8mXCyPoTowSc38+N03Uis4vMVfX8AULSv/+tR7066975+++u/TvHwAAACEMuj/77DM7++yzrVOnTvbnPynTl19+2aZMmRLt8SHGme5kX6M70YJu5nMXb7m4447z5nTff3/p3//tt3unffuatW9f+vcPAACAkAXd77zzjvXo0cMqVKhgX3/9tW3ZssVdvnbtWhs0aFAsxogYoLx8R/5c6HgrL1+/fvuBAoLu4rntNu9U86oXLSq9+x0/3lu2TFMCtGwZAAAAEk+Rg+57773Xhg4dasOGDbP09PScyw899FCbNWtWtMeHGNDcUcrLd57pDrqTdVHMnOmNt3Fjs/r1gx5NfOrUyeyoo8wyMsweeKB07nPbNrN+/bzzV1zhZdwBAACQeIocdM+fP986d+68w+XVqlWzNWvWRGtciKGVK73gQutzE6Rtp6AnNdVs1Sqvi3m8NVEjyx2dbPewYWa//hr7+9P96ABPzZpeIzUAAAAkpiIH3fXq1bOffvpph8s1n7t58+bRGhdKobS8Th2ziGKFpFe+vFmLFvE3r9sPummiVjLKdHfpYrZ16/Z51rGyevX2+1BZuQJvAAAAJKYiB90XX3yxXXvttfbll19aSkqKLV682F599VW74YYb7PLLL4/NKBFVzOfedYl5vMzrVln5F19sL5FG8anyw2+k9uqrZrNnx+6+brrJWw9e68Nfdlns7gcAAADBK1PUP/j3v/9tWVlZ1qVLF9u0aZMrNS9XrpwLuq+++urYjBJRxXzunQfdI0fGT6b755/NVqwwK1vWrEOHoEcT//bf3+z0070lvPRxNmmSN+UgmiZMMHv2We/80KFmZYr8KQwAAIB4UuSvk8pu33rrrbZq1Sr79ttvbdq0abZixQq75557YjNCRB3Lhe26g3m8BN1+afl++5mVKxf0aBKDst0VK2rKjNmLL0a/0/xFF3nnVRh0+OHRvX0AAACET7FzOGXLlrU2bdpYx44drXLlytEdFWKK8vLClZery3vYUVoefeoC7y/fdeONZsuXR++2r7rKq07QfQwZEr3bBQAAQHgVqrCxd+/ehb7BkarNRagRdBesZUuvVHvDBq+Dddh7AxJ0x8a115q9/LLZN9+YnX++2QcfeHO+S+KVV7x1wFWurjnjVatGa7QAAACI+0y3lgMr7IbwI+gumLq5+9nuWDbSioZNm8zmzPHO07k8+vuBgmSV7I8ebfbYYyW7venT1YTSO3/HHWaHHRaVYQIAACBRMt3Dhw+P/UhQamiktnPt25vNmuUFtEUo8ih1M2eaZWZ6c/MbNQp6NIlnn33M/vMfr6HaDTeYtW5t1r170W9HFRMnnWS2ebPZCSdsXw8cAAAAySHKfXkRdvrir6WKhKA7f/vuGx+Z7sjS8pKWPiN/V15pduaZZtu2mZ1yincwpih++cXsiCPMli71gvgRI8zS0mI1WgAAAIRRkRer2W+//VwH87x0Wfny5a1FixZ23nnn2VFHHRWtMSIGWe7y5c2qVw96NOEUL0G337mc0vLY0Ufd8897UzK0fJg+1t591+zoowtXUq5KCf3tnnuaffSRWZUqpTFqAAAAxHWm+5hjjrGff/7ZKlWq5AJrbepevnDhQjvwwANtyZIl1rVrV3vvvfdiM2JEbT432dH8tWvnnS5aZLZ6tYVSdjZN1EqL5nWPGmXWubPZunX6DDS7+26zrVvzv/6WLWYPPOAtB6b3m8rStTY3lSUAAADJqciZ7pUrV9r1119vt99+e67L7733Xvvtt99s7NixNnDgQLdu90mayIhQoYnarqkCoGlTby6u5nUfeaSFjsa2bJnX8KtDh6BHkxz7xJgxZuedZ/bGG2YDB6rXhdccTcF4vXre6/HJJ2YvvOC9PnLyyd7PdCoHAABIXkXOdL/55pt2xhln7HD56aef7n4n+v38+fOjM0JEFU3UCt9MLcwl5n5puUrhK1QIejTJQVMyXnvN23bbzQusb73Vy2hrqTl1JL/zTu/y+vW9oPyddwi4AQAAkl2Rg27N2/788893uFyX6XeSlZWVcx7hQqa7aPO6/SW5wobS8mBoSsbpp3uBtTLYKjVv0sSsYkWvOqJnT+/yBQu8rDhTOAAAAFDk8vKrr77aLrvsMvvqq6/cHG6ZMWOGPfvss3bLLbe4n8eMGWP7+lELQhl0a5kpxG8zNYLuYCnIPvdcbwMAAACiGnTfdttt1qxZM3viiSfs5Zdfdpe1atXKhg0bZmdqbR0zF5RffvnlRb1plAIy3UUrL//uO69hVtmyFhobN24/GEDncgAAACDBgm4566yz3FaQCkwyDS2C7sJRqbDm4qpb9Q8/bO9oHgZffumtG92woVfaDAAAACDBgm7ZunWrLV++3M3fjtS4ceNojAsxWmaKRmqFo7m4KjGfPNnLKocp6J4yxTtVAy/mDAMAAAAJ1khtwYIFdvjhh7tsdpMmTVypubamTZu6U4TXqlXeGsKi7soo3LzuWbMsVPygW92yAQAAACRYpvu8886zMmXK2AcffGD169e3FFJtcVdaXru2WblyQY8m/Pbf3zudOdNCQ2XlfhM1gm4AAAAgAYPu2bNnu87le+21V2xGhJhhPnfR/NOc32W6FeyWKfZkjOjREmYbNphVq2a2995BjwYAAABA1MvL27RpYytXrizqnyEECLqLplUrsypVzP7+22zePAtVafmhh5qlFvndCwAAAKC0Fflr+/3332833XSTTZw40f766y9bt25drg3hRRO1olFQ65eYz5hhofDZZ94ppeUAAABAfChywWzXrl3daZcuXXJdnp2d7eZ3Z2ZmRm90iEmmu0GDoEcSXyXmEyd6QfeFFwbffT6yczkAAACABAy6J0yYEJuRIOYoLy+6jh290+nTgx6J2cKFZsuWmZUta3bAAUGPBgAAAEBMgu4jjjiiwN99++23RbqtwYMH28iRI+2HH35wS5Adcsghrny9lSbT/mPz5s12/fXX2+uvv25btmyxHj162FNPPWV169Yt6tCTHkF38ZupzZ2rfdGsfPngS8t1ICDIcQAAAAAovBK3Ylq/fr0988wz1rFjR2vfvn2R/nbSpEl25ZVX2rRp02zcuHGWkZFh3bt3t40bN+Zc57rrrrP333/f3nrrLXf9xYsXW+/evUs67KRE0F10jRub7bab17189uxgx+IXmXTuHOw4AAAAABResRdBmjx5sj333HP2zjvvWIMGDVwg/OSTTxbpNj7++ONcP7/wwgtWp04dtyRZ586dbe3ate4+RowYYUcffbS7zvDhw61169YuUD/44IOLO/yks3Wr2YoV3nmC7sLTMvTKdo8e7c3rDmqX03zuTz/1zv/zVgAAAACQaEH30qVLXWCsQFidyvv27etKvkeNGuWWEispBdlSs2ZNd6rgW9lvv3mbaH3wxo0b2xdffJFv0K3xaPP5HdV1O9rCyh9brMa4aJH+TbeyZbOtatVtFuKnInQ6dEi10aPT7Msvs+yyy4JpFLhggSoVvNfvwANL//WL9f4JlAT7J8KKfRNhxv6JMMuIk/2zsOMrdNDds2dPl90+/vjj7ZFHHrFjjjnG0tLSbOjQoRYNWVlZ1q9fPzv00ENt7733zgnyy5Yta9WrV891Xc3n1u8Kmid+11137XD52LFjrWLFihZ2KrOPhR9+qKHCZKtefZN99NEnMbmPxKX+AQfbpEkbbfTof9LNpWzMmCZmtq+1bPmXTZgw1RJt/wSigf0TYcW+iTBj/0SYjQv5/rlp06boBt0fffSRXXPNNXb55Zdby5YtLdo0t1uN2Kb4ayIV04ABA6x///65Mt2NGjVyc8WrVq1qYT5Kop2qW7dulp6eHvXb37QpxZ22aFHBjjvuuKjffiJTp/B771WmubIdcshxlucYUKl49dU0d9q7d41AXr9Y759ASbB/IqzYNxFm7J8Is4w42T/9quqoBd0KhlVWvv/++7s51eecc46dfvrpFg1XXXWVffDBBy6T3rBhw5zL69WrZ1u3brU1a9bkynYvW7bM/S4/5cqVc1teerHC/ILFepzLl3unDRumWnp6ifvnJRXNgd9jDy3ZlWJffZVuxxxT+vO5J03yznfrlmbp6V4AHoR4eR8hObF/IqzYNxFm7J8Is/SQ75+FHVuhoy/Nnx42bJgtWbLELr30UreElxqoqSxcRyHUxbyosrOzXcD97rvv2qeffmrNmjXL9XsF+Hog48ePz7ls/vz5tmjRIuvUqVOR7y+Z0bm8ZA491DudGkBl97x53kGTChW2rxsOAAAAID4UOeVZqVIlu+CCC1zme+7cuW4N7SFDhriu4yeeeGKRS8pfeeUV1528SpUqbp62tr///tv9vlq1anbhhRe6cvEJEya4xmrnn3++C7jpXF68oLtBg6BHEt9BdwlnPxSL37X8sMNUyVH69w8AAACg+EpUZ9yqVSt74IEH7I8//rDXXnutyH//9NNPu47lRx55pNWvXz9ne+ONN3Ku8/DDD9sJJ5xgffr0ccuIqax85MiRJRl2UiLTXTIKeOXLLzXHJJj1uY86qnTvFwAAAECA63RHUhfzXr16ua2o5eW7Ur58ebf+d1HXAEduBN0ls9deZjVqmK1ebTZ7trd2d2nYts1s4kTvPEE3AAAAEH/oqJUEdGxj8WLvPEF38aSmmh1ySOmXmE+f7gX6CvhLK9AHAAAAED0E3Ulg7VotGeadZ053yUvMSzPo/ugj77R7d1WUlN79AgAAAIgOgu4kKi1XtlQdsFE8nTt7p1q+KyurdIPuY48tnfsDAAAAEF0E3UmA+dzRofLuSpXM/vrLbO7c2N/f0qVmX33lnS/ttcEBAAAARAdBdxIg6I6O9PTt2W5/Ga9YGjPGO+3Qwaxu3djfHwAAAIDoI+hOAjRRi56jjy69oJvScgAAACD+EXQnATLd0dOly/Z53VrOK1Z022PHeuePOy529wMAAAAgtgi6kyjopnN5ybVv7zWkW7/ebObM0lkq7KCDYnc/AAAAAGKLoDsJkOmO7nrdRx3lnfcz0bHwwQfeKUuFAQAAAPGNoDsJEHRHlz/H+sMPY3P72dlm77zjne/VKzb3AQAAAKB0EHQnOM0NXrbMO0/QHR3+HOsZM7Y/t9H03XdmP/5oVras2fHHR//2AQAAAJQegu4Ep7WelTktU8Zst92CHk1i0Nx4LeOl59XvMB5NfpZbpeVVqkT/9gEAAACUHoLuJCktr1/fm4+M6PAz0LEoMR850jvt0yf6tw0AAACgdBGGJTjmc8fGCSd4p2PGmG3dGr3bVVn5N994zdNOPDF6twsAAAAgGATdCY6gOzYOOMCsXj1v6bBPPone7b788vbS8po1o3e7AAAAAIJB0J3gCLpjQ6X6p5zinX/99ejcZlbW9qD7X/+Kzm0CAAAACBZBd4L74w/vtGHDoEeSeE4/3TsdNcps8+aS396UKWa//eY1TzvppJLfHgAAAIDgEXQnSdDdqFHQI0k8nTp5z6tKzKPRxdzPciuDXqFCyW8PAAAAQPAIuhPc7797p2S6Y1Niftpp3vnXXivZba1du/02KC0HAAAAEgdBdwLTOtKUl8fWmWduLzFfvrz4t/Pii2YbN5q1aWN2xBFRGx4AAACAgBF0J7CVK73lrFJSzBo0CHo0iWm//cw6djTLyDB7/vniN1B74gnv/FVXea8XAAAAgMRA0J0EpeV165qVLRv0aBLXFVd4p0OHmmVmFv3vx441W7DArGpVs3POifrwAAAAAASIoDuBUVpeOvr2NatRw+s8Pnp00acA3Huvd/6CC8wqV47JEAEAAAAEhKA7CTLddC6PLXUav+gi7/zdd3uBdFGy3FOnmpUvb3bjjTEbIgAAAICAEHQnMDLdpeeGG8wqVTKbOdNrqlYYCs5vu217iTrz7gEAAIDEQ9CdwAi6S0+dOmb9+nnnb7+9cHO7hw/3gvSKFc1uvjnmQwQAAAAQAILuBEZ5eem6/nqz6tXNvvvO7PHHd31A5LrrvPN33ukF7QAAAAASD0F3AiPTXbrUTO3++73zAwaYzZmT//W2bfOapq1bZ3bQQWb9+5fqMAEAAACUIoLuBKX5wgTdpU8N1Y45xmzzZrOePc1++SX371V2fvHFZuPGeQ3YtLZ3WlpQowUAAAAQawTdCWrlSrMtW7zzu+8e9GiSR2qq2SuvmLVq5ZX3d+xo9sILZkuWmH32mVmXLt7Put6rr5q1aRP0iAEAAADEUpmY3joC42e569Y1K1s26NEkl1q1zMaP9zLdX39tdv75uX+vLucvvWR28slBjRAAAABAaSHTnaAoLQ+Wqgu0/vaQIWYtW3qX1axpdu653lzv3r2DHiEAAACA0kCmO0HRuTx4mrOtpcC0ZWWZpaR4GwAAAIDkQdCdoMh0h4vmcAMAAABIPoQCCYqgGwAAAACCR9CdoCgvBwAAAIDgEXQnKDLdAAAAABA8gu4ElJ29Pegm0w0AAAAAwSHoTkB//WW2ebN3vkGDoEcDAAAAAMmLoDsB+VnuOnXMypULejQAAAAAkLwIuhMQTdQAAAAAIBwIuhMQTdQAAAAAIBwIuhMQQTcAAAAAhANBdwKivBwAAAAAwoGgOwGR6QYAAACAcCDoTuBMN0E3AAAAAASLoDvBZGVtD7obNw56NAAAAACQ3Ai6E8zy5WZbtpilpJDpBgAAAICgEXQnmEWLvNMGDczS04MeDQAAAAAkN4LuBPPbb95pkyZBjwQAAAAAQNCdoJlugm4AAAAACB5Bd4JmummiBgAAAADBI+hOMJSXAwAAAEB4EHQnaHk5mW4AAAAACB5Bd4Ih0w0AAAAA4UHQnUDWrzdbvdo7T6YbAAAAAIJH0J2ApeXVq5tVrRr0aAAAAAAABN0JhNJyAAAAAAgXgu4EwhrdAAAAABAuBN0JhDW6AQAAACBcCLoTCJluAAAAAAgXgu4EQqYbAAAAAMKFoDuB0EgNAAAAAMIl0KB78uTJ1rNnT2vQoIGlpKTYqFGjcv0+Ozvb7rjjDqtfv75VqFDBunbtagsWLAhsvGGWkWG2eLF3nqAbAAAAAMIh0KB748aN1r59e3vyySfz/f0DDzxgjz32mA0dOtS+/PJLq1SpkvXo0cM2b95c6mMNuz//NMvKMitb1qxOnaBHAwAAAACQMkE+Dccee6zb8qMs9yOPPGK33XabnXTSSe6yl156yerWresy4qeffnopjzY+mqhpPncqkwYAAAAAIBQCDbp35pdffrGlS5e6knJftWrV7KCDDrIvvviiwKB7y5YtbvOtW7fOnWZkZLgtrPyxFXeMCxemuJezUaMsy8jIjPLokOxKun8CscT+ibBi30SYsX8izDLiZP8s7PhCG3Qr4BZltiPpZ/93+Rk8eLDdddddO1w+duxYq1ixooXduHHjivl3e5pZa0tL+91Gj54d9XEBJdk/gdLA/omwYt9EmLF/IszGhXz/3LRpU3wH3cU1YMAA69+/f65Md6NGjax79+5WtWpVC/NREu1U3bp1s/T09CL//TvvpLnTzp0b2nHHNYjBCJHMSrp/ArHE/omwYt9EmLF/Iswy4mT/9Kuq4zborlevnjtdtmyZ617u08/77rtvgX9Xrlw5t+WlFyvML1hJx/nrr95pixZplp7uBeBAtMXL+wjJif0TYcW+iTBj/0SYpYd8/yzs2ELbcqtZs2Yu8B4/fnyuIwnqYt6pU6dAxxZGv/zinTZvHvRIAAAAAAChyHRv2LDBfvrpp1zN02bPnm01a9a0xo0bW79+/ezee++1li1buiD89ttvd2t69+rVK8hhh476