AE-245 - Lista 01 - Q4b.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y1(0) = 0:  True\n",
      "y2(l/2) = 0:  True\n",
      "y1(L/2) = y2(0):  True\n",
      "y'1(L/2) = y'2(0):  True\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import sympy as sym\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "# DECLARA SIMBOLOS\n",
    "x1, x2, L, E, I1, I2, P, C1, C2, C3, C4, C5, C6, C7, C8 = sym.symbols('x1 x2 L E I1 I2 P C1 C2 C3 C4 C5 C6 C7 C8')\n",
    "\n",
    "# POLINOMIOS DE 3º GRAU\n",
    "y1 = C1 + C2*x1 + C3*x1**2 + C4*x1**3\n",
    "y2 = C5 + C6*x2 + C7*x2**2 + C8*x2**3\n",
    "\n",
    "# APLICA CONDIÇÃO DE CONTORNO y1(0) = 0 PARA OBTER C1\n",
    "C1_ = sym.solve(y1.subs(x1,0), C1)[0]\n",
    "y1 = y1.subs(C1,C1_)\n",
    "\n",
    "# APLICA CONDIÇÃO DE CONTORNO y2(l/2) = 0 PARA OBTER C5\n",
    "C5_ = sym.solve(y2.subs(x2,L/2),C5)[0]\n",
    "y2 = y2.subs(C5,C5_)\n",
    "\n",
    "# APLICA CONDIÇÃO DE CONTORNO y1(l/2) = y2(0) PARA OBTER C6\n",
    "C6_ = sym.solve(y1.subs(x1,L/2) - y2.subs(x2,0), C6)[0]\n",
    "y2 = y2.subs(C6,C6_)\n",
    "\n",
    "# APLICA CONDIÇÃO DE CONTORNO dy1(l/2) = dy2(0) PARA OBTER C2\n",
    "C2_ = sym.solve((y1.diff(x1).subs(x1,L/2) - y2.diff(x2).subs(x2,0)), C2)[0]\n",
    "y1 = y1.subs(C2,C2_)\n",
    "y2 = y2.subs(C2,C2_)\n",
    "\n",
    "# SEGUNDA DERIVADA DE y1 E y2\n",
    "ddy1 = y1.diff(x1,x1)\n",
    "ddy2 = y2.diff(x2,x2)\n",
    "\n",
    "# ENERGIA\n",
    "en = sym.integrate((E*I1*ddy1**2)/2, (x1, 0, L/2)) + sym.integrate((E*I2*ddy2**2)/2, (x2, 0, L/2)) - P*y1.subs(x1,L/2)\n",
    "\n",
    "# RESOLVER SISTEMA PARA ENCONTRAR C3, C4, C7 E C8\n",
    "C3_, C4_, C7_, C8_ = sym.solve((en.diff(C3), en.diff(C4), en.diff(C7), en.diff(C8)), C3, C4, C7, C8).values()\n",
    "\n",
    "# EQUAÇÃO DA LINHA ELASTICA FINAL\n",
    "y1 = y1.subs([(C3,C3_), (C4,C4_), (C7,C7_), (C8,C8_)]).simplify()\n",
    "y2 = y2.subs([(C3,C3_), (C4,C4_), (C7,C7_), (C8,C8_)]).simplify()\n",
    "\n",
    "# VALIDA SE TODAS AS CONDIÇÕES DE CONTORNO FORAM SATISFEITAS\n",
    "print(\"y1(0) = 0: \", y1.subs(x1,0).simplify() == 0)\n",
    "print(\"y2(l/2) = 0: \", y2.subs(x2,L/2).simplify() == 0)\n",
    "print(\"y1(L/2) = y2(0): \", y1.subs(x1,L/2).simplify() == y2.subs(x2,0).simplify())\n",
    "print(\"y'1(L/2) = y'2(0): \", y1.diff(x1).subs(x1,L/2).simplify() == y2.diff(x2).subs(x2,0).simplify())\n",
    "\n",
    "# PLOT\n",
    "x_range = np.arange(0.0, 0.51, 0.01)\n",
    "x_1 = list()\n",
    "y_1 = list()\n",
    "x_2 = list()\n",
    "y_2 = list()\n",
    "\n",
    "for i in x_range:\n",
    "    value = y1.subs([(x1, i), (E,5), (L,1), (I1,200), (I2, 50), (P,-100)])\n",
    "    x_1.append(i)\n",
    "    y_1.append(value)\n",
    "        \n",
    "for i in x_range:\n",
    "    value = y2.subs([(x2, i), (E,5), (L,1), (I1,200), (I2, 50), (P,-100)])\n",
    "    x_2.append(i+0.5)\n",
    "    y_2.append(value)\n",
    "\n",
    "plt.plot(x_1, y_1)\n",
    "plt.plot(x_2, y_2)\n",
    "plt.legend([\"$y_1(x_1) = C_{1} + C_{2} x_{1} + C_{3} x_{1}^{2} + C_{4} x_{1}^{3}$\", \"$y_2(x_2) = C_{5} + C_{6} x_{2} + C_{7} x_{2}^{2} + C_{8} x_{2}^{3}$\"])\n",
    "plt.grid()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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