{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Contenido bajo licencia Creative Commons BY 4.0 y código bajo licencia MIT. © Juan Gómez y Nicolás Guarín-Zapata 2020. Este material es parte del curso Modelación Computacional en el programa de Ingeniería Civil de la Universidad EAFIT."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Transformación de coordenadas en cerchas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "IPython console for SymPy 1.4 (Python 3.7.3-64-bit) (ground types: python)\n",
      "\n",
      "These commands were executed:\n",
      ">>> from __future__ import division\n",
      ">>> from sympy import *\n",
      ">>> x, y, z, t = symbols('x y z t')\n",
      ">>> k, m, n = symbols('k m n', integer=True)\n",
      ">>> f, g, h = symbols('f g h', cls=Function)\n",
      ">>> init_printing()\n",
      "\n",
      "Documentation can be found at https://docs.sympy.org/1.4/\n",
      "\n"
     ]
    }
   ],
   "source": [
    "init_session()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Transformación de coordenadas de vectores"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Transformación de las ecuaciones de un elemento cercha"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tomemos un elemento tipo cercha que se encuentra horizontal.\n",
    "\n",
    "<center>\n",
    "    <img src=\"img/cercha_local_horizontal.svg\"\n",
    "         alt=\"Cercha en el sistema de referencia global.\"\n",
    "         style=\"width:450px\">\n",
    "</center>\n",
    "\n",
    "\n",
    "Para este, podemos escribir\n",
    "el balance de fuerzas en componentes como\n",
    "\n",
    "\\begin{align}\n",
    "\\sum F_x = 0\\, ,\\\\\n",
    "\\sum F_y = 0\\, .\n",
    "\\end{align}\n",
    "\n",
    "Y esto podemos escribirlo de forma matricial de la siguiente manera\n",
    "\n",
    "$$\\begin{bmatrix}\n",
    "+k &0 &-k &0\\\\\n",
    " 0 &0 &0  &0\\\\\n",
    "-k &0 &+k &0\\\\\n",
    " 0 &0 &0  &0\\end{bmatrix}\n",
    "\\begin{Bmatrix} u_1\\\\ v_1\\\\ u_2\\\\ v_2\\end{Bmatrix} =\n",
    "\\begin{Bmatrix} -f_1\\\\ 0\\\\ -f_2\\\\ 0\\end{Bmatrix}\\, .$$\n",
    "\n",
    "Ahora, queremos expresar nuestro sistema de ecuaciones en un sistema rotado\n",
    "un ángulo $\\theta$.\n",
    "\n",
    "<center>\n",
    "    <img src=\"img/cercha_global.svg\"\n",
    "         alt=\"Cercha en el sistema de referencia global.\"\n",
    "         style=\"width:400px\">\n",
    "</center>\n",
    "\n",
    "Las componentes estarían dadas por\n",
    "\n",
    "\\begin{align}\n",
    "u_1' = u_1 \\cos\\theta + v_1 \\sin\\theta\\, ,\\\\\n",
    "v_1' = -u_1 \\sin\\theta + v_1 \\cos\\theta\\, ,\\\\\n",
    "u_2' = u_2 \\cos\\theta + v_2 \\sin\\theta\\, ,\\\\\n",
    "v_2' = -u_2 \\sin\\theta + v_2 \\cos\\theta\\, ,\\\\\n",
    "\\end{align}\n",
    "\n",
    "o, matricialmente,\n",
    "\n",
    "$$\\begin{Bmatrix}u_1'\\\\ v_1'\\\\ u_2'\\\\ v_2'\\end{Bmatrix} =\n",
    "\\underbrace{\n",
    "\\begin{bmatrix}\n",
    "\\cos\\theta &\\sin\\theta &0 &0\\\\\n",
    "-\\sin\\theta &\\cos\\theta &0 &0\\\\\n",
    "0 &0 &\\cos\\theta &\\sin\\theta\\\\\n",