xmnJMGnWLOjRAAAAAABCEXTPnDnTjjrqqJyf/bnY5557rr3wwgt20003ubW8L7nkEluzZo0ddthh9vHHH1v58uUDHHX4/Pabllgzq1TJbLfdgh4NAAAAACAUQfeRRx7p1uMuSEpKit19991uQ8F+/nl7aXmKVg4DAAAAAIRCaOd0o+hBN6XlAAAAABAuBN0JgCZqAAAAABBOBN0JgEw3AAAAAIQTQXeCzekGAAAAAIQHQXecUx86gm4AAAAACCeC7ji3erXZunXe+aZNgx4NAAAAACASQXeCNFGrV8+sYsWgRwMAAAAAiETQHedoogYAAAAA4UXQHeeYzw0AAAAA4UXQnSDl5WS6AQAAACB8CLrjHJluAAAAAAgvgu44R9ANAAAAAOFF0B3HMjLMfv3VO0/QDQAAAADhQ9AdxxRwZ2aaVahgtvvuQY8GAAAAAJAXQXcc+/FH77RFC7NUXkkAAAAACB1CtTi2YIF3uueeQY8EAAAAAJAfgu4ECLpbtgx6JAAAAACA/BB0xzGCbgAAAAAIN4LuOEZ5OQAAAACEG0F3nNq82ey337zzZLoBAAAAIJwIuuPUzz+bZWebValiVqdO0KMBAAAAAOSHoDsB5nOnpAQ9GgAAAABAfgi64xTzuQEAAAAg/Ai649T8+d4p87kBAAAAILwIuuPU9997p61bBz0SAAAAAEBBCLrjkBqozZvnnSfoBgAAAIDwIuiOQ8uXm61e7TVQa9Uq6NEAAAAAAApC0B3HpeXNmplVqBD0aAAAAAAABSHojkPM5wYAAACA+EDQHYf8+dxt2gQ9EgAAAADAzhB0xyEy3QAAAAAQHwi64xCdywEAAAAgPhB0x5m1a82WLPHOE3QDAAAAQLgRdMdplrtBA7Nq1YIeDQAAAABgZwi648zcud7p3nsHPRIAAAAAwK4QdMeZOXO80/btgx4JAAAAAGBXCLrjDEE3AAAAAMQPgu44kpVl9s033nmCbgAAAAAIP4LuOPLrr2br15uVLWvWqlXQowEAAAAA7ApBdxyWlrdpY5aeHvRoAAAAAAC7QtAdRygtBwAAAID4QtAdR2iiBgAAAADxhaA7jhB0AwAAAEB8IeiOE3/9Zfbzz975ffcNejQAAAAAgMIg6I4TM2Z4py1bmtWsGfRoAAAAAACFQdAdJ7780js96KCgRwIAAAAAKCyC7jgLug8+OOiRAAAAAAAKi6A7DmRnm02f7p0n0w0AAAAA8YOgOw4sXOg1UitXzqxdu6BHAwAAAAAoLILuOCot79DBrGzZoEcDAAAAACgsgu44MG2ad0ppOQAAAADEF4LuODBhgnd62GFBjwQAAAAAUBQE3SG3dKnZd9+ZpaSYHXVU0KMBAAAAABQFQXfIjR/vne63n1nNmkGPBgAAAABQFATdcRJ0d+kS9EgAAAAAAEVF0B3y9bkJugEAAAAgfhF0h3x97kWLvGXCaKIGAAAAAPGHoDvEPvzQOz3kELNKlYIeDQAAAACgqAi6Q2zkSO+0V6+gRwIAAAAAKA6C7pBatsxsyhTvPEE3AAAAAMQngu6QGjHCLCvL7KCDzJo0CXo0AAAAAIDiIOgOadfy4cO98+eeG/RoAAAAAADFRdAdQpMmmc2da1a+vNlppwU9GgAAAABAQgfdTz75pDVt2tTKly9vBx10kE2fPt0S2ZAh3un555vVrBn0aAAAAAAACRt0v/HGG9a/f38bOHCgzZo1y9q3b289evSw5cuXWyIaOzbFxowxK1PGrH//oEcDAAAAAEjooPuhhx6yiy++2M4//3xr06aNDR061CpWrGjPP/+8JZoVKyrYxRenufNXX23WokXQIwIAAAAAlEQZC7GtW7faV199ZQMGDMi5LDU11bp27WpffPFFvn+zZcsWt/nWrVvnTjMyMtwWRt99ZzZoUIp98MFR9vffKdamTbbdfvs2C+lwkYT8905Y30NIbuyfCCv2TYQZ+yfCLCNO9s/Cji/UQffKlSstMzPT6tatm+ty/fzDDz/k+zeDBw+2u+66a4fLx44d6zLkYfTrr1XsrbeOduebN19j/fpNt8mT/w56WMAOxo0bF/QQgAKxfyKs2DcRZuyfCLNxId8/N23aFP9Bd3EoK6454JGZ7kaNGln37t2tatWqFkZaj3vVqq1WpswMu/ba/axChaOCHhKww1E8feh169bN0tPTgx4OkAv7J8KKfRNhxv6JMMuIk/3Tr6qO66C7du3alpaWZsuWLct1uX6uV69evn9Trlw5t+WlFyvML9htt2XY6NErrUKFcI8TyS3s7yMkN/ZPhBX7JsKM/RNhlh7y/bOwYwt1I7WyZcva/vvvb+PHj8+5LCsry/3cqVOnQMcGAAAAAEBcZ7pFpeLnnnuuHXDAAdaxY0d75JFHbOPGja6bOQAAAAAAYRb6oPu0006zFStW2B133GFLly61fffd1z7++OMdmqsBAAAAABA2oQ+65aqrrnIbAAAAAADxJNRzugEAAAAAiGcE3QAAAAAAxAhBNwAAAAAAMULQDQAAAABAjBB0AwAAAAAQIwTdAAAAAADECEE3AAAAAAAxQtANAAAAAECMEHQDAAAAABAjBN0AAAAAAMQIQTcAAAAAADFSxhJcdna2O123bp2FWUZGhm3atMmNMz09PejhALmwfyLM2D8RVuybCDP2T4RZRpzsn36M6cecSRt0r1+/3p02atQo6KEAAAAAABIw5qxWrVqBv0/J3lVYHueysrJs8eLFVqVKFUtJSbEwHyXRgYHff//dqlatGvRwgFzYPxFm7J8IK/ZNhBn7J8JsXZzsnwqlFXA3aNDAUlNTkzfTrQffsGFDixfaqcK8YyG5sX8izNg/EVbsmwgz9k+EWdU42D93luH20UgNAAAAAIAYIegGAAAAACBGCLpDoly5cjZw4EB3CoQN+yfCjP0TYcW+iTBj/0SYlUuw/TPhG6kBAAAAABAUMt0AAAAAAMQIQTcAAAAAADFC0A0AAAAAQIwQdIfAk08+aU2bNrXy5cvbQQcdZNOnTw96SIANHjzYDjzwQKtSpYrVqVPHevXqZfPnzw96WEC+hgwZYikpKdavX7+ghwI4f/75p5199tlWq1Ytq1Chgu2zzz42c+bMoIcFWGZmpt1+++3WrFkzt2/uscceds899xhtnhCEyZMnW8+ePa1Bgwbu//FRo0bl+r32yzvuuMPq16/v9teuXbvaggULLN4QdAfsjTfesP79+7vufLNmzbL27dtbjx49bPny5UEPDUlu0qRJduWVV9q0adNs3LhxlpGRYd27d7eNGzcGPTQglxkzZtj//d//Wbt27YIeCuCsXr3aDj30UEtPT7ePPvrI5s2bZ//973+tRo0aQQ8NsPvvv9+efvppe+KJJ+z77793Pz/wwAP2+OOPBz00JKGNGze6+EdJyPxo33zsscds6NCh9uWXX1qlSpVcrLR582aLJ3QvD5gy28om6oNPsrKyrFGjRnb11Vfbv//976CHB+RYsWKFy3grGO/cuXPQwwGcDRs2WIcOHeypp56ye++91/bdd1975JFHgh4Wkpz+/546dap99tlnQQ8F2MEJJ5xgdevWteeeey7nsj59+rgs4iuvvBLo2JDcUlJS7N1333XVlaIwVRnw66+/3m644QZ32dq1a93++8ILL9jpp59u8YJMd4C2bt1qX331lSuT8KWmprqfv/jii0DHBuSlDzmpWbNm0EMBcqga4/jjj8/1OQoE7X//+58dcMABduqpp7qDlfvtt58NGzYs6GEBziGHHGLjx4+3H3/80f08Z84cmzJlih177LFBDw3I5ZdffrGlS5fm+j++WrVqLmkZb7FSmaAHkMxWrlzp5tXoaE0k/fzDDz8ENi4gL1VgaK6syiX33nvvoIcDOK+//