    "0 &0 &-\\sin\\theta &\\cos\\theta\n",
    "\\end{bmatrix}}_{T}\n",
    "\\begin{Bmatrix}u_1\\\\ v_1\\\\ u_2\\\\ v_2\\end{Bmatrix}\\, .$$\n",
    "\n",
    "Que podemos escribir de manera abreviada como\n",
    "\n",
    "$$\\{U'\\} = [T]\\{U\\}\\, .$$\n",
    "\n",
    "Tenemos que\n",
    "\n",
    "\\begin{align}\n",
    "&[K] \\{U'\\} = \\{F\\}\\, ,\\\\\n",
    "&[K] [T] \\{U\\} = [T] \\{F\\}\\, .\n",
    "\\end{align}\n",
    "\n",
    "Y, multiplicando ambos lados por $[T]^T$, obtenemos\n",
    "\\begin{align}\n",
    "&[T]^T [K] [T] \\{U\\} = \\underbrace{[T]^T [T]}_{[I]} \\{F\\}\\, ,\\\\\n",
    "&[T]^T [K] [T] \\{U\\} = \\{F\\}\\, .\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ejemplo"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculemos las matrices de rigidez para los elementos de la siguiente\n",
    "cercha.\n",
    "\n",
    "<center>\n",
    "    <img src=\"img/cercha_transformacion.svg\"\n",
    "         alt=\"Cercha ejemplo.\"\n",
    "         style=\"width:400px\">\n",
    "</center>\n",
    "\n",
    "En donde la sección transversal ($A$) y módulo de Young ($E$) son los\n",
    "mismos para todas las barras.\n",
    "\n",
    "Para facilitar los cálculos, haremos uso de SymPy."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "A, E, L, theta = symbols(\"A E L theta\")\n",
    "L0, L1, L2 = symbols(\"L0 L1 L2\")\n",
    "T = Matrix([\n",
    "    [cos(theta), -sin(theta), 0 ,0],\n",
    "    [sin(theta), cos(theta), 0 ,0],\n",
    "    [0, 0, cos(theta), -sin(theta)],\n",
    "    [0, 0, sin(theta), cos(theta)]])\n",
    "rigidez_local = A*E/L * Matrix([\n",
    "    [1, 0, -1, 0],\n",
    "    [0, 0,  0, 0],\n",
    "    [-1, 0, 1, 0],\n",
    "    [0, 0, 0, 0]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Veamos las matrices que definimos."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\cos{\\left(\\theta \\right)} & - \\sin{\\left(\\theta \\right)} & 0 & 0\\\\\\sin{\\left(\\theta \\right)} & \\cos{\\left(\\theta \\right)} & 0 & 0\\\\0 & 0 & \\cos{\\left(\\theta \\right)} & - \\sin{\\left(\\theta \\right)}\\\\0 & 0 & \\sin{\\left(\\theta \\right)} & \\cos{\\left(\\theta \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡cos(θ)  -sin(θ)    0        0   ⎤\n",
       "⎢                                ⎥\n",
       "⎢sin(θ)  cos(θ)     0        0   ⎥\n",
       "⎢                                ⎥\n",
       "⎢  0        0     cos(θ)  -sin(θ)⎥\n",
       "⎢                                ⎥\n",
       "⎣  0        0     sin(θ)  cos(θ) ⎦"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{A E}{L} & 0 & - \\frac{A E}{L} & 0\\\\0 & 0 & 0 & 0\\\\- \\frac{A E}{L} & 0 & \\frac{A E}{L} & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ A⋅E      -A⋅E    ⎤\n",
       "⎢ ───   0  ─────  0⎥\n",
       "⎢  L         L     ⎥\n",
       "⎢                  ⎥\n",
       "⎢  0    0    0    0⎥\n",
       "⎢                  ⎥\n",
       "⎢-A⋅E       A⋅E    ⎥\n",
       "⎢─────  0   ───   0⎥\n",
       "⎢  L         L     ⎥\n",
       "⎢                  ⎥\n",
       "⎣  0    0    0    0⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rigidez_local"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Elemento 0\n",