rqblqPyciBMfv75Z1e+q+ljt9xyi9tHr7nmGitbtqyde+65QQ8PSU6VGOvWrbO99trL0tLS3HfR++67z84666yghwbkooBb8ouV/N/FC4JuAIXKJn777bfuSDgQBr///rtde+21rt+AmlACYTtQqUz3oEGD3M/KdOszVHMSCboRtDfffNNeffVVGzFihLVt29Zmz57tDqyrjJf9E4gNyssDVLt2bXeEcdmyZbku18/16tULbFxApKuuuso++OADmzBhgjVs2DDo4QCOpuao4aTmc5cpU8Zt6jegZis6r8wNEBR12W3Tpk2uy1q3bm2LFi0KbEyA78Ybb3TZbs2HVVf9c845x6677jq3agkQJvX+iYcSIVYi6A6Qysz2339/N68m8ui4fu7UqVOgYwPUvEIBtxpafPrpp25pESAsunTpYnPnznUZGn9TZlHlkTqvA5pAUDQVJ+8Si5o/26RJk8DGBPg2bdrkeghF0memvoMCYdKsWTMXXEfGSpoaoS7m8RYrUV4eMM33UimPvix27NjRdd1V6/zzzz8/6KEhyamkXKVn7733nlur2587owYW6nAKBEn7ZN7+AlpGRGsi03cAQVPWUM2qVF7et29fmz59uj3zzDNuA4KmNZE1h7tx48auvPzrr7+2hx56yC644IKgh4YkXYXkp59+ytU8TQfP1bhX+6imPmh1kpYtW7ogXGvMayqE3+E8XrBkWAhoubAHH3zQBTVa7kblkerKBwS9bEN+hg8fbuedd16pjwfYlSOPPJIlwxAampYzYMAAW7BggfuiqIPsF198cdDDAmz9+vUucFElm6bpKIA544wz7I477nBVmEBpmjhxoh111FE7XK6kpJYFU6g6cOBAd9ByzZo1dthhh7llQvfcc0+LJwTdAAAAAADECHO6AQAAAACIEYJuAAAAAABihKAbAAAAAIAYIegGAAAAACBGCLoBAAAAAIgRgm4AAAAAAGKEoBsAAAAAgBgh6AYAAAAAIEYIugEAiHPnnXee9erVq9Tv94UXXrCUlBS39evXr9Bj9f9m1KhRMR8jAABBKxP0AAAAQMEUnO7MwIED7dFHH7Xs7GwLQtWqVW3+/PlWqVKlQl1fYx0yZIjVr18/5mMDACAMCLoBAAixJUuW5Jx/44037I477nBBrq9y5cpuC/KgQL169Qp9/WrVqrkNAIBkQXk5AAAhpoDW3xSs+kGuvyngzltefuSRR9rVV1/tSr5r1KhhdevWtWHDhtnGjRvt/PPPtypVqliLFi3so48+ynVf3377rR177LHuNvU355xzjq1cubLIY37qqaesZcuWVr58eXc7p5xySlSeCwAA4hFBNwAACejFF1+02rVr2/Tp010Afvnll9upp55qhxxyiM2aNcu6d+/ugupNmza5669Zs8aOPvpo22+//WzmzJn28ccf27Jly6xv375Ful/97TXXXGN33323y8jrdjp37hyjRwkAQPhRXg4AQAJq37693Xbbbe78gAED3DxqBeEXX3yxu0xl6k8//bR98803dvDBB9sTTzzhAu5Bgwbl3Mbzzz9vjRo1sh9//NH23HPPQt3vokWL3PzuE044wWXUmzRp4m4XAIBkRaYbAIAE1K5du5zzaWlpVqtWLdtnn31yLlPZtyxfvtydzpkzxyZMmJAzR1zbXnvt5X63cOHCQt9vt27dXKDdvHlzl0l/9dVXc7LpAAAkI4JuAAASUHp6eq6fNRc88jK/K3pWVpY73bBhg/Xs2dNmz56da1uwYEGRysOV3Vb5+muvveY6lCujrqy7ytcBAEhGlJcDAADr0KGDvfPOO9a0aVMrU6ZkXw/09127dnWbljSrXr26ffrpp9a7d++ojRcAgHhBphsAANiVV15pq1atsjPOOMNmzJjhSsrHjBnjup1nZmYW+nY++OADe+yxx1yW/LfffrOXXnrJZdNbtWoV0/EDABBWBN0AAMAaNGhgU6dOdQG2Optr/reWHFOWOjW18F8XdP2RI0e6TuitW7e2oUOHulLztm3bxnT8AACEVUp2dnZ20IMAAADx54UXXnCBeXHma2tO+bvvvptrfXEAABIRmW4AAFBsa9eudZ3Ob7755kJd/7LLLnPXBwAgWZDpBgAAxbJ+/XpbtmxZTlm51gHfFS1Rtm7dOnde3c21pjcAAImMoBsAAAAAgBihvBwAAAAAgBgh6AYAAAAAIEYIugEAAAAAiBGCbgAAAAAAYoSgGwAAAACAGCHoBgAAAAAgRgi6AQAAAACIEYJuAAAAAABihKAbAAAAAACLjf8HLMIoH3yPwv0AAAAASUVORK5CYII=", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "# Physical parameters\n", "m = 0.2 # mass (kg)\n", "L = 0.5 # pendulum length (m)\n", "b = 0.05 # damping coefficient (N*m*s)\n", "g = 9.81 # gravity (m/s^2)\n", "\n", "# Torque input (step change)\n", "tau_0 = 0.5 # torque applied after t=0 (N*m)\n", "\n", "# Initial conditions\n", "theta_0 = 0.0 # initial angle (rad)\n", "omega_0 = 0.0 # initial angular velocity (rad/s)\n", "\n", "# Derived parameters\n", "I = m * L**2\n", "omega_n = np.sqrt(g / L)\n", "zeta = b / (2 * I * omega_n)\n", "omega_d = omega_n * np.sqrt(1 - zeta**2)\n", "theta_eq = tau_0 / (m * g * L) # new equilibrium position\n", "\n", "# Time vector\n", "t = np.linspace(0, 10, 1000) # 10 seconds, 1000 points\n", "\n", "# Coefficients A and B from general solution\n", "A = theta_0 - theta_eq\n", "B = (omega_0 + zeta * omega_n * A) / omega_d\n", "\n", "# Analytical solution\n", "theta_t = theta_eq + np.exp(-zeta * omega_n * t) * (A * np.cos(omega_d * t) + B * np.sin(omega_d * t))\n", "\n", "#Conversion\n", "rad2deg=180/3.14\n", "\n", "# Plotting\n", "plt.figure(figsize=(10, 5))\n", "plt.plot(t, theta_t*rad2deg, label='θ(t) [rad]', color='blue')\n", "plt.axhline(theta_eq*rad2deg, color='gray', linestyle='--', label='Equilibrium θ_eq')\n", "plt.title('Damped Pendulum Response to Step Torque')\n", "plt.xlabel('Time [s]')\n", "plt.ylabel('Angle [deg]')\n", "plt.grid(True)\n", "plt.legend()\n", "plt.tight_layout()\n", "plt.show()\n" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.13.3" } }, "nbformat": 4, "nbformat_minor": 5 }