    "\n",
    "Para este elemento, tenemos que el ángulo es 0, luego"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{A E}{L_{0}} & 0 & - \\frac{A E}{L_{0}} & 0\\\\0 & 0 & 0 & 0\\\\- \\frac{A E}{L_{0}} & 0 & \\frac{A E}{L_{0}} & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ A⋅E      -A⋅E    ⎤\n",
       "⎢ ───   0  ─────  0⎥\n",
       "⎢  L₀        L₀    ⎥\n",
       "⎢                  ⎥\n",
       "⎢  0    0    0    0⎥\n",
       "⎢                  ⎥\n",
       "⎢-A⋅E       A⋅E    ⎥\n",
       "⎢─────  0   ───   0⎥\n",
       "⎢  L₀        L₀    ⎥\n",
       "⎢                  ⎥\n",
       "⎣  0    0    0    0⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rigidez_0 = T.T.subs(theta, 0) * rigidez_local.subs(L, L0) * T.subs(theta, 0)\n",
    "rigidez_0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Elemento 1\n",
    "\n",
    "Para este elemento $\\theta = 5\\pi/6$, luego"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{3 A E}{4 L_{1}} & \\frac{\\sqrt{3} A E}{4 L_{1}} & - \\frac{3 A E}{4 L_{1}} & - \\frac{\\sqrt{3} A E}{4 L_{1}}\\\\\\frac{\\sqrt{3} A E}{4 L_{1}} & \\frac{A E}{4 L_{1}} & - \\frac{\\sqrt{3} A E}{4 L_{1}} & - \\frac{A E}{4 L_{1}}\\\\- \\frac{3 A E}{4 L_{1}} & - \\frac{\\sqrt{3} A E}{4 L_{1}} & \\frac{3 A E}{4 L_{1}} & \\frac{\\sqrt{3} A E}{4 L_{1}}\\\\- \\frac{\\sqrt{3} A E}{4 L_{1}} & - \\frac{A E}{4 L_{1}} & \\frac{\\sqrt{3} A E}{4 L_{1}} & \\frac{A E}{4 L_{1}}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ 3⋅A⋅E     √3⋅A⋅E   -3⋅A⋅E    -√3⋅A⋅E ⎤\n",
       "⎢ ─────     ──────   ───────   ────────⎥\n",
       "⎢  4⋅L₁      4⋅L₁      4⋅L₁      4⋅L₁  ⎥\n",
       "⎢                                      ⎥\n",
       "⎢ √3⋅A⋅E     A⋅E     -√3⋅A⋅E    -A⋅E   ⎥\n",
       "⎢ ──────     ────    ────────   ─────  ⎥\n",
       "⎢  4⋅L₁      4⋅L₁      4⋅L₁      4⋅L₁  ⎥\n",
       "⎢                                      ⎥\n",
       "⎢-3⋅A⋅E    -√3⋅A⋅E    3⋅A⋅E     √3⋅A⋅E ⎥\n",
       "⎢───────   ────────   ─────     ────── ⎥\n",
       "⎢  4⋅L₁      4⋅L₁      4⋅L₁      4⋅L₁  ⎥\n",
       "⎢                                      ⎥\n",
       "⎢-√3⋅A⋅E    -A⋅E      √3⋅A⋅E     A⋅E   ⎥\n",
       "⎢────────   ─────     ──────     ────  ⎥\n",
       "⎣  4⋅L₁      4⋅L₁      4⋅L₁      4⋅L₁  ⎦"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rigidez_1 = T.T.subs(theta, 5*pi/6) * rigidez_local.subs(L, L1) * T.subs(theta, 5*pi/6)\n",
    "rigidez_1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Elemento 2\n",
    "\n",
    "Para este elemento $\\theta = \\pi/2$, luego"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "rigidez_2 = T.T.subs(theta, pi/2) * rigidez_local.subs(L, L2) * T.subs(theta, pi/2)\n",
    "rigidez_2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Referencias"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Abani Patra. (Marzo 15, 1999). Trusses and Transformations. Fecha de acceso: Octubre 28, 2019, disponible en: http://www.eng.buffalo.edu/~abani/fem/node6.html."
   ]
  }
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